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+arXiv:2301.04627v1 [quant-ph] 11 Jan 2023
+An Improved Approximation for Sparse Fermionic Hamiltonians
+Daniel Hothem∗,
+Ojas Parekh† and Kevin Thompson‡
+Abstract
+We give a classical 1/(qk+1)-approximation for the maximum eigenvalue of k-sparse fermionic
+Hamiltonians with q-local terms as well as a 1/(4k + 1)-approximation when the Hamiltonian
+has both 2-local and 4-local terms.
+We consider approximations for extremal eigenvalues of a k-sparse fermionic Hamiltonian:
+H =
+�
+Γ
+HΓcΓ.
+(1)
+Here H is a fermionic Hamiltonian with real coefficients HΓ, where ignoring phase factors, each
+term cΓ is a product of q Majorana operators (i.e. H is q-local with q even) and each Majorana
+operator appears in at most k non-zero terms (i.e. H is k-sparse). We let m = �
+Γ |HΓ|.
+Herasymenko, Stroeks, Helsen, and Terhal [2] show that λmax(H) ≥ m/Q, where λmax(H) is the
+largest eigenvalue of H and Q = q(q−1)(k−1)2+q(k−1)+2. We demonstrate that this is true with
+Q = qk + 1 and also that Q = 4k + 1 is attainable for k-sparse H with a mix of 2-local and 4-local
+terms. All of these results are obtained by efficient classical algorithms producing descriptions of
+Gaussian states. We refer the reader to [2] for further background, motivation, and applications to
+the SYK model.
+Results of the above flavor were obtained for traceless k-sparse qubit Hamiltonians with constant
+locality by Harrow and Montanaro [1], who show that λmax(H) ≥ Ω(m/k) using product states,
+where m is defined analogously as above. They also give an improved bound with respect to the
+operator norm instead of the maximum eigenvalue: ∥H∥ ≥ Ω(m/
+√
+k). In the fermionic case, we
+give a 2-local example with λmax(H) = ∥H∥ = Θ(m/k), showing that such an improvement is not
+possible.
+We specify our algorithm for the case when H has 2-local and 4-local terms and point out how it
+generalizes when terms are q-local. Concretely, we are given n fermionic modes and corresponding
+traceless Majorana operators {ci}2n
+i=1 satisfying the canonical anticommutation rules {ci, cj} = 2δij.
+We assume that in Equation (1), each Γ corresponds to a subset of [2n]: Γ = {j1, j2, ..., jq} ⊆ [2n]
+with q ∈ {2, 4} and j1 < j2 < ... < jq. The local terms are defined as cΓ = icj1cj2 if q = 2 and
+cΓ = cj1cj2cj3cj4 if q = 4. We let E = {Γ | HΓ ̸= 0}. As noted above, we assume H is k-sparse, i.e.
+for all i ∈ [2n], |{Γ ∈ E | i ∈ Γ}| ≤ k.
+Theorem 1. There is a classical polynomial time algorithm that given as input the weights {HΓ},
+returns a description of a quantum state ρ achieving energy
+Tr(Hρ) ≥
+1
+4k + 1
+�
+Γ
+|HΓ| ≥
+1
+4k + 1λmax(H).
+∗Sandia National Laboratories, email: dhothem@sandia.gov
+†Sandia National Laboratories, email: odparek@sandia.gov
+‡Sandia National Laboratories, email: kevthom@sandia.gov
+1
+
+Proof. Define a graph G = (V, E) with vertices corresponding to the nonzero terms in the Hamilto-
+nian, i.e. V = E. The graph G may contain vertices corresponding to 2-local or 4-local terms. We
+include an edge (vΓ, vΓ′) ∈ E if and only if one of the following conditions is met:
+(i) cΓ and cΓ′ share one or more Marjorana operators, i.e. Γ ∩ Γ′ ̸= ∅, or
+(ii) Γ and Γ′ are disjoint and Γ ∪ Γ′ ∈ E.
+If there are m nonzero terms in the Hamiltonian then the graph G has m vertices, and the degree
+of a vertex in the graph is at most 4k. We can see the latter as follows. Fix some vertex vΓ. By
+construction,
+deg(vΓ) = |{(Γ, Γ′) ∈ E × E | Γ and Γ′ satisfy (i) or (ii)}|.
+(2)
+We consider two cases:
+- Γ is 4-local. Consider an edge (vΓ, vΓ′). As H contains no 6-local or 8-local terms, Γ ∩ Γ′ ̸= ∅.
+As H is k sparse, there are at most 4k Γ′ for which this can occur.
+- Γ is 2-local. Let a equal the number of 4-local Hamiltonian terms overlapping with Γ, and let
+b equal the number of 2-local terms overlapping with Γ. We claim that the degree of vΓ is at
+most 2a + b.
+There are b 2-local Γ′ satisfying (i) with Γ. Each 2-local Γ′ satisfying (ii) results in a unique
+4-local Γ ∪ Γ′ ∈ E overlapping with Γ, hence there at most a such Γ′. Finally, no 4-local Γ′
+may satisfy (ii), and there are a 4-local Γ′ satisfying (i).
+Since Γ overlaps with at most 2k Γ′, we have a + b ≤ 2k so that 2a + b ≤ 4k.
+By Brooks Theorem we can in polynomial time find a coloring of the vertices of G with at
+most 4k + 1 colors. This means we can partition the vertices into at most 4k + 1 independent sets,
+{S1, ..., St}, with one of these sets having at least a 1/(4k + 1) fraction of the sum of the absolute
+values of the weights:
+�
+Γ
+|HΓ| =
+�
+Si
+�
+Γ∈Si
+|HΓ| ≤ (4k + 1) max
+i
+�
+Γ∈Si
+|HΓ|.
+(3)
+It follows from Equation (3) that
+max
+i
+�
+Γ∈Si
+|HΓ| ≥
+1
+(4k + 1)
+�
+Γ
+|HΓ|.
+Define Sj = arg maxj
+�
+Γ∈Sj |HΓ|, and consider the following state:
+ρ = 1
+2n
+�
+Γ∈Sj
+(I + sign(HΓ)cΓ).
+(4)
+We claim that ρ is a valid quantum state and obtains objective �
+Γ∈Sj |HΓ|. The state ρ is
+proportional to a projector on a stabilizer state with stabilizer generators given by cΓ for Γ ∈ Sj:
+Observe that [cΓ, cΓ′] = 0 for all Γ, Γ′ ∈ Sj since Sj is an independent set. Hence, ρ is the product
+of commuting projectors and must be positive semidefinite.
+We expand the product in Equation (4) as a sum and consider products of two or more terms,
+σ = �
+p cΓp for Γp ∈ Sj. If any of the Γp are 4-local or p ≥ 3, σ cannot be proportional to a term of
+2
+
+H since the Γ ∈ Sj are disjoint, and no cancellation in products of Majorona operators can occur.
+The remaining case is a product of two 2-local operators. For any such Γ, Γ′ ∈ Sj, by (ii) and
+because Sj is an independent set, the product cΓcΓ′ cannot be proportional to cΓ′′ for any Γ′′ ∈ E.
+Hence we have
+Tr(Iρ) = 1,
+Tr(cΓρ) = sign(HΓ)
+∀Γ ∈ Sj, and
+Tr(cΓρ) = 0
+∀Γ ∈ E \ Sj.
+This yields the desired claim that ρ is a normalized state for which
+Tr(Hρ) =
+�
+Γ
+HΓTr(cΓρ) =
+�
+Γ∈Sj
+HΓTr(cΓρ) =
+�
+Γ∈Sj
+|HΓ| ≥
+1
+4k + 1
+�
+Γ
+|HΓ|.
+Gaussian states.
+The ρ constructed in Theorem 1 is, in fact, a mixture of Gaussian states. This
+is proven in the following lemma. This implies the existence of a Gaussian state with at least the
+same objective as ρ.
+Lemma 2. The state ρ defined in Equation (4) is a mixture of Gaussian states.
+Proof. For each Γ ∈ Sj let MΓ be the perfect matching of the operators in Γ induced by the lexico-
+graphic ordering of Γ, and let M be a perfect matching of the Majorana operators in {c1, ...c2n}\{ci |
+∃Γ ∈ Sj with i ∈ Γ} induced by the lexicographic ordering. Define the following Gaussian state:
+ρ′(z) = 1
+2n
+�
+Γ∈Sj
+�
+gh∈MΓ
+(I + zgh icgch)
+�
+rs∈M
+(I + zrs icrcs),
+(5)
+where all zgh, zrs ∈ {±1}.
+Consider the state ρ′′ = Ez[ρ′(z)] where for each Γ the set {zgh}gh∈MΓ is uniformly random
+distributed over {±1}|MΓ| subject to the constraint:
+sign
+
+
+
+ �
+gh∈MΓ
+zgh icgch
+
+ cΓ
+
+ = sign(HΓ)
+∀Γ ∈ Sj,
+(6)
+where sign(±I) is defined as ±1. In other words, {zgh}gh∈MΓ is chosen as the uniform distribution
+over strings in {±1}|MΓ| which satisfy Equation (6). We will assume further that {zgh}gh∈MΓ is
+independent of all other {zgh}gh∈MΓ′ and that each zrs for rs ∈ M is uniform and independent of
+all other random variables.
+We claim that ρ = ρ′′. Begin by using independence to push the expectation past the first and
+third products in Equation (5):
+ρ′′ = 1
+2n
+�
+Γ∈Sj
+�
+Ez
+�
+�
+gh∈MΓ
+(I + zgh icgch)
+�� �
+rs∈M
+�
+Ez
+�
+(I + zrs icrcs)
+��
+,
+(7)
+We first focus on the final product. Observe that:
+�
+rs∈M
+�
+Ez
+�
+(I + zrs icrcs)
+��
+= I
+(8)
+3
+
+This follows from the independence of the {zrs | rs ∈ M} and because Ez[zrs] = 0 for all rs ∈ M.
+Hence:
+ρ′′ = 1
+2n
+�
+Γ∈Sj
+�
+Ez
+�
+�
+gh∈MΓ
+(I + zgh icgch)
+��
+.
+(9)
+For fixed Γ ∈ Sj, we claim that:
+Ez
+�
+�
+gh∈MΓ
+(I + zgh icgch)
+�
+= I + sign(HΓ)cΓ.
+(10)
+Lemma 2 follows immediately from Equation (10). For any strict subset Γ′ ⊊ Γ, define
+MΓ′∩Γ := {gh ∈ MΓ : g ∈ Γ′, h ∈ Γ′}.
+We may then expand the left-hand side of Equation (10) as:
+Ez
+�
+�
+gh∈MΓ
+(I + zgh icgch)
+�
+= I +
+�
+Γ′⊊Γ
+Ez
+�
+�
+gh∈MΓ′∩Γ
+zgh icgch
+�
++ Ez
+�
+�
+gh∈MΓ
+zgh icgch
+�
+(11)
+= I + sign(HΓ)cΓ
+(12)
+The final expectation in Equation (11) evaluates to sign(HΓ)cΓ due to constraint 6. The sum of
+expectations in Equation (11) disappears as the marginal distribution of the z when restricted to
+a matching on a strict subset Γ′ ⊊ Γ of size |MΓ′∩Γ| = p is totally uniform over {±1}p. Therefore
+Ez[zgh] = 0 for any such matching.
+Although ρ′(z) in Lemma 2 is a Gaussian state for any z, the state ρ′′ is a mixture of Gaussian
+states by definition. However, we may derandomize the choice of z to obtain a Gaussian state.
+We only require pairwise independence of the elements of z, hence using standard derandomization
+approaches, we can obtain a Gaussian state ρ′(z) in polynomial time such that Tr(Hρ′(z)) ≥
+Tr(Hρ′′).
+Extension to strictly q-local Hamiltonians.
+A simple modification of the proof of Theorem 1
+produces a 1/(qk + 1)-approximation to k-sparse Hamiltonians where each term is q-local. In this
+case we only need to include edges in G between vΓ and vΓ′ precisely when condition (i) holds, since
+(ii) is vacuous. Consequently we may omit the second case below Equation (2) and simply bound
+the degree as qk. We then effectively replace “4” with q in the remaining proof.
+Extension to Hamiltonians with terms of different sizes.
+An additional modification of the
+proof of Theorem 1 produces a 1/O(qk2)-approximation to a k-sparse Hamiltonian with terms of
+different sizes, where q = maxΓ∈E(|Γ|). In this case we need an appropriate generalization of (ii).
+Let us start by defining G using only the condition (i); the maximum possible degree in G is qk.
+The purpose of (ii) in the proof is to ensure that for Γ, Γ′ in the independent set Sj, cΓcΓ′ cannot
+be proportional to cΓ′′ for any Γ′′ ∈ E. Note that if this happens, then Γ′′ must contain both Γ and
+Γ′. Thus it would suffice for our independent set Sj in G to satisfy the additional property that
+no vΓ, vΓ′ ∈ Sj could have a common neighbor vΓ′′ ∈ V with Γ, Γ′ ⊂ Γ′′. We could satisfy this by
+adding an edge in G between all pairs vΓ and vΓ′ with such a common neighbor. By k-sparsity, the
+vertex vΓ has at most k neighbors vΓ′′ in G with Γ ⊂ Γ′′. Since any such vΓ′′ has degree at most
+qk, the degree of vΓ increases by at most k(qk − 1), and maximum degree in the resulting graph G′
+is O(qk2). Applying Brook’s Theorem in G′ produces the desired approximation.
+4
+
+Optimality.
+For k-sparse H where all terms are q-local, since ∥H∥ ≥ λmax(H), our results show
+that
+∥H∥ ≥ λmax(H) ≥
+m
+qk + 1,
+where we recall m = �
+Γ |HΓ|. We give an explicit family of fermionic 2-local n-sparse Hamiltonians
+{Hn}∞
+n=1 demonstrating this bound is asymptotically tight (i.e., cannot be improved for all q and
+k, up to constant factors).
+Each Hn is expressed as a sum of monomials in 2n Majorana operators {c1, c2, ..., c2n} satisfying
+the usual canonical anti-commutation relations. For each n, partition [2n] evenly into A = {1, ..., n}
+and B = {n + 1, ..., 2n}. Then:
+Hn :=
+�
+a∈A,b∈B
+icacb = i
+��
+a∈A
+ca
+� ��
+b∈B
+cb
+�
+.
+The eigenvalues of Hn are easy to determine, define R ∈ O(2n) as some orthogonal matrix
+satisfying:
+Ra,1 = 1/√n
+∀a ∈ A and Rb,2 = 1/√n
+∀b ∈ B.
+Note that this is well defined since the first two columns are orthonormal. We can then define a
+new set of Majorana operators (also satisfying the canonical anti-commutation relations) by:
+˜ci =
+2n
+�
+i=1
+Rj,icj.
+In particular, we have
+˜c1 =
+1
+√n
+�
+a∈A
+ca and ˜c2 =
+1
+√n
+�
+b∈B
+cb,
+so
+H = ni ˜c1 ˜c2.
+Since i ˜c1 ˜c2 is Hermitian and satisfies (i ˜c1 ˜c2)2 = I, it has eigenvalues in {±1}. Thus the eigenvalues
+of Hn are {±n}. Note that Hn is n-sparse, m = n2, and ∥Hn∥ = λmax(Hn) so that
+∥Hn∥ = λmax(Hn) = n = Θ
+�
+n2
+2n + 1
+�
+= Θ
+�
+m
+qk + 1
+�
+.
+Acknowledgements
+We thank Yaroslav Herasymenko for an insightful contribution to Lemma 2.
+This article has been authored by an employee of National Technology & Engineering Solutions
+of Sandia, LLC under Contract No. DE-NA0003525 with the U.S. Department of Energy (DOE).
+The employee owns all right, title and interest in and to the article and is solely responsible for
+its contents. The United States Government retains and the publisher, by accepting the article
+for publication, acknowledges that the United States Government retains a non-exclusive, paid-up,
+irrevocable, world-wide license to publish or reproduce the published form of this article or allow
+others to do so, for United States Government purposes.
+The DOE will provide public access
+to these results of federally sponsored research in accordance with the DOE Public Access Plan
+https://www.energy.gov/downloads/doe-public-access-plan.
+5
+
+This material is based upon work supported by the U.S. Department of Energy, Office of Science,
+Office of Advanced Scientific Computing Research, National Quantum Information Science Research
+Centers, Exploratory Research for Extreme Scale Science program. Support is also acknowledged
+from the Accelerated Research in Quantum Computing program under the same office.
+References
+[1] Aram W. Harrow and Ashley Montanaro. Extremal eigenvalues of local Hamiltonians. Quantum,
+1:6, April 2017. doi:10.22331/q-2017-04-25-6.
+[2] Yaroslav Herasymenko, Maarten Stroeks, Jonas Helsen, and Barbara Terhal. Optimizing sparse
+fermionic hamiltonians. arXiv preprint arXiv:2211.16518, 2022.
+6
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf,len=161
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='04627v1 [quant-ph] 11 Jan 2023 An Improved Approximation for Sparse Fermionic Hamiltonians Daniel Hothem∗, Ojas Parekh† and Kevin Thompson‡ Abstract We give a classical 1/(qk+1)-approximation for the maximum eigenvalue of k-sparse fermionic Hamiltonians with q-local terms as well as a 1/(4k + 1)-approximation when the Hamiltonian has both 2-local and 4-local terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We consider approximations for extremal eigenvalues of a k-sparse fermionic Hamiltonian: H = � Γ HΓcΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' (1) Here H is a fermionic Hamiltonian with real coefficients HΓ, where ignoring phase factors, each term cΓ is a product of q Majorana operators (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' H is q-local with q even) and each Majorana operator appears in at most k non-zero terms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' H is k-sparse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We let m = � Γ |HΓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Herasymenko, Stroeks, Helsen, and Terhal [2] show that λmax(H) ≥ m/Q, where λmax(H) is the largest eigenvalue of H and Q = q(q−1)(k−1)2+q(k−1)+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We demonstrate that this is true with Q = qk + 1 and also that Q = 4k + 1 is attainable for k-sparse H with a mix of 2-local and 4-local terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' All of these results are obtained by efficient classical algorithms producing descriptions of Gaussian states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We refer the reader to [2] for further background, motivation, and applications to the SYK model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Results of the above flavor were obtained for traceless k-sparse qubit Hamiltonians with constant locality by Harrow and Montanaro [1], who show that λmax(H) ≥ Ω(m/k) using product states, where m is defined analogously as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' They also give an improved bound with respect to the operator norm instead of the maximum eigenvalue: ∥H∥ ≥ Ω(m/ √ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' In the fermionic case, we give a 2-local example with λmax(H) = ∥H∥ = Θ(m/k), showing that such an improvement is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We specify our algorithm for the case when H has 2-local and 4-local terms and point out how it generalizes when terms are q-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Concretely, we are given n fermionic modes and corresponding traceless Majorana operators {ci}2n i=1 satisfying the canonical anticommutation rules {ci, cj} = 2δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We assume that in Equation (1), each Γ corresponds to a subset of [2n]: Γ = {j1, j2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=', jq} ⊆ [2n] with q ∈ {2, 4} and j1 < j2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' < jq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' The local terms are defined as cΓ = icj1cj2 if q = 2 and cΓ = cj1cj2cj3cj4 if q = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We let E = {Γ | HΓ ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' As noted above, we assume H is k-sparse, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' for all i ∈ [2n], |{Γ ∈ E | i ∈ Γ}| ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' There is a classical polynomial time algorithm that given as input the weights {HΓ}, returns a description of a quantum state ρ achieving energy Tr(Hρ) ≥ 1 4k + 1 � Γ |HΓ| ≥ 1 4k + 1λmax(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' ∗Sandia National Laboratories, email: dhothem@sandia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='gov †Sandia National Laboratories, email: odparek@sandia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='gov ‡Sandia National Laboratories, email: kevthom@sandia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='gov 1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Define a graph G = (V, E) with vertices corresponding to the nonzero terms in the Hamilto- nian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' V = E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' The graph G may contain vertices corresponding to 2-local or 4-local terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We include an edge (vΓ, vΓ′) ∈ E if and only if one of the following conditions is met: (i) cΓ and cΓ′ share one or more Marjorana operators, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Γ ∩ Γ′ ̸= ∅, or (ii) Γ and Γ′ are disjoint and Γ ∪ Γ′ ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' If there are m nonzero terms in the Hamiltonian then the graph G has m vertices, and the degree of a vertex in the graph is at most 4k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We can see the latter as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Fix some vertex vΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' By construction, deg(vΓ) = |{(Γ, Γ′) ∈ E × E | Γ and Γ′ satisfy (i) or (ii)}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' (2) We consider two cases: Γ is 4-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Consider an edge (vΓ, vΓ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' As H contains no 6-local or 8-local terms, Γ ∩ Γ′ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' As H is k sparse, there are at most 4k Γ′ for which this can occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Γ is 2-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Let a equal the number of 4-local Hamiltonian terms overlapping with Γ, and let b equal the number of 2-local terms overlapping with Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We claim that the degree of vΓ is at most 2a + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' There are b 2-local Γ′ satisfying (i) with Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Each 2-local Γ′ satisfying (ii) results in a unique 4-local Γ ∪ Γ′ ∈ E overlapping with Γ, hence there at most a such Γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Finally, no 4-local Γ′ may satisfy (ii), and there are a 4-local Γ′ satisfying (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Since Γ overlaps with at most 2k Γ′, we have a + b ≤ 2k so that 2a + b ≤ 4k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' By Brooks Theorem we can in polynomial time find a coloring of the vertices of G with at most 4k + 1 colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' This means we can partition the vertices into at most 4k + 1 independent sets, {S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=', St}, with one of these sets having at least a 1/(4k + 1) fraction of the sum of the absolute values of the weights: � Γ |HΓ| = � Si � Γ∈Si |HΓ| ≤ (4k + 1) max i � Γ∈Si |HΓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' (3) It follows from Equation (3) that max i � Γ∈Si |HΓ| ≥ 1 (4k + 1) � Γ |HΓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Define Sj = arg maxj � Γ∈Sj |HΓ|, and consider the following state: ρ = 1 2n � Γ∈Sj (I + sign(HΓ)cΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' (4) We claim that ρ is a valid quantum state and obtains objective � Γ∈Sj |HΓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' The state ρ is proportional to a projector on a stabilizer state with stabilizer generators given by cΓ for Γ ∈ Sj: Observe that [cΓ, cΓ′] = 0 for all Γ, Γ′ ∈ Sj since Sj is an independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Hence, ρ is the product of commuting projectors and must be positive semidefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We expand the product in Equation (4) as a sum and consider products of two or more terms, σ = � p cΓp for Γp ∈ Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' If any of the Γp are 4-local or p ≥ 3, σ cannot be proportional to a term of 2 H since the Γ ∈ Sj are disjoint, and no cancellation in products of Majorona operators can occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' The remaining case is a product of two 2-local operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' For any such Γ, Γ′ ∈ Sj, by (ii) and because Sj is an independent set, the product cΓcΓ′ cannot be proportional to cΓ′′ for any Γ′′ ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Hence we have Tr(Iρ) = 1, Tr(cΓρ) = sign(HΓ) ∀Γ ∈ Sj, and Tr(cΓρ) = 0 ∀Γ ∈ E \\ Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' This yields the desired claim that ρ is a normalized state for which Tr(Hρ) = � Γ HΓTr(cΓρ) = � Γ∈Sj HΓTr(cΓρ) = � Γ∈Sj |HΓ| ≥ 1 4k + 1 � Γ |HΓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Gaussian states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' The ρ constructed in Theorem 1 is, in fact, a mixture of Gaussian states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' This is proven in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' This implies the existence of a Gaussian state with at least the same objective as ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' The state ρ defined in Equation (4) is a mixture of Gaussian states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' For each Γ ∈ Sj let MΓ be the perfect matching of the operators in Γ induced by the lexico- graphic ordering of Γ, and let M be a perfect matching of the Majorana operators in {c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='c2n}\\{ci | ∃Γ ∈ Sj with i ∈ Γ} induced by the lexicographic ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Define the following Gaussian state: ρ′(z) = 1 2n � Γ∈Sj � gh∈MΓ (I + zgh icgch) � rs∈M (I + zrs icrcs), (5) where all zgh, zrs ∈ {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Consider the state ρ′′ = Ez[ρ′(z)] where for each Γ the set {zgh}gh∈MΓ is uniformly random distributed over {±1}|MΓ| subject to the constraint: sign \uf8ee \uf8f0 \uf8eb \uf8ed � gh∈MΓ zgh icgch \uf8f6 \uf8f8 cΓ \uf8f9 \uf8fb = sign(HΓ) ∀Γ ∈ Sj, (6) where sign(±I) is defined as ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' In other words, {zgh}gh∈MΓ is chosen as the uniform distribution over strings in {±1}|MΓ| which satisfy Equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We will assume further that {zgh}gh∈MΓ is independent of all other {zgh}gh∈MΓ′ and that each zrs for rs ∈ M is uniform and independent of all other random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We claim that ρ = ρ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Begin by using independence to push the expectation past the first and third products in Equation (5): ρ′′ = 1 2n � Γ∈Sj � Ez � � gh∈MΓ (I + zgh icgch) �� � rs∈M � Ez � (I + zrs icrcs) �� , (7) We first focus on the final product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Observe that: � rs∈M � Ez � (I + zrs icrcs) �� = I (8) 3 This follows from the independence of the {zrs | rs ∈ M} and because Ez[zrs] = 0 for all rs ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Hence: ρ′′ = 1 2n � Γ∈Sj � Ez � � gh∈MΓ (I + zgh icgch) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' (9) For fixed Γ ∈ Sj, we claim that: Ez � � gh∈MΓ (I + zgh icgch) � = I + sign(HΓ)cΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' (10) Lemma 2 follows immediately from Equation (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' For any strict subset Γ′ ⊊ Γ, define MΓ′∩Γ := {gh ∈ MΓ : g ∈ Γ′, h ∈ Γ′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We may then expand the left-hand side of Equation (10) as: Ez � � gh∈MΓ (I + zgh icgch) � = I + � Γ′⊊Γ Ez � � gh∈MΓ′∩Γ zgh icgch � + Ez � � gh∈MΓ zgh icgch � (11) = I + sign(HΓ)cΓ (12) The final expectation in Equation (11) evaluates to sign(HΓ)cΓ due to constraint 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' The sum of expectations in Equation (11) disappears as the marginal distribution of the z when restricted to a matching on a strict subset Γ′ ⊊ Γ of size |MΓ′∩Γ| = p is totally uniform over {±1}p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Therefore Ez[zgh] = 0 for any such matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Although ρ′(z) in Lemma 2 is a Gaussian state for any z, the state ρ′′ is a mixture of Gaussian states by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' However, we may derandomize the choice of z to obtain a Gaussian state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We only require pairwise independence of the elements of z, hence using standard derandomization approaches, we can obtain a Gaussian state ρ′(z) in polynomial time such that Tr(Hρ′(z)) ≥ Tr(Hρ′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Extension to strictly q-local Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' A simple modification of the proof of Theorem 1 produces a 1/(qk + 1)-approximation to k-sparse Hamiltonians where each term is q-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' In this case we only need to include edges in G between vΓ and vΓ′ precisely when condition (i) holds, since (ii) is vacuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Consequently we may omit the second case below Equation (2) and simply bound the degree as qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We then effectively replace “4” with q in the remaining proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Extension to Hamiltonians with terms of different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' An additional modification of the proof of Theorem 1 produces a 1/O(qk2)-approximation to a k-sparse Hamiltonian with terms of different sizes, where q = maxΓ∈E(|Γ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' In this case we need an appropriate generalization of (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Let us start by defining G using only the condition (i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' the maximum possible degree in G is qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' The purpose of (ii) in the proof is to ensure that for Γ, Γ′ in the independent set Sj, cΓcΓ′ cannot be proportional to cΓ′′ for any Γ′′ ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Note that if this happens, then Γ′′ must contain both Γ and Γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Thus it would suffice for our independent set Sj in G to satisfy the additional property that no vΓ, vΓ′ ∈ Sj could have a common neighbor vΓ′′ ∈ V with Γ, Γ′ ⊂ Γ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We could satisfy this by adding an edge in G between all pairs vΓ and vΓ′ with such a common neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' By k-sparsity, the vertex vΓ has at most k neighbors vΓ′′ in G with Γ ⊂ Γ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Since any such vΓ′′ has degree at most qk, the degree of vΓ increases by at most k(qk − 1), and maximum degree in the resulting graph G′ is O(qk2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Applying Brook’s Theorem in G′ produces the desired approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' 4 Optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' For k-sparse H where all terms are q-local, since ∥H∥ ≥ λmax(H), our results show that ∥H∥ ≥ λmax(H) ≥ m qk + 1, where we recall m = � Γ |HΓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We give an explicit family of fermionic 2-local n-sparse Hamiltonians {Hn}∞ n=1 demonstrating this bound is asymptotically tight (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=', cannot be improved for all q and k, up to constant factors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Each Hn is expressed as a sum of monomials in 2n Majorana operators {c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=', c2n} satisfying the usual canonical anti-commutation relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' For each n, partition [2n] evenly into A = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=', n} and B = {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=', 2n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Then: Hn := � a∈A,b∈B icacb = i �� a∈A ca � �� b∈B cb � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' The eigenvalues of Hn are easy to determine, define R ∈ O(2n) as some orthogonal matrix satisfying: Ra,1 = 1/√n ∀a ∈ A and Rb,2 = 1/√n ∀b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Note that this is well defined since the first two columns are orthonormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We can then define a new set of Majorana operators (also satisfying the canonical anti-commutation relations) by: ˜ci = 2n � i=1 Rj,icj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' In particular, we have ˜c1 = 1 √n � a∈A ca and ˜c2 = 1 √n � b∈B cb, so H = ni ˜c1 ˜c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Since i ˜c1 ˜c2 is Hermitian and satisfies (i ˜c1 ˜c2)2 = I, it has eigenvalues in {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Thus the eigenvalues of Hn are {±n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Note that Hn is n-sparse, m = n2, and ∥Hn∥ = λmax(Hn) so that ∥Hn∥ = λmax(Hn) = n = Θ � n2 2n + 1 � = Θ � m qk + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Acknowledgements We thank Yaroslav Herasymenko for an insightful contribution to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' This article has been authored by an employee of National Technology & Engineering Solutions of Sandia, LLC under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' DE-NA0003525 with the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Department of Energy (DOE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' The employee owns all right, title and interest in and to the article and is solely responsible for its contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this article or allow others to do so, for United States Government purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' The DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='gov/downloads/doe-public-access-plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' 5 This material is based upon work supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, National Quantum Information Science Research Centers, Exploratory Research for Extreme Scale Science program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Support is also acknowledged from the Accelerated Research in Quantum Computing program under the same office.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' References [1] Aram W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Harrow and Ashley Montanaro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Extremal eigenvalues of local Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Quantum, 1:6, April 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='22331/q-2017-04-25-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' [2] Yaroslav Herasymenko, Maarten Stroeks, Jonas Helsen, and Barbara Terhal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Optimizing sparse fermionic hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' arXiv preprint arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='16518, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' 6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
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+arXiv:2301.04204v1 [math.OC] 10 Jan 2023
+A Newton-CG based barrier-augmented Lagrangian method for
+general nonconvex conic optimization
+Chuan He∗
+Heng Huang†
+Zhaosong Lu∗
+January 10, 2023
+Abstract
+In this paper we consider finding an approximate second-order stationary point (SOSP) of general noncon-
+vex conic optimization that minimizes a twice differentiable function subject to nonlinear equality constraints
+and also a convex conic constraint. In particular, we propose a Newton-conjugate gradient (Newton-CG)
+based barrier-augmented Lagrangian method for finding an approximate SOSP of this problem. Under some
+mild assumptions, we show that our method enjoys a total inner iteration complexity of �O(ǫ−11/2) and
+an operation complexity of �O(ǫ−11/2 min{n, ǫ−5/4}) for finding an (ǫ, √ǫ)-SOSP of general nonconvex conic
+optimization with high probability. Moreover, under a constraint qualification, these complexity bounds are
+improved to �O(ǫ−7/2) and �O(ǫ−7/2 min{n, ǫ−3/4}), respectively. To the best of our knowledge, this is the
+first study on the complexity of finding an approximate SOSP of general nonconvex conic optimization. Pre-
+liminary numerical results are presented to demonstrate superiority of the proposed method over first-order
+methods in terms of solution quality.
+Keywords:
+Nonconvex conic optimization, second-order stationary point, augmented Lagrangian method, barrier
+method, Newton-conjugate gradient method, iteration complexity, operation complexity
+Mathematics Subject Classification: 49M05, 49M15, 68Q25, 90C26, 90C30, 90C60
+1
+Introduction
+In this paper we consider the following general nonconvex conic optimization problem:
+min
+x {f(x) : c(x) = 0, x ∈ K},
+(1)
+where K ⊆ Rn is a closed and pointed convex cone with nonempty interior, and f : Rn → R and c : Rn → Rm
+are continuous in K and twice continuously differentiable in the interior of K. Assume that problem (1) has at
+least one optimal solution. Our goal is to propose an implementable method with complexity guarantees for
+finding an approximate second-order stationary point (SOSP) of (1) that will be introduced in Section 3.
+In recent years, there has been considerable research on designing algorithms with complexity guarantees for
+finding an approximate SOSP of nonconvex optimization problems. In particular, numerous algorithms were
+developed for nonconvex unconstrained optimization, such as cubic regularized Newton methods [1, 18, 21, 57],
+trust-region methods [34, 35, 53], quadratic regularization method [14], accelerated gradient method [19, 20],
+second-order line-search method [61], Newton-conjugate gradient (Newton-CG) method [60], and gradient-based
+methods with random perturbations [2, 46, 71]. In addition, several methods with complexity guarantees have
+also been proposed for nonconvex optimization with relatively simple constraints. For example, interior-point
+method [10], log-barrier method [58], and projected gradient descent method [69] were proposed for nonconvex
+∗Department
+of
+Industrial
+and
+Systems
+Engineering,
+University
+of
+Minnesota,
+USA
+(email:
+he000233@umn.edu,
+zhaosong@umn.edu). The work of the last author was partially supported by NSF Award IIS-2211491
+†Department of Electrical and Computer Engineering, University of Pittsburgh, USA (email: heng.huang@pitt.edu). The work
+of this author was partially supported by NSF Award IIS-2211492.
+1
+
+optimization with sign constraints. Besides, the interior-point method [10] was generalized in [42] for nonconvex
+optimization with sign constraints and additional linear equality constraints. Also, a projected gradient descent
+method with random perturbations was proposed in [51] for nonconvex optimization with linear inequality
+constraints.
+Iteration complexity of these methods has been established for finding an approximate SOSP.
+Besides, operation complexity in terms of the total number of fundamental operations has been studied for the
+methods [1, 2, 18, 19, 20, 34, 46, 60, 61, 71].
+Several methods, including trust-region methods [17, 31], sequential quadratic programming method [15],
+two-phase method [11, 27, 30], penalty method [40], and augmented Lagrangian (AL) methods [4, 12, 44, 63, 70],
+were proposed for finding an approximate SOSP of equality constrained optimization:
+min
+x {f(x) : c(x) = 0},
+(2)
+which is special case of (1) with K = Rn. Moreover, total inner iteration complexity and operation complexity,
+which are respectively measured by the total number of iterations of the Newton-CG method in [60] and the
+total number of gradient evaluations and matrix-vector products performed in the method, were established in
+[44, 70] for finding an (ǫ, √ǫ)-SOSP x of (2) which together with some λ ∈ Rm satisfies
+∥c(x)∥ ≤ ǫ, ∥∇f(x) + ∇c(x)λ∥ ≤ ǫ,
+dT (∇2f(x) + �m
+i=1 λi∇2ci(x))d ≥ −√ǫ∥d∥2,
+∀d ∈ {d : ∇c(x)T d = 0},
+where ∇c denotes the transpose of the Jacobian of c. Specifically, under some suitable assumptions, including a
+generalized linear independence constraint qualification (GLICQ), the AL method [70] enjoys a total inner iter-
+ation complexity of �O(ǫ−11/2) and an operation complexity �O(ǫ−11/2 min{n, ǫ−3/4}),1 while the AL method [44]
+achieves a total inner iteration complexity of �O(ǫ−7/2) and an operation complexity of �O(ǫ−7/2 min{n, ǫ−3/4})
+for finding an (ǫ, √ǫ)-SOSP of problem (2) with high probability. On the other hand, when the GLICQ does
+not hold, the AL method [44] has a total inner iteration complexity of �O(ǫ−11/2) and an operation complexity
+of �O(ǫ−11/2 min{n, ǫ−5/4}). Besides, it shall be mentioned that Newton-CG based AL methods were developed
+for efficiently solving a variety of convex optimization problems (e.g., see [72, 73]), though their complexities
+remain unknown.
+In addition, a Newton-CG based barrier method was recently proposed in [43] for finding an approximate
+SOSP of a class of nonconvex conic optimization of the form
+min
+x {f(x) : Ax − b = 0, x ∈ K}
+(3)
+for some A ∈ Rm×n and b ∈ Rm, which is a special case of (1). Iteration and operation complexity of this
+method were established in [43] for finding an (ǫ, √ǫ)-SOSP x of (3) which together with some λ ∈ Rm satisfies
+Ax = b, x ∈ int K, ∇f(x) + AT λ ∈ K∗, ∥∇f(x) + AT λ∥∗
+x ≤ ǫ,
+dT ∇2B(x)−1/2∇2f(x)∇2B(x)−1/2d ≥ −√ǫ∥d∥2,
+∀d ∈ {d : A∇2B(x)−1/2d = 0},
+where int K and K∗ are respectively the interior and dual cone of K, B is a logarithmically homogeneous self-
+concordant barrier function for K, and ∥ · ∥∗
+x is a local norm induced by B at x (see Section 2 for details).
+Under some suitable assumptions, this method achieves an iteration complexity of O(ǫ−3/2) and an operation
+complexity2 of �O(ǫ−3/2 min{n, ǫ−1/4}) for finding an (ǫ, √ǫ)-SOSP with high probability. Besides, a Hessian
+barrier algorithm was proposed in [38] for finding an approximate SOSP of problem (3).
+Given that this
+algorithm requires solving a cubic regularized subproblem exactly per iteration, it is generally not implementable.
+It shall also be mentioned that finding an approximate first-order stationary point of (1) with K = Rn
++ was
+extensively studied in the literature (e.g., [5, 6, 7, 32, 33, 37, 39, 49, 55, 65, 66, 67]). Notably, a hybrid approach
+1In fact, a total inner iteration complexity of �O(ǫ−7) and an operation complexity �O(ǫ−7 min{n, ǫ−1}) were established in [70]
+for finding an (ǫ, ǫ)-SOSP of problem (1) with high probability; see [70, Theorem 4(ii), Corollary 3(ii), Theorem 5]. Nevertheless,
+they can be easily modified to obtain the aforementioned complexity for finding an (ǫ, √ǫ)-SOSP of (1) with high probability.
+2The operation complexity of the barrier method [43] is measured by the amount of fundamental operations consisting of matrix-
+vector products, matrix multiplications, Cholesky factorizations, and backward or forward substitutions to a triangular linear
+system.
+2
+
+by combining barrier and AL methods was commonly used in [5, 6, 7, 33, 37, 39, 49, 55]). However, finding
+an approximate SOSP of (1) by such a hybrid approach has not been considered, even for (1) with K = Rn
++.
+Inspired by these and [43, 44], in this paper we propose a Newton-CG based barrier-AL method for finding an
+approximate SOSP of problem (1) with high probability. Our main contributions are as follows.
+• We study first- and second-order optimality conditions for problem (1) and introduce an approximate
+counterpart of them.
+• We propose an implementable Newton-CG based barrier-AL method for finding an approximate SOSP
+of (1), whose fundamental operations consist of matrix-vector products, Cholesky factorizations, and
+backward or forward substitutions to a triangular linear system.
+• We show that under some mild assumptions, our proposed method has a total inner iteration complex-
+ity of �O(ǫ−11/2) and an operation complexity of �O(ǫ−11/2 min{n, ǫ−5/4}) for finding an (ǫ, √ǫ)-SOSP
+of (1) with high probability. Furthermore, under a constraint qualification, we show that our method
+achieves an improved total inner iteration complexity of �O(ǫ−7/2) and an improved operation complexity
+of �O(ǫ−7/2 min{n, ǫ−3/4}).3 To the best of our knowledge, there was no complexity result for finding an
+approximate SOSP of problem (1) in the literature before.
+The rest of this paper is organized as follows. In Section 2, we introduce some notation. In Section 3, we
+study optimality conditions of problem (1) and introduce an inexact counterpart of them. In Section 4, we
+propose a preconditioned Newton-CG method for solving a barrier problem and study its complexity. We then
+propose a Newton-CG based barrier-AL method for (1) and study its complexity in Section 5. We present in
+Section 6 some preliminary numerical results for the proposed method. In Section 7, we present the proofs of
+the main results. Finally, we make some concluding remarks in Section 8.
+2
+Notation and preliminaries
+Throughout this paper, we let Rn denote the n-dimensional Euclidean space. The symbol ∥ · ∥ stands for the
+Euclidean norm of a vector or the spectral norm of a matrix. The identity matrix is denoted by I. We denote
+by λmin(H) the minimum eigenvalue of a real symmetric matrix H. For any two real symmetric matrices M1
+and M2, M1 ⪯ M2 means that M2 − M1 is positive semidefinite. For any positive semidefinite matrix M, M 1/2
+denotes a positive semidefinite matrix such that M = M 1/2M 1/2. For the closed convex cone K, its interior
+and dual cone are respectively denoted by int K and K∗. For any x ∈ K, the normal cone and tangent cone of
+K at x are denoted by N K(x) and TK(x), respectively. The Euclidean ball centered at the origin with radius
+R ≥ 0 is denoted by BR := {x : ∥x∥ ≤ R}, and we use ΠBR(v) to denote the Euclidean projection of a vector v
+onto BR. For a given finite set A, we let | A | denote its cardinality. For any s ∈ R, we let sgn(s) be 1 if s ≥ 0
+and let it be −1 otherwise. In addition, �O(·) represents O(·) with logarithmic terms omitted.
+Logarithmically homogeneous self-concordant (LHSC) barrier function is a key ingredient in the development
+of interior-point methods for convex programming (see the monograph [56]). It will also play a crucial role in
+the design and analysis of Newton-CG based barrier-AL method for solving problem (1). Throughout this paper,
+we assume that the cone K is equipped with a ϑ-logarithmically homogeneous self-concordant (ϑ-LHSC) barrier
+function B for some ϑ ≥ 1. That is, B : int K → R satisfies the following conditions:
+(i) B is convex and three times continuously differentiable in int K, and moreover, |ψ′′′(0)| ≤ 2(ψ′′(0))3/2
+holds for all x ∈ int K and u ∈ Rn, where ψ(t) = B(x + tu);
+(ii) B is a barrier function for K, that is, B(x) goes to infinity as x approaches the boundary of K;
+(iii) B is logarithmically homogeneous, that is, B(tx) = B(x) − ϑ ln t holds for all x ∈ int K and t > 0.
+For any x ∈ int K, the function B induces the following local norms:
+∥v∥x
+:=
+�
+vT ∇2B(x)v
+�1/2 ,
+∀v ∈ Rn,
+3It shall be mentioned that the total numbers of Cholesky factorizations are only �O(ǫ−7/2) and �O(ǫ−11/2) respectively for the
+case where constraint qualification holds or not. See Subsections 5.3 and 5.4 for details.
+3
+
+∥v∥∗
+x
+:=
+�
+vT ∇2B(x)−1v
+�1/2 ,
+∀v ∈ Rn,
+∥M∥∗
+x
+:=
+max
+∥v∥x≤1 ∥Mv∥∗
+x,
+∀M ∈ Rn×n .
+(4)
+In addition, ∇2B(x)−1 is well-defined only in int K but undefined on the boundary of K. To capture the behavior
+of ∇2B(x)−1 as x approaches the boundary of K, the concept of the limiting inverse of the Hessian of B was
+recently introduced in [43], which can be viewed as a generalization of [∇2B]−1. Specifically, the limiting inverse
+of the Hessian of B is defined as follows:
+∇−2B(x) :=
+�
+M : M = lim
+k→∞ ∇2B(xk)−1 for some {xk} ⊂ int K with xk → x as k → ∞
+�
+,
+∀x ∈ K .
+As established in [43, Theorem 1], the inverse of ∇2B(x) is bounded in any nonempty bounded subset of int K.
+Consequently, ∇−2B(x) ̸= ∅ for all x ∈ K. Moreover, the following property holds for ∇−2B, whose proof can
+be found in [43, Theorem 2].
+Lemma 2.1. For any x ∈ K, it holds that {x + M 1/2d : ∥d∥ < 1} ⊆ K for all M ∈ ∇−2B(x).
+3
+Optimality conditions
+Classical first- and second-order optimality conditions for nonlinear optimization can be specialized to prob-
+lem (1) (e.g., see [62, Theorems 3.38 and 3.46]). However, the inexact counterparts of them are not suitable for
+the design and analysis of a barrier-AL method for solving (1). In this section we study some alternative first-
+and second-order optimality conditions for (1) and also introduce an inexact counterpart of them.
+Suppose that x∗ is a local minimizer of problem (1). To derive optimality conditions, one typically needs
+to impose a constraint qualification (CQ) for x∗. The Robinson’s CQ, {∇c(x∗)T d : d ∈ TK(x∗)} = Rm, is a
+natural and general one (e.g., see [62, Section 3.3.2]). However, verification of Robinson’s CQ may not be easy
+for a general cone K. Thus, we instead consider a more easily verifiable CQ that M 1/2∇c(x∗) has full column
+rank for some M ∈ ∇−2B(x∗), which turns out to be stronger than Robinson’s CQ. Indeed, suppose that such
+a CQ holds at x∗ for some M ∈ ∇−2B(x∗). It then follows from Lemma 2.1 that {M 1/2 ˜d : ∥ ˜d∥ < 1} ⊆ TK(x∗)
+and hence {M 1/2 ˜d : ˜d ∈ Rn} ⊆ TK(x∗). By this and the full column rank of M 1/2∇c(x∗), one has
+{∇c(x∗)T d : d ∈ TK(x∗)} ⊇ {∇c(x∗)T M 1/2 ˜d : ˜d ∈ Rn} = Rm,
+and hence Robinson’s CQ holds at x∗.
+We are now ready to establish some first- and second-order optimality conditions for problem (1) under the
+aforementioned CQ, whose proof is relegated to Section 7.1.
+Theorem 3.1 (first- and second-order optimality conditions). Let x∗ be a local minimizer of problem (1).
+Suppose that f is twice continuously differentiable at x∗ and M 1/2∇c(x∗) has full column rank for some M ∈
+∇−2B(x∗). Then there exists a Lagrangian multiplier λ∗ ∈ Rm such that
+∇f(x∗) + ∇c(x∗)λ∗ ∈ K∗,
+(5)
+M 1/2(∇f(x∗) + ∇c(x∗)λ∗) = 0,
+(6)
+and additionally,
+dT M 1/2
+�
+∇2f(x∗) +
+m
+�
+i=1
+λ∗
+i ∇2ci(x∗)
+�
+M 1/2d ≥ 0,
+∀d ∈ {d : ∇c(x∗)T M 1/2d = 0}.
+(7)
+Remark 3.1. The relations (5) and (6) are the first-order optimality conditions of problem (1), which are
+actually equivalent to the classical optimality condition ∇f(x∗)+∇c(x∗)λ∗ ∈ − N K(x∗) (see [43, Proposition 1]).
+Notice that it is generally impossible to find a point exactly satisfying the above first- and second-order
+optimality conditions. We are instead interested in finding a point satisfying their approximate counterparts.
+To this end, we next introduce the definition of an approximate first-order stationary point (FOSP) and second-
+order stationary point (SOSP) of problem (1).
+4
+
+Definition 3.1 (ǫ1-first-order stationary point). For any ǫ1 > 0, a point x is called an ǫ1-first-order
+stationary point (ǫ1-FOSP) of problem (1) if it, together with some λ ∈ Rm, satisfies
+∥c(x)∥ ≤ ǫ1, x ∈ int K,
+(8)
+∇f(x) + ∇c(x)λ ∈ K∗,
+(9)
+∥∇f(x) + ∇c(x)λ∥∗
+x ≤ ǫ1.
+(10)
+Definition 3.2 ((ǫ1, ǫ2)-second-order stationary point). For any ǫ1, ǫ2 > 0, a point x is called an (ǫ1, ǫ2)-
+second-order stationary point ((ǫ1, ǫ2)-SOSP) of problem (1) if it, together with some λ ∈ Rm, satisfies (8)-(10)
+and additionally
+dT ∇2B(x)−1/2
+�
+∇2f(x) +
+m
+�
+i=1
+λi∇2ci(x)
+�
+∇2B(x)−1/2d ≥ −ǫ2∥d∥2,
+∀d ∈ C(x),
+(11)
+where C(·) is defined as
+C(x) := {d : ∇c(x)T ∇2B(x)−1/2d = 0}.
+(12)
+Remark 3.2. Notice that if the pair (x, λ) satisfies (10) and (11), then it also nearly satisfies (6) and (7) with
+(x∗, λ∗) replaced by (x, λ). Thus, (10) and (11) are indeed inexact counterparts of (6) and (7). Moreover, the
+above definitions of ǫ1-FOSP and (ǫ1, ǫ2)-SOSP are reduced to the ones introduced in [43] for the case where c
+is affine.
+4
+A preconditioned Newton-CG method for barrier problems
+In this section we propose a preconditioned Newton-CG method in Algorithm 1, which is a modification of the
+Newton-CG based barrier method [43, Algorithm 2], for finding an approximate SOSP of the barrier problem
+min
+x
+{φµ(x) := F(x) + µB(x)},
+(13)
+where F : Rn → R is twice continuously differentiable in int K and µ > 0 is a given barrier parameter. Specifically,
+the proposed method finds an (ǫg, ǫH)-SOSP x of problem (13) that satisfies
+∥∇φµ(x)∥∗
+x ≤ ǫg,
+λmin(∇2B(x)−1/2∇2φµ(x)∇2B(x)−1/2) ≥ −ǫH
+(14)
+for any prescribed tolerances ǫg, ǫH ∈ (0, 1). It will be used to solve the subproblems arising in the barrier-AL
+method later.
+Our preconditioned Newton-CG method (Algorithm 1) consists of two main components. The first main
+component is a modified CG method, referred to as capped CG method, which was proposed in [60, Algorithm 1]
+for solving a possibly indefinite linear system
+(H + 2εI) ˆd = −g,
+(15)
+where 0 ̸= g ∈ Rn, ε > 0, and H ∈ Rn×n is a symmetric matrix. The capped CG method terminates within a
+finite number of iterations and returns either an approximate solution ˆd to (15) satisfying ∥(H+2εI) ˆd+g∥ ≤ ˆζ∥g∥
+and ˆdT H ˆd ≥ −ε∥ ˆd∥2 for some ˆζ ∈ (0, 1) or a sufficiently negative curvature direction ˆd of H with ˆdT H ˆd <
+−ε∥ ˆd∥2. The second main component is a minimum eigenvalue oracle. Given a symmetric matrix H ∈ Rn×n
+and ε > 0, this oracle either produces a sufficiently negative curvature direction v of H with ∥v∥ = 1 and
+vT Hv ≤ −ε/2 or certifies that λmin(H) ≥ −ε holds with high probability. For ease of reference, we present
+these two main components in Algorithms 3 and 4 in Appendices A and B, respectively.
+We are now ready to describe our preconditioned Newton-CG method (Algorithm 1) for solving (13). At
+iteration t, if the first relation in (14) is not satisfied at the iterate xt, the capped CG method (Algorithm 3) is
+invoked to find a descent direction for φµ by solving the following damped preconditioned Newton system
+(M T
+t ∇2φµ(xt)Mt + 2ǫHI) ˆd = −M T
+t ∇φµ(xt),
+5
+
+where Mt is a matrix such that
+∇2B(xt)−1 = MtM T
+t .
+(16)
+A line search along this descent direction is then performed to result in a reduction on φµ. Otherwise, the min-
+imum eigenvalue oracle (Algorithm 4) is invoked. This oracle either produces a sufficiently negative curvature
+direction of M T
+t ∇2φµ(xt)Mt along which a line search is performed to result in a reduction on φµ, or certifies
+that the iterate xt also satisfies the second relation in (14) with high probability and terminates the precondi-
+tioned Newton-CG method. The detailed description of our preconditioned Newton-CG method is presented in
+Algorithm 1.
+Algorithm 1 A preconditioned Newton-CG method for problem (13)
+Input: tolerances ǫg, ǫH ∈ (0, 1), backtracking ratio θ ∈ (0, 1), starting point u0 ∈ int K, CG-accuracy parameter ζ ∈ (0, 1),
+maximum step length β ∈ [ǫH, 1), line-search parameter η ∈ (0, 1), probability parameter δ ∈ (0, 1);
+Set x0 = u0;
+for t = 0, 1, 2, . . . do
+if ∥∇φµ(xt)∥∗
+xt > ǫg then
+Call Algorithm 3 (see Appendix A) with H = MT
+t ∇2φµ(xt)Mt, ε = ǫH, g = MT
+t ∇φµ(xt), accuracy parameter ζ,
+and bound U = 0 to obtain outputs ˆdt, d type, where Mt is given in (16);
+if d type=NC then
+dt ← − sgn(( ˆdt)T MT
+t ∇φµ(xt)) min
+�
+|( ˆdt)T MT
+t ∇2φµ(xt)Mt ˆdt|
+∥ ˆdt∥3
+,
+β
+∥ ˆdt∥
+�
+ˆdt;
+(17)
+else {d type=SOL}
+dt ← min
+�
+1,
+β
+∥ ˆdt∥
+�
+ˆdt;
+(18)
+end if
+Go to Line Search;
+else
+Call Algorithm 4 (see Appendix B) with H = MT
+t ∇2φµ(xt)Mt, ε = ǫH, and probability parameter δ;
+if Algorithm 4 certifies that λmin(MT
+t ∇2φµ(xt)Mt) ≥ −ǫH then
+Output xt and terminate;
+else {Sufficiently negative curvature direction v returned by Algorithm 4}
+Set d type=NC and
+dt ← − sgn(vT MT
+t ∇φµ(xt)) min{|vT MT
+t ∇2φµ(xt)Mtv|, β}v;
+(19)
+Go to Line Search;
+end if
+end if
+Line Search:
+if d type=SOL then
+Find αt = θjt, where jt is the smallest nonnegative integer j such that
+φµ(xt + θjMtdt) < φµ(xt) − ηǫHθ2j∥dt∥2;
+(20)
+else {d type=NC}
+Find αt = θjt, where jt is the smallest nonnegative integer j such that
+φµ(xt + θjMtdt) < φµ(xt) − ηθ2j∥dt∥3/2;
+(21)
+end if
+xt+1 = xt + αtMtdt;
+end for
+4.1
+Iteration and operation complexity of Algorithm 1
+In this subsection we study iteration and operation complexity of Algorithm 1.
+To proceed, we make the
+following assumptions on problem (13).
+Assumption 4.1. (a) There exists a finite φlow such that
+φµ(x) ≥ φlow,
+∀x ∈ int K,
+(22)
+S = {x ∈ int K : φµ(x) ≤ φµ(u0)} is bounded,
+(23)
+6
+
+where u0 ∈ int K is the initial point of Algorithm 1 and φµ is given in (13).
+(b) There exists LF
+H > 0 such that
+∥∇2F(y) − ∇2F(x)∥∗
+x ≤ LF
+H∥y − x∥x,
+∀x, y ∈ Ω with ∥y − x∥x ≤ β,
+where Ω ⊂ int K is an open bounded convex neighborhood of S and β ∈ (0, 1) is an input of Algorithm 1.
+(c) The quantities U F
+g and U F
+H are finite, where
+U F
+g := sup
+x∈S
+∥∇F(x)∥∗
+x,
+U F
+H := sup
+x∈S
+∥∇2F(x)∥∗
+x.
+(24)
+Before establishing operation complexity of Algorithm 1, let us make some observations on its fundamental
+operations. Firstly, at iteration t, the main effort of Algorithm 1 is on the execution of Algorithm 3 or 4 with
+H = M T
+t ∇2φµ(xt)Mt. Secondly, the main computational cost of Algorithms 3 and 4 per iteration is on the
+product of H and a vector v. Consequently, it suffices to focus on computing Hv. Indeed, notice from (13) and
+(16) that
+Hv = M T
+t ∇2φµ(xt)Mtv = M T
+t ∇2F(xt)Mtv + µv.
+Thus, computing Hv consists of one Hessian-vector product of F and two matrix-vector products involving Mt
+and M T
+t , respectively. We next discuss how to efficiently compute the product of Mt or M T
+t and a vector.
+• When K is the nonnegative orthant, its associated barrier function is B(x) = − �n
+i=1 ln xi. Notice that
+∇2B(x) is a diagonal matrix and so is Mt. As a result, the operation cost for computing the product of
+Mt or M T
+t and a vector is O(n), which is typically cheaper than the Hessian-vector product of F.
+• When K is a general cone, directly computing Mt may be too expensive. In view of ∇2B(xt) = M −T
+t
+M −1
+t
+(see (16)), one can instead choose M −T
+t
+as the Cholesky factor of ∇2B(xt), which is computed only once
+in each iteration of Algorithm 1. Once M −T
+t
+is available, the product of Mt or M T
+t and a vector can be
+computed by performing backward or forward substitution to a linear system with coefficient matrix M −1
+t
+or M −T
+t
+.
+Based on the above discussion, we conclude that: (i) when K is the nonnegative orthant, the fundamental
+operations of Algorithm 1 consist only of the Hessian-vector products of F; (ii) when K is a general cone, the
+fundamental operations of Algorithm 1 consist of the Hessian-vector products of F, Cholesky factorizations of
+∇2B, and backward or forward substitutions to a triangular linear system.
+The following theorem states the iteration and operation complexity of Algorithm 1, whose proof is deferred
+to Section 7.2.
+Theorem 4.1 (Complexity of Algorithm 1). Suppose that Assumption 4.1 holds. Let
+T1 =
+�
+φhi − φlow
+min{csol, cnc} max{ǫ−2
+g ǫH, ǫ−3
+H }
+�
++
+�φhi − φlow
+cnc
+ǫ−3
+H
+�
++ 1, T2 =
+�φhi − φlow
+cnc
+ǫ−3
+H
+�
++ 1,
+(25)
+where φhi = φµ(u0), φlow is given in (22), and
+csol = η min
+��
+4(1−β)
+4+ζ+√
+(4+ζ)2+8[(1−β)LF
+H+µ(2−β)/(1−β)]
+�2
+,
+�
+min{6(1−η),2}θ
+LF
+H+µ(2−β)/(1−β)2
+�2
+�
+,
+(26)
+cnc =
+η
+16 min
+�
+1,
+�
+min{3(1−η),1}θ
+LF
+H+µ(2−β)/(1−β)2
+�2�
+.
+(27)
+Then the following statements hold.
+(i) The total number of calls of Algorithm 4 in Algorithm 1 is at most T2.
+(ii) The total number of calls of Algorithm 3 in Algorithm 1 is at most T1.
+7
+
+(iii) (iteration complexity) Algorithm 1 terminates in at most T1 + T2 iterations with
+T1 + T2 = O((φhi − φlow)(LF
+H)2 max{ǫ−2
+g ǫH, ǫ−3
+H }).
+(28)
+Moreover, its output xt satisfies the first relation in (14) deterministically and the second relation in (14)
+with probability at least 1 − δ for some 0 ≤ t ≤ T1 + T2.
+(iv) (operation complexity) The total numbers of Cholesky factorizations and other fundamental operations
+consisting of the Hessian-vector products of F and backward or forward substitutions to a triangular linear
+system required by Algorithm 1 are at most T1 + T2 and
+�O((φhi − φlow)(LF
+H)2 max{ǫ−2
+g ǫH, ǫ−3
+H } min{n, (U F
+H/ǫH)1/2}),
+respectively, where U F
+H is given in (24).
+5
+A Newton-CG based barrier-AL method for problem (1)
+In this section we propose a Newton-CG based barrier-AL method for finding a stochastic (ǫ, √ǫ)-SOSP of
+problem (1) for any prescribed tolerance ǫ ∈ (0, 1).
+Recall that B is the ϑ-LHSC barrier function associated with K for some ϑ ≥ 1. We now make the following
+additional assumptions on problem (1).
+Assumption 5.1. (a) An ǫ/2-approximately strictly feasible point zǫ of problem (1), namely satisfying zǫ ∈
+int K and ∥c(zǫ)∥ ≤ ǫ/2, is known.
+(b) There exist constants ¯µ ≥ µ, fhi, flow ∈ R and γ, δf, δc > 0, independent of ǫ, such that
+f(zǫ) + ˜µB(zǫ) ≤ fhi,
+∀˜µ ∈ (0, ¯µ],
+(29)
+f(x) + ˜µB(x) + γ∥c(x)∥2/2 ≥ flow,
+∀˜µ ∈ (0, ¯µ], x ∈ int K,
+(30)
+S(δf, δc) :=
+�
+˜µ∈(0,¯µ]
+{x ∈ int K : f(x) + ˜µB(x) ≤ fhi + δf, ∥c(x)∥ ≤ 1 + δc} is bounded,
+(31)
+where µ = ǫ/(2ϑ1/2 + 2) and zǫ is given in (a).
+(c) There exist Lf
+H, Lc
+H > 0 and β ∈ (0, 1) such that
+∥∇2f(y) − ∇2f(x)∥∗
+x ≤ Lf
+H∥y − x∥x,
+∀x, y ∈ Ω(δf, δc) with ∥y − x∥x ≤ β,
+∥∇2ci(y) − ∇2ci(x)∥∗
+x ≤ Lc
+H∥y − x∥x,
+∀x, y ∈ Ω(δf, δc) with ∥y − x∥x ≤ β, 1 ≤ i ≤ m,
+(32)
+where Ω(δf, δc) ⊂ int K is an open bounded convex neighborhood of S(δf, δc).
+(d) The quantities U f
+g , U c
+g, U f
+H and U c
+H are finite, where
+U f
+g = supx∈Ω(δf ,δc) ∥∇f(x)∥∗
+x,
+U c
+g = supx∈Ω(δf ,δc) max1≤i≤m ∥∇ci(x)∥∗
+x,
+(33)
+U f
+H = supx∈Ω(δf ,δc) ∥∇2f(x)∥∗
+x,
+U c
+H = supx∈Ω(δf,δc) max1≤i≤m ∥∇2ci(x)∥∗
+x.
+(34)
+We next make some remarks about Assumption 5.1.
+Remark 5.1.
+(i) A similar assumption as Assumption 5.1(a) was considered in the study of AL methods
+for nonconvex equality constrained optimization (e.g., see [28, 41, 44, 52, 70]). By imposing Assump-
+tion 5.1(a), we restrict our study on problem (1) for which an ǫ/2-approximately strictly feasible point zǫ
+can be found by an inexpensive procedure. As an example of such problem instances, when the generalized
+LICQ condition λmin(∇c(x)T ∇2B(x)−1∇c(x)) ≥ σ2 > 0 (see Assumption 5.2 below) holds on a level set
+of ∥c(x)∥2 + ˜µB(x) for a sufficiently small ˜µ > 0 and a constant σ, the point zǫ can be found by applying
+our preconditioned Newton-CG method (Algorithm 1) to the barrier problem minx ∥c(x)∥2 + ˜µB(x). As
+observed from Theorem 4.1, the resulting iteration and operation complexity for finding such zǫ are respec-
+tively O(ǫ−3/2) and �O(ǫ−3/2 min{n, ǫ−1/4}), which are negligible compared with those of our barrier-AL
+8
+
+method (see Theorems 5.4 and 5.5 below). In addition, the Newton-CG based barrier AL method (Algo-
+rithm 2) proposed below is a second-order method with the aim to find a second-order stationary point. It
+is more expensive than a first-order method in general. To make best use of such a barrier AL method
+in practice, it is natural to run a first-order method in advance to obtain an ǫ/2-first-order stationary
+point zǫ and then run the barrier AL method using zǫ as an ǫ/2-approximately feasible point. Therefore,
+Assumption 5.1(a) is met in practice, provided that an ǫ/2-first-order stationary point of (1) can be found
+by a first-order method.
+(ii) Assumption 5.1(b) is mild. In particular, the assumption in (29) holds if f(x)+ ¯µ[B(x)]+ is bounded above
+for all x ∈ int K with ∥c(x)∥ ≤ 1. Besides, the function f(x) + ˜µ B(x) + γ∥c(x)∥2/2 is a barrier-quadratic
+penalty function of problem (1) and is typically bounded below on int K. In addition, letting z0 be an
+arbitrary point in int K, it can be shown that S(δf, δc) ⊆ S1 ∪ S2, where
+S1 =
+�
+x ∈ int K : f(x) ≤ fhi + δf + ¯µ + ¯µ[B(z0)]+, B(x) ≥ −1 − [B(z0)]+, ∥c(x)∥ ≤ 1 + δc
+�
+,
+S2 =
+�
+x ∈ int K :
+f(x)
+−B(x) ≤
+[fhi+δf]+
+1+[B(z0)]+ + ¯µ, B(x) ≤ −1 − [B(z0)]+, ∥c(x)∥ ≤ 1 + δc
+�
+,
+and t+ = max{0, t} for all t ∈ R. Thus, the assumption in (31) holds if S1 and S2 are bounded. The latter
+holds, for example, for the problem with f(x) = ℓ(x) + �n
+i=1 xp
+i , B(x) = − �n
+i=1 ln xi and K = Rn
++ studied
+in [42], where ℓ : Rn → R+ is a loss function and p > 0.
+(iii) Assumptions 5.1(c) means that ∇2f and ∇2ci, 1 ≤ i ≤ m, are locally Lipschitz continuous in Ω(δf, δc)
+with respect to the local norms. As pointed out in [43, Section 5], such local Lipschitz continuity is weaker
+than the global Lipschitz continuity of ∇2f and ∇2ci, 1 ≤ i ≤ m, in Ω(δf, δc). Besides, Assumption 5.1(d)
+holds if f and c are twice continuously differentiable in an open set containing K.
+5.1
+A Newton-CG based barrier-AL method
+We now describe our Newton-CG based barrier-AL method (Algorithm 2) for finding a stochastic (ǫ, √ǫ)-SOSP
+of problem (1) for a prescribed tolerance ǫ ∈ (0, 1).
+Instead of solving (1) directly, our method solves the
+following perturbed equality constrained barrier problem
+min
+x
+{f(x) + µB(x) : ˜c(x) := c(x) − c(zǫ) = 0}
+(35)
+with µ = ǫ/(2ϑ1/2+2) and zǫ given in Assumption 5.1(a). It follows a similar AL framework as the one proposed
+in [44]. In particular, at the kth iteration, it first applies the preconditioned Newton-CG method (Algorithm 1)
+to find an approximate stochastic SOSP xk+1 of the subproblem:
+min
+x
+�
+Lµ(x, λk; ρk) := f(x) + µB(x) + (λk)T ˜c(x) + ρk
+2 ∥˜c(x)∥2�
+,
+(36)
+which is an AL subproblem associated with (35). Then the standard multiplier estimate ˜λk+1 is updated by
+the classical scheme (see step 3 of Algorithm 2), and the truncated Lagrangian multiplier λk+1 is updated by
+projecting ˜λk+1 onto a Euclidean ball (see step 5 of Algorithm 2).4 Finally, the penalty parameter ρk+1 is
+adaptively updated according to the improvement on constraint violation (see step 6 of Algorithm 2). This
+update scheme is very practical and widely used in AL type methods (e.g., see [3, 8, 28]).
+Remark 5.2.
+(i) Notice that the starting point x0
+init of Algorithm 2 can be different from zǫ and it may be
+rather infeasible, though zǫ is a nearly feasible point of (1). Besides, zǫ is used to monitor convergence
+of Algorithm 2. Specifically, if the algorithm runs into a “poorly infeasible point” xk, namely satisfying
+Lµ(xk, λk; ρk) > f(zǫ)+µB(zǫ), it will be superseded by zǫ (see (39)), which prevents the iterates {xk} from
+converging to an infeasible point. Yet, xk may be rather infeasible when k is not large. Thus, Algorithm 2
+substantially differs from a funneling or two-phase type algorithm, in which a nearly feasible point is found
+in Phase 1, and then approximate stationarity is sought while near feasibility is maintained throughout
+Phase 2 (e.g., see [11, 16, 22, 23, 24, 25, 26, 36]).
+4The λk+1 is also called a safeguarded Lagrangian multiplier, which has been used in the literature for designing some AL
+methods (e.g., see [3, 13, 44, 47]). It has been shown to enjoy many practical and theoretical advantages (e.g., see [13]).
+9
+
+Algorithm 2 A Newton-CG based barrier-AL method for problem (1)
+Let γ and µ be given in Assumption 5.1.
+Input: ǫ ∈ (0, 1), Λ ≥ 0, x0 ∈ int K, λ0 ∈ BΛ, ρ0 > 2γ, α ∈ (0, 1), r > 1, δ ∈ (0, 1), zǫ given in Assumption 5.1(a), and
+τk = max{µ, rk log µ/ log 2} for all k ≥ 0.
+1: Set k = 0.
+2: Call Algorithm 1 with ǫg = τk, ǫH = √τk and u0 = xk
+init to find an approximate solution xk+1 ∈ int K to
+minx Lµ(x, λk; ρk) such that
+Lµ(xk+1, λk; ρk) ≤ f(zǫ) + µB(zǫ),
+∥∇x Lµ(xk+1, λk; ρk)∥∗
+xk+1 ≤ τk,
+(37)
+λmin(M T
+k+1∇2
+xxLµ(xk+1, λk; ρk)Mk+1) ≥ −√τk with probability at least 1 − δ,
+(38)
+where Mk+1 is defined as in (16) and
+xk
+init =
+� zǫ
+if Lµ(xk, λk; ρk) > f(zǫ) + µB(zǫ),
+xk
+otherwise,
+for k ≥ 0.
+(39)
+3: Set ˜λk+1 = λk + ρk˜c(xk+1).
+4: If τk ≤ µ and ∥c(xk+1)∥ ≤ ǫ, then output (xk+1, ˜λk+1) and terminate.
+5: Set λk+1 = ΠBΛ(˜λk+1).
+6: If k = 0 or ∥˜c(xk+1)∥ > α∥˜c(xk)∥, set ρk+1 = rρk. Otherwise, set ρk+1 = ρk.
+7: Set k ← k + 1, and go to step 2.
+(ii) The choice of ρ0 in Algorithm 2 is mainly for the simplicity of complexity analysis. Yet, it may be overly
+large and lead to highly ill-conditioned AL subproblems in practice. To make Algorithm 2 practically more
+efficient, one can possibly modify it by choosing a relatively small initial penalty parameter, then solving
+the subsequent AL subproblems by a first-order method until an ǫ1-first-order stationary point ˆx of (35)
+along with a Lagrangian multiplier ˆλ is found, and finally performing the steps described in Algorithm 2
+but with x0 = ˆx and λ0 = ΠBΛ(ˆλ).
+(iii) Algorithm 2 can be easily extended to find an (ǫ, √ǫ)-SOSP of a more general conic optimization problem
+of the form minx,y{ ˜f(x, y) : ˜c(x, y) = 0, y ∈ K}.
+Indeed, one can follow almost the same framework
+as Algorithm 2, except that the associated subproblems are solved by a preconditioned Newton-CG method,
+which is a slight modification of Algorithm 1 by choosing the preconditioning matrix �
+Mk as the one satisfying
+�I
+0
+0
+∇2B(yk)
+�−1
+= �
+Mk �
+M T
+k .
+Before analyzing the complexity of Algorithm 2, we first argue that it is well-defined if ρ0 is suitably chosen.
+Specifically, we will show that when ρ0 is sufficiently large, one can apply Algorithm 1 to the subproblem
+minx Lµ(x, λk; ρk) with xk
+init as the initial point to find an xk+1 satisfying (37) and (38). To this end, we start
+by noting from (29), (35), (36) and (39) that
+Lµ(xk
+init, λk; ρk)
+(39)
+≤ max{Lµ(zǫ, λk; ρk), f(zǫ) + µB(zǫ)}
+(35)(36)
+=
+f(zǫ) + µB(zǫ)
+(29)
+≤ fhi.
+(40)
+Based on this observation, we show in the next lemma that when ρ0 is sufficiently large, Lµ(·, λk; ρk) is bounded
+below and its certain level set is bounded, whose proof is deferred to Section 7.2.
+Lemma 5.1 (Properties of Lµ(·, λk; ρk) and L(·, λk; ρk)). Suppose that Assumption 5.1 holds. Let (λk, ρk)
+be generated at the kth iteration of Algorithm 2 for some k ≥ 0, and
+L(x, λk; ρk) := f(x) + (λk)T ˜c(x) + ρk
+2 ∥˜c(x)∥2.
+(41)
+Let S(δf, δc) and xk
+init be respectively defined in (31) and (39) and let δf, δc, µ, fhi, flow, Lf
+H, Lc
+H, U f
+H, U c
+g, U c
+H
+and Ω(δf, δc) be given in Assumption 5.1. Suppose that ρ0 is sufficiently large such that δf,1 ≤ δf and δc,1 ≤ δc,
+where
+δf,1 := Λ2/(2ρ0) and δc,1 :=
+�
+2(fhi − flow + γ)
+ρ0 − 2γ
++
+Λ2
+(ρ0 − 2γ)2 +
+Λ
+ρ0 − 2γ .
+(42)
+10
+
+Then the following statements hold.
+(i) {x ∈ int K : Lµ(x, λk; ρk) ≤ Lµ(xk
+init, λk; ρk)} ⊆ S(δf, δc).
+(ii) infx∈int K Lµ(x, λk; ρk) ≥ flow − γ − Λδc.
+(iii) ∥∇2
+xx L(y, λk; ρk) − ∇2
+xx L(x, λk; ρk)∥∗
+x ≤ Lk,H∥y − x∥x for all x, y ∈ Ω(δf, δc) with ∥y − x∥x ≤ β, where
+Lk,H := Lf
+H + ∥λk∥1Lc
+H + ρkm
+�
+(1 + U c)Lc
+H + U c
+gU c
+H
+1 − β + (2 − β)U c
+gU c
+H
+(1 − β)3
+�
+, U c :=
+sup
+z∈Ω(δf ,δc)
+∥c(z)∥.
+(43)
+(iv) The quantities Uk,g and Uk,H are finite, where
+Uk,g :=
+sup
+x∈S(δf ,δc)
+∥∇x L(x, λk; ρk)∥∗
+x,
+Uk,H :=
+sup
+x∈S(δf,δc)
+∥∇2
+xx L(x, λk; ρk)∥∗
+x.
+Moreover, Uk,H ≤ U f
+H + ∥λk∥1U c
+H + ρk(m(U c
+g)2 + √m(2 + δc)U c
+H).
+In view of (31) and Lemma 5.1(i) and (ii), one can see that the level set {x ∈ int K : Lµ(x, λk; ρk) ≤
+Lµ(xk
+init, λk; ρk)} is bounded and Lµ(x, λk; ρk) is bounded below for all x ∈ int K. By these and Lemma 5.1(iii)
+and (iv), one can further see that Assumption 4.1 holds for F(·) = L(·, λk; ρk) and u0 = xk
+init. Based on this
+and the discussion in Section 4, we can conclude that Algorithm 1, starting with u0 = xk
+init, is applicable to
+the subproblem minx Lµ(x, λk; ρk). Moreover, it follows from Theorem 4.1 that this algorithm with ǫg = τk
+and ǫH = √τk can produce a point xk+1 satisfying (38) and also the second relation in (37). In addition,
+since this algorithm is descent and its starting point is xk
+init, its output xk+1 must satisfy Lµ(xk+1, λk; ρk) ≤
+Lµ(xk
+init, λk; ρk), which along with (40) implies that Lµ(xk+1, λk; ρk) ≤ f(zǫ) + µB(zǫ) and thus xk+1 also
+satisfies the first relation in (37).
+The above discussion leads to the following conclusion concerning the well-definedness of Algorithm 2.
+Theorem 5.1 (Well-definedness of Algorithm 2). Under the same settings as in Lemma 5.1, the precon-
+ditioned Newton-CG method (Algorithm 1), when applied to the subproblem minx Lµ(x, λk; ρk) with u0 = xk
+init,
+can find a point xk+1 satisfying (37) and (38).
+The following theorem characterizes the output of Algorithm 2, whose proof is deferred to Section 7.3.
+Theorem 5.2 (Output of Algorithm 2). Suppose that Assumption 5.1 holds and that ρ0 is sufficiently large
+such that δf,1 ≤ δf and δc,1 ≤ δc, where δf,1 and δc,1 are defined in (42). If Algorithm 2 terminates at some
+iteration k, then its output xk+1 is a deterministic ǫ-FOSP of problem (1), and moreover, it is an (ǫ, √ǫ)-SOSP
+of (1) with probability at least 1 − δ.
+Remark 5.3. As seen from Theorem 5.2, the output of Algorithm 2 is a stochastic (ǫ, √ǫ)-SOSP of problem (1).
+On the other hand, this algorithm can be easily modified to find other approximate solutions of (1) as well. For
+example, if only an ǫ-FOSP of (1) is to be sought, one can remove the condition (38) from Algorithm 2. In
+addition, if one aims to find a deterministic (ǫ, √ǫ)-SOSP of (1), one can replace the condition (38) and
+Algorithm 1 by λmin(M T
+k+1∇2
+xxLµ(xk+1, λk; ρk)Mk+1) ≥ −√τk and a deterministic counterpart, respectively.
+5.2
+Outer iteration complexity of Algorithm 2
+In this subsection we establish outer iteration complexity of Algorithm 2, which measures the number of its
+outer iterations. Notice that τk can be rewritten as
+τk = max{µ, ωk} with ω := rlog µ/ log 2,
+∀k ≥ 0,
+(44)
+where r is an input of Algorithm 2 and µ = ǫ/(2ϑ1/2 + 2) (see Assumption 5.1(b)). By ǫ ∈ (0, 1), ϑ ≥ 1, and
+the definition of µ, one can verify that µ ∈ (0, 1). By this and r > 1, one can see that ω ∈ (0, 1). For notational
+convenience, we introduce the following quantity that will be frequently used later:
+Kǫ :=
+�
+min{k ≥ 0 : ωk ≤ ǫ/(2ϑ1/2 + 2)}
+�
+=
+�
+min{k ≥ 0 : ωk ≤ µ}
+�
+.
+(45)
+11
+
+In view of this and (44), we obtain that τk = µ for all k ≥ Kǫ. This along with the termination criterion of
+Algorithm 2 implies that it runs for at least Kǫ iterations and terminates once ∥c(xk+1)∥ ≤ ǫ for some k ≥ Kǫ.
+Consequently, to establish outer iteration complexity of Algorithm 2, it suffices to bound such k. The resulting
+outer iteration complexity is presented below, whose proof is deferred to Section 7.3.
+Theorem 5.3 (Outer iteration complexity of Algorithm 2). Suppose that Assumption 5.1 holds and that
+ρ0 is sufficiently large such that δf,1 ≤ δf and δc,1 ≤ δc, where δf,1 and δc,1 are defined in (42). Let
+ρǫ,1 := max
+�
+8(fhi − flow + γ)ǫ−2 + 4Λǫ−1 + 2γ, 2ρ0
+�
+,
+(46)
+Kǫ := inf{k ≥ Kǫ : ∥c(xk+1)∥ ≤ ǫ},
+(47)
+where Kǫ is defined in (45), and γ, fhi and flow are given in Assumption 5.1. Then Kǫ is finite, and Algorithm 2
+terminates at iteration Kǫ with
+Kǫ ≤
+�log(ρǫ,1ρ−1
+0 )
+log r
++ 1
+� �����
+log(ǫ(2δc,1)−1)
+log α
+���� + 2
+�
++ 1.
+(48)
+Moreover, ρk ≤ rρǫ,1 holds for 0 ≤ k ≤ Kǫ.
+Remark 5.4 (Upper bounds for Kǫ and {ρk}). As seen from Theorem 5.3, the number of outer iterations
+of Algorithm 2 for finding a stochastic (ǫ, √ǫ)-SOSP of problem (1) is at most of O(| log ǫ|2). In addition, the
+penalty parameters {ρk} generated by this algorithm are at most of O(ǫ−2).
+5.3
+Total inner iteration and operation complexity of Algorithm 2
+In this subsection we present the total inner iteration and operation complexity of Algorithm 2, which measures
+the total number of iterations and fundamental operations performed by Algorithm 1 in Algorithm 2. Its proof
+is deferred to Section 7.3.
+Theorem 5.4 (Total inner iteration and operation complexity of Algorithm 2). Suppose that Assump-
+tion 5.1 holds and that ρ0 is sufficiently large such that δf,1 ≤ δf and δc,1 ≤ δc, where δf,1 and δc,1 are defined
+in (42). Then the following statements hold.
+(i) The total number of inner iterations of Algorithm 2, namely, the total number of iterations of Algorithm 1
+performed in Algorithm 2, is at most �O(ǫ−11/2). If c is further assumed to be affine, it is at most �O(ǫ−3/2).
+(ii) The total numbers of Cholesky factorizations and other fundamental operations consisting of the Hessian-
+vector products of f and c and backward or forward substitutions to a triangular linear system required by
+Algorithm 1 in Algorithm 2 are at most �O(ǫ−11/2) and �O(ǫ−11/2 min{n, ǫ−5/4}), respectively. If c is further
+assumed to be affine, they are at most �O(ǫ−3/2) and �O(ǫ−3/2 min{n, ǫ−5/4}), respectively.
+Remark 5.5. It is worth mentioning that the above complexity results are established without assuming any
+constraint qualification.
+Moreover, when K is the nonnegative orthant, these results match the best known
+ones achieved by a Newton-CG based AL method [44] for nonconvex equality constrained optimization without
+imposing a constraint qualification.
+5.4
+Enhanced complexity of Algorithm 2 under constraint qualification
+In this subsection we study complexity of Algorithm 2 under one additional assumption that a generalized linear
+independence constraint qualification (GLICQ) holds for problem (1), which is introduced below. In particular,
+under GLICQ we will obtain an enhanced total inner iteration complexity of �O(ǫ−7/2) and an enhanced operation
+complexity of �O(ǫ−7/2 min{n, ǫ−3/4}) for Algorithm 2 when the equality constraints in problem (1) are nonlinear,
+which are significantly better than the ones in Theorem 5.4. We now introduce the GLICQ assumption for (1).
+Assumption 5.2 (GLICQ). There exists some σ > 0 such that
+λmin(∇c(x)T ∇2B(x)−1∇c(x)) ≥ σ2,
+∀x ∈ S(δf, δc),
+(49)
+where S(δf, δc) is defined in (31).
+12
+
+The following theorem shows that under Assumption 5.2, the total inner iteration and operation complexity
+results presented in Theorem 5.4 can be significantly improved, whose proof is deferred to Section 7.3.
+Theorem 5.5 (Enhanced total inner iteration and operation complexity of Algorithm 2). Suppose
+that Assumptions 5.1 and 5.2 hold and that ρ0 is sufficiently large such that δf,1 ≤ δf and δc,1 ≤ δc, where δf,1
+and δc,1 are defined in (42). Then the following statements hold.
+(i) The total number of inner iterations of Algorithm 2, namely, the total number of iterations of Algorithm 1
+performed in Algorithm 2, is at most �O(ǫ−7/2). If c is further assumed to be affine, it is at most �O(ǫ−3/2).
+(ii) The total numbers of Cholesky factorizations and other fundamental operations consisting of the Hessian-
+vector products of f and c and backward or forward substitutions to a triangular linear system required by
+Algorithm 1 in Algorithm 2 are at most �O(ǫ−7/2) and �O(ǫ−7/2 min{n, ǫ−3/4}), respectively. If c is further
+assumed to be affine, they are at most �O(ǫ−3/2) and �O(ǫ−3/2 min{n, ǫ−3/4}), respectively.
+Remark 5.6. As seen from Theorem 5.5, under GLICQ and some other suitable assumptions, Algorithm 2
+achieves significantly better complexity bounds than the ones in Theorem 5.4 when the equality constraints in (1)
+are nonlinear. Moreover, when K is the nonnegative orthant, the complexity results in Theorem 5.5 match the
+best known ones achieved by a Newton-CG based AL method [44] for nonconvex equality constrained optimization
+under the constraint qualification that is obtained from the above GLICQ by replacing ∇2B(x) by the identity
+matrix.
+6
+Numerical results
+In this section we conduct some preliminary numerical experiments to test performance of our Newton-CG based
+barrier-AL method (Algorithm 2) for solving a low-rank matrix recovery problem and a simplex-constrained
+nonnegative matrix factorization problem. In our experiments, all the algorithms are coded in Matlab and all
+the computations are performed on a desktop with a 3.79 GHz AMD 3900XT 12-Core processor and 32 GB of
+RAM.
+6.1
+Low-rank matrix recovery
+In this subsection we consider a low-rank matrix recovery problem (e.g., see [9, 29, 59])
+min
+U∈Rn×k
+�1
+2∥ A(UU T ) − y∥2 : ∥U∥2
+F ≤ b
+�
+,
+(50)
+where A : Rn×n → Rm is a linear operator and ∥ · ∥F is the Frobenius norm.
+For each triple (n, k, m), we randomly generate 10 instances of problem (50) in a similar manner as described
+in [9]. In particular, we first randomly generate a linear operator A by setting A(·) = A(vec(·)), where A is
+an m × n2 matrix with all entries chosen from the standard normal distribution, and vec(·) is the vectorization
+of the associated matrix.5 Then we randomly generate the ground-truth low-rank matrix X∗ = �U �U T with all
+entries of �U chosen from the standard normal distribution. We finally set b = ∥ �U|2
+F and y = A(X∗) + e, where
+ei, 1 ≤ i ≤ m, is generated according to the normal distribution with mean zero and standard deviation 0.01.
+Observe that problem (50) is equivalent to
+min
+U,s
+�1
+2∥ A(UU T ) − y∥2 : ∥U∥2
+F + s = b, s ≥ 0
+�
+.
+(51)
+In this experiment, we apply Algorithm 2 to find a (10−4, 10−2)-SOSP of (51) and hence of (50). To ensure
+that the output of Algorithm 2 is a deterministic approximate second-order stationary point, we use a minimum
+eigenvalue oracle that returns a deterministic output in Algorithm 2 instead, which calls the Matlab subroutine
+[v,λ] = eigs(H,1,’smallestreal’) to find the minimum eigenvalue λ and its associated unit eigenvector v of a real
+5The vectorization of a matrix is the column vector obtained by stacking the columns of the matrix on top of one another.
+13
+
+Relative error
+Objective value
+n
+k
+m
+Algorithm 2
+SpaRSA
+Algorithm 2
+SpaRSA
+20
+1
+40
+6.3×10−4
+6.3×10−4
+9.9×10−4
+9.8×10−4
+20
+2
+80
+3.3×10−4
+0.60
+2.0×10−3
+7.8×103
+40
+2
+160
+1.7×10−4
+0.66
+4.2×10−3
+7.1×104
+40
+4
+320
+1.2×10−4
+0.81
+8.0×10−3
+5.5×105
+60
+3
+360
+9.2×10−5
+0.78
+9.3×10−3
+8.4×105
+60
+6
+720
+6.3×10−5
+0.85
+1.9×10−2
+5.1×106
+80
+4
+640
+5.8×10−5
+0.83
+1.6×10−2
+4.4×106
+80
+8
+1280
+3.9×10−5
+0.90
+3.3×10−2
+2.5×107
+100
+5
+1000
+4.2×10−5
+0.89
+2.6×10−2
+1.6×107
+100
+10
+2000
+2.8×10−5
+0.92
+5.2×10−2
+8.1×107
+Table 1: Numerical results for problem (50)
+symmetric matrix H. Besides, we apply [68, Algorithm SpaRSA], which is a nonmonotone proximal gradient
+method, to find a 10−4-FOSP of (50) by generating a sequence {U t} according to
+U t = arg min
+U {∥U − U t−1 + ∇f(U t−1)/αt−1∥F : ∥U∥2
+F ≤ b},
+where f is the objective function of (50) and αt−1 is chosen by a backtracking line search scheme such that
+f(U t) ≤ max[t−M−1]+≤i≤t−1 f(U i) − σαt−1∥U t − U t−1∥2
+F /2 for some σ ∈ (0, 1) and a positive integer M (see
+[68] for details). We terminate SpaRSA once the condition
+∥αt−1(U t − U t−1) + ∇f(U t−1) − ∇f(U t)∥F ≤ 10−4
+is met. It can be verified that such U t is a 10−4-FOSP of (50). We choose the initial point U 0 with all entries
+equal
+�
+b/(2nk) for both methods, s0 = b/2 for Algorithm 2, and set
+• (Λ, ρ0, λ0, α, r) = (103, 102, 0, 0.25, 1.5) for Algorithm 2, and (θ, ζ, η, β) = (0.5, 0.5, 0.01, 0.9) for Algo-
+rithm 1;
+• (σ, M, αmin, αmax, η) = (0.01, 5, 10−30, 1030, 2) for SpaRSA [68].
+Notice that the approximate solution obtained by SpaRSA must be a feasible point of (50), while the one
+found by Algorithm 2 may not be a feasible point of (50). For a fair comparison, we project the latter one
+into the feasible region of (50) to obtain a feasible approximate solution. Then we compare the quality of these
+feasible approximate solutions in terms of objective value and relative error defined as ∥UU T −X∗∥F /∥X∗∥F for
+a given U. The computational results of Algorithm 2 and SpaRSA for the instances randomly generated above
+are presented in Table 1. In detail, the values of n, k and m are listed in the first three columns, respectively.
+For each triple (n, k, m), the average relative error and the average objective value of the feasible approximate
+solutions found by each method over 10 random instances are given in the rest columns. One can observe that
+the approximate SOSP found by Algorithm 2 has significantly lower relative error and objective value than the
+approximate FOSP obtained by SpaRSA.
+6.2
+A simplex-constrained nonnegative matrix factorization
+In this subsection we consider a simplex-constrained nonnegative matrix factorization (e.g., see [45, 50, 54, 64])
+in the form of
+min
+U∈Rn×k,V ∈Rk×m
+�1
+2∥X − UV ∥2
+F + γ(∥U∥2
+F + ∥V ∥2
+F ) : V T ek = em, U ≥ 0, V ≥ 0
+�
+(52)
+for some γ > 0, where ∥ · ∥F is the Frobenius norm and ed is the d-dimensional all-ones vector for any d ≥ 1.
+For each triple (n, k, m), we randomly generate 10 instances of problem (52). In particular, we first randomly
+generate U ∗ with all entries chosen from the uniform distribution over [0, 2]. We next randomly generate �V
+14
+
+Relative error
+Objective value
+n
+k
+m
+Algorithm 2
+SpaRSA
+Algorithm 2
+SpaRSA
+20
+2
+10
+4.8×10−3
+0.15
+0.30
+3.1
+20
+2
+20
+3.6×10−3
+0.16
+0.35
+6.4
+20
+2
+30
+3.2×10−3
+0.16
+0.39
+9.7
+30
+3
+15
+5.8×10−3
+0.16
+0.62
+7.6
+30
+3
+30
+4.3×10−3
+0.17
+0.70
+16.1
+30
+3
+45
+3.6×10−3
+0.17
+0.76
+23.8
+40
+4
+20
+6.3×10−3
+0.15
+1.0
+11.1
+40
+4
+40
+4.6×10−3
+0.15
+1.2
+21.7
+40
+4
+60
+4.0×10−3
+0.15
+1.2
+31.8
+50
+5
+25
+6.8×10−3
+0.14
+1.6
+15.1
+50
+5
+50
+5.0×10−3
+0.14
+1.8
+29.8
+50
+5
+75
+4.3×10−3
+0.14
+1.9
+43.9
+Table 2: Numerical results for problem (52)
+with all entries chosen from the standard uniform distribution and set V ∗ = �V D, where D is a diagonal matrix
+such that (V ∗)T ek = em. In addition, we set γ = 0.005 and X = U ∗V ∗ + E, where the entries of E follow the
+normal distribution with mean zero and standard deviation 0.01.
+Our aim is to apply Algorithm 2 and SpaRSA [68] to solve (52) and compare the solution quality of these
+methods in terms of objective value and relative error defined as ∥UV − U ∗V ∗∥F /∥U ∗V ∗∥F. In particular,
+we first apply Algorithm 2 to find a (10−4, 10−2)-SOSP of (52), in which a minimum eigenvalue oracle that
+returns a deterministic output, namely the Matlab subroutine [v,λ] = eigs(H,1,’smallestreal’) is used. Given
+that the obtained approximate SOSP may not be a feasible point of (52), we post-multiply it by a suitable
+diagonal matrix to obtain a feasible approximate solution of (52). In addition, we apply SpaRSA [68] to find a
+10−4-FOSP of (52) by generating a sequence {(U t, V t)} according to
+(U t, V t) = arg min
+U,V
+�
+∥(U, V ) − (U t−1, V t−1) + ∇f(U t−1, V t−1)/αt−1∥F : V T ek = em, U ≥ 0, V ≥ 0
+�
+,
+where f is the objective function of (52) and αt−1 is chosen by a backtracking line search scheme such that
+f(U t, V t) ≤ max[t−M−1]+≤i≤t−1 f(U i, V i) − σαt−1∥(U t, V t) − (U t−1, V t−1)∥2
+F /2 for some σ ∈ (0, 1) and a
+positive integer M (see [68] for details). We terminate SpaRSA once the condition
+∥αt−1((U t, V t) − (U t−1, V t−1)) + ∇f(U t−1, V t−1) − ∇f(U t, V t)∥F ≤ 10−4
+is met. It can be verified that such (U t, V t) is a 10−4-FOSP of (52). In addition, we choose the initial point U 0
+and V 0 with all entries equal 1 and 1/k respectively for all the methods. We set the parameters for Algorithm 2
+as (Λ, ρ0, α, r) = (103, 102, 0.25, 1.5) and λ0 = (0, . . . , 0)T , and choose the same parameters for Algorithm 1 and
+SpaRSA as the ones described in Subsection 6.1.
+The computational results of Algorithm 2 and SpaRSA [68] for the instances randomly generated above are
+presented in Table 2. In detail, the values of n, k and m are listed in the first three columns, respectively.
+For each triple (n, k, m), the average relative error and the average objective value of the feasible approximate
+solutions found by each method over 10 random instances are given in the rest columns. One can observe that
+the approximate SOSP found by Algorithm 2 has significantly lower relative error and objective value than the
+approximate FOSP obtained by SpaRSA.
+7
+Proof of the main results
+In this section we provide a proof of our main results presented in Sections 3, 4, and 5, which are, particularly,
+Theorems 3.1 and 4.1, Lemma 5.1, and Theorems 5.2, 5.3, 5.4, and 5.5.
+Let us start with the following lemma concerning some properties of the ϑ-LHSC barrier function.
+15
+
+Lemma 7.1. Let x ∈ int K and β ∈ (0, 1) be given. Then the following statements hold for the ϑ-LHSC barrier
+function B.
+(i) (∥∇B(x)∥∗
+x)2 = −xT ∇B(x) = ∥x∥2
+x = ϑ.
+(ii) −∇B(x) ∈ int K∗.
+(iii) {y : ∥y − x∥x < 1} ⊂ int K.
+(iv) For any y satisfying ∥y − x∥x ≤ β, it holds that
+(1 − β)∥v∥x ≤ ∥v∥y ≤ (1 − β)−1∥v∥x,
+∀v ∈ Rn,
+(53)
+(1 − β)∥v∥∗
+x ≤ ∥v∥∗
+y ≤ (1 − β)−1∥v∥∗
+x,
+∀v ∈ Rn .
+(54)
+(v) {s : ∥s + ∇B(x)∥∗
+x ≤ 1} ⊆ K∗.
+(vi) ∥∇2B(y) − ∇2B(x)∥∗
+x ≤
+2−β
+(1−β)2 ∥y − x∥x holds for all y with ∥y − x∥x ≤ β.
+Proof. The proof of statements (i)-(v) can be found in [43, Lemma 1].
+We next prove statement (vi). Let y be such that ∥y − x∥x ≤ β. It follows from [56, Theorem 2.1.1] that
+(1 − ∥y − x∥x)2I ⪯ ∇2B(x)−1/2∇2B(y)∇2B(x)−1/2 ⪯ (1 − ∥y − x∥x)−2I.
+(55)
+By (4), (55), and ∥y − x∥x ≤ β, one has
+∥∇2B(y) − ∇2B(x)∥∗
+x
+(4)
+=
+max∥u∥≤1 ∥∇2B(x)−1/2(∇2B(y) − ∇2B(x))∇2B(x)−1/2u∥
+=
+∥∇2B(x)−1/2∇2B(y)∇2B(x)−1/2 − I∥
+(55)
+≤
+max{1 − (1 − ∥y − x∥x)2, (1 − ∥y − x∥x)−2 − 1} = (1 − ∥y − x∥x)−2 − 1
+=
+2−∥y−x∥x
+(1−∥y−x∥x)2 ∥y − x∥x ≤
+2−β
+(1−β)2 ∥y − x∥x,
+where the last inequality is due to ∥y − x∥x ≤ β. Hence, statement (vi) holds as desired.
+7.1
+Proof of the main results in Section 3
+In this subsection we provide a proof of Theorems 3.1.
+Proof of Theorem 3.1. By M ∈ ∇−2B(x∗), the full column rank of M 1/2∇c(x∗), and also the discussion in
+Section 3, one knows that Robinson’s constraint qualification holds at x∗. Since x∗ is a local minimizer of (1),
+it then follows from [62, Theorem 3.38] that there exists some λ∗ ∈ Rm such that
+∇f(x∗) + ∇c(x∗)λ∗ ∈ − N K(x∗).
+(56)
+Further, by [43, Proposition 1], one knows that (56) holds if and only if (5) and (6) hold. Consequently, (5) and
+(6) hold as desired.
+We next prove (7). It follows from Lemma 2.1 that {x∗ + M 1/2d : ∥d∥ < 1} ⊆ K. Using this and the fact
+that x∗ is a local minimizer of (1), we see that d∗ = 0 is a local minimizer of the problem
+min
+d
+�
+f(x∗ + M 1/2d) : c(x∗ + M 1/2d) = 0
+�
+.
+(57)
+In addition, since M 1/2∇c(x∗) has full column rank, it is clear that LICQ holds at d∗ = 0 for (57). By the first-
+and second-order optimality conditions of (57) at d∗ = 0, there exists some ˜λ∗ ∈ Rm such that
+M 1/2(∇f(x∗) + ∇c(x∗)˜λ∗) = 0,
+(58)
+dT M 1/2
+�
+∇2f(x∗) +
+m
+�
+i=1
+˜λ∗
+i ∇2ci(x∗)
+�
+M 1/2d ≥ 0,
+∀d ∈ {d : ∇c(x∗)T M 1/2d = 0}.
+(59)
+In view of (6), (58), and the fact that M 1/2∇c(x∗) has full column rank, one can see that ˜λ∗ = λ∗. Using this
+and (59), we conclude that (7) holds.
+16
+
+7.2
+Proof of the main results in Section 4
+In this subsection we first establish several technical lemmas and then use them to prove Theorem 4.1.
+As a consequence of Assumption 4.1(b) and Lemma 7.1(vi), one can observe that φµ is locally Lipschitz
+continuous in Ω with respect to the local norms, i.e.,
+∥∇2φµ(y) − ∇2φµ(x)∥∗
+x ≤ Lφ
+H∥y − x∥x,
+∀x, y ∈ Ω with ∥y − x∥x ≤ β,
+(60)
+where
+Lφ
+H := LF
+H + µ(2 − β)/(1 − β)2.
+(61)
+The following lemma directly follows from (60). Its proof can be found in [43, Lemma 3].
+Lemma 7.2. Under Assumption 4.1(b), the following inequalities hold:
+∥∇φµ(y) − ∇φµ(x) − ∇2φµ(x)(y − x)∥∗
+x ≤ 1
+2Lφ
+H∥y − x∥2
+x, ∀x, y ∈ Ω with ∥y − x∥x ≤ β,
+(62)
+φµ(y) ≤ φµ(x)+∇φµ(x)T (y−x)+ 1
+2(y−x)T ∇2φµ(x)(y−x)+ 1
+6Lφ
+H∥y−x∥3
+x, ∀x, y ∈ Ω with ∥y−x∥x ≤ β, (63)
+where Ω and Lφ
+H are given in Assumption 4.1(b) and (61), respectively.
+The following lemma shows that all iterates generated by Algorithm 1 lie in S.
+Lemma 7.3. Suppose that Assumption 4.1 holds. Let {xt}t∈T be all the iterates generated by Algorithm 1,
+where T is a subset of consecutive nonnegative integers starting from 0. Then {xt}t∈T ⊂ S, where S is defined
+in (23).
+Proof. We first prove {xt}t∈T ⊂ int K by induction. Observe from Algorithm 1 that x0 = u0 ∈ int K. Suppose
+that xt ∈ int K is generated at iteration t of Algorithm 1 and xt+1 is generated at iteration t+ 1. We next prove
+xt+1 ∈ int K. Indeed, observe from Algorithm 1 that xt+1 = xt + αtMtdt with αt ∈ (0, 1] and dt given in one of
+(17)-(19). It follows from (17)-(19) that ∥dt∥ ≤ β. By these and (16), one has
+∥xt+1 − xt∥xt = αt∥Mtdt∥xt ≤ ∥Mtdt∥xt
+(16)
+= ∥dt∥ ≤ β,
+(64)
+which, along with xt ∈ int K, β < 1 and Lemma 7.1(iii), implies that xt+1 ∈ int K. Hence, the induction is
+completed, and we have {xt}t∈T ⊂ int K.
+In addition, observe from Algorithm 1 that {φµ(xt)}t∈T is descent. By this, x0 = u0, {xt}t∈T ⊂ int K, and
+(23), one can see that {xt}t∈T ⊂ S.
+The following lemma provides some properties of the output of Algorithm 3, whose proof is similar to the
+ones of [60, Lemma 3] and [58, Lemma 7] and thus omitted here.
+Lemma 7.4. Suppose that Assumption 4.1 holds and the direction dt results from the output ˆdt of Algo-
+rithm 3 with a type specified in d type at some iteration t of Algorithm 1.
+Let Mt be given in (16) and
+γt := max{∥ ˆdt∥/β, 1}. Then the following statements hold.
+(i) If d type=SOL, then dt satisfies
+ǫH∥dt∥2 ≤ (dt)T �
+M T
+t ∇2φµ(xt)Mt + 2ǫHI
+�
+dt,
+(65)
+∥dt∥ ≤ 1.1ǫ−1
+H ∥M T
+t ∇φµ(xt)∥,
+(66)
+(dt)T M T
+t ∇φµ(xt) = −γt(dt)T �
+M T
+t ∇2φµ(xt)Mt + 2ǫHI
+�
+dt.
+(67)
+If ∥ ˆdt∥ ≤ β, then dt also satisfies
+∥(M T
+t ∇2φµ(xt)Mt + 2ǫHI)dt + M T
+t ∇φµ(xt)∥ ≤ ǫHζ∥dt∥/2.
+(68)
+(ii) If d type=NC, then dt satisfies (dt)T M T
+t ∇φµ(xt) ≤ 0 and
+(dt)T M T
+t ∇2φµ(xt)Mtdt
+∥dt∥2
+≤ −∥dt∥ ≤ −ǫH.
+(69)
+17
+
+The following lemma shows that when the search direction dt in Algorithm 1 is of type ‘SOL’, the line search
+step results in a sufficient reduction on φµ.
+Lemma 7.5. Suppose that Assumption 4.1 holds and the direction dt results from the output ˆdt of Algorithm 3
+with d type=SOL at some iteration t of Algorithm 1. Then the following statements hold.
+(i) The step length αt is well-defined, and moreover,
+αt ≥ min
+�
+1,
+�
+min{6(1 − η), 2}
+1.1[LF
+H + µ(2 − β)/(1 − β)2](U F
+g + µ
+√
+ϑ)
+θǫH
+�
+.
+(70)
+(ii) The next iterate xt+1 = xt + αtMtdt satisfies
+φµ(xt) − φµ(xt+1) ≥ csol min{(∥∇φµ(xt+1)∥∗
+xt+1)2ǫ−1
+H , ǫ3
+H},
+(71)
+where Mt and csol are given in (16) and (26), respectively.
+Proof. Notice from Lemma 7.3 that xt ∈ S, that is, xt ∈ int K and φµ(xt) ≤ φµ(u0). It then follows from (16),
+(24) and Lemma 7.1(i) that
+∥M T
+t ∇φµ(xt)∥ = ∥∇φµ(xt)∥∗
+xt ≤ ∥∇F(xt)∥∗
+xt + µ∥∇B(xt)∥∗
+xt ≤ U F
+g + µ
+√
+ϑ.
+(72)
+Since dt results from the output of Algorithm 3 with d type=SOL, one can see that ∥M T
+t ∇φµ(xt)∥ > ǫg and
+the relations (65)-(67) hold. Also, one can observe from Algorithm 3 that its output ˆdt satisfies
+∥(M T
+t ∇2φµ(xt)Mt + 2ǫHI) ˆdt + M T
+t ∇φµ(xt)∥ ≤ ˆζ∥M T
+t ∇φµ(xt)∥
+for some ˆζ ∈ (0, 1/6), which together with ∥M T
+t ∇φµ(xt)∥ > ǫg implies that ˆdt ̸= 0. It then follows from this
+and (18) that dt ̸= 0.
+We first prove statement (i). If (20) holds for j = 0, then αt = 1, which clearly implies that (70) holds. We
+now suppose that (20) fails for j = 0. Claim that for all j ≥ 0 that violate (20), it holds that
+θ2j ≥ min{6(1 − η), 2}ǫH(Lφ
+H)−1∥dt∥−1,
+(73)
+where Lφ
+H is defined in (61). Indeed, we suppose that (20) is violated by some j ≥ 0. We next show that (73)
+holds for such j by considering two separate cases below.
+Case 1) φµ(xt + θjMtdt) > φµ(xt). Let ϕ(α) = φµ(xt + αMtdt). Then ϕ(θj) > ϕ(0). In addition, by (65),
+(67), γt = max{∥ ˆdt∥/β, 1} ≥ 1, and dt ̸= 0, one has
+ϕ′(0) = (dt)T M T
+t ∇φµ(xt)
+(67)
+= −γt(dt)T (M T
+t ∇2φµ(xt)Mt + 2ǫHI)dt (65)
+≤ −γtǫH∥dt∥2 < 0.
+In view of these, we can observe that there exists a local minimizer α∗ ∈ (0, θj) of ϕ such that ϕ(α∗) < ϕ(0)
+and
+ϕ′(α∗) = ∇φµ(xt + α∗Mtdt)T Mtdt = 0.
+(74)
+By φµ(xt) ≤ φµ(u0) and ϕ(α∗) < ϕ(0), one has φµ(xt + α∗Mtdt) < φµ(u0).
+In addition, using (64) and
+0 < α∗ < θj ≤ 1, we have ∥α∗Mtdt∥xt ≤ ∥Mtdt∥xt ≤ β. Hence, (62) holds for x = xt and y = xt + α∗Mtdt. By
+this, (16), (65), (67), (74), 0 < α∗ < 1 and γt ≥ 1, one has
+(α∗)2Lφ
+H
+2
+∥dt∥3
+(16)
+=
+(α∗)2Lφ
+H
+2
+∥dt∥∥Mtdt∥2
+xt
+(62)
+≥ ∥dt∥∥∇φµ(xt + α∗Mtdt) − ∇φµ(xt) − α∗∇2φµ(xt)Mtdt∥∗
+xt
+≥
+(dt)T (M T
+t ∇φµ(xt + α∗Mtdt) − M T
+t ∇φµ(xt) − α∗M T
+t ∇2φµ(xt)Mtdt)
+(74)
+=
+−(dt)T M T
+t ∇φµ(xt) − α∗(dt)T M T
+t ∇2φµ(xt)Mtdt
+(67)
+=
+(γt − α∗)(dt)T (M T
+t ∇2φµ(xt)Mt + 2ǫHI)dt + 2α∗ǫH∥dt∥2
+(65)
+≥
+(γt − α∗)ǫH∥dt∥2 + 2α∗ǫH∥dt∥2 = (γt + α∗)ǫH∥dt∥2 ≥ ǫH∥dt∥2,
+18
+
+which along with dt ̸= 0 implies that (α∗)2 ≥ 2ǫH(Lφ
+H)−1∥dt∥−1. Using this and θj > α∗, we conclude that (73)
+holds in this case.
+Case 2) φµ(xt + θjMtdt) ≤ φµ(xt). By this and φµ(xt) ≤ φµ(u0), one has φµ(xt + θjMtdt) ≤ φµ(u0). Also,
+using (64) and θ ∈ (0, 1), we have ∥θjMtdt∥xt ≤ ∥Mtdt∥xt ≤ β. Hence, (63) holds for x = xt and y = xt+θjMtdt.
+Using this, (16), (65), (67) and the fact that j violates (20), we obtain that
+−ηǫHθ2j∥dt∥2 ≤ φµ(xt + θjMtdt) − φµ(xt)
+(63)
+≤ θj∇φµ(xt)T Mtdt + θ2j
+2 (dt)T M T
+t ∇2φµ(xt)Mtdt + Lφ
+H
+6 θ3j∥Mtdt∥3
+xt
+(16)(67)
+=
+−θjγt(dt)T (M T
+t ∇2φµ(xt)Mt + 2ǫHI)dt + θ2j
+2 (dt)T M T
+t ∇2φµ(xt)Mtdt + Lφ
+H
+6 θ3j∥dt∥3
+= −θj
+�
+γt − θj
+2
+�
+(dt)T (M T
+t ∇2φµ(xt)Mt + 2ǫHI)dt − θ2jǫH∥dt∥2 + Lφ
+H
+6 θ3j∥dt∥3
+(65)
+≤ −θj
+�
+γt − θj
+2
+�
+ǫH∥dt∥2 − θ2jǫH∥dt∥2 + Lφ
+H
+6 θ3j∥dt∥3 ≤ −θjǫHγt∥dt∥2 + Lφ
+H
+6 θ3j∥dt∥3, (75)
+where the first inequality is due to the violation of (20) by such j. Recall that dt ̸= 0. Dividing both sides of
+(75) by Lφ
+Hθj∥dt∥3/6 and using η, θ ∈ (0, 1) and γt ≥ 1, we have
+θ2j ≥ 6(γt − ηθj)ǫH(Lφ
+H)−1∥dt∥−1 ≥ 6(1 − η)ǫH(Lφ
+H)−1∥dt∥−1.
+Hence, (73) also holds in this case.
+Combining the above two cases, we conclude that (73) holds for any j ≥ 0 violating (20). By this and
+θ ∈ (0, 1), one can see that all j ≥ 0 that violate (20) must be bounded above.
+It then follows that the
+step length αt associated with (20) is well-defined. We next prove (70). Observe from the definition of jt in
+Algorithm 1 that j = jt −1 violates (20) and hence (73) holds for j = jt −1. Then, by (61), (73) with j = jt −1,
+and αt = θjt, one has
+αt
+=
+θjt ≥
+�
+min{6(1 − η), 2}ǫH(Lφ
+H)−1θ∥dt∥−1/2
+=
+�
+min{6(1 − η), 2}ǫH[LF
+H + µ(2 − β)/(1 − β)2]−1θ∥dt∥−1/2,
+(76)
+which along with (66) and (72) implies that (70) holds.
+We next prove statement (ii), particularly, (71) by considering three separate cases below.
+Case 1) αt = 1 and ∥ ˆdt∥ ≥ β. It then follows from (18) that dt = β ˆdt/∥ ˆdt∥. Notice from Algorithm 1 that
+β ≥ ǫH. Using this and dt = β ˆdt/∥ ˆdt∥, we see that ∥dt∥ = β ≥ ǫH, which together with (20) and αt = 1 implies
+that (71) holds.
+Case 2) αt = 1 and ∥ ˆdt∥ < β. Notice from αt = 1 that j = 0 is accepted by (20). Then one can see that
+φµ(xt + Mtdt) ≤ φµ(xt) ≤ φµ(u0). Also, observe from (64) that ∥Mtdt∥xt ≤ β. Hence, (62) holds for x = xt
+and y = xt + Mtdt. By these, (54) and (68), one has
+(1 − β)∥∇φµ(xt+1)∥∗
+xt+1
+(54)
+≤ ∥∇φµ(xt+1)∥∗
+xt = ∥∇φµ(xt + Mtdt)∥∗
+xt
+≤ ∥∇φµ(xt + Mtdt) − ∇φµ(xt) − ∇2φµ(xt)Mtdt∥∗
+xt + ∥∇φµ(xt) + ∇2φµ(xt)Mtdt∥∗
+xt
+= ∥∇φµ(xt + Mtdt) − ∇φµ(xt) − ∇2φµ(xt)Mtdt∥∗
+xt + ∥M T
+t (∇φµ(xt) + ∇2φµ(xt)Mtdt)∥
+≤ ∥∇φµ(xt + Mtdt) − ∇φµ(xt) − ∇2φµ(xt)Mtdt∥∗
+xt
++ ∥(M T
+t ∇2φµ(xt)Mt + 2ǫHI)dt + M T
+t ∇φµ(xt)∥ + 2ǫH∥dt∥
+(62)(68)
+≤
+Lφ
+H∥Mtdt∥2
+xt/2 + (4 + ζ)ǫH∥dt∥/2
+(16)
+= Lφ
+H∥dt∥2/2 + (4 + ζ)ǫH∥dt∥/2,
+where the second inequality is due to the triangle inequality, and the second equality follows from (4) and (16).
+19
+
+Solving the above inequality for ∥dt∥ and using (61) and the fact that ∥dt∥ > 0, we obtain that
+∥dt∥
+≥
+−(4+ζ)ǫH+
+�
+(4+ζ)2ǫ2
+H+8(1−β)Lφ
+H∥∇φµ(xt+1)∥∗
+xt+1
+2Lφ
+H
+≥
+−(4+ζ)ǫH+√
+(4+ζ)2ǫ2
+H+8(1−β)Lφ
+Hǫ2
+H
+2Lφ
+H
+min{∥∇φµ(xt+1)∥∗
+xt+1ǫ−2
+H , 1}
+=
+4(1−β)
+4+ζ+√
+(4+ζ)2+8(1−β)Lφ
+H
+min{∥∇φµ(xt+1)∥∗
+xt+1ǫ−1
+H , ǫH}
+(61)
+=
+4(1−β)
+4+ζ+√
+(4+ζ)2+8[(1−β)LF
+H+µ(2−β)/(1−β)] min{∥∇φµ(xt+1)∥∗
+xt+1ǫ−1
+H , ǫH},
+where the second inequality follows from the inequality −a+
+√
+a2 + bs ≥ (−a+
+√
+a2 + b) min{s, 1} for all a, b, s ≥
+0, which can be easily verified by performing a rationalization to the terms −a +
+√
+a2 + bs and −a +
+√
+a2 + b,
+respectively. In view of this, αt = 1, (20) and (26), one can see that (71) holds.
+Case 3) αt < 1. By this, one has that j = 0 violates (20) and hence (73) holds for j = 0. Letting j = 0 in
+(73), we obtain that ∥dt∥ ≥ min{6(1 − η), 2}ǫH/Lφ
+H, which along with (20), (61) and (76) implies that
+φµ(xt) − φµ(xt+1)
+(20)
+≥ ηǫHθ2jt∥dt∥2 ≥ η
+�
+min{6(1 − η), 2}θ
+Lφ
+H
+�2
+ǫ3
+H
+(61)
+= η
+�
+min{6(1 − η), 2}θ
+LF
+H + µ(2 − β)/(1 − β)2
+�2
+ǫ3
+H.
+By this and (26), one can immediately see that (71) also holds in this case.
+The next lemma shows that when the search direction dt in Algorithm 1 is of type ‘NC’, the line search step
+results in a sufficient reduction on φµ as well.
+Lemma 7.6. Suppose that Assumption 4.1 holds and the direction dt results from either the output ˆdt of
+Algorithm 3 with d type=NC or the output v of Algorithm 4 at some iteration t of Algorithm 1. Then the
+following statements hold.
+(i) The step length αt is well-defined, and moreover,
+αt ≥ min
+�
+1,
+min{1, 3(1 − η)}θ
+LF
+H + µ(2 − β)/(1 − β)2
+�
+.
+(77)
+(ii) The next iterate xt+1 = xt + αtMtdt satisfies φµ(xt) − φµ(xt+1) ≥ cncǫ3
+H, where Mt and cnc are given in
+(16) and (27), respectively.
+Proof. It follows from Lemma 7.3 that xt ∈ S, that is, xt ∈ int K and φµ(xt) ≤ φµ(u0). By the assumption on
+dt, one can see from Algorithm 1 that dt is a negative curvature direction given in (17) or (19) and thus dt ̸= 0.
+Also, the vector v satisfies ∥v∥ = 1 whenever it is returned from Algorithm 4. By these, Lemma 7.4(ii), (17)
+and (19), one has
+∇φµ(xt)T Mtdt ≤ 0,
+(dt)T M T
+t ∇2φµ(xt)Mtdt ≤ −∥dt∥3 < 0.
+(78)
+We first prove statement (i). If (21) holds for j = 0, then αt = 1, which clearly implies that (77) holds. We
+now suppose that (21) fails for j = 0. Claim that for all j ≥ 0 that violate (21), it holds that
+θj ≥ min{1, 3(1 − η)}/Lφ
+H,
+(79)
+where Lφ
+H is defined in (61). Indeed, suppose that (21) is violated by some j ≥ 0. We now prove that (79) holds
+for such j by considering two separate cases below.
+Case 1) φµ(xt + θjMtdt) > φµ(xt). Let ϕ(α) = φµ(xt + αMtdt). Then ϕ(θj) > ϕ(0). Also, by (78), one has
+ϕ′(0) = ∇φµ(xt)T Mtdt ≤ 0,
+ϕ′′(0) = (dt)T M T
+t ∇2φµ(xt)Mtdt < 0.
+From these, we can observe that there exists a local minimizer α∗ ∈ (0, θj) of ϕ such that ϕ(α∗) < ϕ(0). By
+the second-order necessary optimality condition of ϕ at α∗, one has
+ϕ′′(α∗) = (dt)T M T
+t ∇2φµ(xt + α∗Mtdt)Mtdt ≥ 0.
+(80)
+20
+
+In addition, by φµ(xt) ≤ φµ(u0) and ϕ(α∗) < ϕ(0), one has φµ(xt + α∗Mtdt) < φµ(u0).
+Using (64) and
+0 < α∗ < θj ≤ 1, we see that ∥α∗Mtdt∥xt ≤ ∥Mtdt∥xt ≤ β. Hence, (60) holds for x = xt and y = xt + α∗Mtdt.
+Using this, (4), (16), (60), (78) and (80), we obtain that
+Lφ
+Hα∗∥dt∥3 (16)
+= Lφ
+Hα∗∥dt∥2∥Mtdt∥xt
+(60)
+≥ ∥dt∥2∥∇2φµ(xt + α∗Mtdt) − ∇2φµ(xt)∥∗
+xt
+(4)
+= ∥dt∥2∥M T
+t (∇2φµ(xt + α∗Mtdt) − ∇2φµ(xt))Mt∥ ≥ (dt)T M T
+t (∇2φµ(xt + α∗Mtdt) − ∇2φµ(xt))Mtdt
+(80)
+≥ −(dt)T M T
+t ∇2φµ(xt)Mtdt (78)
+≥ ∥dt∥3.
+It then follows from this and dt ̸= 0 that α∗ ≥ 1/Lφ
+H, which along with θj > α∗ implies that (79) holds in this
+case.
+Case 2) φµ(xt + θjMtdt) ≤ φµ(xt). By this and φµ(xt) ≤ φµ(u0), one has φµ(xt + θjMtdt) ≤ φµ(u0). In
+addition, it follows from (64) and θ ∈ (0, 1) that ∥θjMtdt∥xt ≤ ∥Mtdt∥xt ≤ β. Hence, (63) holds for x = xt and
+y = xt + θjMtdt. By this, (16), (78) and the fact that j violates (21), one has
+− η
+2θ2j∥dt∥3
+≤ φµ(xt + θjMtdt) − φµ(xt)
+(63)
+≤ θj∇φµ(xt)T Mtdt + θ2j
+2 (dt)T M T
+t ∇2φµ(xt)Mtdt + Lφ
+H
+6 θ3j∥Mtdt∥3
+xt
+(16)(78)
+≤
+− θ2j
+2 ∥dt∥3 + Lφ
+H
+6 θ3j∥dt∥3,
+where the first inequality is due to the violation of (21) by such j.
+Using this and dt ̸= 0, we see that
+θj ≥ 3(1 − η)/Lφ
+H. Hence, (79) also holds in this case.
+Combining the above two cases, we conclude that (79) holds for all j ≥ 0 violating (21).
+By this and
+θ ∈ (0, 1), one can see that all j ≥ 0 that violate (21) must be bounded above. It then follows that the step
+length αt associated with (21) is well-defined. We next derive a lower bound for αt. Notice that j = jt − 1
+violates (21) and hence (79) holds for j = jt − 1. Then by (79) with j = jt − 1 and αt = θjt, one can observe
+that αt = θjt ≥ min{1, 3(1 − η)}θ/Lφ
+H, which along with (61) yields (77) as desired.
+We next prove statement (ii) by considering two separate cases below.
+Case 1) dt results from the output ˆdt of Algorithm 3 with d type=NC. By this and (69), one has ∥dt∥ ≥ ǫH,
+which along with statement (i) and (21) implies that statement (ii) holds.
+Case 2) dt results from the output v of Algorithm 4. Notice from Algorithm 4 that ∥v∥ = 1 and vT M T
+t ∇2φµ(xt)Mtv ≤
+−ǫH/2. It then follows from (19) and β ≥ ǫH that ∥dt∥ ≥ ǫH/2. Using this, (21) and statement (i), we see that
+statement (ii) also holds in this case.
+We are now ready to prove Theorem 4.1.
+Proof of Theorem 4.1. Notice from Lemma 7.3 that all the iterates generated by Algorithm 1 lie in S. By this,
+(4), (16) and (24), one has
+∥M T
+t ∇2φµ(xt)Mt∥
+(4)(16)
+=
+∥∇2φµ(xt)∥∗
+xt ≤ ∥∇2F(xt)∥∗
+xt + µ∥∇2B(xt)∥∗
+xt ≤ U F
+H + µ,
+(81)
+where the last inequality follows from (24) and the fact that ∥∇2B(xt)∥∗
+xt = 1 due to (4) and (16).
+(i) Suppose for contradiction that the total number of calls of Algorithm 4 in Algorithm 1 is more than T2.
+Observe from Algorithm 1 and Lemma 7.6(ii) that each of these calls, except the last one, returns a sufficiently
+negative curvature direction, and each of them results in a reduction on φµ at least by cncǫ3
+H. Using this and
+the fact that x0 = u0, we obtain that
+T2cncǫ3
+H ≤
+�
+t∈T
+[φµ(xt) − φµ(xt+1)] ≤ φµ(x0) − φlow = φhi − φlow,
+where T is given in Lemma 7.3. This contradicts with the definition of T2 given in (25).
+(ii) Suppose for contradiction that the total number of calls of Algorithm 3 in Algorithm 1 is more than T1.
+Note that if Algorithm 3 is called at some iteration t and generates xt+1 satisfying ∥∇φµ(xt+1)∥∗
+xt+1 ≤ ǫg, then
+Algorithm 4 must be called at the next iteration t+1. Using this and statement (i), we see that the total number
+of such iterations t is at most T2. Hence, the total number of iterations t of Algorithm 1 at which Algorithm 3
+21
+
+is called and generates xt+1 satisfying ∥∇φµ(xt+1)∥∗
+xt+1 > ǫg is at least T1 − T2 + 1. Moreover, for each of such
+iterations t, it follows from Lemmas 7.5(ii) and 7.6(ii) that φµ(xt) − φµ(xt+1) ≥ min{csol, cnc} min{ǫ2
+gǫ−1
+H , ǫ3
+H}.
+Thus, one has
+(T1 − T2 + 1) min{csol, cnc} min{ǫ2
+gǫ−1
+H , ǫ3
+H} ≤
+�
+t∈T
+[φµ(xt) − φµ(xt+1)] ≤ φhi − φlow,
+where T is given in Lemma 7.3. This contradicts the definitions of T1 and T2 given in (25).
+(iii) Notice that either Algorithm 3 or Algorithm 4 is called at each iteration of Algorithm 1. By this and
+statements (i) and (ii), one has that the total number of iterations of Algorithm 1 is at most T1+T2. In addition,
+the relation (28) follows from (25), (26) and (27). It is also not hard to see that the output xt of Algorithm 1
+satisfies ∥∇φµ(xt)∥∗
+xt ≤ ǫg deterministically and λmin(M T
+t ∇2φµ(xt)Mt) ≥ −ǫH with probability at least 1 − δ
+for some 0 ≤ t ≤ T1 + T2, where the probability is due to Algorithm 4. Hence, statement (iii) holds.
+(iv) Recall that each iteration of Algorithm 1 requires Cholesky factorization of ∇2B at one point only. This
+together with statement (iii) implies that the total number of Cholesky factorizations required by Algorithm 1
+is at most T1 + T2. By (81) and Theorem A.1 with (H, ε) = (M T
+t ∇2φµ(xt)Mt, ǫH), one can see that the number
+of products of H and a vector v required by each call of Algorithm 3 is at most �O(min{n, [(U F
+H + µ)/ǫH]1/2}).
+In addition, by (81), Theorem B.1 with (H, ε) = (M T
+t ∇2φµ(xt)Mt, ǫH), and the fact that each iteration of the
+Lanczos method requires only one product of H and a vector v, one can observe that the number of products
+of H and a vector v required by each call of Algorithm 4 is also at most �O(min{n, [(U F
+H + µ)/ǫH]1/2}). Recall
+from Section 4.1 that the product of H and a vector v requires at most three fundamental operations, which are
+one Hessian-vector product of F, one backward and forward substitutions to a triangular linear system. Hence,
+each call of Algorithm 3 or 4 requires at most �O(min{n, [(U F
+H + µ)/ǫH]1/2}) fundamental operations. By these
+observations and statement (iii), we conclude that statement (iv) holds.
+7.3
+Proof of the main results in Section 5
+In this subsection we provide a proof of Lemma 5.1 and Theorems 5.2, 5.3, 5.4 and 5.5.
+Before proceeding, we recall that ∥c(zǫ)∥ ≤ ǫ/2 < 1. Using this, (30) and (35), we obtain that
+f(x) + µB(x) + γ∥˜c(x)∥2 ≥ f(x) + µB(x) + γ
+2 ∥c(x)∥2 − γ∥c(zǫ)∥2 ≥ flow − γ,
+∀x ∈ int K .
+(82)
+In addition, by (29) and the first relation in (37), one has
+Lµ(xk+1, λk; ρk) ≤ fhi whenever xk+1 is generated.
+(83)
+We next present an auxiliary lemma that will be frequently used later. Its proof is identical to the one of [44,
+Lemma 5.4] with f replaced by f + µB, and thus omitted here.
+Lemma 7.7. Suppose that Assumption 5.1 holds. Let γ, µ, fhi and flow be given in Assumption 5.1. Assume
+that ρ > 2γ, λ ∈ Rm and x ∈ int K satisfy Lµ(x, λ; ρ) ≤ fhi, where Lµ is defined in (36). Then the following
+statements hold.
+(i) f(x) + µB(x) ≤ fhi + ∥λ∥2/(2ρ).
+(ii) ∥˜c(x)∥ ≤
+�
+2(fhi − flow + γ)/(ρ − 2γ) + ∥λ∥2/(ρ − 2γ)2 + ∥λ∥/(ρ − 2γ).
+(iii) If ρ ≥ ∥λ∥2/(2˜δf) for some ˜δf > 0, then f(x) + µB(x) ≤ fhi + ˜δf.
+(iv) If ρ ≥ 2(fhi − flow + γ)˜δ−2
+c
++ 2∥λ∥˜δ−1
+c
++ 2γ for some ˜δc > 0, then ∥˜c(x)∥ ≤ ˜δc.
+The following lemma establishes the local Lipschitz continuity of ci and ∇ci with respect to the local norm.
+Lemma 7.8. Under Assumption 5.1, the following inequalities hold:
+|ci(y) − ci(x)| ≤
+U c
+g
+1 − β ∥y − x∥x,
+∀x, y ∈ Ω(δf, δc) with ∥y − x∥x ≤ β, 1 ≤ i ≤ m,
+(84)
+∥∇ci(y) − ∇ci(x)∥∗
+x ≤
+U c
+H
+(1 − β)2 ∥y − x∥x,
+∀x, y ∈ Ω(δf, δc) with ∥y − x∥x ≤ β, 1 ≤ i ≤ m,
+(85)
+where Ω(δf, δc) is given in Assumption 5.1, and U c
+g, U c
+H are defined in (33) and (34), respectively.
+22
+
+Proof. Fix any x, y ∈ Ω(δf, δc) with ∥y − x∥x ≤ β and any 1 ≤ i ≤ m. Let zt = x + t(y − x) for all t ∈ [0, 1]. It
+then follows that ∥zt − x∥x ≤ β and zt ∈ Ω(δf, δc). By these, (33), (34), (53) and (54), one has
+∥∇ci(zt)∥∗
+zt ≤ U c
+g,
+∥∇2ci(zt)∥∗
+zt ≤ U c
+H,
+∥v∥zt ≤ (1−β)−1∥v∥x,
+∥v∥∗
+x ≤ (1−β)−1∥v∥∗
+zt,
+∀v ∈ Rn, t ∈ [0, 1].
+By virtue of these and (4), we obtain
+|ci(y) − ci(x)| =
+����
+� 1
+0
+∇ci(zt)T (y − x)dt
+���� ≤
+� 1
+0
+∥∇ci(zt)∥∗
+zt∥y − x∥ztdt ≤
+U c
+g
+1 − β ∥y − x∥x,
+∥∇ci(y) − ∇ci(x)∥∗
+x =
+����
+� 1
+0
+∇2ci(zt)(y − x)dt
+����
+∗
+x
+≤
+� 1
+0
+∥∇2ci(zt)(y − x)∥∗
+xdt ≤
+1
+1 − β
+� 1
+0
+∥∇2ci(zt)(y − x)∥∗
+ztdt
+≤
+1
+1 − β
+� 1
+0
+∥∇2ci(zt)∥∗
+zt∥y − x∥ztdt ≤
+U c
+H
+(1 − β)2 ∥y − x∥x.
+Hence, (84) and (85) hold.
+We are now ready to prove Lemma 5.1.
+Proof of Lemma 5.1. (i) Fix any x ∈ int K satisfying Lµ(x, λk; ρk) ≤ Lµ(xk
+init, λk; ρk). It follows from this and
+(40) that Lµ(x, λk; ρk) ≤ fhi. By this, ∥λk∥ ≤ Λ, ρk ≥ ρ0 > 2γ, δf,1 ≤ δf, δc,1 ≤ δc, and Lemmas 7.7(i) and
+7.7(ii) with (λ, ρ) = (λk, ρk), one has
+f(x) + µB(x) ≤ fhi + ∥λk∥2
+2ρk
+≤ fhi + Λ2
+2ρ0
+(42)
+= fhi + δf,1 ≤ fhi + δf,
+∥˜c(x)∥ ≤
+�
+2(fhi−flow+γ)
+ρk−2γ
++
+∥λk∥2
+(ρk−2γ)2 +
+∥λk∥
+ρk−2γ ≤
+�
+2(fhi−flow+γ)
+ρ0−2γ
++
+Λ2
+(ρ0−2γ)2 +
+Λ
+ρ0−2γ
+(42)
+= δc,1 ≤ δc.
+(86)
+In addition, recall that ∥c(zǫ)∥ ≤ 1, which together with the definition of ˜c in (35) yields ∥c(x)∥ ≤ 1 + ∥˜c(x)∥.
+These along with x ∈ int K, µ ∈ (0, ¯µ] and (31) implies x ∈ S(δf, δc). Hence, statement (i) holds.
+(ii) Observe that
+inf
+x∈int K Lµ(x, λk; ρk) =
+inf
+x∈int K{Lµ(x, λk; ρk) : Lµ(x, λk; ρk) ≤ Lµ(xk
+init, λk; ρk)}.
+Thus, to prove statement (ii), it suffices to show that
+inf
+x∈int K{Lµ(x, λk; ρk) : Lµ(x, λk; ρk) ≤ Lµ(xk
+init, λk; ρk)} ≥ flow − γ − Λδc.
+(87)
+To this end, fix any x ∈ int K satisfying Lµ(x, λk; ρk) ≤ Lµ(xk
+init, λk; ρk).
+It then follows from (86) that
+∥˜c(x)∥ ≤ δc. By this, ∥λk∥ ≤ Λ, ρk > 2γ, µ ∈ (0, ¯µ] and (82), one has
+Lµ(x, λk; ρk)
+=
+f(x) + µB(x) + (λk)T ˜c(x) + ρk
+2 ∥˜c(x)∥2
+≥
+f(x) + µB(x) + γ∥˜c(x)∥2 − Λ∥˜c(x)∥
+(82)
+≥
+flow − γ − Λδc,
+and hence (87) holds as desired.
+(iii) Fix x, y ∈ Ω(δf, δc) with ∥y − x∥x ≤ β and 1 ≤ i ≤ m.
+By this, (32), (34), (35), (43), (84) and
+∥c(zǫ)∥ ≤ 1, one has
+∥˜ci(y)∇2ci(y) − ˜ci(x)∇2ci(x)∥∗
+x = ∥˜ci(y)(∇2ci(y) − ∇2ci(x)) + (˜ci(y) − ˜ci(x))∇2ci(x)∥∗
+x
+≤ |ci(y) − ci(zǫ)|∥∇2ci(y) − ∇2ci(x)∥∗
+x + |ci(y) − ci(x)|∥∇2ci(x)∥∗
+x ≤ (1 + U c)Lc
+H∥y − x∥x + U c
+gU c
+H
+1 − β ∥y − x∥x
+=
+�
+(1 + U c)Lc
+H + U c
+gU c
+H
+1 − β
+�
+∥y − x∥x.
+(88)
+In addition, by (33), (54) and (85), one has
+∥∇ci(y)∇ci(y)T − ∇ci(x)∇ci(x)T ∥∗
+x = ∥∇ci(y)(∇ci(y) − ∇ci(x))T + (∇ci(y) − ∇ci(x))∇ci(x)T ∥∗
+x
+23
+
+≤ ∥∇ci(y)∥∗
+x∥∇ci(y) − ∇ci(x)∥∗
+x + ∥∇ci(x)∥∗
+x∥∇ci(y) − ∇ci(x)∥∗
+x
+≤
+�
+1
+1 − β ∥∇ci(y)∥∗
+y + ∥∇ci(x)∥∗
+x
+�
+∥∇ci(y) − ∇ci(x)∥∗
+x ≤ (2 − β)U c
+gU c
+H
+(1 − β)3
+∥y − x∥x.
+(89)
+In view of (41) and the fact that ∇˜c = ∇c and ∇2˜ci = ∇2ci, 1 ≤ i ≤ m, we see that
+∇2
+xx L(x, λk; ρk) = ∇2f(x) +
+m
+�
+i=1
+λk
+i ∇2ci(x) + ρk
+m
+�
+i=1
+�
+∇ci(x)∇ci(x)T + ˜ci(x)∇2ci(x)
+�
+,
+(90)
+which implies that
+∥∇2
+xx L(y, λk; ρk)−∇2
+xx L(x, λk; ρk)∥∗
+x ≤ ∥∇2f(y) − ∇2f(x)∥∗
+x +
+m
+�
+i=1
+|λk
+i |∥∇2ci(y) − ∇2ci(x)∥∗
+x
++ ρk
+m
+�
+i=1
+�
+∥∇ci(y)∇ci(y)T − ∇ci(x)∇ci(x)T ∥∗
+x + ∥˜ci(y)∇2ci(y) − ˜ci(x)∇2ci(x)∥∗
+x
+�
+.
+Statement (iii) then follows from this, (88) and (89).
+(iv) Notice from (41) and ∇˜c = ∇c that ∇x L(x, λk; ρk) = ∇f(x) + ∇c(x)λk + ρk∇c(x)˜c(x). Also, one can
+see from (31), the definition of ˜c in (35) and ∥c(zǫ)∥ ≤ 1 that ∥˜c(x)∥ ≤ 2 + δc for any x ∈ S(δf, δc). Using these
+and (33), we can see that supx∈S(δf,δc) ∥∇x L(x, λk; ρk)∥∗
+x is finite. In addition, by (4), one can observe that
+∥∇ci(x)∇ci(x)T ∥∗
+x = (∥∇ci(x)∥∗
+x)2 for all i. Using this and (90), we have
+∥∇2
+xx L(x, λk; ρk)∥∗
+x ≤ ∥∇2f(x)∥∗
+x +
+m
+�
+i=1
+|λk
+i |∥∇2ci(x)∥∗
+x + ρk
+m
+�
+i=1
+�
+(∥∇ci(x)∥∗
+x)2 + |˜ci(x)|∥∇2ci(x)∥∗
+x
+�
+,
+which, together with (33), (34) and the fact that ∥˜c(x)∥ ≤ 2 + δc for any x ∈ S(δf, δc), implies that
+Uk,H =
+sup
+x∈S(δf ,δc)
+∥∇2
+xx L(x, λk; ρk)∥∗
+x ≤ U f
+H + ∥λk∥1U c
+H + ρk(m(U c
+g)2 + √m(2 + δc)U c
+H).
+Hence, statement (iv) holds.
+We now prove Theorem 5.2.
+Proof of Theorem 5.2. Suppose that Algorithm 2 terminates at some iteration k, that is, τk ≤ µ and ∥c(xk+1)∥ ≤
+ǫ hold. By τk ≤ µ, ˜λk+1 = λk + ρk˜c(xk+1), ∇˜c = ∇c, and the second relation in (37), one has
+∥∇f(xk+1) + ∇c(xk+1)˜λk+1 + µ∇B(xk+1)∥∗
+xk+1 = ∥∇f(xk+1) + ∇˜c(xk+1)(λk + ρk˜c(xk+1)) + µ∇B(xk+1)∥∗
+xk+1
+= ∥∇x Lµ(xk+1, λk; ρk)∥∗
+xk+1 ≤ τk ≤ µ.
+(91)
+This along with µ > 0 yields that
+∥(∇f(xk+1) + ∇c(xk+1)˜λk+1)/µ + ∇B(xk+1)∥∗
+xk+1 ≤ 1.
+By this, xk+1 ∈ int K and Lemma 7.1(v), one has (∇f(xk+1) + ∇c(xk+1)˜λk+1)/µ ∈ K∗, which implies that
+(9) holds for (xk+1, ˜λk+1).
+We next prove that (10) holds for (xk+1, ˜λk+1) with ǫ1 = ǫ.
+Indeed, by (91),
+µ = ǫ/(2ϑ1/2 + 2), xk+1 ∈ int K and Lemma 7.1(i), one has
+∥∇f(xk+1) + ∇c(xk+1)˜λk+1∥∗
+xk+1 ≤ ∥∇f(xk+1) + ∇c(xk+1)˜λk+1 + µ∇B(xk+1)∥∗
+xk+1 + µ∥∇B(xk+1)∥∗
+xk+1
+≤ µ + µϑ1/2 = ǫ/2 < ǫ,
+and hence (10) holds for (xk+1, ˜λk+1) with ǫ1 = ǫ. In view of these, ∥c(xk+1)∥ ≤ ǫ and xk+1 ∈ int K, we conclude
+that xk+1 is a deterministic ǫ-FOSP of problem (1).
+In addition, by (38) and τk ≤ µ, one can see that λmin(M T
+k+1∇2
+xx Lµ(xk+1, λk; ρk)Mk+1) ≥ −√µ holds
+with probability at least 1 − δ, which implies that ˆdT M T
+k+1∇2
+xx Lµ(xk+1, λk; ρk)Mk+1 ˆd ≥ −√µ∥ ˆd∥2 holds for
+all ˆd ∈ Rn with probability at least 1 − δ. Substituting ˆd = M −1
+k+1∇2B(xk+1)−1/2d in this inequality and using
+24
+
+(16), ˜λk+1 = λk + ρk˜c(xk+1), ∇˜c = ∇c and ∇2˜ci = ∇2ci, 1 ≤ i ≤ m, we obtain that with probability at least
+1 − δ, it holds that
+dT ∇2B(xk+1)−1/2�
+∇2f(xk+1) +
+m
+�
+i=1
+˜λk+1
+i
+∇2ci(xk+1) + ρk∇c(xk+1)∇c(xk+1)T + µ∇2B(xk+1)
+�
+∇2B(xk+1)−1/2d
+≥ −√µ∥M −1
+k+1∇2B(xk+1)−1/2d∥2 (16)
+= −√µ∥d∥2,
+∀d ∈ Rn,
+which together with µ = ǫ/(2ϑ1/2 + 2) ∈ (0, 1) and ϑ ≥ 1 implies that
+dT ∇2B(xk+1)−1/2
+�
+∇2f(xk+1) +
+m
+�
+i=1
+˜λk+1
+i
+∇2ci(xk+1)
+�
+∇2B(xk+1)−1/2d ≥ −(√µ + µ)∥d∥2
+≥ −2√µ∥d∥2 = −2
+�
+ǫ/(2ϑ1/2 + 2)∥d∥2 ≥ −√ǫ∥d∥2,
+∀d ∈ C(xk+1),
+where C(·) is defined in (12). Hence, with probability at least 1 − δ, the relation (11) holds for (xk+1, ˜λk+1) with
+ǫ2 = √ǫ. In view of this and the fact that xk+1 is a deterministic ǫ-FOSP of (1), we conclude that the output
+xk+1 is an (ǫ, √ǫ)-SOSP of (1) with probability at least 1 − δ.
+We next provide a proof for Theorem 5.3.
+Proof of Theorem 5.3. Notice from (46) that ρǫ,1 ≥ 2ρ0, which along with (44) and (45) implies that
+Kǫ
+(45)
+=
+⌈log µ/ log ω⌉
+(44)
+=
+⌈log 2/ log r⌉ ≤ log(ρǫ,1ρ−1
+0 )/ log r + 1.
+(92)
+Since {ρk} is either unchanged or increased by a ratio r as k increases, it follows from (92) that
+max
+0≤k≤Kǫ ρk ≤ rKǫρ0
+(92)
+≤ r
+log(ρǫ,1ρ−1
+0
+)
+log r
++1ρ0 = rρǫ,1.
+(93)
+In addition, observe from Algorithm 2 that ρk > 2γ and ∥λk∥ ≤ Λ. By these, (83), and Lemma 7.7(ii) with
+(x, λ, ρ) = (xk+1, λk, ρk), we obtain that
+∥˜c(xk+1)∥ ≤
+�
+2(fhi − flow + γ)
+ρk − 2γ
++
+∥λk∥2
+(ρk − 2γ)2 +
+∥λk∥
+ρk − 2γ ≤
+�
+2(fhi − flow + γ)
+ρk − 2γ
++
+Λ2
+(ρk − 2γ)2 +
+Λ
+ρk − 2γ . (94)
+On the other hand, notice from ∥c(zǫ)∥ ≤ ǫ/2 and the definition of ˜c in (35) that
+∥c(xk+1)∥ ≤ ∥˜c(xk+1)∥ + ∥c(zǫ)∥ ≤ ∥˜c(xk+1)∥ + ǫ/2.
+(95)
+We now prove that Kǫ is finite. Suppose for contradiction that Kǫ is infinite. By this and (47), one has that
+∥c(xk+1)∥ > ǫ for all k ≥ Kǫ. This along with (95) implies that ∥˜c(xk+1)∥ > ǫ/2 for all k ≥ Kǫ. It then follows
+that ∥˜c(xk+1)∥ > α∥˜c(xk)∥ must hold for infinitely many k’s, which, together with the update scheme on {ρk},
+further implies ρk+1 = rρk holds for infinitely many k’s. Using this and the monotonicity of {ρk}, we see that
+ρk → ∞ as k → ∞. This along with (94) yields that ∥˜c(xk+1)∥ → 0 as k → ∞, which leads to a contradiction
+with the fact that ∥˜c(xk+1)∥ > ǫ/2 for all k ≥ Kǫ. Hence, Kǫ is finite. In addition, notice from τk = max{µ, ωk}
+and (45) that τk = µ for all k ≥ Kǫ. Combining this with the termination criterion of Algorithm 2 and the
+definition of Kǫ, we conclude that Algorithm 2 with τk = max{µ, ωk} must terminate at iteration Kǫ.
+We next prove (48) and that ρk ≤ rρǫ,1 holds for 0 ≤ k ≤ Kǫ by considering two separate cases below.
+Case 1) ∥c(xKǫ+1)∥ ≤ ǫ. It then follows from (47) that Kǫ = Kǫ, and thus (48) holds due to (92). In
+addition, by Kǫ = Kǫ and (93), one has that ρk ≤ rρǫ,1 holds for 0 ≤ k ≤ Kǫ as well.
+Case 2) ∥c(xKǫ+1)∥ > ǫ. It then follows from (47) that Kǫ > Kǫ, and moreover, ∥c(xk+1)∥ > ǫ for all
+Kǫ ≤ k ≤ Kǫ − 1. This along with (95) implies that
+∥˜c(xk+1)∥ > ǫ/2,
+∀Kǫ ≤ k ≤ Kǫ − 1.
+(96)
+25
+
+By this, ∥λk∥ ≤ Λ, (46), (83), and Lemma 7.7(iv) with (x, λ, ρ, ˜δc) = (xk+1, λk, ρk, ǫ/2), one has
+ρk < 8(fhi − flow + γ)ǫ−2 + 4∥λk∥ǫ−1 + 2γ
+≤ 8(fhi − flow + γ)ǫ−2 + 4Λǫ−1 + 2γ
+(46)
+≤ ρǫ,1,
+∀Kǫ ≤ k ≤ Kǫ − 1.
+(97)
+In view of this, (93), and the fact ρKǫ ≤ rρKǫ−1, we obtain that ρk ≤ rρǫ,1 holds for 0 ≤ k ≤ Kǫ. It remains to
+prove (48). To this end, let
+K = {k : ρk+1 = rρk, Kǫ ≤ k ≤ Kǫ − 2}.
+By (97) and the update scheme of ρk, one has r| K |ρKǫ = maxKǫ≤k≤Kǫ−1 ρk ≤ ρǫ,1, which along with ρKǫ ≥ ρ0
+implies that
+| K | ≤ log(ρǫ,1ρ−1
+Kǫ)/ log r ≤ log(ρǫ,1ρ−1
+0 )/ log r.
+(98)
+Let {k1, k2, . . . , k| K |} denote all the elements of K arranged in ascending order, and let k0 = Kǫ and k| K |+1 =
+Kǫ − 1. We next derive an upper bound for kj+1 − kj for j = 0, 1, . . . , | K |. Using the definition of K, we see
+that ρk = ρk′ for kj < k, k′ ≤ kj+1. By this and the update scheme of ρk, one can see that
+∥˜c(xk+1)∥ ≤ α∥˜c(xk)∥,
+∀kj < k < kj+1.
+(99)
+In addition, by (42), (94) and ρk ≥ ρ0, one has ∥˜c(xk+1)∥ ≤ δc,1 for 0 ≤ k ≤ Kǫ. Using this and (96), we obtain
+that
+ǫ/2 < ∥˜c(xk+1)∥ ≤ δc,1,
+∀Kǫ ≤ k ≤ Kǫ − 1.
+(100)
+Now, we notice that either kj+1 − kj = 1 or kj+1 − kj > 1.
+In the latter case, one can apply (99) with
+k = kj+1 − 1, . . . , kj + 1 along with (100) to deduce that
+ǫ/2 < ∥˜c(xkj+1)∥ ≤ α∥˜c(xkj+1−1)∥ ≤ · · · ≤ αkj+1−kj−1∥˜c(xkj+1)∥ ≤ αkj+1−kj−1δc,1,
+∀j = 0, 1, . . . , | K |.
+Combining the two cases, we deduce that
+kj+1 − kj ≤ | log(ǫ(2δc,1)−1)/ log α| + 1,
+∀j = 0, 1, . . ., | K |.
+(101)
+Summing up these inequalities, and using (92), (98), k0 = Kǫ and k| K |+1 = Kǫ − 1, we have
+Kǫ = 1 + k| K |+1 = 1 + k0 + �| K |
+j=0(kj+1 − kj)
+(101)
+≤
+1 + Kǫ + (| K | + 1)
+����
+log(ǫ(2δc,1)−1)
+log α
+��� + 1
+�
+≤ 2 + log(ρǫ,1ρ−1
+0
+)
+log r
++
+�
+log(ρǫ,1ρ−1
+0
+)
+log r
++ 1
+����� log(ǫ(2δc,1)−1)
+log α
+��� + 1
+�
+= 1 +
+�
+log(ρǫ,1ρ−1
+0
+)
+log r
++ 1
+� ���� log(ǫ(2δc,1)−1)
+log α
+��� + 2
+�
+,
+where the second inequality is due to (92) and (98). Hence, (48) holds as well in this case.
+We next prove Theorem 5.4. Before proceeding, we recall from Lemma 5.1 and the discussions in Section 5.1
+that the subproblem minx Lµ(x, λk; ρk) satisfies Assumptions 4.1(b) and 4.1(c) with (F(·), S, Ω, LF
+H, U F
+g , U F
+H) =
+(L(·, λk; ρk), S(δf, δc), Ω(δf, δc), Lk,H, Uk,g, Uk,H). Moreover, in view of the fact that ∥λk∥ ≤ Λ, one can see from
+(43) and Lemma 5.1(iv) that there exist some constants L1, L2, U1 and U2, depending only on f, c, B, β, Λ, m,
+δf and δc, such that
+Lk,H ≤ L1 + ρkL2,
+Uk,H ≤ U1 + ρkU2.
+(102)
+We are now ready to prove Theorem 5.4.
+Proof of Theorem 5.4. Let Tk and Nk denote the number of iterations and fundamental operations performed
+by Algorithm 1 at outer iteration k of Algorithm 2, respectively. It then follows from Theorem 5.3 that the total
+number of iterations of Algorithm 1 performed in Algorithm 2 is �Kǫ
+k=0 Tk, and moreover, the total number
+of Cholesky factorizations and other fundamental operations performed by Algorithm 1 in Algorithm 2 are
+�Kǫ
+k=0 Tk and �Kǫ
+k=0 Nk, respectively. In addition, notice from (46) and Theorem 5.3 that ρǫ,1 = O(ǫ−2) and
+ρk ≤ rρǫ,1, which yield ρk = O(ǫ−2) for all 0 ≤ k ≤ Kǫ.
+(i) Recall from Lemmas 5.1(i) and 5.1(iii) that Lk,H is a Lipschitz constant of ∇2
+xx L(x, λk; ρk) with respect
+to the local norm on an open convex neighborhood of {x ∈ int K : Lµ(x, λk; ρk) ≤ Lµ(xk
+init, λk; ρk)}. In addition,
+26
+
+recall from Lemma 5.1(ii) that infx∈int K Lµ(x, λk; ρk) ≥ flow − γ − Λδc. By these, (40), (102), and Theorem
+4.1(iii) with (φhi, φlow, LF
+H, ǫg, ǫH) = (Lµ(xk
+init, λk; ρk), flow − γ − Λδc, Lk,H, τk, √τk), one has
+Tk = O((fhi − flow + γ + Λδc)L2
+k,Hτ −3/2
+k
+)
+(102)
+=
+O(ρ2
+kτ −3/2
+k
+) = O(ǫ−11/2),
+(103)
+where the last equality follows from τk ≥ µ = ǫ/(2ϑ1/2 +2) and ρk = O(ǫ−2). On the other hand, if c is assumed
+to be affine, namely, c(x) = Ax − b for some A ∈ Rm×n and b ∈ Rm, then ∇c(x) = AT and ∇2ci(x) = 0 for
+1 ≤ i ≤ m. Using these and (90), we observe that Lk,H = O(1). By this and the similar arguments as for (103),
+one has Tk = O(τ −3/2
+k
+) = O(ǫ−3/2). Combining these with (103) and Kǫ = O(| log ǫ|2) (see Remark 5.4), we
+conclude that statement (i) holds.
+(ii) By Lemmas 5.1(i) and 5.1(iv), one has
+Uk,H ≥
+sup
+x∈int K
+{∥∇2
+xx L(x, λk; ρk)∥∗
+x : Lµ(x, λk; ρk) ≤ Lµ(xk
+init, λk; ρk)}.
+In view of this, Lµ(xk
+init, λk; ρk) ≤ fhi, (102), and Theorem 4.1(iv) with (φhi, φlow, LF
+H, U F
+H, ǫg, ǫH) = (Lµ(xk
+init, λk; ρk),
+flow − γ − Λδc, Lk,H, Uk,H, τk, √τk), we obtain that
+Nk
+=
+�O((fhi − flow + γ + Λδc)L2
+k,Hτ −3/2
+k
+min{n, U 1/2
+k,Hτ −1/4
+k
+})
+(102)
+=
+�O(ρ2
+kτ −3/2
+k
+min{n, ρ1/2
+k
+τ −1/4
+k
+}) = �O
+�
+ǫ−11/2 min
+�
+n, ǫ−5/4��
+,
+(104)
+where the last equality follows from τk ≥ µ = ǫ/(2ϑ + 2) and ρk = O(ǫ−2). On the other hand, if c is assumed
+to be affine, it follows from the above discussion that Lk,H = O(1). By this, Uk,H ≤ U1 + ρkU2, and the similar
+arguments as for (104), one has Nk = �O(τ −3/2
+k
+min{n, ρ1/2
+k
+τ −1/4
+k
+}) = �O
+�
+ǫ−3/2 min
+�
+n, ǫ−5/4��
+. Combining these
+with (104) and Kǫ = O(| log ǫ|2) (see Remark 5.4), we conclude that statement (ii) holds.
+We next establish two technical lemmas that will be used to prove Theorem 5.5.
+Lemma 7.9. Suppose that Assumptions 5.1 and 5.2 hold and that ρ0 is sufficiently large such that δf,1 ≤ δf
+and δc,1 ≤ δc, where δf,1 and δc,1 are defined in (42). Let {(xk, λk, ρk)} be generated by Algorithm 2. Suppose
+that
+ρk ≥ max{Λ2(2δf)−1, 2(fhi − flow + γ)δ−2
+c
++ 2Λδ−1
+c
++ 2γ, 2(U f
+g + √mU c
+gΛ +
+√
+ϑ + 1)(σǫ)−1}
+(105)
+for some k ≥ 0, where γ, fhi, flow, δf, δc, U f
+g and U c
+g are given in Assumption 5.1, and σ is given in (49).
+Then it holds that ∥c(xk+1)∥ ≤ ǫ.
+Proof. Using ∥λk∥ ≤ Λ (see step 5 of Algorithm 2) and (105), we have
+ρk ≥ max{∥λk∥2(2δf)−1, 2(fhi − flow + γ)δ−2
+c
++ 2∥λk∥δ−1
+c
++ 2γ}.
+By this, (83), and Lemmas 7.7(iii) and 7.7(iv) with (x, λ, ρ, ˜δf, ˜δc) = (xk+1, λk, ρk, δf, δc), one has f(xk+1) +
+µB(xk+1) ≤ fhi + δf and ∥˜c(xk+1)∥ ≤ δc. Also, notice from ∥c(zǫ)∥ ≤ 1 and the definition of ˜c in (35) that
+∥c(xk+1)∥ ≤ 1 + ∥˜c(xk+1)∥. These along with (31), xk+1 ∈ int K, and µ ∈ (0, ¯µ] yield that xk+1 ∈ S(δf, δc).
+It then follows from (33) that ∥∇f(xk+1)∥∗
+xk+1 ≤ U f
+g and ∥∇ci(xk+1)∥∗
+xk+1 ≤ U c
+g for all 1 ≤ i ≤ m. By these,
+τk ≤ 1, µ ≤ 1, ∥λk∥ ≤ Λ, (35) and the second relation in (37), one has
+ρk∥∇2B(xk+1)−1/2∇c(xk+1)˜c(xk+1)∥ = ρk∥∇c(xk+1)˜c(xk+1)∥∗
+xk+1
+≤ ∥∇f(xk+1) + ∇c(xk+1)λk∥∗
+xk+1 + µ∥∇B(xk+1)∥∗
+xk+1 + ∥∇x Lµ(xk+1, λk; ρk)∥∗
+xk+1
+≤ ∥∇f(xk+1)∥∗
+xk+1 +
+m
+�
+i=1
+|λk
+i |∥∇ci(xk+1)∥∗
+xk+1 + µ
+√
+ϑ + τk ≤ U f
+g + √mU c
+gΛ +
+√
+ϑ + 1,
+(106)
+where the first inequality follows from the triangle inequality, and the second inequality follows from ∥∇B(xk+1)∥∗
+xk+1 =
+√
+ϑ and the second relation in (37). In addition, by xk+1 ∈ S(δf, δc) and (49), one has
+λmin(∇c(xk+1)T ∇2B(xk+1)−1∇c(xk+1)) ≥ σ2, which along with (106) implies that
+∥˜c(xk+1)∥ ≤ ∥[∇c(xk+1)T ∇2B(xk+1)−1∇c(xk+1)]−1∇c(xk+1)T ∇2B(xk+1)−1/2∥∥∇2B(xk+1)−1/2∇c(xk+1)˜c(xk+1)∥
+27
+
+= ∥[∇c(xk+1)T ∇2B(xk+1)−1∇c(xk+1)]−1∥1/2∥∇2B(xk+1)−1/2∇c(xk+1)˜c(xk+1)∥
+= λmin(∇c(xk+1)T ∇2B(xk+1)−1∇c(xk+1))−1/2∥∇2B(xk+1)−1/2∇c(xk+1)˜c(xk+1)∥
+≤ (U f
+g + √mU c
+gΛ +
+√
+ϑ + 1)/(σρk).
+(107)
+Observe from (105) that ρk ≥ 2(U f
+g +√mU c
+gΛ+
+√
+ϑ+1)(σǫ)−1, which along with (107) implies ∥˜c(xk+1)∥ ≤ ǫ/2.
+Combining this with ∥c(zǫ)∥ ≤ ǫ/2 and the definition of ˜c in (35), we obtain ∥c(xk+1)∥ ≤ ǫ as desired.
+The next lemma establishes a stronger upper bound for {ρk} than the one given in Theorem 5.3.
+Lemma 7.10. Suppose that Assumptions 5.1 and 5.2 hold and that ρ0 is sufficiently large such that δf,1 ≤ δf
+and δc,1 ≤ δc, where δf,1 and δc,1 are defined in (42). Let {ρk} be generated by Algorithm 2 and
+ρǫ,2 := max{Λ2(2δf)−1, 2(fhi − flow + γ)δ−2
+c
++ 2Λδ−1
+c
++ 2γ, 2(U f
+g + √mU c
+gΛ +
+√
+ϑ + 1)(σǫ)−1, 2ρ0},
+(108)
+where γ, fhi, flow, δf, δc, U f
+g and U c
+g are given in Assumption 5.1, and σ is given in (49). Then ρk ≤ rρǫ,2
+holds for 0 ≤ k ≤ Kǫ, where Kǫ is defined in (47).
+Proof. Observe from (108) that ρǫ,2 ≥ 2ρ0.
+Using this and similar arguments as for (92), we have Kǫ ≤
+log(ρǫ,2ρ−1
+0 )/ log r + 1, where Kǫ is defined in (45). By this, the update scheme for {ρk}, and similar arguments
+as for (93), one has
+max
+0≤k≤Kǫ ρk ≤ rρǫ,2.
+(109)
+If ∥c(xKǫ+1)∥ ≤ ǫ, it follows from (47) that Kǫ = Kǫ, which along with (109) implies that ρk ≤ rρǫ,2 holds for
+0 ≤ k ≤ Kǫ. On the other hand, if ∥c(xKǫ+1)∥ > ǫ, it follows from (47) that ∥c(xk+1)∥ > ǫ for Kǫ ≤ k ≤ Kǫ − 1,
+which together with Lemma 7.9 and (108) implies that
+ρk < max
+�
+Λ2
+2δf
+, 2(fhi − flow + γ)
+δ2c
++ 2Λ
+δc
++ 2γ, 2(U f
+g + √mU c
+gΛ +
+√
+ϑ + 1)
+σǫ
+�
+(108)
+≤ ρǫ,2,
+∀Kǫ ≤ k ≤ Kǫ − 1.
+Using this, (109), and ρKǫ ≤ rρKǫ−1, we also conclude that ρk ≤ rρǫ,2 holds for 0 ≤ k ≤ Kǫ.
+We now provide a proof for Theorem 5.5.
+Proof of Theorem 5.5. Notice from (108) and Lemma 7.10 that ρǫ,2 = O(ǫ−1) and ρk ≤ rρǫ,2, which imply
+ρk = O(ǫ−1). The rest of the proof follows from the same arguments as for Theorem 5.4 with ρk = O(ǫ−2)
+replaced by ρk = O(ǫ−1).
+8
+Concluding remarks
+In this paper we proposed a Newton-CG based barrier-AL method for finding an approximate SOSP of general
+nonconvex conic optimization problem (1). It can be easily extended to a more general conic optimization
+problem minx,y{ ˜f(x, y) : ˜c(x, y) = 0, y ∈ K}, which includes problem minx{f(x) : c(x) = 0, d(x) ≤ 0} and
+more generally problem minx{f(x) : c(x) = 0, d(x) ∈ K} as special cases. Indeed, the latter problem can be
+equivalently solved as the problem minx,y{f(x) : c(x) = 0, d(x) − y = 0, y ∈ K}.
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+Appendix
+A
+A capped conjugate gradient method
+In this part we present the capped CG method proposed in [60, Algorithm 1] for solving a possibly indefinite
+linear system (15). As briefly discussed in Section 4, the capped CG method finds either an approximate solution
+to (15) or a sufficiently negative curvature direction of the associated matrix H. More details about this method
+can be found in [60, Section 3.1].
+The following theorem presents the iteration complexity of Algorithm 3, whose proof can be found in [44,
+Theorem A.1], and thus omitted here.
+Theorem A.1 (iteration complexity of Algorithm 3). Consider applying Algorithm 3 with the optional
+input U = 0 to the linear system (15) with g ̸= 0, ε > 0, and H being an n × n symmetric matrix. Then the
+number of iterations of Algorithm 3 is �O(min{n,
+�
+∥H∥/ε}).
+32
+
+Algorithm 3 A capped conjugate gradient method
+Input: symmetric matrix H ∈ Rn×n, vector g ̸= 0, damping parameter ε ∈ (0, 1), desired relative accuracy ζ ∈ (0, 1).
+Optional input: scalar U ≥ 0 such that ∥H∥ ≤ U (set to 0 if not provided).
+Outputs: ˆd, d type.
+Secondary outputs: final values of U, κ, ˆζ, τ, and T.
+Set
+¯
+H := H + 2εI,
+κ := U + 2ε
+ε
+,
+ˆζ := ζ
+3κ ,
+τ :=
+√κ
+√κ + 1 ,
+T :=
+4κ4
+(1 − √τ)2 ,
+y0 ← 0, r0 ← g, p0 ← −g, j ← 0.
+if (p0)T ¯
+Hp0 < ε∥p0∥2 then
+Set ˆd ← p0 and terminate with d type = NC;
+else if
+∥Hp0∥ > U∥p0∥ then
+Set U ← ∥Hp0∥/∥p0∥ and update κ, ˆζ, τ, T accordingly;
+end if
+while TRUE do
+αj ← (rj)T rj/(pj)T ¯
+Hpj; {Begin Standard CG Operations}
+yj+1 ← yj + αjpj;
+rj+1 ← rj + αj ¯
+Hpj;
+βj+1 ← ∥rj+1∥2/∥rj∥2;
+pj+1 ← −rj+1 + βj+1pj; {End Standard CG Operations}
+j ← j + 1;
+if ∥Hpj∥ > U∥pj∥ then
+Set U ← ∥Hpj∥/∥pj∥ and update κ, ˆζ, τ, T accordingly;
+end if
+if
+∥Hyj∥ > U∥yj∥ then
+Set U ← ∥Hyj∥/∥yj∥ and update κ, ˆζ, τ, T accordingly;
+end if
+if
+∥Hrj∥ > U∥rj∥ then
+Set U ← ∥Hrj∥/∥rj∥ and update κ, ˆζ, τ, T accordingly;
+end if
+if (yj)T ¯Hyj < ε∥yj∥2 then
+Set ˆd ← yj and terminate with d type = NC;
+else if
+∥rj∥ ≤ ˆζ∥r0∥ then
+Set ˆd ← yj and terminate with d type = SOL;
+else if
+(pj)T ¯
+Hpj < ε∥pj∥2 then
+Set ˆd ← pj and terminate with d type = NC;
+else if
+∥rj∥ >
+√
+Tτ j/2∥r0∥ then
+Compute αj, yj+1 as in the main loop above;
+Find i ∈ {0, . . . , j − 1} such that
+(yj+1 − yi)T ¯
+H(yj+1 − yi) < ε∥yj+1 − yi∥2;
+Set ˆd ← yj+1 − yi and terminate with d type = NC;
+end if
+end while
+B
+A randomized Lanczos based minimum eigenvalue oracle
+In this part we present the randomized Lanczos method proposed in [60, Section 3.2], which can be used as a
+minimum eigenvalue oracle for Algorithm 1. As mentioned in Section 4, this oracle either outputs a sufficiently
+negative curvature direction of H or certifies that H is nearly positive semidefinite with high probability. More
+details about it can be found in [60, Section 3.2].
+The following theorem justifies that Algorithm 4 is a suitable minimum eigenvalue oracle for Algorithm 1.
+Its proof is identical to that of [60, Lemma 2] and thus omitted.
+Theorem B.1 (iteration complexity of Algorithm 4). Consider Algorithm 4 with tolerance ε > 0, prob-
+ability parameter δ ∈ (0, 1), and symmetric matrix H ∈ Rn×n as its input. Then it either finds a sufficiently
+negative curvature direction v satisfying vT Hv ≤ −ε/2 and ∥v∥ = 1 or certifies that λmin(H) ≥ −ε holds with
+probability at least 1 − δ in at most N(ε, δ) iterations, where N(ε, δ) is defined in (110).
+Notice that generally, computing ∥H∥ in Algorithm 4 may not be cheap when n is large. Nevertheless, ∥H∥
+can be efficiently estimated via a randomization scheme with high confidence (e.g., see the discussion in [60,
+Appendix B3]).
+33
+
+Algorithm 4 A randomized Lanczos based minimum eigenvalue oracle
+Input: symmetric matrix H ∈ Rn×n, tolerance ε > 0, and probability parameter δ ∈ (0, 1).
+Output: a sufficiently negative curvature direction v satisfying vT Hv ≤ −ε/2 and ∥v∥ = 1; or a certificate that λmin(H) ≥
+−ε with probability at least 1 − δ.
+Apply the Lanczos method [48] to estimate λmin(H) starting with a random vector uniformly generated on the unit
+sphere, and run it for at most
+N(ε, δ) := min
+�
+n, 1 +
+�
+ln(2.75n/δ2)
+2
+�
+∥H∥
+ε
+��
+(110)
+iterations.
+(i) If it finds a unit vector v such that vT Hv ≤ −ε/2 at some iteration, it terminates immediately and returns v.
+(ii) Otherwise, it certifies that λmin(H) ≥ −ε holds with probability at least 1 − δ.
+34
+
diff --git a/5tE2T4oBgHgl3EQf6wj3/content/tmp_files/load_file.txt b/5tE2T4oBgHgl3EQf6wj3/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..53ea812f3be091dab9721ba463d8fe4a16702f5b
--- /dev/null
+++ b/5tE2T4oBgHgl3EQf6wj3/content/tmp_files/load_file.txt
@@ -0,0 +1,1877 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf,len=1876
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='04204v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='OC] 10 Jan 2023 A Newton-CG based barrier-augmented Lagrangian method for general nonconvex conic optimization Chuan He∗ Heng Huang† Zhaosong Lu∗ January 10, 2023 Abstract In this paper we consider finding an approximate second-order stationary point (SOSP) of general noncon- vex conic optimization that minimizes a twice differentiable function subject to nonlinear equality constraints and also a convex conic constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In particular, we propose a Newton-conjugate gradient (Newton-CG) based barrier-augmented Lagrangian method for finding an approximate SOSP of this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Under some mild assumptions, we show that our method enjoys a total inner iteration complexity of �O(ǫ−11/2) and an operation complexity of �O(ǫ−11/2 min{n, ǫ−5/4}) for finding an (ǫ, √ǫ)-SOSP of general nonconvex conic optimization with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Moreover, under a constraint qualification, these complexity bounds are improved to �O(ǫ−7/2) and �O(ǫ−7/2 min{n, ǫ−3/4}), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' To the best of our knowledge, this is the first study on the complexity of finding an approximate SOSP of general nonconvex conic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Pre- liminary numerical results are presented to demonstrate superiority of the proposed method over first-order methods in terms of solution quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Keywords: Nonconvex conic optimization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' second-order stationary point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' augmented Lagrangian method,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' barrier method,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Newton-conjugate gradient method,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' iteration complexity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' operation complexity Mathematics Subject Classification: 49M05,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 49M15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 68Q25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 90C26,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 90C30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 90C60 1 Introduction In this paper we consider the following general nonconvex conic optimization problem: min x {f(x) : c(x) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' x ∈ K},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (1) where K ⊆ Rn is a closed and pointed convex cone with nonempty interior,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' and f : Rn → R and c : Rn → Rm are continuous in K and twice continuously differentiable in the interior of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Assume that problem (1) has at least one optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Our goal is to propose an implementable method with complexity guarantees for finding an approximate second-order stationary point (SOSP) of (1) that will be introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In recent years, there has been considerable research on designing algorithms with complexity guarantees for finding an approximate SOSP of nonconvex optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In particular, numerous algorithms were developed for nonconvex unconstrained optimization, such as cubic regularized Newton methods [1, 18, 21, 57], trust-region methods [34, 35, 53], quadratic regularization method [14], accelerated gradient method [19, 20], second-order line-search method [61], Newton-conjugate gradient (Newton-CG) method [60], and gradient-based methods with random perturbations [2, 46, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In addition, several methods with complexity guarantees have also been proposed for nonconvex optimization with relatively simple constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' For example, interior-point method [10], log-barrier method [58], and projected gradient descent method [69] were proposed for nonconvex ∗Department of Industrial and Systems Engineering, University of Minnesota, USA (email: he000233@umn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='edu, zhaosong@umn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The work of the last author was partially supported by NSF Award IIS-2211491 †Department of Electrical and Computer Engineering, University of Pittsburgh, USA (email: heng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='huang@pitt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The work of this author was partially supported by NSF Award IIS-2211492.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 1 optimization with sign constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Besides, the interior-point method [10] was generalized in [42] for nonconvex optimization with sign constraints and additional linear equality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Also, a projected gradient descent method with random perturbations was proposed in [51] for nonconvex optimization with linear inequality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Iteration complexity of these methods has been established for finding an approximate SOSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Besides, operation complexity in terms of the total number of fundamental operations has been studied for the methods [1, 2, 18, 19, 20, 34, 46, 60, 61, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Several methods, including trust-region methods [17, 31], sequential quadratic programming method [15], two-phase method [11, 27, 30], penalty method [40], and augmented Lagrangian (AL) methods [4, 12, 44, 63, 70], were proposed for finding an approximate SOSP of equality constrained optimization: min x {f(x) : c(x) = 0}, (2) which is special case of (1) with K = Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' total inner iteration complexity and operation complexity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' which are respectively measured by the total number of iterations of the Newton-CG method in [60] and the total number of gradient evaluations and matrix-vector products performed in the method,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' were established in [44,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 70] for finding an (ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' √ǫ)-SOSP x of (2) which together with some λ ∈ Rm satisfies ∥c(x)∥ ≤ ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ∥∇f(x) + ∇c(x)λ∥ ≤ ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' dT (∇2f(x) + �m i=1 λi∇2ci(x))d ≥ −√ǫ∥d∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ∀d ∈ {d : ∇c(x)T d = 0},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' where ∇c denotes the transpose of the Jacobian of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Specifically, under some suitable assumptions, including a generalized linear independence constraint qualification (GLICQ), the AL method [70] enjoys a total inner iter- ation complexity of �O(ǫ−11/2) and an operation complexity �O(ǫ−11/2 min{n, ǫ−3/4}),1 while the AL method [44] achieves a total inner iteration complexity of �O(ǫ−7/2) and an operation complexity of �O(ǫ−7/2 min{n, ǫ−3/4}) for finding an (ǫ, √ǫ)-SOSP of problem (2) with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' On the other hand, when the GLICQ does not hold, the AL method [44] has a total inner iteration complexity of �O(ǫ−11/2) and an operation complexity of �O(ǫ−11/2 min{n, ǫ−5/4}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Besides, it shall be mentioned that Newton-CG based AL methods were developed for efficiently solving a variety of convex optimization problems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=', see [72, 73]), though their complexities remain unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In addition, a Newton-CG based barrier method was recently proposed in [43] for finding an approximate SOSP of a class of nonconvex conic optimization of the form min x {f(x) : Ax − b = 0, x ∈ K} (3) for some A ∈ Rm×n and b ∈ Rm, which is a special case of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Iteration and operation complexity of this method were established in [43] for finding an (ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' √ǫ)-SOSP x of (3) which together with some λ ∈ Rm satisfies Ax = b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' x ∈ int K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ∇f(x) + AT λ ∈ K∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ∥∇f(x) + AT λ∥∗ x ≤ ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' dT ∇2B(x)−1/2∇2f(x)∇2B(x)−1/2d ≥ −√ǫ∥d∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ∀d ∈ {d : A∇2B(x)−1/2d = 0},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' where int K and K∗ are respectively the interior and dual cone of K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' B is a logarithmically homogeneous self- concordant barrier function for K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' and ∥ · ∥∗ x is a local norm induced by B at x (see Section 2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Under some suitable assumptions, this method achieves an iteration complexity of O(ǫ−3/2) and an operation complexity2 of �O(ǫ−3/2 min{n, ǫ−1/4}) for finding an (ǫ, √ǫ)-SOSP with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Besides, a Hessian barrier algorithm was proposed in [38] for finding an approximate SOSP of problem (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Given that this algorithm requires solving a cubic regularized subproblem exactly per iteration, it is generally not implementable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It shall also be mentioned that finding an approximate first-order stationary point of (1) with K = Rn + was extensively studied in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=', [5, 6, 7, 32, 33, 37, 39, 49, 55, 65, 66, 67]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Notably, a hybrid approach 1In fact, a total inner iteration complexity of �O(ǫ−7) and an operation complexity �O(ǫ−7 min{n, ǫ−1}) were established in [70] for finding an (ǫ, ǫ)-SOSP of problem (1) with high probability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' see [70, Theorem 4(ii), Corollary 3(ii), Theorem 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Nevertheless, they can be easily modified to obtain the aforementioned complexity for finding an (ǫ, √ǫ)-SOSP of (1) with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 2The operation complexity of the barrier method [43] is measured by the amount of fundamental operations consisting of matrix- vector products, matrix multiplications, Cholesky factorizations, and backward or forward substitutions to a triangular linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 2 by combining barrier and AL methods was commonly used in [5, 6, 7, 33, 37, 39, 49, 55]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' However, finding an approximate SOSP of (1) by such a hybrid approach has not been considered, even for (1) with K = Rn +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Inspired by these and [43, 44], in this paper we propose a Newton-CG based barrier-AL method for finding an approximate SOSP of problem (1) with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Our main contributions are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We study first- and second-order optimality conditions for problem (1) and introduce an approximate counterpart of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We propose an implementable Newton-CG based barrier-AL method for finding an approximate SOSP of (1), whose fundamental operations consist of matrix-vector products, Cholesky factorizations, and backward or forward substitutions to a triangular linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We show that under some mild assumptions, our proposed method has a total inner iteration complex- ity of �O(ǫ−11/2) and an operation complexity of �O(ǫ−11/2 min{n, ǫ−5/4}) for finding an (ǫ, √ǫ)-SOSP of (1) with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Furthermore, under a constraint qualification, we show that our method achieves an improved total inner iteration complexity of �O(ǫ−7/2) and an improved operation complexity of �O(ǫ−7/2 min{n, ǫ−3/4}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3 To the best of our knowledge, there was no complexity result for finding an approximate SOSP of problem (1) in the literature before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In Section 2, we introduce some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In Section 3, we study optimality conditions of problem (1) and introduce an inexact counterpart of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In Section 4, we propose a preconditioned Newton-CG method for solving a barrier problem and study its complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We then propose a Newton-CG based barrier-AL method for (1) and study its complexity in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We present in Section 6 some preliminary numerical results for the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In Section 7, we present the proofs of the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Finally, we make some concluding remarks in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 2 Notation and preliminaries Throughout this paper, we let Rn denote the n-dimensional Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The symbol ∥ · ∥ stands for the Euclidean norm of a vector or the spectral norm of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The identity matrix is denoted by I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We denote by λmin(H) the minimum eigenvalue of a real symmetric matrix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' For any two real symmetric matrices M1 and M2, M1 ⪯ M2 means that M2 − M1 is positive semidefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' For any positive semidefinite matrix M, M 1/2 denotes a positive semidefinite matrix such that M = M 1/2M 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' For the closed convex cone K, its interior and dual cone are respectively denoted by int K and K∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' For any x ∈ K, the normal cone and tangent cone of K at x are denoted by N K(x) and TK(x), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The Euclidean ball centered at the origin with radius R ≥ 0 is denoted by BR := {x : ∥x∥ ≤ R}, and we use ΠBR(v) to denote the Euclidean projection of a vector v onto BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' For a given finite set A, we let | A | denote its cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' For any s ∈ R, we let sgn(s) be 1 if s ≥ 0 and let it be −1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In addition, �O(·) represents O(·) with logarithmic terms omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Logarithmically homogeneous self-concordant (LHSC) barrier function is a key ingredient in the development of interior-point methods for convex programming (see the monograph [56]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It will also play a crucial role in the design and analysis of Newton-CG based barrier-AL method for solving problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Throughout this paper, we assume that the cone K is equipped with a ϑ-logarithmically homogeneous self-concordant (ϑ-LHSC) barrier function B for some ϑ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' That is, B : int K → R satisfies the following conditions: (i) B is convex and three times continuously differentiable in int K, and moreover, |ψ′′′(0)| ≤ 2(ψ′′(0))3/2 holds for all x ∈ int K and u ∈ Rn, where ψ(t) = B(x + tu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (ii) B is a barrier function for K, that is, B(x) goes to infinity as x approaches the boundary of K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (iii) B is logarithmically homogeneous, that is, B(tx) = B(x) − ϑ ln t holds for all x ∈ int K and t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' For any x ∈ int K, the function B induces the following local norms: ∥v∥x := � vT ∇2B(x)v �1/2 , ∀v ∈ Rn, 3It shall be mentioned that the total numbers of Cholesky factorizations are only �O(ǫ−7/2) and �O(ǫ−11/2) respectively for the case where constraint qualification holds or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' See Subsections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='4 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 3 ∥v∥∗ x := � vT ∇2B(x)−1v �1/2 , ∀v ∈ Rn, ∥M∥∗ x := max ∥v∥x≤1 ∥Mv∥∗ x, ∀M ∈ Rn×n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (4) In addition, ∇2B(x)−1 is well-defined only in int K but undefined on the boundary of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' To capture the behavior of ∇2B(x)−1 as x approaches the boundary of K, the concept of the limiting inverse of the Hessian of B was recently introduced in [43], which can be viewed as a generalization of [∇2B]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Specifically, the limiting inverse of the Hessian of B is defined as follows: ∇−2B(x) := � M : M = lim k→∞ ∇2B(xk)−1 for some {xk} ⊂ int K with xk → x as k → ∞ � , ∀x ∈ K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' As established in [43, Theorem 1], the inverse of ∇2B(x) is bounded in any nonempty bounded subset of int K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Consequently, ∇−2B(x) ̸= ∅ for all x ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Moreover, the following property holds for ∇−2B, whose proof can be found in [43, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' For any x ∈ K, it holds that {x + M 1/2d : ∥d∥ < 1} ⊆ K for all M ∈ ∇−2B(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 3 Optimality conditions Classical first- and second-order optimality conditions for nonlinear optimization can be specialized to prob- lem (1) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=', see [62, Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='38 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='46]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' However, the inexact counterparts of them are not suitable for the design and analysis of a barrier-AL method for solving (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In this section we study some alternative first- and second-order optimality conditions for (1) and also introduce an inexact counterpart of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Suppose that x∗ is a local minimizer of problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' To derive optimality conditions, one typically needs to impose a constraint qualification (CQ) for x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The Robinson’s CQ, {∇c(x∗)T d : d ∈ TK(x∗)} = Rm, is a natural and general one (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=', see [62, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' However, verification of Robinson’s CQ may not be easy for a general cone K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Thus, we instead consider a more easily verifiable CQ that M 1/2∇c(x∗) has full column rank for some M ∈ ∇−2B(x∗), which turns out to be stronger than Robinson’s CQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Indeed, suppose that such a CQ holds at x∗ for some M ∈ ∇−2B(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It then follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 that {M 1/2 ˜d : ∥ ˜d∥ < 1} ⊆ TK(x∗) and hence {M 1/2 ˜d : ˜d ∈ Rn} ⊆ TK(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By this and the full column rank of M 1/2∇c(x∗), one has {∇c(x∗)T d : d ∈ TK(x∗)} ⊇ {∇c(x∗)T M 1/2 ˜d : ˜d ∈ Rn} = Rm, and hence Robinson’s CQ holds at x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We are now ready to establish some first- and second-order optimality conditions for problem (1) under the aforementioned CQ, whose proof is relegated to Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 (first- and second-order optimality conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Let x∗ be a local minimizer of problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Suppose that f is twice continuously differentiable at x∗ and M 1/2∇c(x∗) has full column rank for some M ∈ ∇−2B(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Then there exists a Lagrangian multiplier λ∗ ∈ Rm such that ∇f(x∗) + ∇c(x∗)λ∗ ∈ K∗, (5) M 1/2(∇f(x∗) + ∇c(x∗)λ∗) = 0, (6) and additionally, dT M 1/2 � ∇2f(x∗) + m � i=1 λ∗ i ∇2ci(x∗) � M 1/2d ≥ 0, ∀d ∈ {d : ∇c(x∗)T M 1/2d = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (7) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The relations (5) and (6) are the first-order optimality conditions of problem (1), which are actually equivalent to the classical optimality condition ∇f(x∗)+∇c(x∗)λ∗ ∈ − N K(x∗) (see [43, Proposition 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Notice that it is generally impossible to find a point exactly satisfying the above first- and second-order optimality conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We are instead interested in finding a point satisfying their approximate counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' To this end, we next introduce the definition of an approximate first-order stationary point (FOSP) and second- order stationary point (SOSP) of problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 4 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 (ǫ1-first-order stationary point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' For any ǫ1 > 0, a point x is called an ǫ1-first-order stationary point (ǫ1-FOSP) of problem (1) if it, together with some λ ∈ Rm, satisfies ∥c(x)∥ ≤ ǫ1, x ∈ int K, (8) ∇f(x) + ∇c(x)λ ∈ K∗, (9) ∥∇f(x) + ∇c(x)λ∥∗ x ≤ ǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (10) Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2 ((ǫ1, ǫ2)-second-order stationary point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' For any ǫ1, ǫ2 > 0, a point x is called an (ǫ1, ǫ2)- second-order stationary point ((ǫ1, ǫ2)-SOSP) of problem (1) if it, together with some λ ∈ Rm, satisfies (8)-(10) and additionally dT ∇2B(x)−1/2 � ∇2f(x) + m � i=1 λi∇2ci(x) � ∇2B(x)−1/2d ≥ −ǫ2∥d∥2, ∀d ∈ C(x), (11) where C(·) is defined as C(x) := {d : ∇c(x)T ∇2B(x)−1/2d = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (12) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Notice that if the pair (x, λ) satisfies (10) and (11), then it also nearly satisfies (6) and (7) with (x∗, λ∗) replaced by (x, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Thus, (10) and (11) are indeed inexact counterparts of (6) and (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Moreover, the above definitions of ǫ1-FOSP and (ǫ1, ǫ2)-SOSP are reduced to the ones introduced in [43] for the case where c is affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 4 A preconditioned Newton-CG method for barrier problems In this section we propose a preconditioned Newton-CG method in Algorithm 1, which is a modification of the Newton-CG based barrier method [43, Algorithm 2], for finding an approximate SOSP of the barrier problem min x {φµ(x) := F(x) + µB(x)}, (13) where F : Rn → R is twice continuously differentiable in int K and µ > 0 is a given barrier parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Specifically, the proposed method finds an (ǫg, ǫH)-SOSP x of problem (13) that satisfies ∥∇φµ(x)∥∗ x ≤ ǫg, λmin(∇2B(x)−1/2∇2φµ(x)∇2B(x)−1/2) ≥ −ǫH (14) for any prescribed tolerances ǫg, ǫH ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It will be used to solve the subproblems arising in the barrier-AL method later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Our preconditioned Newton-CG method (Algorithm 1) consists of two main components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The first main component is a modified CG method, referred to as capped CG method, which was proposed in [60, Algorithm 1] for solving a possibly indefinite linear system (H + 2εI) ˆd = −g, (15) where 0 ̸= g ∈ Rn, ε > 0, and H ∈ Rn×n is a symmetric matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The capped CG method terminates within a finite number of iterations and returns either an approximate solution ˆd to (15) satisfying ∥(H+2εI) ˆd+g∥ ≤ ˆζ∥g∥ and ˆdT H ˆd ≥ −ε∥ ˆd∥2 for some ˆζ ∈ (0, 1) or a sufficiently negative curvature direction ˆd of H with ˆdT H ˆd < −ε∥ ˆd∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The second main component is a minimum eigenvalue oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Given a symmetric matrix H ∈ Rn×n and ε > 0, this oracle either produces a sufficiently negative curvature direction v of H with ∥v∥ = 1 and vT Hv ≤ −ε/2 or certifies that λmin(H) ≥ −ε holds with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' For ease of reference, we present these two main components in Algorithms 3 and 4 in Appendices A and B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We are now ready to describe our preconditioned Newton-CG method (Algorithm 1) for solving (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' At iteration t, if the first relation in (14) is not satisfied at the iterate xt, the capped CG method (Algorithm 3) is invoked to find a descent direction for φµ by solving the following damped preconditioned Newton system (M T t ∇2φµ(xt)Mt + 2ǫHI) ˆd = −M T t ∇φµ(xt), 5 where Mt is a matrix such that ∇2B(xt)−1 = MtM T t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (16) A line search along this descent direction is then performed to result in a reduction on φµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Otherwise, the min- imum eigenvalue oracle (Algorithm 4) is invoked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' This oracle either produces a sufficiently negative curvature direction of M T t ∇2φµ(xt)Mt along which a line search is performed to result in a reduction on φµ, or certifies that the iterate xt also satisfies the second relation in (14) with high probability and terminates the precondi- tioned Newton-CG method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The detailed description of our preconditioned Newton-CG method is presented in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Algorithm 1 A preconditioned Newton-CG method for problem (13) Input: tolerances ǫg, ǫH ∈ (0, 1), backtracking ratio θ ∈ (0, 1), starting point u0 ∈ int K, CG-accuracy parameter ζ ∈ (0, 1), maximum step length β ∈ [ǫH, 1), line-search parameter η ∈ (0, 1), probability parameter δ ∈ (0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Set x0 = u0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' for t = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' do if ∥∇φµ(xt)∥∗ xt > ǫg then Call Algorithm 3 (see Appendix A) with H = MT t ∇2φµ(xt)Mt, ε = ǫH, g = MT t ∇φµ(xt), accuracy parameter ζ, and bound U = 0 to obtain outputs ˆdt, d type, where Mt is given in (16);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' if d type=NC then dt ← − sgn(( ˆdt)T MT t ∇φµ(xt)) min � |( ˆdt)T MT t ∇2φµ(xt)Mt ˆdt| ∥ ˆdt∥3 , β ∥ ˆdt∥ � ˆdt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (17) else {d type=SOL} dt ← min � 1, β ∥ ˆdt∥ � ˆdt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (18) end if Go to Line Search;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' else Call Algorithm 4 (see Appendix B) with H = MT t ∇2φµ(xt)Mt, ε = ǫH, and probability parameter δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' if Algorithm 4 certifies that λmin(MT t ∇2φµ(xt)Mt) ≥ −ǫH then Output xt and terminate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' else {Sufficiently negative curvature direction v returned by Algorithm 4} Set d type=NC and dt ← − sgn(vT MT t ∇φµ(xt)) min{|vT MT t ∇2φµ(xt)Mtv|, β}v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (19) Go to Line Search;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' end if end if Line Search: if d type=SOL then Find αt = θjt, where jt is the smallest nonnegative integer j such that φµ(xt + θjMtdt) < φµ(xt) − ηǫHθ2j∥dt∥2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (20) else {d type=NC} Find αt = θjt, where jt is the smallest nonnegative integer j such that φµ(xt + θjMtdt) < φµ(xt) − ηθ2j∥dt∥3/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (21) end if xt+1 = xt + αtMtdt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' end for 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 Iteration and operation complexity of Algorithm 1 In this subsection we study iteration and operation complexity of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' To proceed, we make the following assumptions on problem (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (a) There exists a finite φlow such that φµ(x) ≥ φlow, ∀x ∈ int K, (22) S = {x ∈ int K : φµ(x) ≤ φµ(u0)} is bounded, (23) 6 where u0 ∈ int K is the initial point of Algorithm 1 and φµ is given in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (b) There exists LF H > 0 such that ∥∇2F(y) − ∇2F(x)∥∗ x ≤ LF H∥y − x∥x, ∀x, y ∈ Ω with ∥y − x∥x ≤ β, where Ω ⊂ int K is an open bounded convex neighborhood of S and β ∈ (0, 1) is an input of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (c) The quantities U F g and U F H are finite, where U F g := sup x∈S ∥∇F(x)∥∗ x, U F H := sup x∈S ∥∇2F(x)∥∗ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (24) Before establishing operation complexity of Algorithm 1, let us make some observations on its fundamental operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Firstly, at iteration t, the main effort of Algorithm 1 is on the execution of Algorithm 3 or 4 with H = M T t ∇2φµ(xt)Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Secondly, the main computational cost of Algorithms 3 and 4 per iteration is on the product of H and a vector v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Consequently, it suffices to focus on computing Hv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Indeed, notice from (13) and (16) that Hv = M T t ∇2φµ(xt)Mtv = M T t ∇2F(xt)Mtv + µv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Thus, computing Hv consists of one Hessian-vector product of F and two matrix-vector products involving Mt and M T t , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We next discuss how to efficiently compute the product of Mt or M T t and a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' When K is the nonnegative orthant, its associated barrier function is B(x) = − �n i=1 ln xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Notice that ∇2B(x) is a diagonal matrix and so is Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' As a result, the operation cost for computing the product of Mt or M T t and a vector is O(n), which is typically cheaper than the Hessian-vector product of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' When K is a general cone, directly computing Mt may be too expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In view of ∇2B(xt) = M −T t M −1 t (see (16)), one can instead choose M −T t as the Cholesky factor of ∇2B(xt), which is computed only once in each iteration of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Once M −T t is available, the product of Mt or M T t and a vector can be computed by performing backward or forward substitution to a linear system with coefficient matrix M −1 t or M −T t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Based on the above discussion, we conclude that: (i) when K is the nonnegative orthant, the fundamental operations of Algorithm 1 consist only of the Hessian-vector products of F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (ii) when K is a general cone, the fundamental operations of Algorithm 1 consist of the Hessian-vector products of F, Cholesky factorizations of ∇2B, and backward or forward substitutions to a triangular linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The following theorem states the iteration and operation complexity of Algorithm 1, whose proof is deferred to Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 (Complexity of Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Suppose that Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Let T1 = � φhi − φlow min{csol, cnc} max{ǫ−2 g ǫH, ǫ−3 H } � + �φhi − φlow cnc ǫ−3 H � + 1, T2 = �φhi − φlow cnc ǫ−3 H � + 1, (25) where φhi = φµ(u0), φlow is given in (22), and csol = η min �� 4(1−β) 4+ζ+√ (4+ζ)2+8[(1−β)LF H+µ(2−β)/(1−β)] �2 , � min{6(1−η),2}θ LF H+µ(2−β)/(1−β)2 �2 � , (26) cnc = η 16 min � 1, � min{3(1−η),1}θ LF H+µ(2−β)/(1−β)2 �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (27) Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (i) The total number of calls of Algorithm 4 in Algorithm 1 is at most T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (ii) The total number of calls of Algorithm 3 in Algorithm 1 is at most T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 7 (iii) (iteration complexity) Algorithm 1 terminates in at most T1 + T2 iterations with T1 + T2 = O((φhi − φlow)(LF H)2 max{ǫ−2 g ǫH, ǫ−3 H }).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (28) Moreover, its output xt satisfies the first relation in (14) deterministically and the second relation in (14) with probability at least 1 − δ for some 0 ≤ t ≤ T1 + T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (iv) (operation complexity) The total numbers of Cholesky factorizations and other fundamental operations consisting of the Hessian-vector products of F and backward or forward substitutions to a triangular linear system required by Algorithm 1 are at most T1 + T2 and �O((φhi − φlow)(LF H)2 max{ǫ−2 g ǫH, ǫ−3 H } min{n, (U F H/ǫH)1/2}), respectively, where U F H is given in (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 5 A Newton-CG based barrier-AL method for problem (1) In this section we propose a Newton-CG based barrier-AL method for finding a stochastic (ǫ, √ǫ)-SOSP of problem (1) for any prescribed tolerance ǫ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Recall that B is the ϑ-LHSC barrier function associated with K for some ϑ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We now make the following additional assumptions on problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (a) An ǫ/2-approximately strictly feasible point zǫ of problem (1), namely satisfying zǫ ∈ int K and ∥c(zǫ)∥ ≤ ǫ/2, is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (b) There exist constants ¯µ ≥ µ, fhi, flow ∈ R and γ, δf, δc > 0, independent of ǫ, such that f(zǫ) + ˜µB(zǫ) ≤ fhi, ∀˜µ ∈ (0, ¯µ], (29) f(x) + ˜µB(x) + γ∥c(x)∥2/2 ≥ flow, ∀˜µ ∈ (0, ¯µ], x ∈ int K, (30) S(δf, δc) := � ˜µ∈(0,¯µ] {x ∈ int K : f(x) + ˜µB(x) ≤ fhi + δf, ∥c(x)∥ ≤ 1 + δc} is bounded, (31) where µ = ǫ/(2ϑ1/2 + 2) and zǫ is given in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (c) There exist Lf H, Lc H > 0 and β ∈ (0, 1) such that ∥∇2f(y) − ∇2f(x)∥∗ x ≤ Lf H∥y − x∥x, ∀x, y ∈ Ω(δf, δc) with ∥y − x∥x ≤ β, ∥∇2ci(y) − ∇2ci(x)∥∗ x ≤ Lc H∥y − x∥x, ∀x, y ∈ Ω(δf, δc) with ∥y − x∥x ≤ β, 1 ≤ i ≤ m, (32) where Ω(δf, δc) ⊂ int K is an open bounded convex neighborhood of S(δf, δc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (d) The quantities U f g , U c g, U f H and U c H are finite, where U f g = supx∈Ω(δf ,δc) ∥∇f(x)∥∗ x, U c g = supx∈Ω(δf ,δc) max1≤i≤m ∥∇ci(x)∥∗ x, (33) U f H = supx∈Ω(δf ,δc) ∥∇2f(x)∥∗ x, U c H = supx∈Ω(δf,δc) max1≤i≤m ∥∇2ci(x)∥∗ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (34) We next make some remarks about Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (i) A similar assumption as Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(a) was considered in the study of AL methods for nonconvex equality constrained optimization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=', see [28, 41, 44, 52, 70]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By imposing Assump- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(a), we restrict our study on problem (1) for which an ǫ/2-approximately strictly feasible point zǫ can be found by an inexpensive procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' As an example of such problem instances, when the generalized LICQ condition λmin(∇c(x)T ∇2B(x)−1∇c(x)) ≥ σ2 > 0 (see Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2 below) holds on a level set of ∥c(x)∥2 + ˜µB(x) for a sufficiently small ˜µ > 0 and a constant σ, the point zǫ can be found by applying our preconditioned Newton-CG method (Algorithm 1) to the barrier problem minx ∥c(x)∥2 + ˜µB(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' As observed from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1, the resulting iteration and operation complexity for finding such zǫ are respec- tively O(ǫ−3/2) and �O(ǫ−3/2 min{n, ǫ−1/4}), which are negligible compared with those of our barrier-AL 8 method (see Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='5 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In addition, the Newton-CG based barrier AL method (Algo- rithm 2) proposed below is a second-order method with the aim to find a second-order stationary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It is more expensive than a first-order method in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' To make best use of such a barrier AL method in practice, it is natural to run a first-order method in advance to obtain an ǫ/2-first-order stationary point zǫ and then run the barrier AL method using zǫ as an ǫ/2-approximately feasible point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Therefore, Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(a) is met in practice, provided that an ǫ/2-first-order stationary point of (1) can be found by a first-order method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (ii) Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(b) is mild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In particular, the assumption in (29) holds if f(x)+ ¯µ[B(x)]+ is bounded above for all x ∈ int K with ∥c(x)∥ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Besides, the function f(x) + ˜µ B(x) + γ∥c(x)∥2/2 is a barrier-quadratic penalty function of problem (1) and is typically bounded below on int K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In addition, letting z0 be an arbitrary point in int K, it can be shown that S(δf, δc) ⊆ S1 ∪ S2, where S1 = � x ∈ int K : f(x) ≤ fhi + δf + ¯µ + ¯µ[B(z0)]+, B(x) ≥ −1 − [B(z0)]+, ∥c(x)∥ ≤ 1 + δc � , S2 = � x ∈ int K : f(x) −B(x) ≤ [fhi+δf]+ 1+[B(z0)]+ + ¯µ, B(x) ≤ −1 − [B(z0)]+, ∥c(x)∥ ≤ 1 + δc � , and t+ = max{0, t} for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Thus, the assumption in (31) holds if S1 and S2 are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The latter holds, for example, for the problem with f(x) = ℓ(x) + �n i=1 xp i , B(x) = − �n i=1 ln xi and K = Rn + studied in [42], where ℓ : Rn → R+ is a loss function and p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (iii) Assumptions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(c) means that ∇2f and ∇2ci, 1 ≤ i ≤ m, are locally Lipschitz continuous in Ω(δf, δc) with respect to the local norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' As pointed out in [43, Section 5], such local Lipschitz continuity is weaker than the global Lipschitz continuity of ∇2f and ∇2ci, 1 ≤ i ≤ m, in Ω(δf, δc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Besides, Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(d) holds if f and c are twice continuously differentiable in an open set containing K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 A Newton-CG based barrier-AL method We now describe our Newton-CG based barrier-AL method (Algorithm 2) for finding a stochastic (ǫ, √ǫ)-SOSP of problem (1) for a prescribed tolerance ǫ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Instead of solving (1) directly, our method solves the following perturbed equality constrained barrier problem min x {f(x) + µB(x) : ˜c(x) := c(x) − c(zǫ) = 0} (35) with µ = ǫ/(2ϑ1/2+2) and zǫ given in Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It follows a similar AL framework as the one proposed in [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In particular, at the kth iteration, it first applies the preconditioned Newton-CG method (Algorithm 1) to find an approximate stochastic SOSP xk+1 of the subproblem: min x � Lµ(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) := f(x) + µB(x) + (λk)T ˜c(x) + ρk 2 ∥˜c(x)∥2� , (36) which is an AL subproblem associated with (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Then the standard multiplier estimate ˜λk+1 is updated by the classical scheme (see step 3 of Algorithm 2), and the truncated Lagrangian multiplier λk+1 is updated by projecting ˜λk+1 onto a Euclidean ball (see step 5 of Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='4 Finally, the penalty parameter ρk+1 is adaptively updated according to the improvement on constraint violation (see step 6 of Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' This update scheme is very practical and widely used in AL type methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=', see [3, 8, 28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (i) Notice that the starting point x0 init of Algorithm 2 can be different from zǫ and it may be rather infeasible, though zǫ is a nearly feasible point of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Besides, zǫ is used to monitor convergence of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Specifically, if the algorithm runs into a “poorly infeasible point” xk, namely satisfying Lµ(xk, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) > f(zǫ)+µB(zǫ), it will be superseded by zǫ (see (39)), which prevents the iterates {xk} from converging to an infeasible point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Yet, xk may be rather infeasible when k is not large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Thus, Algorithm 2 substantially differs from a funneling or two-phase type algorithm, in which a nearly feasible point is found in Phase 1, and then approximate stationarity is sought while near feasibility is maintained throughout Phase 2 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=', see [11, 16, 22, 23, 24, 25, 26, 36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 4The λk+1 is also called a safeguarded Lagrangian multiplier, which has been used in the literature for designing some AL methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=', see [3, 13, 44, 47]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It has been shown to enjoy many practical and theoretical advantages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=', see [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 9 Algorithm 2 A Newton-CG based barrier-AL method for problem (1) Let γ and µ be given in Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Input: ǫ ∈ (0, 1), Λ ≥ 0, x0 ∈ int K, λ0 ∈ BΛ, ρ0 > 2γ, α ∈ (0, 1), r > 1, δ ∈ (0, 1), zǫ given in Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(a), and τk = max{µ, rk log µ/ log 2} for all k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 1: Set k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 2: Call Algorithm 1 with ǫg = τk, ǫH = √τk and u0 = xk init to find an approximate solution xk+1 ∈ int K to minx Lµ(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) such that Lµ(xk+1, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) ≤ f(zǫ) + µB(zǫ), ∥∇x Lµ(xk+1, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk)∥∗ xk+1 ≤ τk, (37) λmin(M T k+1∇2 xxLµ(xk+1, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk)Mk+1) ≥ −√τk with probability at least 1 − δ, (38) where Mk+1 is defined as in (16) and xk init = � zǫ if Lµ(xk, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) > f(zǫ) + µB(zǫ), xk otherwise, for k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (39) 3: Set ˜λk+1 = λk + ρk˜c(xk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 4: If τk ≤ µ and ∥c(xk+1)∥ ≤ ǫ, then output (xk+1, ˜λk+1) and terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 5: Set λk+1 = ΠBΛ(˜λk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 6: If k = 0 or ∥˜c(xk+1)∥ > α∥˜c(xk)∥, set ρk+1 = rρk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Otherwise, set ρk+1 = ρk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 7: Set k ← k + 1, and go to step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (ii) The choice of ρ0 in Algorithm 2 is mainly for the simplicity of complexity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Yet, it may be overly large and lead to highly ill-conditioned AL subproblems in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' To make Algorithm 2 practically more efficient, one can possibly modify it by choosing a relatively small initial penalty parameter, then solving the subsequent AL subproblems by a first-order method until an ǫ1-first-order stationary point ˆx of (35) along with a Lagrangian multiplier ˆλ is found, and finally performing the steps described in Algorithm 2 but with x0 = ˆx and λ0 = ΠBΛ(ˆλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (iii) Algorithm 2 can be easily extended to find an (ǫ, √ǫ)-SOSP of a more general conic optimization problem of the form minx,y{ ˜f(x, y) : ˜c(x, y) = 0, y ∈ K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Indeed, one can follow almost the same framework as Algorithm 2, except that the associated subproblems are solved by a preconditioned Newton-CG method, which is a slight modification of Algorithm 1 by choosing the preconditioning matrix � Mk as the one satisfying �I 0 0 ∇2B(yk) �−1 = � Mk � M T k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Before analyzing the complexity of Algorithm 2, we first argue that it is well-defined if ρ0 is suitably chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Specifically, we will show that when ρ0 is sufficiently large, one can apply Algorithm 1 to the subproblem minx Lµ(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) with xk init as the initial point to find an xk+1 satisfying (37) and (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' To this end, we start by noting from (29), (35), (36) and (39) that Lµ(xk init, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) (39) ≤ max{Lµ(zǫ, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk), f(zǫ) + µB(zǫ)} (35)(36) = f(zǫ) + µB(zǫ) (29) ≤ fhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (40) Based on this observation, we show in the next lemma that when ρ0 is sufficiently large, Lµ(·, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) is bounded below and its certain level set is bounded, whose proof is deferred to Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 (Properties of Lµ(·, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) and L(·, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Suppose that Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Let (λk, ρk) be generated at the kth iteration of Algorithm 2 for some k ≥ 0, and L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) := f(x) + (λk)T ˜c(x) + ρk 2 ∥˜c(x)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (41) Let S(δf, δc) and xk init be respectively defined in (31) and (39) and let δf, δc, µ, fhi, flow, Lf H, Lc H, U f H, U c g, U c H and Ω(δf, δc) be given in Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Suppose that ρ0 is sufficiently large such that δf,1 ≤ δf and δc,1 ≤ δc, where δf,1 := Λ2/(2ρ0) and δc,1 := � 2(fhi − flow + γ) ρ0 − 2γ + Λ2 (ρ0 − 2γ)2 + Λ ρ0 − 2γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (42) 10 Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (i) {x ∈ int K : Lµ(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) ≤ Lµ(xk init, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk)} ⊆ S(δf, δc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (ii) infx∈int K Lµ(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) ≥ flow − γ − Λδc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (iii) ∥∇2 xx L(y, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) − ∇2 xx L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk)∥∗ x ≤ Lk,H∥y − x∥x for all x, y ∈ Ω(δf, δc) with ∥y − x∥x ≤ β, where Lk,H := Lf H + ∥λk∥1Lc H + ρkm � (1 + U c)Lc H + U c gU c H 1 − β + (2 − β)U c gU c H (1 − β)3 � , U c := sup z∈Ω(δf ,δc) ∥c(z)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (43) (iv) The quantities Uk,g and Uk,H are finite, where Uk,g := sup x∈S(δf ,δc) ∥∇x L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk)∥∗ x, Uk,H := sup x∈S(δf,δc) ∥∇2 xx L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk)∥∗ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Moreover, Uk,H ≤ U f H + ∥λk∥1U c H + ρk(m(U c g)2 + √m(2 + δc)U c H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In view of (31) and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(i) and (ii), one can see that the level set {x ∈ int K : Lµ(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) ≤ Lµ(xk init, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk)} is bounded and Lµ(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) is bounded below for all x ∈ int K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By these and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(iii) and (iv), one can further see that Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 holds for F(·) = L(·, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) and u0 = xk init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Based on this and the discussion in Section 4, we can conclude that Algorithm 1, starting with u0 = xk init, is applicable to the subproblem minx Lµ(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Moreover, it follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 that this algorithm with ǫg = τk and ǫH = √τk can produce a point xk+1 satisfying (38) and also the second relation in (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In addition, since this algorithm is descent and its starting point is xk init, its output xk+1 must satisfy Lµ(xk+1, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) ≤ Lµ(xk init, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk), which along with (40) implies that Lµ(xk+1, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) ≤ f(zǫ) + µB(zǫ) and thus xk+1 also satisfies the first relation in (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The above discussion leads to the following conclusion concerning the well-definedness of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 (Well-definedness of Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Under the same settings as in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1, the precon- ditioned Newton-CG method (Algorithm 1), when applied to the subproblem minx Lµ(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) with u0 = xk init, can find a point xk+1 satisfying (37) and (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The following theorem characterizes the output of Algorithm 2, whose proof is deferred to Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2 (Output of Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Suppose that Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 holds and that ρ0 is sufficiently large such that δf,1 ≤ δf and δc,1 ≤ δc, where δf,1 and δc,1 are defined in (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' If Algorithm 2 terminates at some iteration k, then its output xk+1 is a deterministic ǫ-FOSP of problem (1), and moreover, it is an (ǫ, √ǫ)-SOSP of (1) with probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' As seen from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2, the output of Algorithm 2 is a stochastic (ǫ, √ǫ)-SOSP of problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' On the other hand, this algorithm can be easily modified to find other approximate solutions of (1) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' For example, if only an ǫ-FOSP of (1) is to be sought, one can remove the condition (38) from Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In addition, if one aims to find a deterministic (ǫ, √ǫ)-SOSP of (1), one can replace the condition (38) and Algorithm 1 by λmin(M T k+1∇2 xxLµ(xk+1, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk)Mk+1) ≥ −√τk and a deterministic counterpart, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2 Outer iteration complexity of Algorithm 2 In this subsection we establish outer iteration complexity of Algorithm 2, which measures the number of its outer iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Notice that τk can be rewritten as τk = max{µ, ωk} with ω := rlog µ/ log 2, ∀k ≥ 0, (44) where r is an input of Algorithm 2 and µ = ǫ/(2ϑ1/2 + 2) (see Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By ǫ ∈ (0, 1), ϑ ≥ 1, and the definition of µ, one can verify that µ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By this and r > 1, one can see that ω ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' For notational convenience, we introduce the following quantity that will be frequently used later: Kǫ := � min{k ≥ 0 : ωk ≤ ǫ/(2ϑ1/2 + 2)} � = � min{k ≥ 0 : ωk ≤ µ} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (45) 11 In view of this and (44), we obtain that τk = µ for all k ≥ Kǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' This along with the termination criterion of Algorithm 2 implies that it runs for at least Kǫ iterations and terminates once ∥c(xk+1)∥ ≤ ǫ for some k ≥ Kǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Consequently, to establish outer iteration complexity of Algorithm 2, it suffices to bound such k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The resulting outer iteration complexity is presented below, whose proof is deferred to Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3 (Outer iteration complexity of Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Suppose that Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 holds and that ρ0 is sufficiently large such that δf,1 ≤ δf and δc,1 ≤ δc, where δf,1 and δc,1 are defined in (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Let ρǫ,1 := max � 8(fhi − flow + γ)ǫ−2 + 4Λǫ−1 + 2γ, 2ρ0 � , (46) Kǫ := inf{k ≥ Kǫ : ∥c(xk+1)∥ ≤ ǫ}, (47) where Kǫ is defined in (45), and γ, fhi and flow are given in Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Then Kǫ is finite, and Algorithm 2 terminates at iteration Kǫ with Kǫ ≤ �log(ρǫ,1ρ−1 0 ) log r + 1 � ����� log(ǫ(2δc,1)−1) log α ���� + 2 � + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (48) Moreover, ρk ≤ rρǫ,1 holds for 0 ≤ k ≤ Kǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='4 (Upper bounds for Kǫ and {ρk}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' As seen from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3, the number of outer iterations of Algorithm 2 for finding a stochastic (ǫ, √ǫ)-SOSP of problem (1) is at most of O(| log ǫ|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In addition, the penalty parameters {ρk} generated by this algorithm are at most of O(ǫ−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3 Total inner iteration and operation complexity of Algorithm 2 In this subsection we present the total inner iteration and operation complexity of Algorithm 2, which measures the total number of iterations and fundamental operations performed by Algorithm 1 in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Its proof is deferred to Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='4 (Total inner iteration and operation complexity of Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Suppose that Assump- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 holds and that ρ0 is sufficiently large such that δf,1 ≤ δf and δc,1 ≤ δc, where δf,1 and δc,1 are defined in (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (i) The total number of inner iterations of Algorithm 2, namely, the total number of iterations of Algorithm 1 performed in Algorithm 2, is at most �O(ǫ−11/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' If c is further assumed to be affine, it is at most �O(ǫ−3/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (ii) The total numbers of Cholesky factorizations and other fundamental operations consisting of the Hessian- vector products of f and c and backward or forward substitutions to a triangular linear system required by Algorithm 1 in Algorithm 2 are at most �O(ǫ−11/2) and �O(ǫ−11/2 min{n, ǫ−5/4}), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' If c is further assumed to be affine, they are at most �O(ǫ−3/2) and �O(ǫ−3/2 min{n, ǫ−5/4}), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It is worth mentioning that the above complexity results are established without assuming any constraint qualification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Moreover, when K is the nonnegative orthant, these results match the best known ones achieved by a Newton-CG based AL method [44] for nonconvex equality constrained optimization without imposing a constraint qualification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='4 Enhanced complexity of Algorithm 2 under constraint qualification In this subsection we study complexity of Algorithm 2 under one additional assumption that a generalized linear independence constraint qualification (GLICQ) holds for problem (1), which is introduced below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In particular, under GLICQ we will obtain an enhanced total inner iteration complexity of �O(ǫ−7/2) and an enhanced operation complexity of �O(ǫ−7/2 min{n, ǫ−3/4}) for Algorithm 2 when the equality constraints in problem (1) are nonlinear, which are significantly better than the ones in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We now introduce the GLICQ assumption for (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2 (GLICQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' There exists some σ > 0 such that λmin(∇c(x)T ∇2B(x)−1∇c(x)) ≥ σ2, ∀x ∈ S(δf, δc), (49) where S(δf, δc) is defined in (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 12 The following theorem shows that under Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2, the total inner iteration and operation complexity results presented in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='4 can be significantly improved, whose proof is deferred to Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='5 (Enhanced total inner iteration and operation complexity of Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Suppose that Assumptions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2 hold and that ρ0 is sufficiently large such that δf,1 ≤ δf and δc,1 ≤ δc, where δf,1 and δc,1 are defined in (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (i) The total number of inner iterations of Algorithm 2, namely, the total number of iterations of Algorithm 1 performed in Algorithm 2, is at most �O(ǫ−7/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' If c is further assumed to be affine, it is at most �O(ǫ−3/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (ii) The total numbers of Cholesky factorizations and other fundamental operations consisting of the Hessian- vector products of f and c and backward or forward substitutions to a triangular linear system required by Algorithm 1 in Algorithm 2 are at most �O(ǫ−7/2) and �O(ǫ−7/2 min{n, ǫ−3/4}), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' If c is further assumed to be affine, they are at most �O(ǫ−3/2) and �O(ǫ−3/2 min{n, ǫ−3/4}), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' As seen from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='5, under GLICQ and some other suitable assumptions, Algorithm 2 achieves significantly better complexity bounds than the ones in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='4 when the equality constraints in (1) are nonlinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Moreover, when K is the nonnegative orthant, the complexity results in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='5 match the best known ones achieved by a Newton-CG based AL method [44] for nonconvex equality constrained optimization under the constraint qualification that is obtained from the above GLICQ by replacing ∇2B(x) by the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 6 Numerical results In this section we conduct some preliminary numerical experiments to test performance of our Newton-CG based barrier-AL method (Algorithm 2) for solving a low-rank matrix recovery problem and a simplex-constrained nonnegative matrix factorization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In our experiments, all the algorithms are coded in Matlab and all the computations are performed on a desktop with a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='79 GHz AMD 3900XT 12-Core processor and 32 GB of RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 Low-rank matrix recovery In this subsection we consider a low-rank matrix recovery problem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=', see [9, 29, 59]) min U∈Rn×k �1 2∥ A(UU T ) − y∥2 : ∥U∥2 F ≤ b � , (50) where A : Rn×n → Rm is a linear operator and ∥ · ∥F is the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' For each triple (n, k, m), we randomly generate 10 instances of problem (50) in a similar manner as described in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In particular, we first randomly generate a linear operator A by setting A(·) = A(vec(·)), where A is an m × n2 matrix with all entries chosen from the standard normal distribution, and vec(·) is the vectorization of the associated matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='5 Then we randomly generate the ground-truth low-rank matrix X∗ = �U �U T with all entries of �U chosen from the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We finally set b = ∥ �U|2 F and y = A(X∗) + e, where ei, 1 ≤ i ≤ m, is generated according to the normal distribution with mean zero and standard deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Observe that problem (50) is equivalent to min U,s �1 2∥ A(UU T ) − y∥2 : ∥U∥2 F + s = b, s ≥ 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (51) In this experiment, we apply Algorithm 2 to find a (10−4, 10−2)-SOSP of (51) and hence of (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' To ensure that the output of Algorithm 2 is a deterministic approximate second-order stationary point, we use a minimum eigenvalue oracle that returns a deterministic output in Algorithm 2 instead, which calls the Matlab subroutine [v,λ] = eigs(H,1,’smallestreal’) to find the minimum eigenvalue λ and its associated unit eigenvector v of a real 5The vectorization of a matrix is the column vector obtained by stacking the columns of the matrix on top of one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 13 Relative error Objective value n k m Algorithm 2 SpaRSA Algorithm 2 SpaRSA 20 1 40 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3×10−4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3×10−4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='9×10−4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='8×10−4 20 2 80 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3×10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='60 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='0×10−3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='8×103 40 2 160 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='7×10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='66 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2×10−3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1×104 40 4 320 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2×10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='81 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='0×10−3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='5×105 60 3 360 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2×10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='78 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3×10−3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='4×105 60 6 720 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3×10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='9×10−2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1×106 80 4 640 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='8×10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='83 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='6×10−2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='4×106 80 8 1280 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='9×10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='90 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3×10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='5×107 100 5 1000 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2×10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='89 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='6×10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='6×107 100 10 2000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='8×10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='92 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2×10−2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1×107 Table 1: Numerical results for problem (50) symmetric matrix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Besides, we apply [68, Algorithm SpaRSA], which is a nonmonotone proximal gradient method, to find a 10−4-FOSP of (50) by generating a sequence {U t} according to U t = arg min U {∥U − U t−1 + ∇f(U t−1)/αt−1∥F : ∥U∥2 F ≤ b}, where f is the objective function of (50) and αt−1 is chosen by a backtracking line search scheme such that f(U t) ≤ max[t−M−1]+≤i≤t−1 f(U i) − σαt−1∥U t − U t−1∥2 F /2 for some σ ∈ (0, 1) and a positive integer M (see [68] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We terminate SpaRSA once the condition ∥αt−1(U t − U t−1) + ∇f(U t−1) − ∇f(U t)∥F ≤ 10−4 is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It can be verified that such U t is a 10−4-FOSP of (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We choose the initial point U 0 with all entries equal � b/(2nk) for both methods, s0 = b/2 for Algorithm 2, and set (Λ, ρ0, λ0, α, r) = (103, 102, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='25, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='5) for Algorithm 2, and (θ, ζ, η, β) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='9) for Algo- rithm 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (σ, M, αmin, αmax, η) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='01, 5, 10−30, 1030, 2) for SpaRSA [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Notice that the approximate solution obtained by SpaRSA must be a feasible point of (50), while the one found by Algorithm 2 may not be a feasible point of (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' For a fair comparison, we project the latter one into the feasible region of (50) to obtain a feasible approximate solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Then we compare the quality of these feasible approximate solutions in terms of objective value and relative error defined as ∥UU T −X∗∥F /∥X∗∥F for a given U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The computational results of Algorithm 2 and SpaRSA for the instances randomly generated above are presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In detail, the values of n, k and m are listed in the first three columns, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' For each triple (n, k, m), the average relative error and the average objective value of the feasible approximate solutions found by each method over 10 random instances are given in the rest columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' One can observe that the approximate SOSP found by Algorithm 2 has significantly lower relative error and objective value than the approximate FOSP obtained by SpaRSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2 A simplex-constrained nonnegative matrix factorization In this subsection we consider a simplex-constrained nonnegative matrix factorization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=', see [45, 50, 54, 64]) in the form of min U∈Rn×k,V ∈Rk×m �1 2∥X − UV ∥2 F + γ(∥U∥2 F + ∥V ∥2 F ) : V T ek = em, U ≥ 0, V ≥ 0 � (52) for some γ > 0, where ∥ · ∥F is the Frobenius norm and ed is the d-dimensional all-ones vector for any d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' For each triple (n, k, m), we randomly generate 10 instances of problem (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In particular, we first randomly generate U ∗ with all entries chosen from the uniform distribution over [0, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We next randomly generate �V 14 Relative error Objective value n k m Algorithm 2 SpaRSA Algorithm 2 SpaRSA 20 2 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='8×10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 20 2 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='6×10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='35 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='4 20 2 30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2×10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='39 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='7 30 3 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='8×10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='62 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='6 30 3 30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3×10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='70 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 30 3 45 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='6×10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='76 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='8 40 4 20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3×10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 40 4 40 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='6×10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='7 40 4 60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='0×10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='8 50 5 25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='8×10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='6 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 50 5 50 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='0×10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='8 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='8 50 5 75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3×10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='9 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='9 Table 2: Numerical results for problem (52) with all entries chosen from the standard uniform distribution and set V ∗ = �V D, where D is a diagonal matrix such that (V ∗)T ek = em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In addition, we set γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='005 and X = U ∗V ∗ + E, where the entries of E follow the normal distribution with mean zero and standard deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Our aim is to apply Algorithm 2 and SpaRSA [68] to solve (52) and compare the solution quality of these methods in terms of objective value and relative error defined as ∥UV − U ∗V ∗∥F /∥U ∗V ∗∥F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In particular, we first apply Algorithm 2 to find a (10−4, 10−2)-SOSP of (52), in which a minimum eigenvalue oracle that returns a deterministic output, namely the Matlab subroutine [v,λ] = eigs(H,1,’smallestreal’) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Given that the obtained approximate SOSP may not be a feasible point of (52), we post-multiply it by a suitable diagonal matrix to obtain a feasible approximate solution of (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In addition, we apply SpaRSA [68] to find a 10−4-FOSP of (52) by generating a sequence {(U t, V t)} according to (U t, V t) = arg min U,V � ∥(U, V ) − (U t−1, V t−1) + ∇f(U t−1, V t−1)/αt−1∥F : V T ek = em, U ≥ 0, V ≥ 0 � , where f is the objective function of (52) and αt−1 is chosen by a backtracking line search scheme such that f(U t, V t) ≤ max[t−M−1]+≤i≤t−1 f(U i, V i) − σαt−1∥(U t, V t) − (U t−1, V t−1)∥2 F /2 for some σ ∈ (0, 1) and a positive integer M (see [68] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We terminate SpaRSA once the condition ∥αt−1((U t, V t) − (U t−1, V t−1)) + ∇f(U t−1, V t−1) − ∇f(U t, V t)∥F ≤ 10−4 is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It can be verified that such (U t, V t) is a 10−4-FOSP of (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In addition, we choose the initial point U 0 and V 0 with all entries equal 1 and 1/k respectively for all the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We set the parameters for Algorithm 2 as (Λ, ρ0, α, r) = (103, 102, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='25, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='5) and λ0 = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' , 0)T , and choose the same parameters for Algorithm 1 and SpaRSA as the ones described in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The computational results of Algorithm 2 and SpaRSA [68] for the instances randomly generated above are presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In detail, the values of n, k and m are listed in the first three columns, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' For each triple (n, k, m), the average relative error and the average objective value of the feasible approximate solutions found by each method over 10 random instances are given in the rest columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' One can observe that the approximate SOSP found by Algorithm 2 has significantly lower relative error and objective value than the approximate FOSP obtained by SpaRSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 7 Proof of the main results In this section we provide a proof of our main results presented in Sections 3, 4, and 5, which are, particularly, Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1, and Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='4, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Let us start with the following lemma concerning some properties of the ϑ-LHSC barrier function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 15 Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Let x ∈ int K and β ∈ (0, 1) be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Then the following statements hold for the ϑ-LHSC barrier function B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (i) (∥∇B(x)∥∗ x)2 = −xT ∇B(x) = ∥x∥2 x = ϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (ii) −∇B(x) ∈ int K∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (iii) {y : ∥y − x∥x < 1} ⊂ int K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (iv) For any y satisfying ∥y − x∥x ≤ β, it holds that (1 − β)∥v∥x ≤ ∥v∥y ≤ (1 − β)−1∥v∥x, ∀v ∈ Rn, (53) (1 − β)∥v∥∗ x ≤ ∥v∥∗ y ≤ (1 − β)−1∥v∥∗ x, ∀v ∈ Rn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (54) (v) {s : ∥s + ∇B(x)∥∗ x ≤ 1} ⊆ K∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (vi) ∥∇2B(y) − ∇2B(x)∥∗ x ≤ 2−β (1−β)2 ∥y − x∥x holds for all y with ∥y − x∥x ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The proof of statements (i)-(v) can be found in [43, Lemma 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We next prove statement (vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Let y be such that ∥y − x∥x ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It follows from [56, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1] that (1 − ∥y − x∥x)2I ⪯ ∇2B(x)−1/2∇2B(y)∇2B(x)−1/2 ⪯ (1 − ∥y − x∥x)−2I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (55) By (4), (55), and ∥y − x∥x ≤ β, one has ∥∇2B(y) − ∇2B(x)∥∗ x (4) = max∥u∥≤1 ∥∇2B(x)−1/2(∇2B(y) − ∇2B(x))∇2B(x)−1/2u∥ = ∥∇2B(x)−1/2∇2B(y)∇2B(x)−1/2 − I∥ (55) ≤ max{1 − (1 − ∥y − x∥x)2, (1 − ∥y − x∥x)−2 − 1} = (1 − ∥y − x∥x)−2 − 1 = 2−∥y−x∥x (1−∥y−x∥x)2 ∥y − x∥x ≤ 2−β (1−β)2 ∥y − x∥x, where the last inequality is due to ∥y − x∥x ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Hence, statement (vi) holds as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 Proof of the main results in Section 3 In this subsection we provide a proof of Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By M ∈ ∇−2B(x∗), the full column rank of M 1/2∇c(x∗), and also the discussion in Section 3, one knows that Robinson’s constraint qualification holds at x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Since x∗ is a local minimizer of (1), it then follows from [62, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='38] that there exists some λ∗ ∈ Rm such that ∇f(x∗) + ∇c(x∗)λ∗ ∈ − N K(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (56) Further, by [43, Proposition 1], one knows that (56) holds if and only if (5) and (6) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Consequently, (5) and (6) hold as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We next prove (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 that {x∗ + M 1/2d : ∥d∥ < 1} ⊆ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Using this and the fact that x∗ is a local minimizer of (1), we see that d∗ = 0 is a local minimizer of the problem min d � f(x∗ + M 1/2d) : c(x∗ + M 1/2d) = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (57) In addition, since M 1/2∇c(x∗) has full column rank, it is clear that LICQ holds at d∗ = 0 for (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By the first- and second-order optimality conditions of (57) at d∗ = 0, there exists some ˜λ∗ ∈ Rm such that M 1/2(∇f(x∗) + ∇c(x∗)˜λ∗) = 0, (58) dT M 1/2 � ∇2f(x∗) + m � i=1 ˜λ∗ i ∇2ci(x∗) � M 1/2d ≥ 0, ∀d ∈ {d : ∇c(x∗)T M 1/2d = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (59) In view of (6), (58), and the fact that M 1/2∇c(x∗) has full column rank, one can see that ˜λ∗ = λ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Using this and (59), we conclude that (7) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 16 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2 Proof of the main results in Section 4 In this subsection we first establish several technical lemmas and then use them to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' As a consequence of Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(b) and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(vi), one can observe that φµ is locally Lipschitz continuous in Ω with respect to the local norms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=', ∥∇2φµ(y) − ∇2φµ(x)∥∗ x ≤ Lφ H∥y − x∥x, ∀x, y ∈ Ω with ∥y − x∥x ≤ β, (60) where Lφ H := LF H + µ(2 − β)/(1 − β)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (61) The following lemma directly follows from (60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Its proof can be found in [43, Lemma 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Under Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(b), the following inequalities hold: ∥∇φµ(y) − ∇φµ(x) − ∇2φµ(x)(y − x)∥∗ x ≤ 1 2Lφ H∥y − x∥2 x, ∀x, y ∈ Ω with ∥y − x∥x ≤ β, (62) φµ(y) ≤ φµ(x)+∇φµ(x)T (y−x)+ 1 2(y−x)T ∇2φµ(x)(y−x)+ 1 6Lφ H∥y−x∥3 x, ∀x, y ∈ Ω with ∥y−x∥x ≤ β, (63) where Ω and Lφ H are given in Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(b) and (61), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The following lemma shows that all iterates generated by Algorithm 1 lie in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Suppose that Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Let {xt}t∈T be all the iterates generated by Algorithm 1, where T is a subset of consecutive nonnegative integers starting from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Then {xt}t∈T ⊂ S, where S is defined in (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We first prove {xt}t∈T ⊂ int K by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Observe from Algorithm 1 that x0 = u0 ∈ int K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Suppose that xt ∈ int K is generated at iteration t of Algorithm 1 and xt+1 is generated at iteration t+ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We next prove xt+1 ∈ int K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Indeed, observe from Algorithm 1 that xt+1 = xt + αtMtdt with αt ∈ (0, 1] and dt given in one of (17)-(19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It follows from (17)-(19) that ∥dt∥ ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By these and (16), one has ∥xt+1 − xt∥xt = αt∥Mtdt∥xt ≤ ∥Mtdt∥xt (16) = ∥dt∥ ≤ β, (64) which, along with xt ∈ int K, β < 1 and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(iii), implies that xt+1 ∈ int K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Hence, the induction is completed, and we have {xt}t∈T ⊂ int K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In addition, observe from Algorithm 1 that {φµ(xt)}t∈T is descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By this, x0 = u0, {xt}t∈T ⊂ int K, and (23), one can see that {xt}t∈T ⊂ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The following lemma provides some properties of the output of Algorithm 3, whose proof is similar to the ones of [60, Lemma 3] and [58, Lemma 7] and thus omitted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Suppose that Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 holds and the direction dt results from the output ˆdt of Algo- rithm 3 with a type specified in d type at some iteration t of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Let Mt be given in (16) and γt := max{∥ ˆdt∥/β, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (i) If d type=SOL, then dt satisfies ǫH∥dt∥2 ≤ (dt)T � M T t ∇2φµ(xt)Mt + 2ǫHI � dt, (65) ∥dt∥ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1ǫ−1 H ∥M T t ∇φµ(xt)∥, (66) (dt)T M T t ∇φµ(xt) = −γt(dt)T � M T t ∇2φµ(xt)Mt + 2ǫHI � dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (67) If ∥ ˆdt∥ ≤ β, then dt also satisfies ∥(M T t ∇2φµ(xt)Mt + 2ǫHI)dt + M T t ∇φµ(xt)∥ ≤ ǫHζ∥dt∥/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (68) (ii) If d type=NC, then dt satisfies (dt)T M T t ∇φµ(xt) ≤ 0 and (dt)T M T t ∇2φµ(xt)Mtdt ∥dt∥2 ≤ −∥dt∥ ≤ −ǫH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (69) 17 The following lemma shows that when the search direction dt in Algorithm 1 is of type ‘SOL’, the line search step results in a sufficient reduction on φµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Suppose that Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 holds and the direction dt results from the output ˆdt of Algorithm 3 with d type=SOL at some iteration t of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (i) The step length αt is well-defined, and moreover, αt ≥ min � 1, � min{6(1 − η), 2} 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1[LF H + µ(2 − β)/(1 − β)2](U F g + µ √ ϑ) θǫH � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (70) (ii) The next iterate xt+1 = xt + αtMtdt satisfies φµ(xt) − φµ(xt+1) ≥ csol min{(∥∇φµ(xt+1)∥∗ xt+1)2ǫ−1 H , ǫ3 H}, (71) where Mt and csol are given in (16) and (26), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Notice from Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3 that xt ∈ S, that is, xt ∈ int K and φµ(xt) ≤ φµ(u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It then follows from (16), (24) and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(i) that ∥M T t ∇φµ(xt)∥ = ∥∇φµ(xt)∥∗ xt ≤ ∥∇F(xt)∥∗ xt + µ∥∇B(xt)∥∗ xt ≤ U F g + µ √ ϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (72) Since dt results from the output of Algorithm 3 with d type=SOL, one can see that ∥M T t ∇φµ(xt)∥ > ǫg and the relations (65)-(67) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Also, one can observe from Algorithm 3 that its output ˆdt satisfies ∥(M T t ∇2φµ(xt)Mt + 2ǫHI) ˆdt + M T t ∇φµ(xt)∥ ≤ ˆζ∥M T t ∇φµ(xt)∥ for some ˆζ ∈ (0, 1/6), which together with ∥M T t ∇φµ(xt)∥ > ǫg implies that ˆdt ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It then follows from this and (18) that dt ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We first prove statement (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' If (20) holds for j = 0, then αt = 1, which clearly implies that (70) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We now suppose that (20) fails for j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Claim that for all j ≥ 0 that violate (20), it holds that θ2j ≥ min{6(1 − η), 2}ǫH(Lφ H)−1∥dt∥−1, (73) where Lφ H is defined in (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Indeed, we suppose that (20) is violated by some j ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We next show that (73) holds for such j by considering two separate cases below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Case 1) φµ(xt + θjMtdt) > φµ(xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Let ϕ(α) = φµ(xt + αMtdt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Then ϕ(θj) > ϕ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In addition, by (65), (67), γt = max{∥ ˆdt∥/β, 1} ≥ 1, and dt ̸= 0, one has ϕ′(0) = (dt)T M T t ∇φµ(xt) (67) = −γt(dt)T (M T t ∇2φµ(xt)Mt + 2ǫHI)dt (65) ≤ −γtǫH∥dt∥2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In view of these, we can observe that there exists a local minimizer α∗ ∈ (0, θj) of ϕ such that ϕ(α∗) < ϕ(0) and ϕ′(α∗) = ∇φµ(xt + α∗Mtdt)T Mtdt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (74) By φµ(xt) ≤ φµ(u0) and ϕ(α∗) < ϕ(0), one has φµ(xt + α∗Mtdt) < φµ(u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In addition, using (64) and 0 < α∗ < θj ≤ 1, we have ∥α∗Mtdt∥xt ≤ ∥Mtdt∥xt ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Hence, (62) holds for x = xt and y = xt + α∗Mtdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By this,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (16),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (65),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (67),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (74),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 0 < α∗ < 1 and γt ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' one has (α∗)2Lφ H 2 ∥dt∥3 (16) = (α∗)2Lφ H 2 ∥dt∥∥Mtdt∥2 xt (62) ≥ ∥dt∥∥∇φµ(xt + α∗Mtdt) − ∇φµ(xt) − α∗∇2φµ(xt)Mtdt∥∗ xt ≥ (dt)T (M T t ∇φµ(xt + α∗Mtdt) − M T t ∇φµ(xt) − α∗M T t ∇2φµ(xt)Mtdt) (74) = −(dt)T M T t ∇φµ(xt) − α∗(dt)T M T t ∇2φµ(xt)Mtdt (67) = (γt − α∗)(dt)T (M T t ∇2φµ(xt)Mt + 2ǫHI)dt + 2α∗ǫH∥dt∥2 (65) ≥ (γt − α∗)ǫH∥dt∥2 + 2α∗ǫH∥dt∥2 = (γt + α∗)ǫH∥dt∥2 ≥ ǫH∥dt∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 18 which along with dt ̸= 0 implies that (α∗)2 ≥ 2ǫH(Lφ H)−1∥dt∥−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Using this and θj > α∗, we conclude that (73) holds in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Case 2) φµ(xt + θjMtdt) ≤ φµ(xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By this and φµ(xt) ≤ φµ(u0), one has φµ(xt + θjMtdt) ≤ φµ(u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Also, using (64) and θ ∈ (0, 1), we have ∥θjMtdt∥xt ≤ ∥Mtdt∥xt ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Hence, (63) holds for x = xt and y = xt+θjMtdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Using this,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (16),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (65),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (67) and the fact that j violates (20),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' we obtain that −ηǫHθ2j∥dt∥2 ≤ φµ(xt + θjMtdt) − φµ(xt) (63) ≤ θj∇φµ(xt)T Mtdt + θ2j 2 (dt)T M T t ∇2φµ(xt)Mtdt + Lφ H 6 θ3j∥Mtdt∥3 xt (16)(67) = −θjγt(dt)T (M T t ∇2φµ(xt)Mt + 2ǫHI)dt + θ2j 2 (dt)T M T t ∇2φµ(xt)Mtdt + Lφ H 6 θ3j∥dt∥3 = −θj � γt − θj 2 � (dt)T (M T t ∇2φµ(xt)Mt + 2ǫHI)dt − θ2jǫH∥dt∥2 + Lφ H 6 θ3j∥dt∥3 (65) ≤ −θj � γt − θj 2 � ǫH∥dt∥2 − θ2jǫH∥dt∥2 + Lφ H 6 θ3j∥dt∥3 ≤ −θjǫHγt∥dt∥2 + Lφ H 6 θ3j∥dt∥3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (75) where the first inequality is due to the violation of (20) by such j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Recall that dt ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Dividing both sides of (75) by Lφ Hθj∥dt∥3/6 and using η, θ ∈ (0, 1) and γt ≥ 1, we have θ2j ≥ 6(γt − ηθj)ǫH(Lφ H)−1∥dt∥−1 ≥ 6(1 − η)ǫH(Lφ H)−1∥dt∥−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Hence, (73) also holds in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Combining the above two cases, we conclude that (73) holds for any j ≥ 0 violating (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By this and θ ∈ (0, 1), one can see that all j ≥ 0 that violate (20) must be bounded above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It then follows that the step length αt associated with (20) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We next prove (70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Observe from the definition of jt in Algorithm 1 that j = jt −1 violates (20) and hence (73) holds for j = jt −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Then, by (61), (73) with j = jt −1, and αt = θjt, one has αt = θjt ≥ � min{6(1 − η), 2}ǫH(Lφ H)−1θ∥dt∥−1/2 = � min{6(1 − η), 2}ǫH[LF H + µ(2 − β)/(1 − β)2]−1θ∥dt∥−1/2, (76) which along with (66) and (72) implies that (70) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We next prove statement (ii), particularly, (71) by considering three separate cases below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Case 1) αt = 1 and ∥ ˆdt∥ ≥ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It then follows from (18) that dt = β ˆdt/∥ ˆdt∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Notice from Algorithm 1 that β ≥ ǫH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Using this and dt = β ˆdt/∥ ˆdt∥, we see that ∥dt∥ = β ≥ ǫH, which together with (20) and αt = 1 implies that (71) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Case 2) αt = 1 and ∥ ˆdt∥ < β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Notice from αt = 1 that j = 0 is accepted by (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Then one can see that φµ(xt + Mtdt) ≤ φµ(xt) ≤ φµ(u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Also, observe from (64) that ∥Mtdt∥xt ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Hence, (62) holds for x = xt and y = xt + Mtdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By these,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (54) and (68),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' one has (1 − β)∥∇φµ(xt+1)∥∗ xt+1 (54) ≤ ∥∇φµ(xt+1)∥∗ xt = ∥∇φµ(xt + Mtdt)∥∗ xt ≤ ∥∇φµ(xt + Mtdt) − ∇φµ(xt) − ∇2φµ(xt)Mtdt∥∗ xt + ∥∇φµ(xt) + ∇2φµ(xt)Mtdt∥∗ xt = ∥∇φµ(xt + Mtdt) − ∇φµ(xt) − ∇2φµ(xt)Mtdt∥∗ xt + ∥M T t (∇φµ(xt) + ∇2φµ(xt)Mtdt)∥ ≤ ∥∇φµ(xt + Mtdt) − ∇φµ(xt) − ∇2φµ(xt)Mtdt∥∗ xt + ∥(M T t ∇2φµ(xt)Mt + 2ǫHI)dt + M T t ∇φµ(xt)∥ + 2ǫH∥dt∥ (62)(68) ≤ Lφ H∥Mtdt∥2 xt/2 + (4 + ζ)ǫH∥dt∥/2 (16) = Lφ H∥dt∥2/2 + (4 + ζ)ǫH∥dt∥/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' where the second inequality is due to the triangle inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' and the second equality follows from (4) and (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 19 Solving the above inequality for ∥dt∥ and using (61) and the fact that ∥dt∥ > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' we obtain that ∥dt∥ ≥ −(4+ζ)ǫH+ � (4+ζ)2ǫ2 H+8(1−β)Lφ H∥∇φµ(xt+1)∥∗ xt+1 2Lφ H ≥ −(4+ζ)ǫH+√ (4+ζ)2ǫ2 H+8(1−β)Lφ Hǫ2 H 2Lφ H min{∥∇φµ(xt+1)∥∗ xt+1ǫ−2 H ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 1} = 4(1−β) 4+ζ+√ (4+ζ)2+8(1−β)Lφ H min{∥∇φµ(xt+1)∥∗ xt+1ǫ−1 H ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ǫH} (61) = 4(1−β) 4+ζ+√ (4+ζ)2+8[(1−β)LF H+µ(2−β)/(1−β)] min{∥∇φµ(xt+1)∥∗ xt+1ǫ−1 H ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ǫH},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' where the second inequality follows from the inequality −a+ √ a2 + bs ≥ (−a+ √ a2 + b) min{s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 1} for all a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' s ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' which can be easily verified by performing a rationalization to the terms −a + √ a2 + bs and −a + √ a2 + b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In view of this, αt = 1, (20) and (26), one can see that (71) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Case 3) αt < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By this, one has that j = 0 violates (20) and hence (73) holds for j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Letting j = 0 in (73), we obtain that ∥dt∥ ≥ min{6(1 − η), 2}ǫH/Lφ H, which along with (20), (61) and (76) implies that φµ(xt) − φµ(xt+1) (20) ≥ ηǫHθ2jt∥dt∥2 ≥ η � min{6(1 − η), 2}θ Lφ H �2 ǫ3 H (61) = η � min{6(1 − η), 2}θ LF H + µ(2 − β)/(1 − β)2 �2 ǫ3 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By this and (26), one can immediately see that (71) also holds in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The next lemma shows that when the search direction dt in Algorithm 1 is of type ‘NC’, the line search step results in a sufficient reduction on φµ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Suppose that Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 holds and the direction dt results from either the output ˆdt of Algorithm 3 with d type=NC or the output v of Algorithm 4 at some iteration t of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (i) The step length αt is well-defined, and moreover, αt ≥ min � 1, min{1, 3(1 − η)}θ LF H + µ(2 − β)/(1 − β)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (77) (ii) The next iterate xt+1 = xt + αtMtdt satisfies φµ(xt) − φµ(xt+1) ≥ cncǫ3 H, where Mt and cnc are given in (16) and (27), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It follows from Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3 that xt ∈ S, that is, xt ∈ int K and φµ(xt) ≤ φµ(u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By the assumption on dt, one can see from Algorithm 1 that dt is a negative curvature direction given in (17) or (19) and thus dt ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Also, the vector v satisfies ∥v∥ = 1 whenever it is returned from Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By these, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='4(ii), (17) and (19), one has ∇φµ(xt)T Mtdt ≤ 0, (dt)T M T t ∇2φµ(xt)Mtdt ≤ −∥dt∥3 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (78) We first prove statement (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' If (21) holds for j = 0, then αt = 1, which clearly implies that (77) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We now suppose that (21) fails for j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Claim that for all j ≥ 0 that violate (21), it holds that θj ≥ min{1, 3(1 − η)}/Lφ H, (79) where Lφ H is defined in (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Indeed, suppose that (21) is violated by some j ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We now prove that (79) holds for such j by considering two separate cases below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Case 1) φµ(xt + θjMtdt) > φµ(xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Let ϕ(α) = φµ(xt + αMtdt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Then ϕ(θj) > ϕ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Also, by (78), one has ϕ′(0) = ∇φµ(xt)T Mtdt ≤ 0, ϕ′′(0) = (dt)T M T t ∇2φµ(xt)Mtdt < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' From these, we can observe that there exists a local minimizer α∗ ∈ (0, θj) of ϕ such that ϕ(α∗) < ϕ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By the second-order necessary optimality condition of ϕ at α∗, one has ϕ′′(α∗) = (dt)T M T t ∇2φµ(xt + α∗Mtdt)Mtdt ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (80) 20 In addition, by φµ(xt) ≤ φµ(u0) and ϕ(α∗) < ϕ(0), one has φµ(xt + α∗Mtdt) < φµ(u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Using (64) and 0 < α∗ < θj ≤ 1, we see that ∥α∗Mtdt∥xt ≤ ∥Mtdt∥xt ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Hence, (60) holds for x = xt and y = xt + α∗Mtdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Using this, (4), (16), (60), (78) and (80), we obtain that Lφ Hα∗∥dt∥3 (16) = Lφ Hα∗∥dt∥2∥Mtdt∥xt (60) ≥ ∥dt∥2∥∇2φµ(xt + α∗Mtdt) − ∇2φµ(xt)∥∗ xt (4) = ∥dt∥2∥M T t (∇2φµ(xt + α∗Mtdt) − ∇2φµ(xt))Mt∥ ≥ (dt)T M T t (∇2φµ(xt + α∗Mtdt) − ∇2φµ(xt))Mtdt (80) ≥ −(dt)T M T t ∇2φµ(xt)Mtdt (78) ≥ ∥dt∥3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It then follows from this and dt ̸= 0 that α∗ ≥ 1/Lφ H, which along with θj > α∗ implies that (79) holds in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Case 2) φµ(xt + θjMtdt) ≤ φµ(xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By this and φµ(xt) ≤ φµ(u0), one has φµ(xt + θjMtdt) ≤ φµ(u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In addition, it follows from (64) and θ ∈ (0, 1) that ∥θjMtdt∥xt ≤ ∥Mtdt∥xt ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Hence, (63) holds for x = xt and y = xt + θjMtdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By this, (16), (78) and the fact that j violates (21), one has − η 2θ2j∥dt∥3 ≤ φµ(xt + θjMtdt) − φµ(xt) (63) ≤ θj∇φµ(xt)T Mtdt + θ2j 2 (dt)T M T t ∇2φµ(xt)Mtdt + Lφ H 6 θ3j∥Mtdt∥3 xt (16)(78) ≤ − θ2j 2 ∥dt∥3 + Lφ H 6 θ3j∥dt∥3, where the first inequality is due to the violation of (21) by such j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Using this and dt ̸= 0, we see that θj ≥ 3(1 − η)/Lφ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Hence, (79) also holds in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Combining the above two cases, we conclude that (79) holds for all j ≥ 0 violating (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By this and θ ∈ (0, 1), one can see that all j ≥ 0 that violate (21) must be bounded above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It then follows that the step length αt associated with (21) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We next derive a lower bound for αt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Notice that j = jt − 1 violates (21) and hence (79) holds for j = jt − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Then by (79) with j = jt − 1 and αt = θjt, one can observe that αt = θjt ≥ min{1, 3(1 − η)}θ/Lφ H, which along with (61) yields (77) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We next prove statement (ii) by considering two separate cases below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Case 1) dt results from the output ˆdt of Algorithm 3 with d type=NC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By this and (69), one has ∥dt∥ ≥ ǫH, which along with statement (i) and (21) implies that statement (ii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Case 2) dt results from the output v of Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Notice from Algorithm 4 that ∥v∥ = 1 and vT M T t ∇2φµ(xt)Mtv ≤ −ǫH/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It then follows from (19) and β ≥ ǫH that ∥dt∥ ≥ ǫH/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Using this, (21) and statement (i), we see that statement (ii) also holds in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We are now ready to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Notice from Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3 that all the iterates generated by Algorithm 1 lie in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By this, (4), (16) and (24), one has ∥M T t ∇2φµ(xt)Mt∥ (4)(16) = ∥∇2φµ(xt)∥∗ xt ≤ ∥∇2F(xt)∥∗ xt + µ∥∇2B(xt)∥∗ xt ≤ U F H + µ, (81) where the last inequality follows from (24) and the fact that ∥∇2B(xt)∥∗ xt = 1 due to (4) and (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (i) Suppose for contradiction that the total number of calls of Algorithm 4 in Algorithm 1 is more than T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Observe from Algorithm 1 and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='6(ii) that each of these calls, except the last one, returns a sufficiently negative curvature direction, and each of them results in a reduction on φµ at least by cncǫ3 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Using this and the fact that x0 = u0, we obtain that T2cncǫ3 H ≤ � t∈T [φµ(xt) − φµ(xt+1)] ≤ φµ(x0) − φlow = φhi − φlow, where T is given in Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' This contradicts with the definition of T2 given in (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (ii) Suppose for contradiction that the total number of calls of Algorithm 3 in Algorithm 1 is more than T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Note that if Algorithm 3 is called at some iteration t and generates xt+1 satisfying ∥∇φµ(xt+1)∥∗ xt+1 ≤ ǫg, then Algorithm 4 must be called at the next iteration t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Using this and statement (i), we see that the total number of such iterations t is at most T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Hence, the total number of iterations t of Algorithm 1 at which Algorithm 3 21 is called and generates xt+1 satisfying ∥∇φµ(xt+1)∥∗ xt+1 > ǫg is at least T1 − T2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Moreover, for each of such iterations t, it follows from Lemmas 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='5(ii) and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='6(ii) that φµ(xt) − φµ(xt+1) ≥ min{csol, cnc} min{ǫ2 gǫ−1 H , ǫ3 H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Thus, one has (T1 − T2 + 1) min{csol, cnc} min{ǫ2 gǫ−1 H , ǫ3 H} ≤ � t∈T [φµ(xt) − φµ(xt+1)] ≤ φhi − φlow, where T is given in Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' This contradicts the definitions of T1 and T2 given in (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (iii) Notice that either Algorithm 3 or Algorithm 4 is called at each iteration of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By this and statements (i) and (ii), one has that the total number of iterations of Algorithm 1 is at most T1+T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In addition, the relation (28) follows from (25), (26) and (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It is also not hard to see that the output xt of Algorithm 1 satisfies ∥∇φµ(xt)∥∗ xt ≤ ǫg deterministically and λmin(M T t ∇2φµ(xt)Mt) ≥ −ǫH with probability at least 1 − δ for some 0 ≤ t ≤ T1 + T2, where the probability is due to Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Hence, statement (iii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (iv) Recall that each iteration of Algorithm 1 requires Cholesky factorization of ∇2B at one point only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' This together with statement (iii) implies that the total number of Cholesky factorizations required by Algorithm 1 is at most T1 + T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By (81) and Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 with (H, ε) = (M T t ∇2φµ(xt)Mt, ǫH), one can see that the number of products of H and a vector v required by each call of Algorithm 3 is at most �O(min{n, [(U F H + µ)/ǫH]1/2}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In addition, by (81), Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 with (H, ε) = (M T t ∇2φµ(xt)Mt, ǫH), and the fact that each iteration of the Lanczos method requires only one product of H and a vector v, one can observe that the number of products of H and a vector v required by each call of Algorithm 4 is also at most �O(min{n, [(U F H + µ)/ǫH]1/2}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Recall from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 that the product of H and a vector v requires at most three fundamental operations, which are one Hessian-vector product of F, one backward and forward substitutions to a triangular linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Hence, each call of Algorithm 3 or 4 requires at most �O(min{n, [(U F H + µ)/ǫH]1/2}) fundamental operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By these observations and statement (iii), we conclude that statement (iv) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3 Proof of the main results in Section 5 In this subsection we provide a proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 and Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Before proceeding, we recall that ∥c(zǫ)∥ ≤ ǫ/2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Using this, (30) and (35), we obtain that f(x) + µB(x) + γ∥˜c(x)∥2 ≥ f(x) + µB(x) + γ 2 ∥c(x)∥2 − γ∥c(zǫ)∥2 ≥ flow − γ, ∀x ∈ int K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (82) In addition, by (29) and the first relation in (37), one has Lµ(xk+1, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) ≤ fhi whenever xk+1 is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (83) We next present an auxiliary lemma that will be frequently used later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Its proof is identical to the one of [44, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='4] with f replaced by f + µB, and thus omitted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Suppose that Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Let γ, µ, fhi and flow be given in Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Assume that ρ > 2γ, λ ∈ Rm and x ∈ int K satisfy Lµ(x, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρ) ≤ fhi, where Lµ is defined in (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (i) f(x) + µB(x) ≤ fhi + ∥λ∥2/(2ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (ii) ∥˜c(x)∥ ≤ � 2(fhi − flow + γ)/(ρ − 2γ) + ∥λ∥2/(ρ − 2γ)2 + ∥λ∥/(ρ − 2γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (iii) If ρ ≥ ∥λ∥2/(2˜δf) for some ˜δf > 0, then f(x) + µB(x) ≤ fhi + ˜δf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (iv) If ρ ≥ 2(fhi − flow + γ)˜δ−2 c + 2∥λ∥˜δ−1 c + 2γ for some ˜δc > 0, then ∥˜c(x)∥ ≤ ˜δc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The following lemma establishes the local Lipschitz continuity of ci and ∇ci with respect to the local norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Under Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1, the following inequalities hold: |ci(y) − ci(x)| ≤ U c g 1 − β ∥y − x∥x, ∀x, y ∈ Ω(δf, δc) with ∥y − x∥x ≤ β, 1 ≤ i ≤ m, (84) ∥∇ci(y) − ∇ci(x)∥∗ x ≤ U c H (1 − β)2 ∥y − x∥x, ∀x, y ∈ Ω(δf, δc) with ∥y − x∥x ≤ β, 1 ≤ i ≤ m, (85) where Ω(δf, δc) is given in Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1, and U c g, U c H are defined in (33) and (34), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 22 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Fix any x, y ∈ Ω(δf, δc) with ∥y − x∥x ≤ β and any 1 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Let zt = x + t(y − x) for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It then follows that ∥zt − x∥x ≤ β and zt ∈ Ω(δf, δc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By these, (33), (34), (53) and (54), one has ∥∇ci(zt)∥∗ zt ≤ U c g, ∥∇2ci(zt)∥∗ zt ≤ U c H, ∥v∥zt ≤ (1−β)−1∥v∥x, ∥v∥∗ x ≤ (1−β)−1∥v∥∗ zt, ∀v ∈ Rn, t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By virtue of these and (4), we obtain |ci(y) − ci(x)| = ���� � 1 0 ∇ci(zt)T (y − x)dt ���� ≤ � 1 0 ∥∇ci(zt)∥∗ zt∥y − x∥ztdt ≤ U c g 1 − β ∥y − x∥x, ∥∇ci(y) − ∇ci(x)∥∗ x = ���� � 1 0 ∇2ci(zt)(y − x)dt ���� ∗ x ≤ � 1 0 ∥∇2ci(zt)(y − x)∥∗ xdt ≤ 1 1 − β � 1 0 ∥∇2ci(zt)(y − x)∥∗ ztdt ≤ 1 1 − β � 1 0 ∥∇2ci(zt)∥∗ zt∥y − x∥ztdt ≤ U c H (1 − β)2 ∥y − x∥x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Hence, (84) and (85) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We are now ready to prove Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (i) Fix any x ∈ int K satisfying Lµ(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) ≤ Lµ(xk init, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It follows from this and (40) that Lµ(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) ≤ fhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By this, ∥λk∥ ≤ Λ, ρk ≥ ρ0 > 2γ, δf,1 ≤ δf, δc,1 ≤ δc, and Lemmas 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='7(i) and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='7(ii) with (λ, ρ) = (λk, ρk), one has f(x) + µB(x) ≤ fhi + ∥λk∥2 2ρk ≤ fhi + Λ2 2ρ0 (42) = fhi + δf,1 ≤ fhi + δf, ∥˜c(x)∥ ≤ � 2(fhi−flow+γ) ρk−2γ + ∥λk∥2 (ρk−2γ)2 + ∥λk∥ ρk−2γ ≤ � 2(fhi−flow+γ) ρ0−2γ + Λ2 (ρ0−2γ)2 + Λ ρ0−2γ (42) = δc,1 ≤ δc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (86) In addition, recall that ∥c(zǫ)∥ ≤ 1, which together with the definition of ˜c in (35) yields ∥c(x)∥ ≤ 1 + ∥˜c(x)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' These along with x ∈ int K, µ ∈ (0, ¯µ] and (31) implies x ∈ S(δf, δc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Hence, statement (i) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (ii) Observe that inf x∈int K Lµ(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) = inf x∈int K{Lµ(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) : Lµ(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) ≤ Lµ(xk init, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Thus, to prove statement (ii), it suffices to show that inf x∈int K{Lµ(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) : Lµ(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) ≤ Lµ(xk init, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk)} ≥ flow − γ − Λδc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (87) To this end, fix any x ∈ int K satisfying Lµ(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) ≤ Lµ(xk init, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It then follows from (86) that ∥˜c(x)∥ ≤ δc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By this, ∥λk∥ ≤ Λ, ρk > 2γ, µ ∈ (0, ¯µ] and (82), one has Lµ(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) = f(x) + µB(x) + (λk)T ˜c(x) + ρk 2 ∥˜c(x)∥2 ≥ f(x) + µB(x) + γ∥˜c(x)∥2 − Λ∥˜c(x)∥ (82) ≥ flow − γ − Λδc, and hence (87) holds as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (iii) Fix x, y ∈ Ω(δf, δc) with ∥y − x∥x ≤ β and 1 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By this, (32), (34), (35), (43), (84) and ∥c(zǫ)∥ ≤ 1, one has ∥˜ci(y)∇2ci(y) − ˜ci(x)∇2ci(x)∥∗ x = ∥˜ci(y)(∇2ci(y) − ∇2ci(x)) + (˜ci(y) − ˜ci(x))∇2ci(x)∥∗ x ≤ |ci(y) − ci(zǫ)|∥∇2ci(y) − ∇2ci(x)∥∗ x + |ci(y) − ci(x)|∥∇2ci(x)∥∗ x ≤ (1 + U c)Lc H∥y − x∥x + U c gU c H 1 − β ∥y − x∥x = � (1 + U c)Lc H + U c gU c H 1 − β � ∥y − x∥x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (88) In addition, by (33), (54) and (85), one has ∥∇ci(y)∇ci(y)T − ∇ci(x)∇ci(x)T ∥∗ x = ∥∇ci(y)(∇ci(y) − ∇ci(x))T + (∇ci(y) − ∇ci(x))∇ci(x)T ∥∗ x 23 ≤ ∥∇ci(y)∥∗ x∥∇ci(y) − ∇ci(x)∥∗ x + ∥∇ci(x)∥∗ x∥∇ci(y) − ∇ci(x)∥∗ x ≤ � 1 1 − β ∥∇ci(y)∥∗ y + ∥∇ci(x)∥∗ x � ∥∇ci(y) − ∇ci(x)∥∗ x ≤ (2 − β)U c gU c H (1 − β)3 ∥y − x∥x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (89) In view of (41) and the fact that ∇˜c = ∇c and ∇2˜ci = ∇2ci, 1 ≤ i ≤ m, we see that ∇2 xx L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) = ∇2f(x) + m � i=1 λk i ∇2ci(x) + ρk m � i=1 � ∇ci(x)∇ci(x)T + ˜ci(x)∇2ci(x) � , (90) which implies that ∥∇2 xx L(y, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk)−∇2 xx L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk)∥∗ x ≤ ∥∇2f(y) − ∇2f(x)∥∗ x + m � i=1 |λk i |∥∇2ci(y) − ∇2ci(x)∥∗ x + ρk m � i=1 � ∥∇ci(y)∇ci(y)T − ∇ci(x)∇ci(x)T ∥∗ x + ∥˜ci(y)∇2ci(y) − ˜ci(x)∇2ci(x)∥∗ x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Statement (iii) then follows from this, (88) and (89).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (iv) Notice from (41) and ∇˜c = ∇c that ∇x L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) = ∇f(x) + ∇c(x)λk + ρk∇c(x)˜c(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Also, one can see from (31), the definition of ˜c in (35) and ∥c(zǫ)∥ ≤ 1 that ∥˜c(x)∥ ≤ 2 + δc for any x ∈ S(δf, δc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Using these and (33), we can see that supx∈S(δf,δc) ∥∇x L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk)∥∗ x is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In addition, by (4), one can observe that ∥∇ci(x)∇ci(x)T ∥∗ x = (∥∇ci(x)∥∗ x)2 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Using this and (90), we have ∥∇2 xx L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk)∥∗ x ≤ ∥∇2f(x)∥∗ x + m � i=1 |λk i |∥∇2ci(x)∥∗ x + ρk m � i=1 � (∥∇ci(x)∥∗ x)2 + |˜ci(x)|∥∇2ci(x)∥∗ x � , which, together with (33), (34) and the fact that ∥˜c(x)∥ ≤ 2 + δc for any x ∈ S(δf, δc), implies that Uk,H = sup x∈S(δf ,δc) ∥∇2 xx L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk)∥∗ x ≤ U f H + ∥λk∥1U c H + ρk(m(U c g)2 + √m(2 + δc)U c H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Hence, statement (iv) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We now prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Suppose that Algorithm 2 terminates at some iteration k, that is, τk ≤ µ and ∥c(xk+1)∥ ≤ ǫ hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By τk ≤ µ, ˜λk+1 = λk + ρk˜c(xk+1), ∇˜c = ∇c, and the second relation in (37), one has ∥∇f(xk+1) + ∇c(xk+1)˜λk+1 + µ∇B(xk+1)∥∗ xk+1 = ∥∇f(xk+1) + ∇˜c(xk+1)(λk + ρk˜c(xk+1)) + µ∇B(xk+1)∥∗ xk+1 = ∥∇x Lµ(xk+1, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk)∥∗ xk+1 ≤ τk ≤ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (91) This along with µ > 0 yields that ∥(∇f(xk+1) + ∇c(xk+1)˜λk+1)/µ + ∇B(xk+1)∥∗ xk+1 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By this, xk+1 ∈ int K and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(v), one has (∇f(xk+1) + ∇c(xk+1)˜λk+1)/µ ∈ K∗, which implies that (9) holds for (xk+1, ˜λk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We next prove that (10) holds for (xk+1, ˜λk+1) with ǫ1 = ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Indeed, by (91), µ = ǫ/(2ϑ1/2 + 2), xk+1 ∈ int K and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(i), one has ∥∇f(xk+1) + ∇c(xk+1)˜λk+1∥∗ xk+1 ≤ ∥∇f(xk+1) + ∇c(xk+1)˜λk+1 + µ∇B(xk+1)∥∗ xk+1 + µ∥∇B(xk+1)∥∗ xk+1 ≤ µ + µϑ1/2 = ǫ/2 < ǫ, and hence (10) holds for (xk+1, ˜λk+1) with ǫ1 = ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In view of these, ∥c(xk+1)∥ ≤ ǫ and xk+1 ∈ int K, we conclude that xk+1 is a deterministic ǫ-FOSP of problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In addition, by (38) and τk ≤ µ, one can see that λmin(M T k+1∇2 xx Lµ(xk+1, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk)Mk+1) ≥ −√µ holds with probability at least 1 − δ, which implies that ˆdT M T k+1∇2 xx Lµ(xk+1, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk)Mk+1 ˆd ≥ −√µ∥ ˆd∥2 holds for all ˆd ∈ Rn with probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Substituting ˆd = M −1 k+1∇2B(xk+1)−1/2d in this inequality and using 24 (16),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ˜λk+1 = λk + ρk˜c(xk+1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ∇˜c = ∇c and ∇2˜ci = ∇2ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 1 ≤ i ≤ m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' we obtain that with probability at least 1 − δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' it holds that dT ∇2B(xk+1)−1/2� ∇2f(xk+1) + m � i=1 ˜λk+1 i ∇2ci(xk+1) + ρk∇c(xk+1)∇c(xk+1)T + µ∇2B(xk+1) � ∇2B(xk+1)−1/2d ≥ −√µ∥M −1 k+1∇2B(xk+1)−1/2d∥2 (16) = −√µ∥d∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ∀d ∈ Rn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' which together with µ = ǫ/(2ϑ1/2 + 2) ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 1) and ϑ ≥ 1 implies that dT ∇2B(xk+1)−1/2 � ∇2f(xk+1) + m � i=1 ˜λk+1 i ∇2ci(xk+1) � ∇2B(xk+1)−1/2d ≥ −(√µ + µ)∥d∥2 ≥ −2√µ∥d∥2 = −2 � ǫ/(2ϑ1/2 + 2)∥d∥2 ≥ −√ǫ∥d∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ∀d ∈ C(xk+1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' where C(·) is defined in (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Hence, with probability at least 1 − δ, the relation (11) holds for (xk+1, ˜λk+1) with ǫ2 = √ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In view of this and the fact that xk+1 is a deterministic ǫ-FOSP of (1), we conclude that the output xk+1 is an (ǫ, √ǫ)-SOSP of (1) with probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We next provide a proof for Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Notice from (46) that ρǫ,1 ≥ 2ρ0, which along with (44) and (45) implies that Kǫ (45) = ⌈log µ/ log ω⌉ (44) = ⌈log 2/ log r⌉ ≤ log(ρǫ,1ρ−1 0 )/ log r + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (92) Since {ρk} is either unchanged or increased by a ratio r as k increases, it follows from (92) that max 0≤k≤Kǫ ρk ≤ rKǫρ0 (92) ≤ r log(ρǫ,1ρ−1 0 ) log r +1ρ0 = rρǫ,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (93) In addition, observe from Algorithm 2 that ρk > 2γ and ∥λk∥ ≤ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By these, (83), and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='7(ii) with (x, λ, ρ) = (xk+1, λk, ρk), we obtain that ∥˜c(xk+1)∥ ≤ � 2(fhi − flow + γ) ρk − 2γ + ∥λk∥2 (ρk − 2γ)2 + ∥λk∥ ρk − 2γ ≤ � 2(fhi − flow + γ) ρk − 2γ + Λ2 (ρk − 2γ)2 + Λ ρk − 2γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (94) On the other hand, notice from ∥c(zǫ)∥ ≤ ǫ/2 and the definition of ˜c in (35) that ∥c(xk+1)∥ ≤ ∥˜c(xk+1)∥ + ∥c(zǫ)∥ ≤ ∥˜c(xk+1)∥ + ǫ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (95) We now prove that Kǫ is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Suppose for contradiction that Kǫ is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By this and (47), one has that ∥c(xk+1)∥ > ǫ for all k ≥ Kǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' This along with (95) implies that ∥˜c(xk+1)∥ > ǫ/2 for all k ≥ Kǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It then follows that ∥˜c(xk+1)∥ > α∥˜c(xk)∥ must hold for infinitely many k’s, which, together with the update scheme on {ρk}, further implies ρk+1 = rρk holds for infinitely many k’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Using this and the monotonicity of {ρk}, we see that ρk → ∞ as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' This along with (94) yields that ∥˜c(xk+1)∥ → 0 as k → ∞, which leads to a contradiction with the fact that ∥˜c(xk+1)∥ > ǫ/2 for all k ≥ Kǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Hence, Kǫ is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In addition, notice from τk = max{µ, ωk} and (45) that τk = µ for all k ≥ Kǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Combining this with the termination criterion of Algorithm 2 and the definition of Kǫ, we conclude that Algorithm 2 with τk = max{µ, ωk} must terminate at iteration Kǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We next prove (48) and that ρk ≤ rρǫ,1 holds for 0 ≤ k ≤ Kǫ by considering two separate cases below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Case 1) ∥c(xKǫ+1)∥ ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It then follows from (47) that Kǫ = Kǫ, and thus (48) holds due to (92).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In addition, by Kǫ = Kǫ and (93), one has that ρk ≤ rρǫ,1 holds for 0 ≤ k ≤ Kǫ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Case 2) ∥c(xKǫ+1)∥ > ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It then follows from (47) that Kǫ > Kǫ, and moreover, ∥c(xk+1)∥ > ǫ for all Kǫ ≤ k ≤ Kǫ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' This along with (95) implies that ∥˜c(xk+1)∥ > ǫ/2, ∀Kǫ ≤ k ≤ Kǫ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (96) 25 By this, ∥λk∥ ≤ Λ, (46), (83), and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='7(iv) with (x, λ, ρ, ˜δc) = (xk+1, λk, ρk, ǫ/2), one has ρk < 8(fhi − flow + γ)ǫ−2 + 4∥λk∥ǫ−1 + 2γ ≤ 8(fhi − flow + γ)ǫ−2 + 4Λǫ−1 + 2γ (46) ≤ ρǫ,1, ∀Kǫ ≤ k ≤ Kǫ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (97) In view of this, (93), and the fact ρKǫ ≤ rρKǫ−1, we obtain that ρk ≤ rρǫ,1 holds for 0 ≤ k ≤ Kǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It remains to prove (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' To this end, let K = {k : ρk+1 = rρk, Kǫ ≤ k ≤ Kǫ − 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By (97) and the update scheme of ρk, one has r| K |ρKǫ = maxKǫ≤k≤Kǫ−1 ρk ≤ ρǫ,1, which along with ρKǫ ≥ ρ0 implies that | K | ≤ log(ρǫ,1ρ−1 Kǫ)/ log r ≤ log(ρǫ,1ρ−1 0 )/ log r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (98) Let {k1, k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' , k| K |} denote all the elements of K arranged in ascending order, and let k0 = Kǫ and k| K |+1 = Kǫ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We next derive an upper bound for kj+1 − kj for j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' , | K |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Using the definition of K, we see that ρk = ρk′ for kj < k, k′ ≤ kj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By this and the update scheme of ρk, one can see that ∥˜c(xk+1)∥ ≤ α∥˜c(xk)∥, ∀kj < k < kj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (99) In addition, by (42), (94) and ρk ≥ ρ0, one has ∥˜c(xk+1)∥ ≤ δc,1 for 0 ≤ k ≤ Kǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Using this and (96), we obtain that ǫ/2 < ∥˜c(xk+1)∥ ≤ δc,1, ∀Kǫ ≤ k ≤ Kǫ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (100) Now, we notice that either kj+1 − kj = 1 or kj+1 − kj > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In the latter case, one can apply (99) with k = kj+1 − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' , kj + 1 along with (100) to deduce that ǫ/2 < ∥˜c(xkj+1)∥ ≤ α∥˜c(xkj+1−1)∥ ≤ · · · ≤ αkj+1−kj−1∥˜c(xkj+1)∥ ≤ αkj+1−kj−1δc,1, ∀j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' , | K |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Combining the two cases, we deduce that kj+1 − kj ≤ | log(ǫ(2δc,1)−1)/ log α| + 1, ∀j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=', | K |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (101) Summing up these inequalities, and using (92), (98), k0 = Kǫ and k| K |+1 = Kǫ − 1, we have Kǫ = 1 + k| K |+1 = 1 + k0 + �| K | j=0(kj+1 − kj) (101) ≤ 1 + Kǫ + (| K | + 1) ���� log(ǫ(2δc,1)−1) log α ��� + 1 � ≤ 2 + log(ρǫ,1ρ−1 0 ) log r + � log(ρǫ,1ρ−1 0 ) log r + 1 ����� log(ǫ(2δc,1)−1) log α ��� + 1 � = 1 + � log(ρǫ,1ρ−1 0 ) log r + 1 � ���� log(ǫ(2δc,1)−1) log α ��� + 2 � , where the second inequality is due to (92) and (98).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Hence, (48) holds as well in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We next prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Before proceeding, we recall from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 and the discussions in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 that the subproblem minx Lµ(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) satisfies Assumptions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(b) and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(c) with (F(·), S, Ω, LF H, U F g , U F H) = (L(·, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk), S(δf, δc), Ω(δf, δc), Lk,H, Uk,g, Uk,H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Moreover, in view of the fact that ∥λk∥ ≤ Λ, one can see from (43) and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(iv) that there exist some constants L1, L2, U1 and U2, depending only on f, c, B, β, Λ, m, δf and δc, such that Lk,H ≤ L1 + ρkL2, Uk,H ≤ U1 + ρkU2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (102) We are now ready to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Let Tk and Nk denote the number of iterations and fundamental operations performed by Algorithm 1 at outer iteration k of Algorithm 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It then follows from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3 that the total number of iterations of Algorithm 1 performed in Algorithm 2 is �Kǫ k=0 Tk, and moreover, the total number of Cholesky factorizations and other fundamental operations performed by Algorithm 1 in Algorithm 2 are �Kǫ k=0 Tk and �Kǫ k=0 Nk, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In addition, notice from (46) and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3 that ρǫ,1 = O(ǫ−2) and ρk ≤ rρǫ,1, which yield ρk = O(ǫ−2) for all 0 ≤ k ≤ Kǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (i) Recall from Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(i) and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(iii) that Lk,H is a Lipschitz constant of ∇2 xx L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) with respect to the local norm on an open convex neighborhood of {x ∈ int K : Lµ(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) ≤ Lµ(xk init, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In addition, 26 recall from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(ii) that infx∈int K Lµ(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) ≥ flow − γ − Λδc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By these, (40), (102), and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(iii) with (φhi, φlow, LF H, ǫg, ǫH) = (Lµ(xk init, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk), flow − γ − Λδc, Lk,H, τk, √τk), one has Tk = O((fhi − flow + γ + Λδc)L2 k,Hτ −3/2 k ) (102) = O(ρ2 kτ −3/2 k ) = O(ǫ−11/2), (103) where the last equality follows from τk ≥ µ = ǫ/(2ϑ1/2 +2) and ρk = O(ǫ−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' On the other hand, if c is assumed to be affine, namely, c(x) = Ax − b for some A ∈ Rm×n and b ∈ Rm, then ∇c(x) = AT and ∇2ci(x) = 0 for 1 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Using these and (90), we observe that Lk,H = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By this and the similar arguments as for (103), one has Tk = O(τ −3/2 k ) = O(ǫ−3/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Combining these with (103) and Kǫ = O(| log ǫ|2) (see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='4), we conclude that statement (i) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (ii) By Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(i) and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(iv), one has Uk,H ≥ sup x∈int K {∥∇2 xx L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk)∥∗ x : Lµ(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) ≤ Lµ(xk init, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In view of this, Lµ(xk init, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk) ≤ fhi, (102), and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1(iv) with (φhi, φlow, LF H, U F H, ǫg, ǫH) = (Lµ(xk init, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk), flow − γ − Λδc, Lk,H, Uk,H, τk, √τk), we obtain that Nk = �O((fhi − flow + γ + Λδc)L2 k,Hτ −3/2 k min{n, U 1/2 k,Hτ −1/4 k }) (102) = �O(ρ2 kτ −3/2 k min{n, ρ1/2 k τ −1/4 k }) = �O � ǫ−11/2 min � n, ǫ−5/4�� , (104) where the last equality follows from τk ≥ µ = ǫ/(2ϑ + 2) and ρk = O(ǫ−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' On the other hand, if c is assumed to be affine, it follows from the above discussion that Lk,H = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By this, Uk,H ≤ U1 + ρkU2, and the similar arguments as for (104), one has Nk = �O(τ −3/2 k min{n, ρ1/2 k τ −1/4 k }) = �O � ǫ−3/2 min � n, ǫ−5/4�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Combining these with (104) and Kǫ = O(| log ǫ|2) (see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='4), we conclude that statement (ii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We next establish two technical lemmas that will be used to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Suppose that Assumptions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2 hold and that ρ0 is sufficiently large such that δf,1 ≤ δf and δc,1 ≤ δc, where δf,1 and δc,1 are defined in (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Let {(xk, λk, ρk)} be generated by Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Suppose that ρk ≥ max{Λ2(2δf)−1, 2(fhi − flow + γ)δ−2 c + 2Λδ−1 c + 2γ, 2(U f g + √mU c gΛ + √ ϑ + 1)(σǫ)−1} (105) for some k ≥ 0, where γ, fhi, flow, δf, δc, U f g and U c g are given in Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1, and σ is given in (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Then it holds that ∥c(xk+1)∥ ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Using ∥λk∥ ≤ Λ (see step 5 of Algorithm 2) and (105), we have ρk ≥ max{∥λk∥2(2δf)−1, 2(fhi − flow + γ)δ−2 c + 2∥λk∥δ−1 c + 2γ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By this, (83), and Lemmas 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='7(iii) and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='7(iv) with (x, λ, ρ, ˜δf, ˜δc) = (xk+1, λk, ρk, δf, δc), one has f(xk+1) + µB(xk+1) ≤ fhi + δf and ∥˜c(xk+1)∥ ≤ δc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Also, notice from ∥c(zǫ)∥ ≤ 1 and the definition of ˜c in (35) that ∥c(xk+1)∥ ≤ 1 + ∥˜c(xk+1)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' These along with (31), xk+1 ∈ int K, and µ ∈ (0, ¯µ] yield that xk+1 ∈ S(δf, δc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It then follows from (33) that ∥∇f(xk+1)∥∗ xk+1 ≤ U f g and ∥∇ci(xk+1)∥∗ xk+1 ≤ U c g for all 1 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By these, τk ≤ 1, µ ≤ 1, ∥λk∥ ≤ Λ, (35) and the second relation in (37), one has ρk∥∇2B(xk+1)−1/2∇c(xk+1)˜c(xk+1)∥ = ρk∥∇c(xk+1)˜c(xk+1)∥∗ xk+1 ≤ ∥∇f(xk+1) + ∇c(xk+1)λk∥∗ xk+1 + µ∥∇B(xk+1)∥∗ xk+1 + ∥∇x Lµ(xk+1, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' ρk)∥∗ xk+1 ≤ ∥∇f(xk+1)∥∗ xk+1 + m � i=1 |λk i |∥∇ci(xk+1)∥∗ xk+1 + µ √ ϑ + τk ≤ U f g + √mU c gΛ + √ ϑ + 1, (106) where the first inequality follows from the triangle inequality, and the second inequality follows from ∥∇B(xk+1)∥∗ xk+1 = √ ϑ and the second relation in (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In addition, by xk+1 ∈ S(δf, δc) and (49), one has λmin(∇c(xk+1)T ∇2B(xk+1)−1∇c(xk+1)) ≥ σ2, which along with (106) implies that ∥˜c(xk+1)∥ ≤ ∥[∇c(xk+1)T ∇2B(xk+1)−1∇c(xk+1)]−1∇c(xk+1)T ∇2B(xk+1)−1/2∥∥∇2B(xk+1)−1/2∇c(xk+1)˜c(xk+1)∥ 27 = ∥[∇c(xk+1)T ∇2B(xk+1)−1∇c(xk+1)]−1∥1/2∥∇2B(xk+1)−1/2∇c(xk+1)˜c(xk+1)∥ = λmin(∇c(xk+1)T ∇2B(xk+1)−1∇c(xk+1))−1/2∥∇2B(xk+1)−1/2∇c(xk+1)˜c(xk+1)∥ ≤ (U f g + √mU c gΛ + √ ϑ + 1)/(σρk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (107) Observe from (105) that ρk ≥ 2(U f g +√mU c gΛ+ √ ϑ+1)(σǫ)−1, which along with (107) implies ∥˜c(xk+1)∥ ≤ ǫ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Combining this with ∥c(zǫ)∥ ≤ ǫ/2 and the definition of ˜c in (35), we obtain ∥c(xk+1)∥ ≤ ǫ as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The next lemma establishes a stronger upper bound for {ρk} than the one given in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Suppose that Assumptions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2 hold and that ρ0 is sufficiently large such that δf,1 ≤ δf and δc,1 ≤ δc, where δf,1 and δc,1 are defined in (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Let {ρk} be generated by Algorithm 2 and ρǫ,2 := max{Λ2(2δf)−1, 2(fhi − flow + γ)δ−2 c + 2Λδ−1 c + 2γ, 2(U f g + √mU c gΛ + √ ϑ + 1)(σǫ)−1, 2ρ0}, (108) where γ, fhi, flow, δf, δc, U f g and U c g are given in Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1, and σ is given in (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Then ρk ≤ rρǫ,2 holds for 0 ≤ k ≤ Kǫ, where Kǫ is defined in (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Observe from (108) that ρǫ,2 ≥ 2ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Using this and similar arguments as for (92), we have Kǫ ≤ log(ρǫ,2ρ−1 0 )/ log r + 1, where Kǫ is defined in (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' By this, the update scheme for {ρk}, and similar arguments as for (93), one has max 0≤k≤Kǫ ρk ≤ rρǫ,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (109) If ∥c(xKǫ+1)∥ ≤ ǫ, it follows from (47) that Kǫ = Kǫ, which along with (109) implies that ρk ≤ rρǫ,2 holds for 0 ≤ k ≤ Kǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' On the other hand, if ∥c(xKǫ+1)∥ > ǫ, it follows from (47) that ∥c(xk+1)∥ > ǫ for Kǫ ≤ k ≤ Kǫ − 1, which together with Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='9 and (108) implies that ρk < max � Λ2 2δf , 2(fhi − flow + γ) δ2c + 2Λ δc + 2γ, 2(U f g + √mU c gΛ + √ ϑ + 1) σǫ � (108) ≤ ρǫ,2, ∀Kǫ ≤ k ≤ Kǫ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Using this, (109), and ρKǫ ≤ rρKǫ−1, we also conclude that ρk ≤ rρǫ,2 holds for 0 ≤ k ≤ Kǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' We now provide a proof for Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Notice from (108) and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='10 that ρǫ,2 = O(ǫ−1) and ρk ≤ rρǫ,2, which imply ρk = O(ǫ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The rest of the proof follows from the same arguments as for Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='4 with ρk = O(ǫ−2) replaced by ρk = O(ǫ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 8 Concluding remarks In this paper we proposed a Newton-CG based barrier-AL method for finding an approximate SOSP of general nonconvex conic optimization problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' It can be easily extended to a more general conic optimization problem minx,y{ ˜f(x, y) : ˜c(x, y) = 0, y ∈ K}, which includes problem minx{f(x) : c(x) = 0, d(x) ≤ 0} and more generally problem minx{f(x) : c(x) = 0, d(x) ∈ K} as special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Indeed, the latter problem can be equivalently solved as the problem minx,y{f(x) : c(x) = 0, d(x) − y = 0, y ∈ K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' References [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Agarwal, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Allen-Zhu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Bullins, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Hazan, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Finding approximate local minima faster than gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 1195–1199, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' [2] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Mart´ınez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
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+page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
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+page_content=' Byrd, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Schnabel, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Shultz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
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+page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
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+page_content=' Duchi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Gradient descent finds the cubic-regularized nonconvex newton step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
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+page_content=' [19] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Carmon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Duchi, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Hinder, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Sidford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' “Convex until proven guilty”: Dimension-free accelera- tion of gradient descent on non-convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In International Conference on Machine Learning, pages 654–663.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' PMLR, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' [20] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Carmon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Duchi, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Hinder, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Sidford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Accelerated methods for nonconvex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=', 28(2):1751–1772, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' [21] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Cartis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Gould, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Toint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Adaptive cubic regularisation methods for unconstrained optimiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Part I: motivation, convergence and numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=', 127(2):245–295, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' [22] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Cartis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Gould, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Toint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' On the evaluation complexity of cubic regularization methods for potentially rank-deficient nonlinear least-squares problems and its relevance to constrained nonlinear optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=', 23(3):1553–1574, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 29 [23] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Cartis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Gould, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Toint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' On the complexity of finding first-order critical points in constrained nonlinear optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=', 144(1):93–106, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' [24] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Cartis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Gould, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Toint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' On the complexity of finding first-order critical points in constrained nonlinear optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=', 144(1):93–106, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' [25] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Cartis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Gould, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Toint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' On the evaluation complexity of constrained nonlinear least- squares and general constrained nonlinear optimization using second-order methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=', 53(2):836–851, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' [26] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Cartis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Gould, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Toint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Evaluation complexity bounds for smooth constrained nonlinear op- timization using scaled KKT conditions, high-order models and the criticality measure χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' In Approximation and Optimization, pages 5–26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Springer, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' [27] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Cartis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Gould, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Toint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Optimality of orders one to three and beyond: characterization and evaluation complexity in constrained nonconvex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=', 53:68–94, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
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+page_content=' arXiv preprint arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='15989, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' [70] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
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+page_content=' Wright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Complexity of proximal augmented Lagrangian for nonconvex optimization with nonlinear equality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
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+page_content=' Jin, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' NEON+: Accelerated gradient methods for extracting negative curvature for non-convex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' arXiv preprint arXiv:1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='01033, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' [72] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Yang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Sun, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Toh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' SDPNAL+: A majorized semismooth Newton-CG augmented Lagrangian method for semidefinite programming with nonnegative constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=', 7(3):331– 366, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' [73] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Zhao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Sun, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Toh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' A Newton-CG augmented Lagrangian method for semidefinite program- ming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=', 20(4):1737–1765, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Appendix A A capped conjugate gradient method In this part we present the capped CG method proposed in [60, Algorithm 1] for solving a possibly indefinite linear system (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' As briefly discussed in Section 4, the capped CG method finds either an approximate solution to (15) or a sufficiently negative curvature direction of the associated matrix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' More details about this method can be found in [60, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The following theorem presents the iteration complexity of Algorithm 3, whose proof can be found in [44, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1], and thus omitted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 (iteration complexity of Algorithm 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Consider applying Algorithm 3 with the optional input U = 0 to the linear system (15) with g ̸= 0, ε > 0, and H being an n × n symmetric matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Then the number of iterations of Algorithm 3 is �O(min{n, � ∥H∥/ε}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 32 Algorithm 3 A capped conjugate gradient method Input: symmetric matrix H ∈ Rn×n, vector g ̸= 0, damping parameter ε ∈ (0, 1), desired relative accuracy ζ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Optional input: scalar U ≥ 0 such that ∥H∥ ≤ U (set to 0 if not provided).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Outputs: ˆd, d type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Secondary outputs: final values of U, κ, ˆζ, τ, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Set ¯ H := H + 2εI, κ := U + 2ε ε , ˆζ := ζ 3κ , τ := √κ √κ + 1 , T := 4κ4 (1 − √τ)2 , y0 ← 0, r0 ← g, p0 ← −g, j ← 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' if (p0)T ¯ Hp0 < ε∥p0∥2 then Set ˆd ← p0 and terminate with d type = NC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' else if ∥Hp0∥ > U∥p0∥ then Set U ← ∥Hp0∥/∥p0∥ and update κ, ˆζ, τ, T accordingly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' end if while TRUE do αj ← (rj)T rj/(pj)T ¯ Hpj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' {Begin Standard CG Operations} yj+1 ← yj + αjpj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' rj+1 ← rj + αj ¯ Hpj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' βj+1 ← ∥rj+1∥2/∥rj∥2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' pj+1 ← −rj+1 + βj+1pj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' {End Standard CG Operations} j ← j + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' if ∥Hpj∥ > U∥pj∥ then Set U ← ∥Hpj∥/∥pj∥ and update κ, ˆζ, τ, T accordingly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' end if if ∥Hyj∥ > U∥yj∥ then Set U ← ∥Hyj∥/∥yj∥ and update κ, ˆζ, τ, T accordingly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' end if if ∥Hrj∥ > U∥rj∥ then Set U ← ∥Hrj∥/∥rj∥ and update κ, ˆζ, τ, T accordingly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' end if if (yj)T ¯Hyj < ε∥yj∥2 then Set ˆd ← yj and terminate with d type = NC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' else if ∥rj∥ ≤ ˆζ∥r0∥ then Set ˆd ← yj and terminate with d type = SOL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' else if (pj)T ¯ Hpj < ε∥pj∥2 then Set ˆd ← pj and terminate with d type = NC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' else if ∥rj∥ > √ Tτ j/2∥r0∥ then Compute αj, yj+1 as in the main loop above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Find i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' , j − 1} such that (yj+1 − yi)T ¯ H(yj+1 − yi) < ε∥yj+1 − yi∥2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Set ˆd ← yj+1 − yi and terminate with d type = NC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' end if end while B A randomized Lanczos based minimum eigenvalue oracle In this part we present the randomized Lanczos method proposed in [60, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2], which can be used as a minimum eigenvalue oracle for Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' As mentioned in Section 4, this oracle either outputs a sufficiently negative curvature direction of H or certifies that H is nearly positive semidefinite with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' More details about it can be found in [60, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' The following theorem justifies that Algorithm 4 is a suitable minimum eigenvalue oracle for Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Its proof is identical to that of [60, Lemma 2] and thus omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='1 (iteration complexity of Algorithm 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Consider Algorithm 4 with tolerance ε > 0, prob- ability parameter δ ∈ (0, 1), and symmetric matrix H ∈ Rn×n as its input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Then it either finds a sufficiently negative curvature direction v satisfying vT Hv ≤ −ε/2 and ∥v∥ = 1 or certifies that λmin(H) ≥ −ε holds with probability at least 1 − δ in at most N(ε, δ) iterations, where N(ε, δ) is defined in (110).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Notice that generally, computing ∥H∥ in Algorithm 4 may not be cheap when n is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Nevertheless, ∥H∥ can be efficiently estimated via a randomization scheme with high confidence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=', see the discussion in [60, Appendix B3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 33 Algorithm 4 A randomized Lanczos based minimum eigenvalue oracle Input: symmetric matrix H ∈ Rn×n, tolerance ε > 0, and probability parameter δ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Output: a sufficiently negative curvature direction v satisfying vT Hv ≤ −ε/2 and ∥v∥ = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' or a certificate that λmin(H) ≥ −ε with probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' Apply the Lanczos method [48] to estimate λmin(H) starting with a random vector uniformly generated on the unit sphere, and run it for at most N(ε, δ) := min � n, 1 + � ln(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content='75n/δ2) 2 � ∥H∥ ε �� (110) iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (i) If it finds a unit vector v such that vT Hv ≤ −ε/2 at some iteration, it terminates immediately and returns v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' (ii) Otherwise, it certifies that λmin(H) ≥ −ε holds with probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
+page_content=' 34' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQf6wj3/content/2301.04204v1.pdf'}
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+arXiv:2301.02398v1 [gr-qc] 6 Jan 2023
+Glitch subtraction from gravitational wave data
+using adaptive spline fitting
+Soumya D. Mohanty, Mohammad A. T. Chowdhury
+Department of Physics and Astronomy, University of Texas Rio Grande Valley, One
+West University Blvd., Brownsville, Texas 78520, USA
+E-mail: soumya.mohanty@utrgv.edu
+E-mail: mohammadabuthaher.chowdhury01@utrgv.edu
+Abstract.
+Transient signals of instrumental and environmental origins (“glitches”)
+in gravitational wave data elevate the false alarm rate of searches for astrophysical
+signals and reduce their sensitivity. Glitches that directly overlap astrophysical signals
+hinder their detection and worsen parameter estimation errors.
+As the fraction of
+data occupied by detectable astrophysical signals will be higher in next generation
+detectors, such problematic overlaps could become more frequent. These adverse effects
+of glitches can be mitigated by estimating and subtracting them out from the data,
+but their unpredictable waveforms and large morphological diversity pose a challenge.
+Subtraction of glitches using data from auxiliary sensors as predictors works but not
+for the majority of cases. Thus, there is a need for nonparametric glitch mitigation
+methods that do not require auxiliary data, work for a large variety of glitches, and have
+minimal effect on astrophysical signals in the case of overlaps. In order to cope with
+the high rate of glitches, it is also desirable that such methods be computationally fast.
+We show that adaptive spline fitting, in which the placement of free knots is optimized
+to estimate both smooth and non-smooth curves in noisy data, offers a promising
+approach to satisfying these requirements for broadband short-duration glitches, the
+type that appear quite frequently. The method is demonstrated on glitches drawn
+from three distinct classes in the Gravity Spy database as well as on the glitch that
+overlapped the double neutron star signal GW170817. The impact of glitch subtraction
+on the GW170817 signal, or those like it injected into the data, is seen to be negligible.
+1. Introduction
+In a fairly short time since the first direct detection of a gravitational wave (GW) signal
+(GW150914) in 2015 [1] by the twin LIGO [2] detectors, GW astronomy has emerged as
+an information-rich field that will revolutionize our understanding of compact objects
+such as black holes and neutron stars. By now, the network of LIGO and Virgo [3]
+detectors has reported 90 confirmed detections of GW signals from compact binary
+coalescences (CBCs) across the first observing run (O1) [4] to the third (O3) [5]. The
+majority of these are binary black hole (BBH) mergers but the haul also includes a
+double neutron star (DNS) system (GW170817) [6].
+
+Adaptive spline glitch removal
+2
+The rate of detectable GW signals will grow as more detectors, namely KAGRA [7]
+and LIGO-India [8], join the network and increase its distance reach for GW sources.
+Design studies are already underway for the successors to the current generation of GW
+detectors [9, 10, 11] with the goal of achieving an order of magnitude improvement in
+sensitivity across the current operational frequency band. In addition, next-generation
+detectors will seek to expand the operational range to lower frequencies (≈ 1 Hz),
+thereby increasing the duration of in-band GW signals across the board: for example, a
+DNS signal starting at ≈ 10 Hz will last for days compared to the ≈ 1 min for GW170817.
+Thus, future detectors will not only see a higher rate but also longer signals, raising the
+prospect [12] that there will be no data segment free of detectable GW signals.
+The false alarm rate, hence the sensitivity, of searches for CBCs as well as generic
+short duration GW signals, or bursts, is dominated [13] by transient non-GW signals
+of instrumental or environmental origins, commonly called glitches.
+This is because
+glitches that populate the same frequency band as CBC or burst signals and happen to
+be transient in duration can falsely trigger the respective search pipelines. A glitch has
+a particularly adverse effect if it overlaps with a GW signal, as happened in the case of
+GW170817 [6], and causes the glitch rejection step of a search pipeline to also discard
+the signal. Even a non-overlapping glitch can severely degrade parameter estimation
+if it is close enough to a GW signal [14]. In the third observing run of the LIGO and
+Virgo detectors, ≈ 20% of detected GW signals overlapped with glitches [15] due to the
+high rate of the latter. For future detectors, the frequency of chance overlaps will be
+enhanced by the higher rate of detectable GW signals as well as, for CBC signals, their
+longer durations.
+Glitches have dissimilar and unpredictable waveforms but many of the observed ones
+tend to fall into distinct morphological classes. This has motivated the investigation
+of automated glitch classification using machine learning where a range of different
+methods have been proposed, such as Support Vector Machine [16], t-Sne [17], random
+forests [16], S-means [18], and Deep Convolutional Neural Networks [19]. The Gravity
+Spy [20] project uses a citizen science approach to engage the lay public in labeling
+glitches by visual inspection of their constant Q-transform [21, 22] images. This has
+created a high quality training dataset for machine learning methods. By now, there
+exist more than 20 named glitch classes in the Gravity Spy database, collected over
+multiple observing runs of the LIGO detectors [17].
+Several different approaches have been developed to mitigate the adverse effects
+of glitches on GW searches.
+GW search pipelines typically compute secondary
+functionals, called vetoes, of the data besides the primary detection statistic that help
+in distinguishing genuine GW signals from glitches. A well-known example is the Chi-
+square [23] veto used in CBC search pipelines. For LIGO-Virgo data, a set of Data
+Quality flags have been developed that use information from a large number of auxiliary
+sensors to quantify the safety of analyzing a given segment of GW strain data [24]. For
+glitches that overlap a GW signal, the gating [25] method excises the rectangular time-
+frequency block, or just the time interval, containing an identified glitch from the data.
+
+Adaptive spline glitch removal
+3
+Cross-channel regression using data from auxiliary sensors
+[26, 27, 28, 29] has been
+used to reduce excess broadband noise and a few types of glitches [30].
+A relatively recent approach is that of estimating the waveform of a glitch
+from the data time series itself and subtracting it out.
+Glitch subtraction was of
+critical importance in the case of GW170817 and has been shown to be an important
+requirement in reducing bias in the estimation of GW signal parameters [31].
+The
+GW170817 glitch subtraction was carried out using the multi-detector BayesWave
+pipeline [32, 33], which has also been used for other types of glitches [15]. Another
+method, Glitschen [34], follows the approach of constructing parametrized waveform
+models for identified glitch classes using principal component analysis of training sets.
+A strong motivation for developing glitch estimation and subtraction methods is that
+one could, in principle, preprocess the data to clean out every sufficiently loud glitch of
+a known type and make glitch rejection in all downstream GW searches safer.
+In this paper, we present a method for the estimation and subtraction of broadband,
+short-duration glitches that have appeared frequently in the observation runs of the
+LIGO detectors.
+The method is computationally cheap, works with single-detector
+data, does not require a training set of pre-identified glitches, and is not predicated on
+auxiliary sensor data. The core component of the method is SHAPES (Swarm Heuristics
+based Adaptive and Penalized Estimation of Splines), an adaptive spline curve fitting
+algorithm introduced in [35]‡. SHAPES uses splines with free placement of knots to fit
+both smooth and non-smooth curves in noisy data. In particular, point discontinuities
+in the curve or its derivatives (up to some order) can be accommodated in the fit by
+allowing knots to merge. The ability to handle both sharp and slow changes in a curve
+is a built-in form of multiresolution analysis in SHAPES and a critical requirement for
+effective estimation of broadband glitches. We examine the performance of our glitch
+subtraction method on the GW170817 glitch in LIGO-Livingston data and instances of
+glitches from three morphologically distinct classes, namely, Blip, Koi Fish, and Tomte,
+in the Gravity Spy database. In each of the latter three cases, we inject a DNS signal
+overlapping with the glitch to mimic the case of GW170817. We find that the impact
+of glitch subtraction on the signals, real or injected, is negligible.
+The rest of the paper is organized as follows. Sec. 2 reviews SHAPES with the
+goal of providing a self-contained description of the algorithm that is pertinent to this
+paper. Further details, such as the motivation and justification for certain features of
+the algorithm, can be found in [35]. Sec. 3 describes the dataset used in this paper and
+the details of how SHAPES is used for glitch subtraction. Sec. 4 presents the results.
+Our conclusions and discussion of future work are presented in Sec. 5.
+‡ The SHAPES code is available from the Github repository mohanty-sd/SHAPES.git.
+
+Adaptive spline glitch removal
+4
+2. Adaptive spline fitting: the SHAPES algorithm
+SHAPES is derived under the following models for the noisy data, y, and the signal
+s(θ).
+y = s(θ) + ǫ ,
+(1)
+where y, s, and ǫ are row vectors with N elements, yi = y(ti) and si(θ) = s(ti; θ),
+i = 0, 1, . . . , N −1, are samples taken at ti = i/fs with fs being the sampling frequency,
+and θ denotes the set of signal parameters that need to be estimated from the data.
+The noise samples, ǫi, are drawn independently from the zero mean and unit variance
+normal (Gaussian) probability density function N(0, 1). This assumption, namely, that
+of a white Gaussian noise process does not entail a loss of generalization since GW data
+can always be whitened using the estimated noise power spectral density (PSD).
+The signal s(t; θ) is assumed to be a spline of polynomial order k and, as such, can
+be represented by a linear combination of B-spline functions [36],
+s (t; θ = {α, τ}) =
+P −k−1
+�
+j=0
+αjBj,k(t; τ) ,
+(2)
+where α = (α0, α1, . . . , αP −k−1), and τ = (τ0, τ1, . . . , τP −1), τi+1 ≥ τi, is a sequence of P
+knots that marks the end points of the contiguous intervals containing the polynomial
+pieces of the spline. Note that knots are allowed to be equal, leading to knots with
+multiplicity higher than one. Repeating knots create discontinuity in either the value
+of a B-spline function or its derivatives (up to order k − 2). This allows the s(t; θ) in
+Eq. 2 to model signals with point discontinuities in value or derivatives. In the rest of
+the paper, we will set k = 4, making s(t; θ) a cubic spline.
+The best fit spline parameters, �α and �τ, are the ones that minimize a penalized
+least-squares function,
+Lλ(α, τ) = L(α, τ) + λR(α) ,
+(3)
+L(α, τ) =
+N−1
+�
+i=0
+(yi − si(α, τ))2 ,
+(4)
+where the penalty term,
+R(α) =
+P −k−1
+�
+j=0
+α2
+j ,
+(5)
+is found to be useful in the suppression of spurious clustering of the knots.
+These
+clusters are observed when the method tries to minimize Lλ(α, τ) by fitting out outlier
+data points arising from the noise alone. The strength of the penalty is controlled by
+the gain factor λ, with higher values of λ leading to smoother estimates.
+The optimization of Lλ(α, τ) over the non-linear parameters τ has been a long-
+standing computational barrier [37, 38, 39, 40] for adaptive spline fitting. At the same
+time, the benefits of optimizing the placement of knots have also been demonstrated
+extensively [38, 41]. It was shown in [42], and independently in [43], that Particle Swarm
+Optimization (PSO) [44, 45], a widely used nature-inspired metaheuristic for global
+
+Adaptive spline glitch removal
+5
+optimization, has good performance on the free knot placement problem. Moreover,
+being a continuous optimization method, PSO can explore all arrangements of knots,
+including the ones where knots are sufficiently close to be merged into a single knot of
+higher multiplicity. This allows the fitting of functions that have a mix of smooth and
+non-smooth parts.
+There are many variations [46] among the algorithms that fall under the umbrella
+of the PSO metaheuristic but they all share the following common features. (i) The
+function to be optimized, called the fitness function, is sampled at multiple locations,
+called particles, that move iteratively to explore the domain, called the search space, over
+which the the global optimum of the fitness is to be found. The set of particles is called
+a swarm. (ii) The location of each particle is updated following a dynamical rule that
+incorporates randomness. The rule typically uses the best location found by a particle
+in its history, called its personal best, and the best location found by the particles in
+its neighborhood, called its local best. Here, the fitness value at a location defines how
+good it is: for a minimization problem, the lower the fitness, the better the location.
+(iii) Each particle explores the search space independently but is constantly attracted
+towards the personal and local bests. This leads to a form of communication between
+the particles that speeds up convergence to a promising region, followed by refinement
+of the solution until the iterations are terminated.
+The best location among all the particles at termination is the final solution found
+by the swarm for the global optimum. While there is no guarantee that the final solution
+is the true global optimum, the probability of successful convergence can be boosted
+exponentially by running multiple independent runs of PSO and picking the one with
+the best final solution. Most of the parameters involved in the PSO algorithm, such
+as the number of particles or the weights attached to the attractive forces, have very
+robust values across a wide variety of benchmark optimization problems [47] and rarely
+need to be changed. In our experience, there are typically only two quantities that
+need tuning: the number of iterations, Niter, to termination and the hyper-parameter
+Nruns, the number of independent PSO runs. In this paper, we fix Niter = 2000 and
+Nruns = 8 throughout. The number of particles is always set to 40 and the settings for
+the remaining parameters, as well as the definition of the neighborhood used for the
+local best, are described in [35].
+The description above was for the case where the number of knots, P, is fixed. The
+complete SHAPES algorithm incorporates model selection using the Akaike Information
+Criterion (AIC) [48], where the optimum number of knots minimizes,
+AIC = 4P + Lλ(�α, �τ) .
+(6)
+While, given sufficient computing resources, model selection could be performed over all
+values of P until the minimum value of AIC is found, practical considerations dictate
+that the set of knot numbers used be a finite and small one. In this paper, for example,
+we use knot numbers in the set starting at 5 and ending at 60 in increments of 5. It is
+important to note that this restriction of knot numbers is not a fundamental limitation
+
+Adaptive spline glitch removal
+6
+but a technical one meant to manage the computational burden of model selection.
+Thus, the only significant free parameter that needs to be set by the user in the current
+version of SHAPES is λ.
+Since SHAPES assumes that the noise in the data is white, GW strain data must
+be whitened prior to glitch estimation and subtraction. The data conditioning steps
+involved are as follows (in sequential order).
+(a) Suppression of the seismic noise
+below 10 Hz, (b) robust estimation of the power spectral density (PSD) noise floor,
+(c) whitening of the noise floor using the estimated PSD [49], and (d) automated
+identification of high-power narrowband noise features (“lines”) and their suppression
+using notch filters. These steps are common to all GW search pipelines, so they do not
+need to be elaborated further here.
+3. Demonstration data
+The glitches considered in this paper for demonstrating the performance of SHAPES are
+listed in Table 1. The corresponding GW strain data files can be located and downloaded
+from the Gravitational Wave Open Science Center (GWOSC) [50] using the information
+provided in this table. We have used the standard 4096 sec long GWOSC data files
+sampled at 4 kHz.
+The GW170817 glitch presents a particularly interesting example of the deleterious
+effect of glitches on GW searches. The GW signal appeared in both LIGO-Hanford (H1)
+and LIGO-Livingston (L1) with a combined network signal to noise ratio (SNR) of 32.4.
+Such a strong signal would have been detected easily in coincidence across L1 and H1
+by the GW search pipelines in operation at the time. However, a coincident detection
+was prevented by a large overlapping glitch in L1 causing the release of only an unusual
+single-detector GW detection alert to the astronomical community. About 11 hours
+elapsed between the initial alert and the release of the skymap localizing GW170817, a
+process that included the subtraction of the glitch using BayesWave.
+In addition to the GW170817 glitch, we have taken three representative glitches
+from the Blip, Koi Fish, and Tomte, classes in the Gravity Spy [20] database [51].
+These glitches were selected by taking the loudest 5 events, in terms of their signal-to-
+noise ratio (SNR) as given in the Gravity Spy database, for each class and then picking
+the first one in this list for which the corresponding GWOSC file had 100% science data
+that was also reasonably stationary. As can be seen from Table 1, this results in the
+selected glitches spanning a wide range in SNR.
+After conditioning the data, we use the start time of a glitch, recorded in Table 1,
+to locate the glitch. Starting from the peak of the glitch, the data time series is scanned
+visually in both directions to identify a segment, containing the glitch, that tapers off
+at both its boundaries to the general noise level of the conditioned data.
+To mimic the case of GW170817 and to study the effect of glitch subtraction on
+an overlapping GW signal, we injected a whitened restricted-2PN circularized binary
+inspiral signal with equal 1.4 M⊙ components in the conditioned data. The SNR (in
+
+Adaptive spline glitch removal
+7
+Glitch Name
+GPS start (sec)
+SNR
+Detector
+run
+GW170817 glitch
+1187008880
+–
+L1
+O2
+Blip
+1182397347
+109.1
+H1
+O2
+Koi Fish
+1169847108
+608.1
+H1
+O2
+Tomte
+1173086299
+19.6
+H1
+O2
+Table 1. Glitches considered in this paper along with their GPS start times, SNRs,
+the detectors in which they appeared, and the observation runs. For the Blip, Koi Fish,
+and Tomte glitches, the start times are taken from the Gravity Spy database. To the
+best of our knowledge, there is no SNR available in the literature for the GW170817
+glitch.
+white noise with unit variance) of the injected signal is set at 37.3, which is an ad
+hoc factor of
+√
+2 higher than the observed SNR of 26.4 of GW170817 in L1 [6]. The
+enhancement in SNR allows clearer visibility of the signal in time-frequency images
+while also posing a stronger challenge to SHAPES in terms how well it ignores the GW
+signal when estimating a glitch. The segment containing the glitch, taken from the
+conditioned data with the injected signal, is passed to SHAPES for estimation of the
+glitch waveform followed by its subtraction.
+4. Results
+In common with other papers on glitch estimation and subtraction, we present our
+results in the form of constant Q-transform (CQT) time-frequency images and time
+series plots. These are obtained by taking projections of the data on a set of windowed
+sinusoids. The width of the window decreases with an increase in the carrier frequency,
+fc, such that Q = fc/∆f, where ∆f is the −3 dB bandwidth of the Fourier transform
+of the window, remains constant. We use the CQT code provided in the librosa [52]
+Python package for audio processing. For each glitch, we show CQTs of the conditioned
+data with injected signal and the residual after subtraction of the glitch estimate.
+Fig. 1 shows the data segments that were processed using SHAPES and the
+corresponding estimated glitch waveforms. Except for GW170817, each segment was
+processed as a whole to obtain the glitch estimate. In the case of GW170817, SHAPES
+was applied independently to three separate but contiguous time intervals to estimate
+the complete glitch. This was necessitated by the presence of extended wings, preceding
+and trailing the core broadband (and rapidly varying) part in the middle, that dominate
+the conditioned data for ≈ 0.5 sec on each side. Applying SHAPES to the complete
+segment would have required using a very large number of knots (> 60), making it
+unnecessarily expensive computationally given that splitting the segment achieves a
+good solution.
+As mentioned in Sec. 2, the penalty gain λ controls the smoothness of the estimate
+and is a user-specified parameter of the SHAPES algorithm. Typically, when a glitch
+is loud and has a complex shape, λ = 0.01 allows SHAPES to provide a better fit. For
+
+Adaptive spline glitch removal
+8
+932.67
+932.68
+932.69
+932.7
+932.71
+-60
+-40
+-20
+0
+20
+40
+60
+80
+Data
+Estimate
+839.1
+839.12
+839.14
+839.16
+Time (sec)
+-100
+-50
+0
+50
+100
+150
+Whitened Strain
+Data
+Estimate
+93.04
+93.06
+93.08
+93.1
+93.12
+93.14
+Time (sec)
+-4
+-2
+0
+2
+4
+Data
+Estimate
+370
+370.5
+371
+-100
+-50
+0
+50
+100
+Whitened Strain
+Data
+Estimate
+Figure 1. The conditioned strain data and the glitch waveform estimated by SHAPES
+for each of the glitches considered in this paper. Top row: GW170817 (left) and Blip
+(right). Bottom row: Koi Fish (left) and Tomte (right). The X-axis in each plot shows
+the time (sec) since the start of the open data file containing the glitch as provided
+by GWOSC. For GW170817, the dashed vertical lines demarcate the three adjacent
+segments that were analyzed separately.
+low SNR and simple glitch waveforms, or if the data is just plain white noise, λ = 0.1
+does an adequate job. In general, estimates from SHAPES are not sensitive to small
+variations of λ around these values because the model selection is able to compensate
+for a lower value of λ by selecting a higher knot number and vice versa. Without much
+fine tuning, we found that the values of λ listed in Table 2 work well for the glitches
+studied in this paper. We have also listed in this table the number of knots for the best
+fit models selected by the AIC.
+Fig. 2 to Fig. 5 show the CQTs of the conditioned data and residuals after glitch
+subtraction for the glitches in the sequence GW170817, Blip, Koi Fish, and Tomte,
+respectively. In all cases, we see that the subtraction of the glitch does not affect the
+overlapping GW signal (real or injected) in any significant way. Some overfitting to the
+data, seen as very small CQT values, is visible in the residual for the GW170817 glitch at
+frequencies below ≈ 32 Hz but this band has no overlap with the signal. The overfitted
+parts are the two wings of the GW170817 glitch mentioned earlier. The CQTs of the
+residuals for the Blip and Tomte glitches show near perfect removal of the glitch. (For
+
+Adaptive spline glitch removal
+9
+Glitch Name
+Penalty gain (λ)
+Number of knots
+GW170817 glitch
+0.1, 0.01, 0.1
+60,40,50
+Blip
+0.01
+15
+Koi Fish
+0.01
+30
+Tomte
+0.1
+15
+Table 2. The penalty gain λ used for the glitches and the number of knots in the best
+fit model. For the GW170817 glitch, there are three segments with the middle one
+containing the principal glitch and adjacent ones containing the wings. The penalty
+gains and best fit model are listed for all three segments in sequential order from left
+to right.
+Tomte, the coalescence time of the GW signal was kept further away from the glitch
+in order to create an overlap between the signal track and the glitch.) The residual for
+Koi Fish shows effective removal of the glitch with the exception of a transient and low
+frequency narrowband component. This leftover component does not overlap with the
+signal.
+The principal computational cost in SHAPES is the global optimization of the
+fitness function in Eq. 3. The time taken by the current MATLAB [53] code for a single
+PSO run on a segment with ≈ 300 samples and knot numbers P ∈ [10, 60] (in steps of
+5) is < 10 min on an Intel Xeon E5 processor (clock rate 3 GHz). The runtime increases
+with the number of knots used, mainly due to an increase in the number of B-spline
+functions that need to be computed. With a code currently under construction in the
+C language, and implementation of further hardware acceleration (e.g., using Graphics
+Processing Units), the runtime is expected to decrease substantially. We also note that
+the segments containing glitches can be processed in parallel since SHAPES is a purely
+time-domain method. Hence, the computational cost will scale slower than linearly with
+the number of glitches when analyzing data containing multiple glitches.
+5. Discussion and Conclusions
+We have presented a new approach to glitch subtraction using an adaptive spline fitting
+method called SHAPES. The method was demonstrated on the GW170817 glitch as
+well as other representative short duration and broadband glitches. In a single detector
+and in the absence of strong prior information about the signal, it is not possible to
+distinguish a GW signal from a glitch in the part where they overlap.
+Hence, it is
+expected that the signal power will be removed in that part along with the glitch when
+the latter is estimated and subtracted out. Nonetheless, as far as the DNS signal used
+in this paper is concerned, we observe very little impact on the signal across a wide
+range of glitch SNRs. While this conclusion will be quantified in future studies using a
+much larger number of glitches, it is clear that SHAPES is effective at addressing glitch
+subtraction.
+SHAPES is not well adapted to fitting highly oscillatory waveforms since these
+
+Adaptive spline glitch removal
+10
+0
+1.5
+3
+4.5
+6
+7.5
+9
+10.5
+12
+Time (sec)
+16
+32
+64
+128
+256
+512
+Frequency (Hz)
+ 1
+ 2
+ 3
+ 4
+ 5
+0
+1.5
+3
+4.5
+6
+7.5
+9
+10.5
+12
+Time (sec)
+16
+32
+64
+128
+256
+512
+Frequency (Hz)
+ 1
+ 2
+ 3
+ 4
+Figure 2. Subtraction of the GW170817 Glitch. The top and bottom panels show
+the CQT of the data and residual, respectively. The glitch is the vertical feature at
+≈ 10.5 sec. In order to show both the glitch and the signal in the same image, a
+threshold has been applied to the CQT as indicated by the maximum value in the
+colorbar of the top panel.
+are are not represented well by splines without using an inordinate number of knots.
+Therefore, the direct use of SHAPES for glitches in the Gravity Spy database such as
+whistlers or wandering lines is not viable. However, chirp signals such as these could be
+estimated using the method proposed in [54, 55], where adaptive splines figure indirectly
+
+Adaptive spline glitch removal
+11
+0
+1
+2
+3
+4
+5
+6
+7
+8
+Time (sec)
+16
+32
+64
+128
+256
+512
+Frequency (Hz)
+ 1
+ 2
+ 3
+ 4
+ 5
+0
+1
+2
+3
+4
+5
+6
+7
+8
+Time (sec)
+16
+32
+64
+128
+256
+512
+Frequency (Hz)
+ 1
+ 2
+ 3
+ 4
+ 5
+Figure 3. Subtraction of the Blip Glitch. The top and bottom panels show the CQT
+of the data and residual, respectively. The glitch is the vertical feature at ≈ 6 sec. In
+order to show both the glitch and the signal in the same image, a threshold has been
+applied to the CQT as indicated by the maximum value in the colorbar of the top
+panel.
+in a non-linear signal model. This is an interesting direction that will be pursued in
+future work.
+Other current limitations of SHAPES, which are technical in nature, are that the
+penalty gain parameter λ as well as the segment length to be processed must be specified
+
+Adaptive spline glitch removal
+12
+0
+1.5
+3
+4.5
+6
+7.5
+9
+10.5
+Time (sec)
+16
+32
+64
+128
+256
+512
+Frequency (Hz)
+ 1
+ 2
+ 3
+ 4
+ 5
+0
+1.5
+3
+4.5
+6
+7.5
+9
+10.5
+Time (sec)
+16
+32
+64
+128
+256
+512
+Frequency (Hz)
+ 1
+ 2
+ 3
+ 4
+ 5
+Figure 4.
+Subtraction of the Koi Fish glitch.
+The top and bottom panels show
+the CQT of the data and residual, respectively. The glitch is the vertical feature at
+≈ 9.0 sec.
+In order to show both the glitch and the signal in the same image, a
+threshold has been applied to the CQT as indicated by the maximum value in the
+colorbar of the top panel.
+by the user. The choice of the latter, along with the nature of the data, influences the
+number of knots used in the fit and led to the necessity of breaking up the data for the
+GW170817 glitch into three ad hoc parts. Work is in progress to address both of these
+limitations.
+
+Adaptive spline glitch removal
+13
+0
+2
+4
+6
+8
+10
+12
+14
+Time (sec)
+16
+32
+64
+128
+256
+512
+Frequency (Hz)
+ 1
+ 2
+ 3
+ 4
+ 5
+0
+5
+10
+15
+Time (sec)
+16
+32
+64
+128
+256
+512
+Frequency (Hz)
+ 1
+ 2
+ 3
+ 4
+ 5
+Figure 5. Subtraction of the Tomte Glitch. The top and bottom panels show the CQT
+of the data and residual, respectively. The glitch is the vertical feature at ≈ 8.0 sec.
+In order to show both the glitch and the signal in the same image, a threshold has
+been applied to the CQT as indicated by the maximum value in the colorbar of the
+top panel.
+Our results show that SHAPES is a promising addition to the toolbox of glitch
+subtraction methods that will become increasingly important as GW detectors become
+more sensitive. SHAPES is computationally inexpensive, taking on the order of a few
+minutes for each glitch, and will be made much faster by planned code improvements.
+
+Adaptive spline glitch removal
+14
+This could allow, in principle, the subtraction of a large number of broadband glitches
+of known types as part of data conditioning and provide significantly cleaner data for
+any type of GW search.
+Acknowledgments
+S.D.M is supported by U.S. National Science Foundation (NSF) grant PHY-2207935 and
+partially supported by the U.S. Department of Defense grant W911NF2110169. MATC
+acknowledges support from the Presidential Graduate Research Award at the University
+of Texas Rio Grande Valley. We acknowledge the Texas Advanced Computing Center
+(TACC) at the University of Texas at Austin (www.tacc.utexas.edu) for providing high
+performance computing resources.
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+
diff --git a/6tE0T4oBgHgl3EQffABm/content/tmp_files/load_file.txt b/6tE0T4oBgHgl3EQffABm/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf,len=561
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='02398v1 [gr-qc] 6 Jan 2023 Glitch subtraction from gravitational wave data using adaptive spline fitting Soumya D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Mohanty, Mohammad A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Chowdhury Department of Physics and Astronomy, University of Texas Rio Grande Valley, One West University Blvd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=', Brownsville, Texas 78520, USA E-mail: soumya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='mohanty@utrgv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='edu E-mail: mohammadabuthaher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='chowdhury01@utrgv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='edu Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Transient signals of instrumental and environmental origins (“glitches”) in gravitational wave data elevate the false alarm rate of searches for astrophysical signals and reduce their sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Glitches that directly overlap astrophysical signals hinder their detection and worsen parameter estimation errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' As the fraction of data occupied by detectable astrophysical signals will be higher in next generation detectors, such problematic overlaps could become more frequent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' These adverse effects of glitches can be mitigated by estimating and subtracting them out from the data, but their unpredictable waveforms and large morphological diversity pose a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Subtraction of glitches using data from auxiliary sensors as predictors works but not for the majority of cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Thus, there is a need for nonparametric glitch mitigation methods that do not require auxiliary data, work for a large variety of glitches, and have minimal effect on astrophysical signals in the case of overlaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' In order to cope with the high rate of glitches, it is also desirable that such methods be computationally fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' We show that adaptive spline fitting, in which the placement of free knots is optimized to estimate both smooth and non-smooth curves in noisy data, offers a promising approach to satisfying these requirements for broadband short-duration glitches, the type that appear quite frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The method is demonstrated on glitches drawn from three distinct classes in the Gravity Spy database as well as on the glitch that overlapped the double neutron star signal GW170817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The impact of glitch subtraction on the GW170817 signal, or those like it injected into the data, is seen to be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Introduction In a fairly short time since the first direct detection of a gravitational wave (GW) signal (GW150914) in 2015 [1] by the twin LIGO [2] detectors, GW astronomy has emerged as an information-rich field that will revolutionize our understanding of compact objects such as black holes and neutron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' By now, the network of LIGO and Virgo [3] detectors has reported 90 confirmed detections of GW signals from compact binary coalescences (CBCs) across the first observing run (O1) [4] to the third (O3) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The majority of these are binary black hole (BBH) mergers but the haul also includes a double neutron star (DNS) system (GW170817) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Adaptive spline glitch removal 2 The rate of detectable GW signals will grow as more detectors, namely KAGRA [7] and LIGO-India [8], join the network and increase its distance reach for GW sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Design studies are already underway for the successors to the current generation of GW detectors [9, 10, 11] with the goal of achieving an order of magnitude improvement in sensitivity across the current operational frequency band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' In addition, next-generation detectors will seek to expand the operational range to lower frequencies (≈ 1 Hz), thereby increasing the duration of in-band GW signals across the board: for example, a DNS signal starting at ≈ 10 Hz will last for days compared to the ≈ 1 min for GW170817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Thus, future detectors will not only see a higher rate but also longer signals, raising the prospect [12] that there will be no data segment free of detectable GW signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The false alarm rate, hence the sensitivity, of searches for CBCs as well as generic short duration GW signals, or bursts, is dominated [13] by transient non-GW signals of instrumental or environmental origins, commonly called glitches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' This is because glitches that populate the same frequency band as CBC or burst signals and happen to be transient in duration can falsely trigger the respective search pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' A glitch has a particularly adverse effect if it overlaps with a GW signal, as happened in the case of GW170817 [6], and causes the glitch rejection step of a search pipeline to also discard the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Even a non-overlapping glitch can severely degrade parameter estimation if it is close enough to a GW signal [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' In the third observing run of the LIGO and Virgo detectors, ≈ 20% of detected GW signals overlapped with glitches [15] due to the high rate of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' For future detectors, the frequency of chance overlaps will be enhanced by the higher rate of detectable GW signals as well as, for CBC signals, their longer durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Glitches have dissimilar and unpredictable waveforms but many of the observed ones tend to fall into distinct morphological classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' This has motivated the investigation of automated glitch classification using machine learning where a range of different methods have been proposed, such as Support Vector Machine [16], t-Sne [17], random forests [16], S-means [18], and Deep Convolutional Neural Networks [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The Gravity Spy [20] project uses a citizen science approach to engage the lay public in labeling glitches by visual inspection of their constant Q-transform [21, 22] images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' This has created a high quality training dataset for machine learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' By now, there exist more than 20 named glitch classes in the Gravity Spy database, collected over multiple observing runs of the LIGO detectors [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Several different approaches have been developed to mitigate the adverse effects of glitches on GW searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' GW search pipelines typically compute secondary functionals, called vetoes, of the data besides the primary detection statistic that help in distinguishing genuine GW signals from glitches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' A well-known example is the Chi- square [23] veto used in CBC search pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' For LIGO-Virgo data, a set of Data Quality flags have been developed that use information from a large number of auxiliary sensors to quantify the safety of analyzing a given segment of GW strain data [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' For glitches that overlap a GW signal, the gating [25] method excises the rectangular time- frequency block, or just the time interval, containing an identified glitch from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Adaptive spline glitch removal 3 Cross-channel regression using data from auxiliary sensors [26, 27, 28, 29] has been used to reduce excess broadband noise and a few types of glitches [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' A relatively recent approach is that of estimating the waveform of a glitch from the data time series itself and subtracting it out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Glitch subtraction was of critical importance in the case of GW170817 and has been shown to be an important requirement in reducing bias in the estimation of GW signal parameters [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The GW170817 glitch subtraction was carried out using the multi-detector BayesWave pipeline [32, 33], which has also been used for other types of glitches [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Another method, Glitschen [34], follows the approach of constructing parametrized waveform models for identified glitch classes using principal component analysis of training sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' A strong motivation for developing glitch estimation and subtraction methods is that one could, in principle, preprocess the data to clean out every sufficiently loud glitch of a known type and make glitch rejection in all downstream GW searches safer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' In this paper, we present a method for the estimation and subtraction of broadband, short-duration glitches that have appeared frequently in the observation runs of the LIGO detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The method is computationally cheap, works with single-detector data, does not require a training set of pre-identified glitches, and is not predicated on auxiliary sensor data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The core component of the method is SHAPES (Swarm Heuristics based Adaptive and Penalized Estimation of Splines), an adaptive spline curve fitting algorithm introduced in [35]‡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' SHAPES uses splines with free placement of knots to fit both smooth and non-smooth curves in noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' In particular, point discontinuities in the curve or its derivatives (up to some order) can be accommodated in the fit by allowing knots to merge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The ability to handle both sharp and slow changes in a curve is a built-in form of multiresolution analysis in SHAPES and a critical requirement for effective estimation of broadband glitches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' We examine the performance of our glitch subtraction method on the GW170817 glitch in LIGO-Livingston data and instances of glitches from three morphologically distinct classes, namely, Blip, Koi Fish, and Tomte, in the Gravity Spy database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' In each of the latter three cases, we inject a DNS signal overlapping with the glitch to mimic the case of GW170817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' We find that the impact of glitch subtraction on the signals, real or injected, is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' 2 reviews SHAPES with the goal of providing a self-contained description of the algorithm that is pertinent to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Further details, such as the motivation and justification for certain features of the algorithm, can be found in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' 3 describes the dataset used in this paper and the details of how SHAPES is used for glitch subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' 4 presents the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Our conclusions and discussion of future work are presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' ‡ The SHAPES code is available from the Github repository mohanty-sd/SHAPES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='git.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Adaptive spline glitch removal 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Adaptive spline fitting: the SHAPES algorithm SHAPES is derived under the following models for the noisy data, y, and the signal s(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' y = s(θ) + ǫ , (1) where y, s, and ǫ are row vectors with N elements, yi = y(ti) and si(θ) = s(ti;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' θ), i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' , N −1, are samples taken at ti = i/fs with fs being the sampling frequency, and θ denotes the set of signal parameters that need to be estimated from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The noise samples, ǫi, are drawn independently from the zero mean and unit variance normal (Gaussian) probability density function N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' This assumption, namely, that of a white Gaussian noise process does not entail a loss of generalization since GW data can always be whitened using the estimated noise power spectral density (PSD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The signal s(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' θ) is assumed to be a spline of polynomial order k and, as such, can be represented by a linear combination of B-spline functions [36], s (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' θ = {α, τ}) = P −k−1 � j=0 αjBj,k(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' τ) , (2) where α = (α0, α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' , αP −k−1), and τ = (τ0, τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' , τP −1), τi+1 ≥ τi, is a sequence of P knots that marks the end points of the contiguous intervals containing the polynomial pieces of the spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Note that knots are allowed to be equal, leading to knots with multiplicity higher than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Repeating knots create discontinuity in either the value of a B-spline function or its derivatives (up to order k − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' This allows the s(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' θ) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' 2 to model signals with point discontinuities in value or derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' In the rest of the paper, we will set k = 4, making s(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' θ) a cubic spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The best fit spline parameters, �α and �τ, are the ones that minimize a penalized least-squares function, Lλ(α, τ) = L(α, τ) + λR(α) , (3) L(α, τ) = N−1 � i=0 (yi − si(α, τ))2 , (4) where the penalty term, R(α) = P −k−1 � j=0 α2 j , (5) is found to be useful in the suppression of spurious clustering of the knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' These clusters are observed when the method tries to minimize Lλ(α, τ) by fitting out outlier data points arising from the noise alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The strength of the penalty is controlled by the gain factor λ, with higher values of λ leading to smoother estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The optimization of Lλ(α, τ) over the non-linear parameters τ has been a long- standing computational barrier [37, 38, 39, 40] for adaptive spline fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' At the same time, the benefits of optimizing the placement of knots have also been demonstrated extensively [38, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' It was shown in [42], and independently in [43], that Particle Swarm Optimization (PSO) [44, 45], a widely used nature-inspired metaheuristic for global Adaptive spline glitch removal 5 optimization, has good performance on the free knot placement problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Moreover, being a continuous optimization method, PSO can explore all arrangements of knots, including the ones where knots are sufficiently close to be merged into a single knot of higher multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' This allows the fitting of functions that have a mix of smooth and non-smooth parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' There are many variations [46] among the algorithms that fall under the umbrella of the PSO metaheuristic but they all share the following common features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' (i) The function to be optimized, called the fitness function, is sampled at multiple locations, called particles, that move iteratively to explore the domain, called the search space, over which the the global optimum of the fitness is to be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The set of particles is called a swarm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' (ii) The location of each particle is updated following a dynamical rule that incorporates randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The rule typically uses the best location found by a particle in its history, called its personal best, and the best location found by the particles in its neighborhood, called its local best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Here, the fitness value at a location defines how good it is: for a minimization problem, the lower the fitness, the better the location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' (iii) Each particle explores the search space independently but is constantly attracted towards the personal and local bests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' This leads to a form of communication between the particles that speeds up convergence to a promising region, followed by refinement of the solution until the iterations are terminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The best location among all the particles at termination is the final solution found by the swarm for the global optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' While there is no guarantee that the final solution is the true global optimum, the probability of successful convergence can be boosted exponentially by running multiple independent runs of PSO and picking the one with the best final solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Most of the parameters involved in the PSO algorithm, such as the number of particles or the weights attached to the attractive forces, have very robust values across a wide variety of benchmark optimization problems [47] and rarely need to be changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' In our experience, there are typically only two quantities that need tuning: the number of iterations, Niter, to termination and the hyper-parameter Nruns, the number of independent PSO runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' In this paper, we fix Niter = 2000 and Nruns = 8 throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The number of particles is always set to 40 and the settings for the remaining parameters, as well as the definition of the neighborhood used for the local best, are described in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The description above was for the case where the number of knots, P, is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The complete SHAPES algorithm incorporates model selection using the Akaike Information Criterion (AIC) [48], where the optimum number of knots minimizes, AIC = 4P + Lλ(�α, �τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' (6) While, given sufficient computing resources, model selection could be performed over all values of P until the minimum value of AIC is found, practical considerations dictate that the set of knot numbers used be a finite and small one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' In this paper, for example, we use knot numbers in the set starting at 5 and ending at 60 in increments of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' It is important to note that this restriction of knot numbers is not a fundamental limitation Adaptive spline glitch removal 6 but a technical one meant to manage the computational burden of model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Thus, the only significant free parameter that needs to be set by the user in the current version of SHAPES is λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Since SHAPES assumes that the noise in the data is white, GW strain data must be whitened prior to glitch estimation and subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The data conditioning steps involved are as follows (in sequential order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' (a) Suppression of the seismic noise below 10 Hz, (b) robust estimation of the power spectral density (PSD) noise floor, (c) whitening of the noise floor using the estimated PSD [49], and (d) automated identification of high-power narrowband noise features (“lines”) and their suppression using notch filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' These steps are common to all GW search pipelines, so they do not need to be elaborated further here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Demonstration data The glitches considered in this paper for demonstrating the performance of SHAPES are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The corresponding GW strain data files can be located and downloaded from the Gravitational Wave Open Science Center (GWOSC) [50] using the information provided in this table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' We have used the standard 4096 sec long GWOSC data files sampled at 4 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The GW170817 glitch presents a particularly interesting example of the deleterious effect of glitches on GW searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The GW signal appeared in both LIGO-Hanford (H1) and LIGO-Livingston (L1) with a combined network signal to noise ratio (SNR) of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Such a strong signal would have been detected easily in coincidence across L1 and H1 by the GW search pipelines in operation at the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' However, a coincident detection was prevented by a large overlapping glitch in L1 causing the release of only an unusual single-detector GW detection alert to the astronomical community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' About 11 hours elapsed between the initial alert and the release of the skymap localizing GW170817, a process that included the subtraction of the glitch using BayesWave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' In addition to the GW170817 glitch, we have taken three representative glitches from the Blip, Koi Fish, and Tomte, classes in the Gravity Spy [20] database [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' These glitches were selected by taking the loudest 5 events, in terms of their signal-to- noise ratio (SNR) as given in the Gravity Spy database, for each class and then picking the first one in this list for which the corresponding GWOSC file had 100% science data that was also reasonably stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' As can be seen from Table 1, this results in the selected glitches spanning a wide range in SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' After conditioning the data, we use the start time of a glitch, recorded in Table 1, to locate the glitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Starting from the peak of the glitch, the data time series is scanned visually in both directions to identify a segment, containing the glitch, that tapers off at both its boundaries to the general noise level of the conditioned data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' To mimic the case of GW170817 and to study the effect of glitch subtraction on an overlapping GW signal, we injected a whitened restricted-2PN circularized binary inspiral signal with equal 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='4 M⊙ components in the conditioned data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The SNR (in Adaptive spline glitch removal 7 Glitch Name GPS start (sec) SNR Detector run GW170817 glitch 1187008880 – L1 O2 Blip 1182397347 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='1 H1 O2 Koi Fish 1169847108 608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='1 H1 O2 Tomte 1173086299 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='6 H1 O2 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Glitches considered in this paper along with their GPS start times, SNRs, the detectors in which they appeared, and the observation runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' For the Blip, Koi Fish, and Tomte glitches, the start times are taken from the Gravity Spy database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' To the best of our knowledge, there is no SNR available in the literature for the GW170817 glitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' white noise with unit variance) of the injected signal is set at 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='3, which is an ad hoc factor of √ 2 higher than the observed SNR of 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='4 of GW170817 in L1 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The enhancement in SNR allows clearer visibility of the signal in time-frequency images while also posing a stronger challenge to SHAPES in terms how well it ignores the GW signal when estimating a glitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The segment containing the glitch, taken from the conditioned data with the injected signal, is passed to SHAPES for estimation of the glitch waveform followed by its subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Results In common with other papers on glitch estimation and subtraction, we present our results in the form of constant Q-transform (CQT) time-frequency images and time series plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' These are obtained by taking projections of the data on a set of windowed sinusoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The width of the window decreases with an increase in the carrier frequency, fc, such that Q = fc/∆f, where ∆f is the −3 dB bandwidth of the Fourier transform of the window, remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' We use the CQT code provided in the librosa [52] Python package for audio processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' For each glitch, we show CQTs of the conditioned data with injected signal and the residual after subtraction of the glitch estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' 1 shows the data segments that were processed using SHAPES and the corresponding estimated glitch waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Except for GW170817, each segment was processed as a whole to obtain the glitch estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' In the case of GW170817, SHAPES was applied independently to three separate but contiguous time intervals to estimate the complete glitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' This was necessitated by the presence of extended wings, preceding and trailing the core broadband (and rapidly varying) part in the middle, that dominate the conditioned data for ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='5 sec on each side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Applying SHAPES to the complete segment would have required using a very large number of knots (> 60), making it unnecessarily expensive computationally given that splitting the segment achieves a good solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' As mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' 2, the penalty gain λ controls the smoothness of the estimate and is a user-specified parameter of the SHAPES algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Typically, when a glitch is loud and has a complex shape, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='01 allows SHAPES to provide a better fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' For Adaptive spline glitch removal 8 932.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='67 932.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='68 932.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='69 932.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='7 932.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='71 60 40 20 0 20 40 60 80 Data Estimate 839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='1 839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='12 839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='14 839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='16 Time (sec) 100 50 0 50 100 150 Whitened Strain Data Estimate 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='04 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='06 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='08 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='1 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='12 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='14 Time (sec) 4 2 0 2 4 Data Estimate 370 370.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='5 371 100 50 0 50 100 Whitened Strain Data Estimate Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The conditioned strain data and the glitch waveform estimated by SHAPES for each of the glitches considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Top row: GW170817 (left) and Blip (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Bottom row: Koi Fish (left) and Tomte (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The X-axis in each plot shows the time (sec) since the start of the open data file containing the glitch as provided by GWOSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' For GW170817, the dashed vertical lines demarcate the three adjacent segments that were analyzed separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' low SNR and simple glitch waveforms, or if the data is just plain white noise, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='1 does an adequate job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' In general, estimates from SHAPES are not sensitive to small variations of λ around these values because the model selection is able to compensate for a lower value of λ by selecting a higher knot number and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Without much fine tuning, we found that the values of λ listed in Table 2 work well for the glitches studied in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' We have also listed in this table the number of knots for the best fit models selected by the AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' 2 to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' 5 show the CQTs of the conditioned data and residuals after glitch subtraction for the glitches in the sequence GW170817, Blip, Koi Fish, and Tomte, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' In all cases, we see that the subtraction of the glitch does not affect the overlapping GW signal (real or injected) in any significant way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Some overfitting to the data, seen as very small CQT values, is visible in the residual for the GW170817 glitch at frequencies below ≈ 32 Hz but this band has no overlap with the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The overfitted parts are the two wings of the GW170817 glitch mentioned earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The CQTs of the residuals for the Blip and Tomte glitches show near perfect removal of the glitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' (For Adaptive spline glitch removal 9 Glitch Name Penalty gain (λ) Number of knots GW170817 glitch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='1 60,40,50 Blip 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='01 15 Koi Fish 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='01 30 Tomte 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='1 15 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The penalty gain λ used for the glitches and the number of knots in the best fit model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' For the GW170817 glitch, there are three segments with the middle one containing the principal glitch and adjacent ones containing the wings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The penalty gains and best fit model are listed for all three segments in sequential order from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Tomte, the coalescence time of the GW signal was kept further away from the glitch in order to create an overlap between the signal track and the glitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=') The residual for Koi Fish shows effective removal of the glitch with the exception of a transient and low frequency narrowband component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' This leftover component does not overlap with the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The principal computational cost in SHAPES is the global optimization of the fitness function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The time taken by the current MATLAB [53] code for a single PSO run on a segment with ≈ 300 samples and knot numbers P ∈ [10, 60] (in steps of 5) is < 10 min on an Intel Xeon E5 processor (clock rate 3 GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The runtime increases with the number of knots used, mainly due to an increase in the number of B-spline functions that need to be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' With a code currently under construction in the C language, and implementation of further hardware acceleration (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=', using Graphics Processing Units), the runtime is expected to decrease substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' We also note that the segments containing glitches can be processed in parallel since SHAPES is a purely time-domain method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Hence, the computational cost will scale slower than linearly with the number of glitches when analyzing data containing multiple glitches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Discussion and Conclusions We have presented a new approach to glitch subtraction using an adaptive spline fitting method called SHAPES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The method was demonstrated on the GW170817 glitch as well as other representative short duration and broadband glitches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' In a single detector and in the absence of strong prior information about the signal, it is not possible to distinguish a GW signal from a glitch in the part where they overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Hence, it is expected that the signal power will be removed in that part along with the glitch when the latter is estimated and subtracted out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Nonetheless, as far as the DNS signal used in this paper is concerned, we observe very little impact on the signal across a wide range of glitch SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' While this conclusion will be quantified in future studies using a much larger number of glitches, it is clear that SHAPES is effective at addressing glitch subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' SHAPES is not well adapted to fitting highly oscillatory waveforms since these Adaptive spline glitch removal 10 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='5 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='5 6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='5 9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='5 12 Time (sec) 16 32 64 128 256 512 Frequency (Hz) 1 2 3 4 5 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='5 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='5 6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='5 9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='5 12 Time (sec) 16 32 64 128 256 512 Frequency (Hz) 1 2 3 4 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Subtraction of the GW170817 Glitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The top and bottom panels show the CQT of the data and residual, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The glitch is the vertical feature at ≈ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='5 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' In order to show both the glitch and the signal in the same image, a threshold has been applied to the CQT as indicated by the maximum value in the colorbar of the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' are are not represented well by splines without using an inordinate number of knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Therefore, the direct use of SHAPES for glitches in the Gravity Spy database such as whistlers or wandering lines is not viable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' However, chirp signals such as these could be estimated using the method proposed in [54, 55], where adaptive splines figure indirectly Adaptive spline glitch removal 11 0 1 2 3 4 5 6 7 8 Time (sec) 16 32 64 128 256 512 Frequency (Hz) 1 2 3 4 5 0 1 2 3 4 5 6 7 8 Time (sec) 16 32 64 128 256 512 Frequency (Hz) 1 2 3 4 5 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Subtraction of the Blip Glitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The top and bottom panels show the CQT of the data and residual, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The glitch is the vertical feature at ≈ 6 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' In order to show both the glitch and the signal in the same image, a threshold has been applied to the CQT as indicated by the maximum value in the colorbar of the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' in a non-linear signal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' This is an interesting direction that will be pursued in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Other current limitations of SHAPES, which are technical in nature, are that the penalty gain parameter λ as well as the segment length to be processed must be specified Adaptive spline glitch removal 12 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='5 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='5 6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='5 9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='5 Time (sec) 16 32 64 128 256 512 Frequency (Hz) 1 2 3 4 5 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='5 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='5 6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='5 9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='5 Time (sec) 16 32 64 128 256 512 Frequency (Hz) 1 2 3 4 5 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Subtraction of the Koi Fish glitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The top and bottom panels show the CQT of the data and residual, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The glitch is the vertical feature at ≈ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='0 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' In order to show both the glitch and the signal in the same image, a threshold has been applied to the CQT as indicated by the maximum value in the colorbar of the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The choice of the latter, along with the nature of the data, influences the number of knots used in the fit and led to the necessity of breaking up the data for the GW170817 glitch into three ad hoc parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Work is in progress to address both of these limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Adaptive spline glitch removal 13 0 2 4 6 8 10 12 14 Time (sec) 16 32 64 128 256 512 Frequency (Hz) 1 2 3 4 5 0 5 10 15 Time (sec) 16 32 64 128 256 512 Frequency (Hz) 1 2 3 4 5 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Subtraction of the Tomte Glitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The top and bottom panels show the CQT of the data and residual, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' The glitch is the vertical feature at ≈ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='0 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' In order to show both the glitch and the signal in the same image, a threshold has been applied to the CQT as indicated by the maximum value in the colorbar of the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Our results show that SHAPES is a promising addition to the toolbox of glitch subtraction methods that will become increasingly important as GW detectors become more sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' SHAPES is computationally inexpensive, taking on the order of a few minutes for each glitch, and will be made much faster by planned code improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Adaptive spline glitch removal 14 This could allow, in principle, the subtraction of a large number of broadband glitches of known types as part of data conditioning and provide significantly cleaner data for any type of GW search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Acknowledgments S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='M is supported by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' National Science Foundation (NSF) grant PHY-2207935 and partially supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Department of Defense grant W911NF2110169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' MATC acknowledges support from the Presidential Graduate Research Award at the University of Texas Rio Grande Valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' We acknowledge the Texas Advanced Computing Center (TACC) at the University of Texas at Austin (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='tacc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='edu) for providing high performance computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
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+page_content='mathworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content='com/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Adaptive spline glitch removal 17 [54] Soumya D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Mohanty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Spline based search method for unmodeled transient gravitational wave chirps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Physical Review D, 96:102008, Nov 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' [55] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Mohanty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' Detection and estimation of unmodeled chirps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' In 2018 26th European Signal Processing Conference (EUSIPCO), pages 2643–2647, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQffABm/content/2301.02398v1.pdf'}
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+arXiv:2301.04645v1 [math.CA] 11 Jan 2023
+VERTICAL PROJECTIONS IN THE HEISENBERG GROUP FOR
+SETS OF DIMENSION BETWEEN 2 AND 3
+TERENCE L. J. HARRIS
+Abstract. It is shown that vertical projections in the Heisenberg group al-
+most surely do not decrease Hausdorff dimension for Borel sets of dimension
+between 2 and 3.
+The proof uses the method of point-plate incidences in-
+troduced by F¨assler and Orponen, and uses a similar approach to a recent
+theorem of Zahl.
+1. Introduction
+Let H be the Heisenberg group, identified as a set with C × R, and equipped
+with the group law
+(z, t) ∗ (ζ, τ) =
+�
+z + ζ, t + τ + 1
+2ω(z, ζ)
+�
+,
+where
+ω(x + iy, u + iv) := xv − uy.
+For each θ ∈ [0, π), let
+Vθ =
+��
+λeiθ, 0
+�
+: λ ∈ R
+�
+,
+and let V⊥
+θ be the Euclidean orthogonal complement of Vθ. Each (z, t) ∈ H can be
+uniquely decomposed as a product
+(z, t) = PV⊥
+θ (z, t) ∗ PVθ(z, t),
+of an element of V⊥
+θ on the left, with an element of Vθ on the right. This defines
+the vertical projection maps PV⊥
+θ . Let dH be the Kor´anyi metric on H, given by
+dH((z, t), (ζ, τ)) =
+��(ζ, τ)−1 ∗ (z, t)
+��
+H ,
+where
+∥(z, t)∥H =
+�
+|z|4 + 16t2�1/4 .
+The Kor´anyi metric is bi-Lipschitz equivalent to the more natural Carnot-Carath´eodory
+metric on H, and thus induces the same Hausdorff dimension. Let dim refer to the
+Hausdorff dimension of a set in H with respect to the Kor´anyi metric. This work
+gives a proof of the following theorem.
+Theorem 1. Let A be an analytic subset of H. Then
+dim PV⊥
+θ (A) ≥ min{dim A, 3}
+for a.e. θ ∈ [0, π).
+2020 Mathematics Subject Classification. 28A78; 28A80.
+Key words and phrases. Heisenberg group, Hausdorff dimension, vertical projections.
+1
+
+2
+TERENCE L. J. HARRIS
+This was first conjectured by Balogh, Durand-Caragena, F¨assler, Mattila and
+Tyson [1, Conjecture 1.5], who proved the conjecture in the range dim A ≤ 1.
+Recently, this conjecture was proved for dim A ∈ [0, 2]∪{3} by F¨assler and Orponen
+(and thus also for dim A > 3, though Conjecture 1.5 in [1] also predicts positive area
+in this range). They introduced a method of point-plate incidences, and proved
+(1) in the case dim A = 3 by using a square function estimate for the cone of
+Guth, Wang and Zhang [4] to control the average L2 norm of pushforwards of 3-
+dimensional measures. The point-line duality principle they used is due to Liu [5].
+Theorem 1 resolves the conjecture in the remaining range dim A ∈ (2, 3). The proof
+of Theorem 1 uses the incidence approach of F¨assler and Orponen, but rather than
+using the square function estimate for the cone, it uses a broad-narrow approach
+to Kakeya-type inequalities for tubes arranged in fractal families of planks. This is
+based on recent work of Zahl [6], which used a broad-narrow approach to Kakeya-
+type inequalities for fractal families of tubes.
+Acknowledgements
+I thank Shaoming Guo for some discussion in the earlier stages of working on
+this problem, when I visited UW Madison in October 2022. I also thank UW for
+their hospitality.
+2. Preliminaries
+For each θ ∈ [0, π] let H2 be the 2-dimensional Lebesgue measure on V⊥
+θ . A line
+ℓ in H is called horizontal if it is a left translate of a horizontal subgroup Vθ for
+some θ ∈ [0, π); meaning that there exists p ∈ H such that ℓ = p ∗ Vθ. For each
+horizontal line ℓ ⊆ H, let H1 be the Lebesgue measure on ℓ with respect to the
+Euclidean metric. Given a non-negative Borel function f and a horizontal line ℓ,
+define
+Xf(ℓ) =
+�
+ℓ
+f dH1.
+Given a measure µ on H, let
+cα(µ) =
+sup
+x∈H,r>0
+µ (BH(x, r))
+rα
+.
+More generally, given δ > 0, define
+cα,δ(µ) =
+sup
+x∈H,r>δ
+µ (BH(x, r))
+rα
+.
+Definition 1. Define ℓ∗ : H → P(R3) by
+ℓ∗(x, y, t) = (0, x, t − xy/2) + Ly,
+where
+Ly =
+�
+λ(1, −y, y2/2) : λ ∈ R
+�
+.
+Define ℓ : R3 → P(H) by
+ℓ(a, b, c) = {(as + b, s, (bs)/2 + c) : s ∈ R} .
+The following lemma is the point-line duality principle.
+Lemma 1 ([3, Lemma 4.11]). Let p∗ ∈ H and let p ∈ R3. Then
+p ∈ ℓ∗(p∗)
+if and only if
+p∗ ∈ ℓ(p).
+
+VERTICAL PROJECTIONS IN THE HEISENBERG GROUP
+3
+Lemma 2. Given a non-negative continuous function f supported in the unit ball
+of H, let µf be the measure whose Radon-Nikodym derivative with respect to the
+Lebesgue measure on H is equal to f. Then for any q ∈ (1, ∞) and ε > 0,
+� π−ε
+ε
+���PV⊥
+θ #µf
+���
+q
+Lq(H2) dθ ∼q,ε
+�
+Uǫ
+|Xf(ℓ(p))|q dH3(p),
+where Uǫ is the set of p ∈ R3 such that ℓ(p) is a horizontal line with corresponding
+angle in [ε, π − ε].
+Proof. This is outlined in [3, Eq. 4.15] in the case q = 2. One version of the proof
+uses the coarea formula, and the formula for L2 norms in terms of the distribution
+function, which naturally extends to the case q ∈ (1, ∞).
+□
+Lemma 3. Let f be a non-negative continuous function supported in the unit ball
+of H, let µf be the measure whose Radon-Nikodym derivative with respect to the
+Lebesgue measure on H is equal to f. Then for any q ∈ (1, ∞) and any p ∈ BH(0, 1),
+� π
+0
+���PV⊥
+θ # (Lp#µf)
+���
+q
+Lq(H2) dθ
+� π
+0
+���PV⊥
+θ #µf
+���
+q
+Lq(H2) dθ ∼q,
+where Lp(z, t) = p ∗ (z, t).
+Proof. The Radon-Nikodym derivative of Lp#µf is f ◦ L−1
+p , since left translation
+has Jacobian equal to 1. Hence
+� π
+0
+���PV⊥
+θ # (Lp#µf)
+���
+q
+Lq(H2) dθ ∼
+� ����
+�
+ℓ
+(f ◦ L−1
+p ) dH1
+����
+q
+dh(ℓ),
+where h is the natural left-invariant measure on the set of horizontal lines; see [3,
+Eq. 4.14]. If ℓ is such that the integrand is nonzero, then ℓ = (z, t) ∗ Vθ for some
+θ ∈ [0, π) and (z, t) ∈ V⊥
+θ with dH((z, t), 0) ≲ 1 (since f is supported in the unit
+ball), which implies that the Euclidean measure H1 is equivalent to the Heisenberg
+Hausdorff measure H1
+H on ℓ. Similarly, these two measures are equivalent on p ∗ ℓ
+(which is also a horizontal line). Since H1
+H is left-invariant, and h is left-invariant,
+it follows that
+� ����
+�
+ℓ
+(f ◦ L−1
+p ) dH1
+����
+q
+dh(ℓ) ∼
+� ����
+�
+ℓ
+f dH1
+����
+q
+dh(ℓ) ∼
+� π
+0
+���PV⊥
+θ # (µf)
+���
+q
+Lq(H2) dθ.
+□
+3. Main results
+For the statement of the following theorem, let dθ refer to the Lebesgue measure
+on [0, π). Let
+�
+∗ refer to the lower integral (this is only used to avoid measurability
+issues in the statement).
+Theorem 2. Let t ∈ [0, 3], and let µ be a Borel measure supported in the unit
+Heisenberg ball, with ct(µ) ≤ 1. Let ǫ > 0. Let δ > 0, and suppose that for each θ,
+Dθ is a disjoint collection of at most δ
+√ǫ−tµ(H) Heisenberg δ-balls in V⊥
+θ . Then
+�
+∗
+�
+PV⊥
+θ #µ
+� � �
+D∈Dθ
+D
+�
+dθ ≲ǫ δǫµ(H).
+Before proving Theorem 2, it will be shown that it implies Theorem 1.
+
+4
+TERENCE L. J. HARRIS
+Proof of Theorem 1. Measurability issues will be ignored since they can be easily
+adjusted for. By scaling it may be assumed that A is contained in the unit ball.
+Let ǫ > 0, and (using Frostman’s lemma) let µ be a Borel probability measure on
+A with cα(µ) < ∞, where α = dim A − ǫ. Let E ⊆ [0, π) be a compact set such
+that dim PV⊥
+θ (supp µ) < α − ǫ for all θ ∈ E. Let ε > 0, and for each θ ∈ E,
+let Dθ = {BH(pj(θ), rj(θ))}j be a finitely overlapping cover of PV⊥
+θ (supp µ) by
+Heisenberg balls of dyadic radii at most ε, such that
+(1)
+�
+j
+rj(θ)α−ǫ < cα(µ)−1.
+For each k, let Dθ,k be the subcollection of balls in Dθ with dyadic radii equal to
+2−k. Then for each θ ∈ E,
+1 ≤
+�
+k
+�
+PV⊥
+θ #µ
+�
+
+
+�
+D∈Dθ,k
+D
+
+ .
+Integrating over E gives
+H1(E) ≤
+�
+k
+�
+E
+�
+PV⊥
+θ #µ
+�
+
+
+�
+D∈Dθ,k
+D
+
+ dθ.
+By (1), each set Dθ,k satisfies
+|Dθ,k| ≲ 2k(α−ǫ)cα(µ)−1.
+By applying Theorem 2 and summing the geometric series, this yields
+H1(E) ≤
+�
+k
+�
+E
+�
+PV⊥
+θ #µ
+�
+
+
+�
+D∈Dθ,k
+D
+
+ dθ ≲ cα(µ)
+�
+k≥| log2 ε|
+2−kǫ2 ≲ cα(µ)εǫ2,
+Letting ε → 0 gives H1(E) = 0. By inner regularity of the Lebesgue measure on
+[0, π), it follows that
+dim PV⊥
+θ (A) ≥ dim PV⊥
+θ (supp µ) ≥ α − ǫ ≥ dim A − 2ǫ.
+for a.e. θ ∈ [0, π). Since the outer parts of this inequality hold for any ǫ > 0, this
+implies the theorem.
+□
+It remains to prove Theorem 2.
+Proof of Theorem 2. Let φ be a fixed non-negative bump function supported in the
+unit Euclidean ball of H around the origin, such that
+�
+φ = 1 and such that φ ≳ 1
+on BE(0, 1/10). For each λ > 0 let φλ = λ−3φ(x/λ). Define ν by the Euclidean
+convolution ν = µ ∗ φδ2, and let
+q = 3 + t
+1 + t ∈ [3/2, 3].
+By H¨older’s inequality with respect to the Lebesgue measure on each plane V⊥
+θ ,
+and the fact that the vertical projections are uniformly 1
+2-H¨older continuous when
+considered as maps from (H, dE) to (H, dH),
+�
+∗
+�
+PV⊥
+θ #µ
+� � �
+D∈Dθ
+D
+�
+dθ ≲ µ(H)1− 1
+q δ(3+√ǫ−t)(1− 1
+q)
+�� π
+0
+���PV⊥
+θ #ν
+���
+q
+Lq(H2)
+�1/q
+.
+
+VERTICAL PROJECTIONS IN THE HEISENBERG GROUP
+5
+Therefore, it suffices to prove that
+(2)
+� π
+0
+���PV⊥
+θ #ν
+���
+q
+Lq(H2) dθ ≲ǫ ν(H)ct,δ(ν)q−1δ−ǫ−(3−t)(q−1),
+for any δ > 0, whenever ν = µ ∗ φδ2 for some finite Borel measure µ supported in
+the unit Heisenberg ball. This will be shown via induction on δ. Thus, let δ > 0
+be given and assume that (2) holds for all �δ < δ1−ǫ2, for any finite Borel measure
+µ supported in the unit Heisenberg ball.
+It will now be shown that (2) holds for δ. By scaling, it may be assumed that ν
+is a probability measure. By rotational symmetry, it may be assumed that
+� π
+0
+���PV⊥
+θ #ν
+���
+q
+Lq(H2) dθ ≲
+� 3π/4
+π/4
+���PV⊥
+θ #ν
+���
+q
+Lq(H2) dθ.
+By pigeonholing, it may be assumed that
+ν =
+1
+ω3|B|δ6
+�
+B∈B
+χB,
+where B is a disjoint collection of Euclidean δ2-balls in BH(0, c), for a small constant
+c (to be chosen in a moment) and ω3 = 4π
+3 is the volume of the unit Euclidean ball
+in R3. Following the argument in [3, Proof of Theorem 5.2]:
+� 3π/4
+π/4
+���PV⊥
+θ #ν
+���
+q
+Lq(H2) dθ ∼
+�
+L∠
+|Xν(ℓ)|q dm(ℓ)
+∼
+1
+|B|qδ6q
+�
+L∠
+�����
+�
+B∈B
+H1(ℓ ∩ B)
+�����
+q
+dm(ℓ)
+≲
+1
+|B|qδ4q
+�
+{p:ℓ(p)∈L∠}
+|{B ∈ B : B ∩ ℓ(p) ̸= ∅}|q dH3(p)
+(3)
+∼
+1
+|B|q δ4q
+�����
+�
+B∈B
+χℓ∗(B)
+�����
+q
+Lq(B(0,1))
+,
+where the integration restricts to B(0, 1) if c is small enough. Let ρ = δǫ2. Let {τ}
+be a covering of Γ by boxes of dimensions ρ × ρ2 × 1 in the standard way, where
+Γ = {(ξ, |ξ|) ∈ R3 : ξ ∈ B2(0, 1)}.
+Call x ∈ B(0, 1) “narrow” if there is a 2-dimensional subspace V of R3 (depending
+on x), such that at least half of the tubes ℓ∗(B) passing through x have direction
+vectors in a ρ2-neighbourhood of V . If x is narrow, then because of the curvature
+of Γ,
+�
+B∈B
+χℓ∗(B)(x) ≲
+��
+τ
+|{B ∈ B : dir(ℓ∗(B)) ∈ τ and x ∈ ℓ∗(B)}|q
+�1/q
+.
+
+6
+TERENCE L. J. HARRIS
+If x is not narrow then it is called “broad”. If x is a broad point, then
+�
+B∈B
+χℓ∗(B)(x) ≲ ρ−100×
+� �
+B1∈B
+�
+B2∈B
+�
+B3∈B
+χℓ∗(B1)χℓ∗(B2)χℓ∗(B3)
+��uℓ∗(B1) ∧ uℓ∗(B2) ∧ uℓ∗(B3)
+��
+�1/3
+.
+Clearly
+(4)
+�����
+�
+B∈B
+χℓ∗(B)
+�����
+Lq(B(0,1))
+≲
+�����χbroad
+�
+B∈B
+χℓ∗(B)
+�����
+Lq(B(0,1))
++
+�����χnarrow
+�
+B∈B
+χℓ∗(B)
+�����
+Lq(B(0,1))
+.
+If the broad part dominates in (4), then using q ≥ 3/2 and applying the trilinear
+Kakeya theorem (see e.g. [2, Theorem 1]) gives
+�����χbroad
+�
+B∈B
+χℓ∗(B)
+�����
+q
+Lq(R3)
+≲ ρ−100
+�
+
+�
+B1,B2,B3∈B
+χℓ∗(B1)χℓ∗(B2)χℓ∗(B3) |u(ℓ∗(B1)) ∧ u(ℓ∗(B2)) ∧ u(ℓ∗(B3))|
+
+
+q/3
+≲ ρ−100 |B|q− 3
+2
+�
+
+�
+B1,B2,B3∈B
+χℓ∗(B1)χℓ∗(B2)χℓ∗(B3) |u(ℓ∗(B1)) ∧ u(ℓ∗(B2)) ∧ u(ℓ∗(B3))|
+
+
+1/2
+,
+≲ ρ−100 |B|q δ6−ǫ
+= ρ−100 |B|q δ4q−(q−1)(3−t)−ǫ,
+by the definition of q. That ν is a probability measure implies that ct,δ(ν) ≳ 1, so
+this proves (2) when the broad part dominates.
+If the narrow part dominates in (4), then
+�����
+�
+B∈B
+χℓ∗(B)
+�����
+q
+Lq(R3)
+≲
+�
+τ
+������
+�
+B∈B:dir(ℓ∗(B))∈τ
+χℓ∗(B)
+������
+q
+q
+.
+For each cap τ, let Tτ be a finitely overlapping cover of physical space by ∼ ρ×ρ2×1
+planks dual to τ. If δ is smaller than some absolute constant, then each δ2-tube
+ℓ∗(B) with dir(ℓ∗(B)) ∈ τ is contained in at least 1 plank from Tτ, and intersects
+≲ 1 planks from Tτ. We associate each ℓ∗(B) with exactly one plank T ∈ Tτ such
+that dir(ℓ∗(B)) ∈ τ and ℓ∗(B) ⊆ T , and abbreviate this by writing ℓ∗(B) ≤ T .
+Then
+�
+τ
+������
+�
+B∈B:dir(ℓ∗(B))∈τ
+χℓ∗(B)
+������
+q
+q
+≲
+�
+τ
+�
+T ∈Tτ
+������
+�
+B∈B:ℓ∗(B)≤T
+χℓ∗(B)
+������
+q
+q
+.
+
+VERTICAL PROJECTIONS IN THE HEISENBERG GROUP
+7
+The point-line duality argument at Eq. (3) is reversible provided the Euclidean
+balls are enlarged by a factor of 2. This gives
+� 3π/4
+π/4
+���PV⊥
+θ #ν
+���
+q
+Lq(H2) dθ ≲
+�
+τ
+�
+T ∈Tτ
+� 3π/4
+π/4
+���PV⊥
+θ #νT
+���
+q
+Lq(H2) dθ,
+where
+νT =
+1
+|B|6δ6
+�
+B∈B:ℓ∗(B)≤T
+χ2B.
+Each measure νT is essentially the restriction of ν to a Heisenberg ball of radius
+ρ. By left translation (using Lemma 3), followed by a Heisenberg dilation (which
+commutes with vertical projections), and then by applying the induction hypothesis,
+�
+τ
+�
+T ∈Tτ
+� 3π/4
+π/4
+���PV⊥
+θ #νT
+���
+q
+Lq(H2) dθ
+≲
+�
+τ
+�
+T ∈Tτ
+ρ(t−3)(q−1)ct,δ(ν)q−1ν
+
+
+�
+B∈B:ℓ∗(B)≤T
+B
+
+ (δ/ρ)−ǫ−(3−t)(q−1)
+≲ ρǫν(H)ct,δ(ν)q−1δ−ǫ−(3−t)(q−1).
+The power of ρ is positive, so the induction closes provided δ is sufficiently small
+(depending only on ǫ). For the rescaled measures, the Euclidean δ2-balls are sent
+to δ2/ρ × δ2/ρ × δ2/ρ2 ellipsoids, which are essentially unions of (δ/ρ)2-balls. This
+means that the rescaled measures can be convolved with φδ/ρ without significantly
+affecting their properties (at least on Heisenberg balls of radius ≥ δ/ρ).
+□
+References
+[1] Balogh, Z. M., Durand-Cartagena, E, F¨assler, K., Mattila, P., Tyson, J. T.: The effect of
+projections on dimension in the Heisenberg group. Rev. Mat. Iberoam. 29, 381–432 (2013)
+[2] Carbery, A., Valdimarsson, S. I.: The endpoint multilinear Kakeya theorem via the Borsuk-
+Ulam theorem. J. Funct. Anal. 264, 1643–1663 (2013)
+[3] F¨assler, K., Orponen, T.: Vertical projections in the Heisenberg group via cinematic functions
+and point-plate incidences. arXiv:2210.00458v2 (2022)
+[4] Guth. L., Wang. H., Zhang, R.: A sharp square function estimate for the cone in R3. Ann. of
+Math. (2) 192, 551–581 (2020)
+[5] Liu,
+J.:
+On
+the
+dimension
+of
+Kakeya
+sets
+in
+the
+first
+Heisenberg
+group.
+Proc. Amer. Math. Soc. 150, 3445–3455 (2022)
+[6] Zahl, J.: Unions of lines in Rn. To appear in Mathematika. arXiv:2208.02913v1 (2022)
+Department of Mathematics, Cornell University, Ithaca, NY 14853, USA
+Email address: tlh236@cornell.edu
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf,len=184
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='04645v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='CA] 11 Jan 2023 VERTICAL PROJECTIONS IN THE HEISENBERG GROUP FOR SETS OF DIMENSION BETWEEN 2 AND 3 TERENCE L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' HARRIS Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' It is shown that vertical projections in the Heisenberg group al- most surely do not decrease Hausdorff dimension for Borel sets of dimension between 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' The proof uses the method of point-plate incidences in- troduced by F¨assler and Orponen, and uses a similar approach to a recent theorem of Zahl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Introduction Let H be the Heisenberg group, identified as a set with C × R, and equipped with the group law (z, t) ∗ (ζ, τ) = � z + ζ, t + τ + 1 2ω(z, ζ) � , where ω(x + iy, u + iv) := xv − uy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' For each θ ∈ [0, π), let Vθ = �� λeiθ, 0 � : λ ∈ R � , and let V⊥ θ be the Euclidean orthogonal complement of Vθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Each (z, t) ∈ H can be uniquely decomposed as a product (z, t) = PV⊥ θ (z, t) ∗ PVθ(z, t), of an element of V⊥ θ on the left, with an element of Vθ on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' This defines the vertical projection maps PV⊥ θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let dH be the Kor´anyi metric on H, given by dH((z, t), (ζ, τ)) = ��(ζ, τ)−1 ∗ (z, t) �� H , where ∥(z, t)∥H = � |z|4 + 16t2�1/4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' The Kor´anyi metric is bi-Lipschitz equivalent to the more natural Carnot-Carath´eodory metric on H, and thus induces the same Hausdorff dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let dim refer to the Hausdorff dimension of a set in H with respect to the Kor´anyi metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' This work gives a proof of the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let A be an analytic subset of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Then dim PV⊥ θ (A) ≥ min{dim A, 3} for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' θ ∈ [0, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' 28A78;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' 28A80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Heisenberg group, Hausdorff dimension, vertical projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' 1 2 TERENCE L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' HARRIS This was first conjectured by Balogh, Durand-Caragena, F¨assler, Mattila and Tyson [1, Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='5], who proved the conjecture in the range dim A ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Recently, this conjecture was proved for dim A ∈ [0, 2]∪{3} by F¨assler and Orponen (and thus also for dim A > 3, though Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='5 in [1] also predicts positive area in this range).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' They introduced a method of point-plate incidences, and proved (1) in the case dim A = 3 by using a square function estimate for the cone of Guth, Wang and Zhang [4] to control the average L2 norm of pushforwards of 3- dimensional measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' The point-line duality principle they used is due to Liu [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Theorem 1 resolves the conjecture in the remaining range dim A ∈ (2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' The proof of Theorem 1 uses the incidence approach of F¨assler and Orponen, but rather than using the square function estimate for the cone, it uses a broad-narrow approach to Kakeya-type inequalities for tubes arranged in fractal families of planks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' This is based on recent work of Zahl [6], which used a broad-narrow approach to Kakeya- type inequalities for fractal families of tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Acknowledgements I thank Shaoming Guo for some discussion in the earlier stages of working on this problem, when I visited UW Madison in October 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' I also thank UW for their hospitality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Preliminaries For each θ ∈ [0, π] let H2 be the 2-dimensional Lebesgue measure on V⊥ θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' A line ℓ in H is called horizontal if it is a left translate of a horizontal subgroup Vθ for some θ ∈ [0, π);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' meaning that there exists p ∈ H such that ℓ = p ∗ Vθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' For each horizontal line ℓ ⊆ H, let H1 be the Lebesgue measure on ℓ with respect to the Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Given a non-negative Borel function f and a horizontal line ℓ, define Xf(ℓ) = � ℓ f dH1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Given a measure µ on H, let cα(µ) = sup x∈H,r>0 µ (BH(x, r)) rα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' More generally, given δ > 0, define cα,δ(µ) = sup x∈H,r>δ µ (BH(x, r)) rα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Define ℓ∗ : H → P(R3) by ℓ∗(x, y, t) = (0, x, t − xy/2) + Ly, where Ly = � λ(1, −y, y2/2) : λ ∈ R � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Define ℓ : R3 → P(H) by ℓ(a, b, c) = {(as + b, s, (bs)/2 + c) : s ∈ R} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' The following lemma is the point-line duality principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Lemma 1 ([3, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let p∗ ∈ H and let p ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Then p ∈ ℓ∗(p∗) if and only if p∗ ∈ ℓ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' VERTICAL PROJECTIONS IN THE HEISENBERG GROUP 3 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Given a non-negative continuous function f supported in the unit ball of H, let µf be the measure whose Radon-Nikodym derivative with respect to the Lebesgue measure on H is equal to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Then for any q ∈ (1, ∞) and ε > 0, � π−ε ε ���PV⊥ θ #µf ��� q Lq(H2) dθ ∼q,ε � Uǫ |Xf(ℓ(p))|q dH3(p), where Uǫ is the set of p ∈ R3 such that ℓ(p) is a horizontal line with corresponding angle in [ε, π − ε].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' This is outlined in [3, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='15] in the case q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' One version of the proof uses the coarea formula, and the formula for L2 norms in terms of the distribution function, which naturally extends to the case q ∈ (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let f be a non-negative continuous function supported in the unit ball of H, let µf be the measure whose Radon-Nikodym derivative with respect to the Lebesgue measure on H is equal to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Then for any q ∈ (1, ∞) and any p ∈ BH(0, 1), � π 0 ���PV⊥ θ # (Lp#µf) ��� q Lq(H2) dθ � π 0 ���PV⊥ θ #µf ��� q Lq(H2) dθ ∼q, where Lp(z, t) = p ∗ (z, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' The Radon-Nikodym derivative of Lp#µf is f ◦ L−1 p , since left translation has Jacobian equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Hence � π 0 ���PV⊥ θ # (Lp#µf) ��� q Lq(H2) dθ ∼ � ���� � ℓ (f ◦ L−1 p ) dH1 ���� q dh(ℓ), where h is the natural left-invariant measure on the set of horizontal lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' see [3, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' If ℓ is such that the integrand is nonzero, then ℓ = (z, t) ∗ Vθ for some θ ∈ [0, π) and (z, t) ∈ V⊥ θ with dH((z, t), 0) ≲ 1 (since f is supported in the unit ball), which implies that the Euclidean measure H1 is equivalent to the Heisenberg Hausdorff measure H1 H on ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Similarly, these two measures are equivalent on p ∗ ℓ (which is also a horizontal line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Since H1 H is left-invariant, and h is left-invariant, it follows that � ���� � ℓ (f ◦ L−1 p ) dH1 ���� q dh(ℓ) ∼ � ���� � ℓ f dH1 ���� q dh(ℓ) ∼ � π 0 ���PV⊥ θ # (µf) ��� q Lq(H2) dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Main results For the statement of the following theorem, let dθ refer to the Lebesgue measure on [0, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let � ∗ refer to the lower integral (this is only used to avoid measurability issues in the statement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let t ∈ [0, 3], and let µ be a Borel measure supported in the unit Heisenberg ball, with ct(µ) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let δ > 0, and suppose that for each θ, Dθ is a disjoint collection of at most δ √ǫ−tµ(H) Heisenberg δ-balls in V⊥ θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Then � ∗ � PV⊥ θ #µ � � � D∈Dθ D � dθ ≲ǫ δǫµ(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Before proving Theorem 2, it will be shown that it implies Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' 4 TERENCE L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' HARRIS Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Measurability issues will be ignored since they can be easily adjusted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' By scaling it may be assumed that A is contained in the unit ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let ǫ > 0, and (using Frostman’s lemma) let µ be a Borel probability measure on A with cα(µ) < ∞, where α = dim A − ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let E ⊆ [0, π) be a compact set such that dim PV⊥ θ (supp µ) < α − ǫ for all θ ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let ε > 0, and for each θ ∈ E, let Dθ = {BH(pj(θ), rj(θ))}j be a finitely overlapping cover of PV⊥ θ (supp µ) by Heisenberg balls of dyadic radii at most ε, such that (1) � j rj(θ)α−ǫ < cα(µ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' For each k, let Dθ,k be the subcollection of balls in Dθ with dyadic radii equal to 2−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Then for each θ ∈ E, 1 ≤ � k � PV⊥ θ #µ � \uf8eb \uf8ed � D∈Dθ,k D \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Integrating over E gives H1(E) ≤ � k � E � PV⊥ θ #µ � \uf8eb \uf8ed � D∈Dθ,k D \uf8f6 \uf8f8 dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' By (1), each set Dθ,k satisfies |Dθ,k| ≲ 2k(α−ǫ)cα(µ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' By applying Theorem 2 and summing the geometric series, this yields H1(E) ≤ � k � E � PV⊥ θ #µ � \uf8eb \uf8ed � D∈Dθ,k D \uf8f6 \uf8f8 dθ ≲ cα(µ) � k≥| log2 ε| 2−kǫ2 ≲ cα(µ)εǫ2, Letting ε → 0 gives H1(E) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' By inner regularity of the Lebesgue measure on [0, π), it follows that dim PV⊥ θ (A) ≥ dim PV⊥ θ (supp µ) ≥ α − ǫ ≥ dim A − 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' θ ∈ [0, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Since the outer parts of this inequality hold for any ǫ > 0, this implies the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' □ It remains to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let φ be a fixed non-negative bump function supported in the unit Euclidean ball of H around the origin, such that � φ = 1 and such that φ ≳ 1 on BE(0, 1/10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' For each λ > 0 let φλ = λ−3φ(x/λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Define ν by the Euclidean convolution ν = µ ∗ φδ2, and let q = 3 + t 1 + t ∈ [3/2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' By H¨older’s inequality with respect to the Lebesgue measure on each plane V⊥ θ , and the fact that the vertical projections are uniformly 1 2-H¨older continuous when considered as maps from (H, dE) to (H, dH), � ∗ � PV⊥ θ #µ � � � D∈Dθ D � dθ ≲ µ(H)1− 1 q δ(3+√ǫ−t)(1− 1 q) �� π 0 ���PV⊥ θ #ν ��� q Lq(H2) �1/q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' VERTICAL PROJECTIONS IN THE HEISENBERG GROUP 5 Therefore, it suffices to prove that (2) � π 0 ���PV⊥ θ #ν ��� q Lq(H2) dθ ≲ǫ ν(H)ct,δ(ν)q−1δ−ǫ−(3−t)(q−1), for any δ > 0, whenever ν = µ ∗ φδ2 for some finite Borel measure µ supported in the unit Heisenberg ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' This will be shown via induction on δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Thus, let δ > 0 be given and assume that (2) holds for all �δ < δ1−ǫ2, for any finite Borel measure µ supported in the unit Heisenberg ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' It will now be shown that (2) holds for δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' By scaling, it may be assumed that ν is a probability measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' By rotational symmetry, it may be assumed that � π 0 ���PV⊥ θ #ν ��� q Lq(H2) dθ ≲ � 3π/4 π/4 ���PV⊥ θ #ν ��� q Lq(H2) dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' By pigeonholing, it may be assumed that ν = 1 ω3|B|δ6 � B∈B χB, where B is a disjoint collection of Euclidean δ2-balls in BH(0, c), for a small constant c (to be chosen in a moment) and ω3 = 4π 3 is the volume of the unit Euclidean ball in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Following the argument in [3, Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='2]: � 3π/4 π/4 ���PV⊥ θ #ν ��� q Lq(H2) dθ ∼ � L∠ |Xν(ℓ)|q dm(ℓ) ∼ 1 |B|qδ6q � L∠ ����� � B∈B H1(ℓ ∩ B) ����� q dm(ℓ) ≲ 1 |B|qδ4q � {p:ℓ(p)∈L∠} |{B ∈ B : B ∩ ℓ(p) ̸= ∅}|q dH3(p) (3) ∼ 1 |B|q δ4q ����� � B∈B χℓ∗(B) ����� q Lq(B(0,1)) , where the integration restricts to B(0, 1) if c is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let ρ = δǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let {τ} be a covering of Γ by boxes of dimensions ρ × ρ2 × 1 in the standard way, where Γ = {(ξ, |ξ|) ∈ R3 : ξ ∈ B2(0, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Call x ∈ B(0, 1) “narrow” if there is a 2-dimensional subspace V of R3 (depending on x), such that at least half of the tubes ℓ∗(B) passing through x have direction vectors in a ρ2-neighbourhood of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' If x is narrow, then because of the curvature of Γ, � B∈B χℓ∗(B)(x) ≲ �� τ |{B ∈ B : dir(ℓ∗(B)) ∈ τ and x ∈ ℓ∗(B)}|q �1/q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' 6 TERENCE L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' HARRIS If x is not narrow then it is called “broad”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' If x is a broad point, then � B∈B χℓ∗(B)(x) ≲ ρ−100× � � B1∈B � B2∈B � B3∈B χℓ∗(B1)χℓ∗(B2)χℓ∗(B3) ��uℓ∗(B1) ∧ uℓ∗(B2) ∧ uℓ∗(B3) �� �1/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Clearly (4) ����� � B∈B χℓ∗(B) ����� Lq(B(0,1)) ≲ �����χbroad � B∈B χℓ∗(B) ����� Lq(B(0,1)) + �����χnarrow � B∈B χℓ∗(B) ����� Lq(B(0,1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' If the broad part dominates in (4), then using q ≥ 3/2 and applying the trilinear Kakeya theorem (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' [2, Theorem 1]) gives �����χbroad � B∈B χℓ∗(B) ����� q Lq(R3) ≲ ρ−100 � \uf8eb \uf8ed � B1,B2,B3∈B χℓ∗(B1)χℓ∗(B2)χℓ∗(B3) |u(ℓ∗(B1)) ∧ u(ℓ∗(B2)) ∧ u(ℓ∗(B3))| \uf8f6 \uf8f8 q/3 ≲ ρ−100 |B|q− 3 2 � \uf8eb \uf8ed � B1,B2,B3∈B χℓ∗(B1)χℓ∗(B2)χℓ∗(B3) |u(ℓ∗(B1)) ∧ u(ℓ∗(B2)) ∧ u(ℓ∗(B3))| \uf8f6 \uf8f8 1/2 , ≲ ρ−100 |B|q δ6−ǫ = ρ−100 |B|q δ4q−(q−1)(3−t)−ǫ, by the definition of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' That ν is a probability measure implies that ct,δ(ν) ≳ 1, so this proves (2) when the broad part dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' If the narrow part dominates in (4), then ����� � B∈B χℓ∗(B) ����� q Lq(R3) ≲ � τ ������ � B∈B:dir(ℓ∗(B))∈τ χℓ∗(B) ������ q q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' For each cap τ, let Tτ be a finitely overlapping cover of physical space by ∼ ρ×ρ2×1 planks dual to τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' If δ is smaller than some absolute constant, then each δ2-tube ℓ∗(B) with dir(ℓ∗(B)) ∈ τ is contained in at least 1 plank from Tτ, and intersects ≲ 1 planks from Tτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' We associate each ℓ∗(B) with exactly one plank T ∈ Tτ such that dir(ℓ∗(B)) ∈ τ and ℓ∗(B) ⊆ T , and abbreviate this by writing ℓ∗(B) ≤ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Then � τ ������ � B∈B:dir(ℓ∗(B))∈τ χℓ∗(B) ������ q q ≲ � τ � T ∈Tτ ������ � B∈B:ℓ∗(B)≤T χℓ∗(B) ������ q q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' VERTICAL PROJECTIONS IN THE HEISENBERG GROUP 7 The point-line duality argument at Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' (3) is reversible provided the Euclidean balls are enlarged by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' This gives � 3π/4 π/4 ���PV⊥ θ #ν ��� q Lq(H2) dθ ≲ � τ � T ∈Tτ � 3π/4 π/4 ���PV⊥ θ #νT ��� q Lq(H2) dθ, where νT = 1 |B|6δ6 � B∈B:ℓ∗(B)≤T χ2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Each measure νT is essentially the restriction of ν to a Heisenberg ball of radius ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' By left translation (using Lemma 3), followed by a Heisenberg dilation (which commutes with vertical projections), and then by applying the induction hypothesis, � τ � T ∈Tτ � 3π/4 π/4 ���PV⊥ θ #νT ��� q Lq(H2) dθ ≲ � τ � T ∈Tτ ρ(t−3)(q−1)ct,δ(ν)q−1ν \uf8eb \uf8ed � B∈B:ℓ∗(B)≤T B \uf8f6 \uf8f8 (δ/ρ)−ǫ−(3−t)(q−1) ≲ ρǫν(H)ct,δ(ν)q−1δ−ǫ−(3−t)(q−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' The power of ρ is positive, so the induction closes provided δ is sufficiently small (depending only on ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' For the rescaled measures, the Euclidean δ2-balls are sent to δ2/ρ × δ2/ρ × δ2/ρ2 ellipsoids, which are essentially unions of (δ/ρ)2-balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' This means that the rescaled measures can be convolved with φδ/ρ without significantly affecting their properties (at least on Heisenberg balls of radius ≥ δ/ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' □ References [1] Balogh, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=', Durand-Cartagena, E, F¨assler, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=', Mattila, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
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+page_content=': The effect of projections on dimension in the Heisenberg group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
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+page_content=' Iberoam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
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+page_content=': The endpoint multilinear Kakeya theorem via the Borsuk- Ulam theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
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+page_content=', Orponen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=': Vertical projections in the Heisenberg group via cinematic functions and point-plate incidences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='00458v2 (2022) [4] Guth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
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+page_content=', Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=', Zhang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=': A sharp square function estimate for the cone in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' (2) 192, 551–581 (2020) [5] Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=': On the dimension of Kakeya sets in the first Heisenberg group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
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+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
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+page_content=' 150, 3445–3455 (2022) [6] Zahl, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=': Unions of lines in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' To appear in Mathematika.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='02913v1 (2022) Department of Mathematics, Cornell University, Ithaca, NY 14853, USA Email address: tlh236@cornell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf,len=1489
+page_content='EXIF as Language: Learning Cross-Modal Associations Between Images and Camera Metadata Chenhao Zheng Ayush Shrivastava Andrew Owens University of Michigan https://hellomuffin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='io/exif-as-language Patch Embedding Manipulated image Patch embeddings Detected splice Ground truth EXIF Text Embedding (a) Multimodal image-metadata embeddings Photo EXIF Metadata (b) Detecting spliced images by spotting inconsistent patch embeddings Model: NIKON D3200 Exposure Time: 1/500 Flash: Fired Focal Length: 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='0mm Exposure Program: Aperture Contrastive Learning Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' (a) We learn a joint embedding between image patches and the EXIF metadata that cameras automatically insert into image files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Our model treats this metadata as a language-like modality: we convert the EXIF tags to text, concatenate them together, and then processes the result with a transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' (b) We apply our representation to tasks that require understanding camera properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' For example, we can detect image splicing “zero shot” (and without metadata at test time) by finding inconsistent embeddings within an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We show a manipulated image that contains content from two source photos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Since these photos were captured with different cameras, the two regions have dissimilar embeddings (visualized by PCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We localize the splice by clustering the image’s patch embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Abstract We learn a visual representation that captures informa- tion about the camera that recorded a given photo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' To do this, we train a multimodal embedding between image patches and the EXIF metadata that cameras automatically insert into image files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Our model represents this meta- data by simply converting it to text and then processing it with a transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' The features that we learn significantly outperform other self-supervised and supervised features on downstream image forensics and calibration tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' In particular, we successfully localize spliced image regions “zero shot” by clustering the visual embeddings for all of the patches within an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Introduction A major goal of the computer vision community has been to use cross-modal associations to learn concepts that would be hard to glean from images alone [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' A particular focus has been on learning high level semantics, such as objects, from other rich sensory signals, like language and sound [59, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' By design, the representations learned by these approaches typically discard imaging properties, such as the type of camera that shot the photo, its lens, and the exposure settings, which are not useful for their cross-modal prediction tasks [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We argue that obtaining a complete understanding of an image requires both capabilities — for our models to per- ceive not only the semantic content of a scene, but also the properties of the camera that captured it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' This type of low level understanding has proven crucial for a variety of tasks, from image forensics [34, 53, 81] to 3D reconstruc- tion [35, 36], yet it has not typically been a focus of rep- resentation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' It is also widely used in image gen- eration, such as when users of text-to-image tools specify camera properties with phrases like “DSLR photo” [60,64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We propose to learn low level imaging properties from the abundantly available (but often neglected) camera meta- data that is added to the image file at the moment of cap- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' This metadata is typically represented as dozens of arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='04647v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='CV] 11 Jan 2023 Nikon 3200 NIKOn AF-S DXExchangeable Image File Format (EXIF) tags that describe the camera, its settings, and postprocessing operations that were applied to the image: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=', Model: “iPhone 4s” or Focal Length: “35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='0 mm”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We train a joint embed- ding through contrastive learning that puts image patches into correspondence with camera metadata (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Our model processes the metadata with a transformer [76] after converting it to a language-like representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' To do this conversion, we take advantage of the fact that EXIF tags are typically stored in a human-readable (and text-based) format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We convert each tag to text, and then concate- nate them together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Our model thus closely resembles con- trastive vision-and-language models, such as CLIP [63], but with EXIF-derived text in place of natural language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We show that our model can successfully estimate cam- era properties solely from images, and that it provides a useful representation for a variety of image forensics and camera calibration tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Our approaches to these tasks do not require camera metadata at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Instead, camera properties are estimated implicitly from image content via multimodal embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We evaluate the learned feature of our model on two clas- sification tasks that benefit from a low-level understanding of images: estimating an image’s radial distortion param- eter, and distinguishing real and manipulated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We find that our features significantly outperform alternative supervised and self-supervised feature sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We also show that our embeddings can be used to de- tect image splicing “zero shot” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=', without labeled data), drawing on recent work [8,34,55] that detects inconsisten- cies in camera fingerprints hidden within image patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Spliced images contain content from multiple real images, each potentially captured with a different camera and imag- ing pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Thus, the embeddings that our model assigns to their patches, which convey camera properties, will have less consistency than those of real images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We detect ma- nipulations by flagging images whose patch embeddings do not fit into a single, compact cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We also localize spliced regions by clustering the embeddings within an im- age (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We show through our experiments that: Camera metadata provides supervision for self- supervised representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Image patches can be successfully associated with camera metadata via joint embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Image-metadata embeddings are a useful representation for forensics and camera understanding tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Image manipulations can be identified “zero shot” by identifying inconsistencies in patch embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Related Work Estimating camera properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Camera metadata has been used for a range of tasks in computer vision, such Patch Encoder “Make: NIKON, Model: NIKON “Make: NIKON, Model: NIKON D3200, Flash: Fired, Exposure Time: 1/500, Focal Length: 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='0mm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Exposure Program: Aperture,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Components Configuration: YCbCr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Scene Capture Type: Standard,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' … ” ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='Patch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='Encoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
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+page_content='EXIF Text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='Encoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='“Make: NIKON,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Model: NIKON D3200,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Flash: Fired,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Exposure Time: 1/500,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Focal Length: 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='0mm, Exposure Program: Aperture, Components Configuration: YCbCr, Scene Capture Type: Standard, … ” Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Cross-modal image and camera metadata model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We use contrastive learning to associate each image patch with the EXIF metadata that was extracted from its image file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We repre- sent the metadata as text, which is obtained by concatenating the EXIF tags together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We then process it using a transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' as for predicting focal length [1, 32, 51, 72], performing white balancing [21, 50] and estimating camera models [7,40,75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' It has also been used as extra input for recogni- tion tasks [20,73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Instead of estimating camera properties directly (which can be highly error prone [34]), our model predicts an embedding that distinguishes a patch’s camera properties from that of other patches in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Image forensics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Early work used physically motivated cues, such as misaligned JPEG blocks [6, 22], color filter array mismatches [3,4,24,79], inconsistencies in noise pat- terns [38, 48, 49, 62], and compression or boundary arti- facts [5, 27, 33, 83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Other works use supervised learning methods [41,54,65,67,77,78,80–82,86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' The challenge of collecting large datasets of fake images has led to alterna- tive approaches, such as synthetic examples [54,80,82,86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Other work uses self-supervised learning, such as methods based on denoising [14], or that detect image manipulations by identifying image content that appears to come from dif- ferent camera models [7, 9, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Huh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' [34] learned a patch similarity metric in two steps: they determined which EXIF tags are shared between the patches, then use these bi- nary predictions as features for a second classifier that pre- dicts whether two patches come from the same (or different) images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' In contrast, we obtain a visual similarity metric that is well-suited to splice localization directly from our multi- modal embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Language supervision in vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Recent works have ob- tained visual supervision from language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' The formula- tion includes specific keyword prediction [56], bag-of-word multilabel classification [37], n-gram classification [47] and autoregressive language models [17,69,85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Recently, Rad- ford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' [63] obtained strong performance by training a contrastive model on a large image-and-language dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Our technical approach is similar, but uses text from cam- era metadata in lieu of image captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Work in text-to-image synthesis often exploits camera information through prompting, such as by adding text like “DSLR photo of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='..” or “Sigma 500mm f/5” to prompts [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' These methods, however, learn these camera associations through the (relatively rare) descriptions of cameras pro- vided by humans, while ours learns them from an abundant and complementary learning signal, camera metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Associating Images with Camera Metadata We desire a visual representation that captures low level imaging properties, such as the settings of the camera that were used to shoot the photo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We then apply this learned representation to downstream tasks that require an under- standing of camera properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Learning Cross-Modal Embeddings We train a model to predict camera metadata from im- age content, thereby obtaining a representation that con- veys camera properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Following previous work in mul- timodal contrastive learning [63], we train a joint embed- ding between the two modalities, allowing our model to avoid the (error prone) task of directly predicting the at- tributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Specifically, we want to jointly learn an image encoder and metadata encoder such that, given N images and N pieces of metadata information, the corresponding image–metadata pairs can be recognized by the model by maximizing embedding similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We use full-resolution image patches rather than resized images, so that our model can analyze low-level details that may be lost during down- sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Given a dataset of image patches and their corresponding camera metadata {(vi, mi)}N i=1, we learn visual and EXIF representations fθ(v) and gφ(m) by jointly training fθ and gφ using a contrastive loss [58]: LV,M i = − log exp (fθ(vi) · gφ(mi)/τ) �N j=1 exp(fθ(vi) · gφ(mj)/τ) , (1) where τ is a small constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Following prior work [63], we define an analogous loss LM,V that sums over visual (rather than metadata) examples in the denominator, and minimize a combined loss L = LV,M + LM,V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Representing the Camera Metadata This formulation raises a natural question: how should we represent the metadata?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' The metadata within photos is stored as a set of EXIF tags, each indicating a different im- age property as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' EXIF tags span a range of formats and data types, and the set of tags that are present in a given photo can be highly inconsistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Previous works that predict camera properties from images typically extract attributes of interest from the EXIF tags, and cast them to an EXIF tag Example values Camera Make Canon, NIKON Corporation, Apple Camera Model NIKON D90, Canon EOS 7 Software Picasa, Adobe Photoshop, QuickTime Exposure Time 1/60 sec, 1/125 sec, 1/250 sec Focal Length 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='0 mm, 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='0 mm, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='3 mm Aperture Value F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='8, F4, F5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='6, F3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='5 Scene Capture Type Landscape, Portrait, Night Scene Exposure Program Aperture priority, Manual control White Balance Mode Auto, Manual Thumbnail Compression JPEG, Uncompressed Digital Zoom Ratio 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='5, 2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='2 ISO speed Ratings 100, 400, 300 Shutter Speed Value 1/60 sec, 1/63 sec, 1/124 sec Date/Time Digitized 2013:03:28 04:20:46 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' What information is contained within photo EXIF metadata?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We list several of the most common EXIF tags, as well as the common values they contain in the YFCC100M dataset [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' appropriate data format — e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=', extracting a scalar-valued focal length category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' This tag-specific processing limits the amount of metadata information that can be used as part of learning, and requires special-purpose architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We exploit the fact that EXIF tags are typically stored in a human-readable format and can be straightforwardly con- verted to text (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' This allows us to directly process camera metadata using models from natural language pro- cessing — an approach that has successfully been applied to processing various text-like inputs other than language, such as math [46] and code [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Specifically, we create a long piece of text from a photo’s metadata by convert- ing each tag’s name and value to strings, and concatenating them together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We separate each tag name and value with a colon and space, and separate different tags with a space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We evaluate a number of design decisions for this model in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='4, such as the text format, choice of tags, and network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Application: Zero-shot Image Forensics After learning cross-modal representations from images and camera metadata, we can use them for downstream tasks that require an understanding of camera properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' One way to do this is by using the learned visual network features as a representation for classification tasks, fol- lowing other work in self-supervised representation learn- ing [12,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We can also use our learned visual embeddings to perform “zero shot” image splice detection, by detecting inconsistencies in an input image’s imputed camera proper- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Spliced images are composed of regions from multiple real images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Since they are typically designed to fool hu- mans, forensic models need to rely more on subtle (often non-semantic) cues to detect them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We got inspiration from Huh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' [34], which predicts whether two image patches Similarity Aggregation N M M x N Spectral Clustering Patch Encoder M x N Patch Affinity Matrix Similarity Map Patch Embeddings Input Image Detected Splice ⊙ Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Zero shot splice localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Given a spliced image (left), we compute our cross-modal embeddings for each image patch, which we visualize here using projections onto the top 3 principal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We then compute the affinity matrix by taking dot product for every pair of patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We localize the spliced region by clustering these embedding vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' share the same camera properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' If two patches are pre- dicted to have very different camera properties, then this provides evidence that they come from different images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' In our work, we can naturally obtain this patch similarity by computing the dot product between two patches’ em- beddings, since they have been trained to convey camera properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We note that, unlike Huh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' [34], we do not train a second, special-purpose classifier for this task, nor do we use augmentation to provide synthetic training ex- amples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=', by applying different types of compression to the patches).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' To determine whether an image is likely to contain a splice, we first compute an affinity matrix Aij = fθ(vi) · fθ(vj) whose entries are the dot product between patches’ normalized embedding vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We score an image v us- ing the sum of the exponentiated dot products between em- beddings, φ(v) = � i,j exp(Aij/τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' This score indicates the likelihood that the image is unmodified, since high dot products indicate high similarity in imputed camera prop- erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' To localize the spliced image regions within an im- age, we aggregate the similarity scores in Aij by clustering the rows using mean shift, following [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' This results in a similarity map indicating the likelihood that each patch was extracted from the largest source photo that was used to create the composite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Alternatively, we can visualize the spliced region by performing spectral clustering via normal- ized cuts [34, 70], using Aij as an affinity matrix between patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We visualize this approach in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Implementation Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We use ResNet-50 pretrained on ImageNet as our image encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We found that the text encoder in models trained on captioning, such as CLIP [63], were not well-suited to our task, since they place low limits on the number of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' For the EXIF text encoder, we use Dis- tilBERT [68] pretrained on Wikipedia and the Toronto Book Corpus [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We compute the feature representation of the EXIF as the activations for the end-of-sentence token from the last layer which is layer normalized and then linearly projected into multi-modal embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' To train our model, we use 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='5M full-resolution images and EXIF pairs from a subset of YFCC100M [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We discard images that have less than 10 of the EXIF tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Because many images only have a small number of EXIF tags available, we only use tags that are present in more than half of these images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' This results in 44 EXIF tags (see Ta- ble 6 in supplementary for the complete list).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' In contrast to other work [34], we do not rebalance the images to increase the rate of rare tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' During training, we randomly crop 124×124 patches from high-resolution images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We use the AdamW optimizer [39] with a learning rate of 10−4, weight decay of 10−3, and mixed precision training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We use a co- sine annealing learning rate schedule [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' The batch size is set to 1024, and we train our model for 50 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Other model variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' To study the importance of metadata supervision on the learned representation, we train a similar model that performs contrastive learning but does not use metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' The model resembles image-image con- trastive learning [12,30,34,88], which has been shown to be highly effective for representation learning, and which may learn low-level camera information [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Different from typical contrastive learning approaches, we use strict crop- ping augmentation so that the views for our model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 1) come from different crops of the same image, to encour- age it to learn low-level image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We call this model CropCLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Additionally, we evaluate a number of ablations of our model, including models that are trained with indi- vidual EXIF tags, that use different formats for the EXIF- derived text, and different network architectures (Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Evaluating the learned features First, we want to measure how well the learned fea- tures convey camera properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Since EXIF file is already Models Forensics Radial Distortion CASIA I CASIA II Dresden RAISE resize resize crop crop resize resize crop crop resize resize resize resize ImageNet pretrained 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
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+page_content='24 MoCo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
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+page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
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+page_content='28 CLIP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
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+page_content='22 Ours - CropCLR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
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+page_content='32 Ours - Full 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
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+page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
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+page_content='35 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We do linear probing on top of learned representation to predict two camera related properties that are not presented in EXIF files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' The good performance indicates that our model learns general imaging properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' resize and crop denote the image preprocessing applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' embedded with a lot of camera properties such as camera model, focal length, shuttle speed, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=', it should be unsur- prising if we can predict those properties from images (we provide such results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Instead, we want to study if the feature learned by the model can be generalized to other imaging properties that are not provided in the EXIF file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Specifically, we fit a linear classifier to our learned fea- tures on two prediction tasks: radial distortion estimation and forensic feature evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We compare the features from our image encoder with several other approaches, including supervised ImageNet pretraining [66], a state-of-the-art self-supervised model MoCo [30], CLIP [63], which obtains strong semantic rep- resentations using natural language supervision (rather than EXIF supervision), and finally the CropCLR variation of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' To ensure a fair comparison, the backbone ar- chitectures for all approaches are the same (ResNet-50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Radial distortion estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Imperfections in camera lens production often lead to radial distortion artifacts in captured images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' These artifacts are often removed as part of multi-view 3D reconstruction [28, 71], using methods that model distortion as a 4th-order polynomial of pixel po- sition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Radial distortion is not typically specified directly by the camera metadata, and is thus often must be estimated through calibration [10], bundle adjustment [71], or from monocular cues [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We followed the evaluation setup of Lopez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' [51], which estimates the quadratic term of the radial distor- tion model, k1, directly from synthetically distorted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' This term can be used to provide an estimate of radial dis- tortion that is sufficient for many tasks [51, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We syn- thesized the 512 × 512 images from the Dresden Image Database [26] and RAISE dataset [15] using k1 parame- ters uniformly sampled in the range [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='4, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' To predict k1, we used a regression-by-classification approach, quan- tizing the values of k1 into 20 bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We extracted features from different models, and trained a linear classifier on this 20-way classification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We provided them with a 512×512 image as input, and obtained image features from the global average pooling layer after the final convolutional layer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=', the penultimate layer of a typical ResNet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Representation learning for image forensics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We evalu- ate our model’s ability to distinguish real and manipulated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' This is a task that requires a broader understanding of low level imaging properties, such as spotting unusual image statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We use the CASIA I [19] and CASIA II [44] datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' The former contains only spliced fakes, while the latter contains a wider variety of manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We again perform linear classification using the features provided by different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We evaluate two types of preprocessing, resizing and center cropping, to test whether this low level task is sensitive to these details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' In both tasks, we found that our model’s features signif- icantly outperformed those of the other models (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Our method achieves much better performance than tradi- tional representational learning methods [30, 31, 63], per- haps because these models are encouraged to discard low- level details, while for our training task they are crucially important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Interestingly, the variation of our model that does not use EXIF metadata, CropCLR, outperforms the su- pervised [31] and self-supervised baselines [30], but signifi- cantly lags behind our full method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' This is perhaps because it often suffices to use high-level cues (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' color histograms and object co-occurrence) to solve CropCLR’s pretext task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' This suggests metadata supervision is an important learning signal and can effectively guide our model to learn general imaging information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Zero Shot Splice Detection and Localization We evaluate our model on the task of detecting spliced images without any labeled training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' This is in contrast to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='2, which used labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We perform both splice detection (distinguish an image being spliced or not) and splice localization (localize spliced region within an image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' For fair evaluation, we closely follow the approach of Huh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Given an image, we sample patches in a grid, using a stride such that the number of patches sampled along the longest image dimension is 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' To increase the spatial resolution of each similarity map, we average the predictions of overlapping patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We consider the smaller of the two detected regions to be the splice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' In splice localization task, we compare our model to a variety of forensics methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' These in- clude traditional methods that use handcrafted features [25, 53, 84], supervised methods [42, 80, 81], and self- supervised approaches [14, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' The datasets we use in- clude Columbia [57], DSO [16], Realistic Tampering (RT) [43], In-the-Wild [34] and Hays and Efros inpaint- ing images [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Columbia and DSO are created purely Style Method Columbia [57] DSO [16] RT [43] In-the-Wild [34] Hays [29] p-mAP p-mAP cIoU cIoU p-mAP p-mAP cIoU cIoU p-mAP p-mAP cIoU cIoU p-mAP p-mAP cIoU cIoU p-mAP p-mAP cIoU cIoU Handcrafted CFA [25] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
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+page_content='58 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Zero shot splice localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We evaluate our model on several datasets using permutation-invariant mean average precision (p-mAP) over pixels and class-balanced IOU (cIoU) with optimal threshold selected per image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' The result indicates that our model is comparable to state-of-the-art methods, although not specially optimized for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Dataset Columbia [57] DSO [16] RT [43] CFA [25] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
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+page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='53 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Zero-shot splice detection: We compare our splice de- tection accuracy on 3 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We measure the mean average pre- cision (mAP) of detecting whether an image has been spliced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' via image splicing, while Realistic Tampering contains a di- verse set of manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' In-the-Wild is a splicing image dataset composed of internet images, which may also con- tain a variety of other manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Hays and Efros [29] perform data-driven image inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' The quantitative comparison in terms of permuted-mAP (p-mAP p-mAP) and class- balanced IoU (cIoU cIoU) following [34] are presented in Ta- ble 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We also include splice image detection result in Ta- ble 4, where we compare our model to methods that enable splice detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Our model ranks first or second place for metrics in most datasets, and obtains performance comparable to top self- supervised methods that are specially designed for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' In particular, our model significantly outperforms the most related technique, EXIF-SC [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We note that both our method and EXIF-SC get relatively low performance on the Realistic Tampering dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' This may be due to the fact that this dataset contains manipulations such as copy- move that we do not expect to detect (since both regions share the same camera properties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' In contrast to meth- ods based on segmentation [14, 80, 81], we do not aim to have spatially precise matches, and output relatively low- resolution localization maps based on large patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Conse- quently, our model is not well-suited to detecting very small splices, which also commonly occur in the Realistic Tam- pering dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 4, we show qualitative results, including both similarity maps and spectral clustering results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 6, we compare our model with those of several other techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Interestingly, EXIF-SC has false positives in overexposed regions (as pointed out by [34]), since its classifier cannot infer whether these regions are (or are not) part of the rest of the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' In contrast, our model successfully handles these regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' CropCLR incorrectly flags regions that are semantically different from the background, because this is a strong indication that the patches come from different im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' In contrast, we successfully handle these cases, since our model has no such “shortcut” in its learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Ablation Study To help understand which aspects of our approach are responsible for its performance, we evaluated a variety of variations of our model, including different training super- vision, representations for the camera metadata, and net- work architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We evaluated each model’s features quality using linear probing on the radial distortion estimation and splice detec- tion task (same as Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' As an additional evaluation, we classify the values of common EXIF tags by applying lin- ear classifiers to our visual representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We convert the values of each EXIF tag into discrete categories, by quan- tizing common values and removing examples that do not fit into any category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We average prediction accuracies over 44 EXIF tags to obtain overall accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We provide more details in the supplementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' All models were trained for 30 epochs on 800K images on a subset of YFCC100M dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' The associated texts are obtained from the image descrip- tions and EXIF data provided by the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Image Similarity Map Normalized Cut Ground Truth Image Similarity Map Normalized Cut Ground Truth Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Qualitative visualization of splice localization result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We also include two typical scenarios where our model fails: copy-move tampering and very small splicing region (last two rows in right column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Method EXIF Radial Forens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Supervision All EXIF tags 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='85 CropCLR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='84 “Camera Model” tag only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='80 “Color Space” tag only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='61 YFCC image descriptions 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='70 Tag format Fixed order, w/ tag name 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='85 Fixed order, w/o tag name 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='86 Rand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' order, w/ tag name 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='77 Rand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' order, w/o tag name 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='76 Architecture DistilBERT, w/ pretrained 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='85 DistilBERT, w/o pretrained 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='77 ALBERT, w/ pretrained 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='84 ALBERT, w/o pretrained 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='79 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Model ablations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Downstream accuracy for versions of the model trained with different text supervision, representations of camera metadata, and architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We use linear probing to evaluate the average prediction accuracy of EXIF tag values on our YFCC test set, radial distortion estimation on Dresden dataset, and real-or-fake classification on CASIA I dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Rows with gray background (replicated for ease of comparison) represent the same model which is our “full” model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Metadata supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We evaluate a variation of our model that trains using the image descriptions provided by YFCC100M in lieu of camera metadata, as well as mod- els supervised by individual EXIF tags (Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' For the variations supervised by a single EXIF tag, we chose 14 common tags for this experiment, training a separate net- All Camera Model X Resolution Y Resolution EXIF Version Focal Length Aperture Value Exposure Program Scene Capture Type Make Exposure Mode Flash File Source Color Space Sensing Method Forensics Task Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
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+page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='61 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Per-tag forensics task accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We train various mod- els supervised by individual EXIF tags, then evaluate the learned representations for splice detection task on CASIA I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' work for each one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' The results of the per-tag evaluation is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' These results suggest that having ac- cess to the full metadata provides significantly better per- formance than using individual tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Moreover, there is a wide variation in the performance of models that use dif- ferent tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' This may be because the high performing tags, such as Camera Model, convey significantly more infor- mation about the full range of camera properties than oth- ers, such as Color Space and Sensing Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' These results suggest that a model that simply uses the full range of tags can extract significantly more camera information from the metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We also found that the variation trained on image descriptions (rather than EXIF text) performed significantly worse than other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Tag format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Since EXIF does not have a natural order of tags, we ask what will happen if we randomize the EXIF tag order during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Table 5 shows the performance drops 91I85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='LOTTA ACCADEMIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='EXIF-SC Ground Truth Image CropCLR Noiseprint OSN Ours Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Qualitative comparison to other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Our method can correctly localize splices in many scenarios where other methods fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' For example, EXIF-SC [34] fails on overexposed image regions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' OSN [80] and CropCLR (variation of our model) often segment scenes based on semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' for all three evaluations in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' This may be due to the fact that the Transformer model is forced to learn meaning- ful positional embeddings corresponding to each EXIF tag if their order keeps on changing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We also tried removing the tag names from the camera metadata and just provide the values for those keys, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' replacing Make: Apple with Apple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Interestingly, this model performs on par with the model that has tag names, suggesting that the network can discern information about the tags from the values alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Text encoder architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' To test whether performance of our model tied to a specific transformer architecture, we experimented with two different transformer models, Dis- tilBERT [68] and ALBERT [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We see that both archi- tectures obtain similar performance on all three tasks with DistilBERT slightly outperforming ALBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We also test how much pretraining the text encoder helps with the per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' From Table 5, we can see pretraining improves performance on the radial distortion and forensics tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Discussion In this paper, we proposed to learn camera properties by training models to find cross-modal correspondences be- tween images and camera metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' To achieve this, we created a model that exploits the fact that EXIF metadata can easily be represented and processed as text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Our model achieves strong performance amongst self-supervised meth- ods on a variety of downstream tasks that require un- derstanding camera properties, including zero shot image forensics and radial distortion estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We see our work opening several possible directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' First, it opens the pos- sibility of creating multimodal learning systems that use camera metadata as another form of supervision, providing complementary information to high-level modalities like language and sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Second, it opens applications that require an understanding of low level sensor information, which may benefit from our feature sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Limitations and Broader Impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We have shown that our learned features are useful for image forensics, which has potential to reduce the spread of disinformation [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' The model that we will release may not be fully representa- tive of the cameras in the wild, since it was trained only on photos available in the YFCC100M datatset [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
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+page_content=' Ieee, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 5, 6 [85] Yuhao Zhang, Hang Jiang, Yasuhide Miura, Christopher D Manning, and Curtis P Langlotz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Contrastive learning of medical visual representations from paired images and text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' arXiv preprint arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='00747, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 2 [86] Peng Zhou, Xintong Han, Vlad I Morariu, and Larry S Davis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Learning rich features for image manipulation detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1053–1061, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 2 [87] Yukun Zhu, Ryan Kiros, Rich Zemel, Ruslan Salakhutdinov, Raquel Urtasun, Antonio Torralba, and Sanja Fidler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Align- ing books and movies: Towards story-like visual explana- tions by watching movies and reading books.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' In Proceed- ings of the IEEE international conference on computer vi- sion, pages 19–27, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 4 [88] Daniel Zoran, Phillip Isola, Dilip Krishnan, and William T Freeman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Learning ordinal relationships for mid-level vi- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' In Proceedings of the IEEE international conference on computer vision, pages 388–396, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' List of EXIF tags We present the complete list of EXIF tags being used by our model in Table 6, along with representative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Experimental Details We provide additional experimental details about the downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Radial Distortion Model We follow the radial distortion model proposed in Lopez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Let (x, y) represent the normalized image pixel coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Radial distortion can be modeled as scaling the normalized coordinates by a factor of d, which is a function of the distance from pixel location to center of image r and distortion parameters k1 and k2: d = 1 + k1r2 + k2r4 (2) and set (xd, yd) = (dx, dy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We note that the relationship between k1 and k2 can be approximated modeled as: k2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='019k1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='805k2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' (3) Thus, we only aim to predict k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' EXIF Prediction Application We provide implementation details for the downstream application of predicting EXIF tags from visual features (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' To formulate the problem as a classification task, we convert the values of each EXIF tag into discrete cate- gories, using the following rules: if an EXIF tag has less than 20 distinct values, we use each value as a class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' For example, the white balance mode tag has only two values auto, manual, each of which becomes a cate- gory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' If an EXIF tag has continuous values (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=', focal length) or more than 20 discrete value (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=', camera model), we will quantize its common values to a set of bins using hand-chosen rules, and remove examples that do not fit into any category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' For example, for the camera model tag, which holds a sparse set of camera models, we merge their value according to their brand (value NIKON EXIF tag Example values Aperture Value F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='8, F4, F5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='6, F3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='5 Camera Make Canon, NIKON Corporation, Apple Camera Model NIKON D90, Canon EOS 7 Color Space sRGB, Undefined Components Configuration YCbCr, CrCbY Compressed Bits 4 bits per pixel Custom Rendered Custom process, Normal process Data/Time 2013:03:28 04:20:46 Data/Time Digitized 2013:03:28 04:20:46 Data/Time Original 2013:03:28 04:20:46 Digital Zoom Ratio 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='5, 2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='2 Exif Image Height 2592 pixels, 2304 pixels Exif Image Width 2592 pixels, 2408 pixels Exif Version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='21, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='20, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='30 Exposure Bias Value 0 EV, -1 EV, 1 EV Exposure Mode Auto exposure, Manual exposure Exposure Program Aperture priority, Manual control Exposure Time 1/60 sec, 1/125 sec, 1/250 sec F-Number F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='8, F5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='6, F4 File Source Digital Still Camera, Print Scanner Flash Unfired, Fired(red-eye reduction) FlashPix Version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='00, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='10, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='01 Focal Length 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='0 mm, 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='0 mm, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='3 mm Focal Place X Resolution 292 dots per inch Focal Place Y Resolution 292 dots per inch Gain Control Low, High ISO Speed Ratings 100, 400, 300 Interoperability Index 0, unknown Interoperability Version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='00, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='10, 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='00 Max Aperture Value F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='8, F3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' F4 Metering Mode Multi-segment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Spot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' average Orientation Top,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' right side (Mirror horizontal) Resolution Unit Inch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' cm Scene Capture Type Landscape,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Portrait,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Night Scene Sensing Method One-Chip color area sensor Shutter Speed Value 1/60 sec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 1/63 sec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 1/124 sec Software Picasa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Adobe Photoshop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' QuickTime Thumbnail Compression JPEG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Uncompressed Thumbnail Length 0 bytes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 16712 bytes Thumbnail Offset 5108 bytes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 9716 bytes White Balance Mode Auto,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Manual X Resolution 72 dots per inch Y Resolution 72 dots per inch YCbCr Positioning datum point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Center of pixel array Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Full list of EXIF tags being used in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' This extends the list from Table 1 in the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' D90 will fall into NIKON category).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We define common values to be those that occur with probability greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='1% in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Linear Probing Implementation Details The linear probing experiment set up is as follows: We follow the approach from Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We use Adam optimizer with no weight decay, and set learning rate to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='01 and optimizer momentum to be β1, β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We also normalize the image features before providing them to the linear classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We use a batch size of 1024, and we train the classifier for 20 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Camera Model Exposure X Resolution Y Resolution Make EXIF Version Aperture Focal Length Exposure Mode Scene Capture Type Flash Camera Model Exposure X Resolution Y Resolution Make EXIF Version Aperture Focal Length Exposure Mode Scene Capture Type Flash Chance Performance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='6 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Confusion matrix of EXIF tags’ prediction accuracy for per-tag model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Each column represents the prediction accuracy of a specific tag using models trained under different tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Confusion Plot It is interesting to know the performance of a model trained using a specific tag when asked to predict the value of other tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' This may indicate the generalizability of the training tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' We therefore take the per-tag models (same as fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 5) and measure their prediction accuracies of different tag values (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' The result shows models trained on some tags may contain useful information to be generalized to other tags, such as the model trained with ”camera make” performs well in ”camera model” and ”aperture” predic- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' In contrast, that model trained on tags that don’t have rich values or information (like Flash) can not generalize to other tags well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Additional Qualitative Results We provide additional qualitative results for our zero- shot splice localization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 8 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 9 (left), we show accurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' 9 (right), we show failure cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Image Similarity Map Normalized Cut Ground Truth Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Additional zero-shot splice localization results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' elhineeineOUIJA YES NO UVWX NOPQ 12 06829 BYE PARKEREROTHERSOUIJA YES NO UVWX NOPQI 12 06829 GOOD BYE PARKER EROTHERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='INLNITAOX T ANT 人区 iellino1922 INALSKOLLISTER 1922 INALS379 PPO 川川379 CladtOUIJA YES NO 12 06829 GOODBYE PHRKER EROTHERSM 379 PO1922 INALSImage Similarity Map Normalized Cut Ground Truth Image Similarity Map Normalized Cut Ground Truth Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' Additional zero-shot splice localization results: success cases (left) and failure cases (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' syco M250LREVESREVESAPDAPDLAPDREVESEnjog!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='VentureKayaksEnjog!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='Enjog,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='Venfure KayaksABetter Construction JobForBrian DEMOCRATSABetter Construction JobForBrian D DEMOCRATSABetter construction lobForBnianRESTAURANTV:Francesco Barbieri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='born inCento,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content="madeit-1623 nillais remembered as Saint Peter's daughter (perhaps in a mo ore her married to Flacco," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='a Roman noble she did not wish to IforanaltaroftheBasilicaof SaintPeter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='was commissioned to crived in Rome,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='by George Bush XV in 1923and it was comp tom part there is a representation of the burial of the young wor omen in tears,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='a boy and a young man holding a church candle urban,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='an old man (Saint Peter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=") and a young man wearing the upper part the glorification of the saint, kneeling beforeXV:FrancescoBarbieriborninCentomadeit-I623 nillaisrememberedas SaintPeter'sdaughter (perhapsin amo orehermarried to Flacco,a Romannoble shedid not wish to IforanaltaroftheBasilicaofSaintPeterwascommissioned to rivedinRome,by GeorgeBushXVin 9 23anditwascomp om part there is a representation of the burial of the young wo omenintears,aboyandayoungman holdingachurchcandle urban,an old man (Saint Peter?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=' )and a young man wearing the upper part the glorification of the saint,kneeling beforerancescoBarbierl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='borninCento.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content='madeit- 623 nillaisremembered as Saint Peters daughter (perhapsin amo ore hermarried to Flacco,aRoman noble she did not wish to IforanaltaroftheBasilica of SaintPeter,wascommissioned to rrived in Rome, by Ceorge Bush XV in 1923 and it was comp tom part there is a representation of the burial of the young wor omen in tears,a boy and ayoung man holding achurch candle turban, an old man (Saint Peter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
+page_content=') and a young man wearing the upper part the glorification of the saint, kneeling before' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfqApm/content/2301.04647v1.pdf'}
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+Wave function-based emulation for nucleon-nucleon scattering in momentum space
+A. J. Garcia
+,1, ∗ C. Drischler
+,2, 3, † R. J. Furnstahl
+,1, ‡ J. A. Melendez
+,1, § and Xilin Zhang
+3, ¶
+1Department of Physics, The Ohio State University, Columbus, OH 43210, USA
+2Department of Physics and Astronomy and Institute of Nuclear and Particle Physics, Ohio University, Athens, OH 45701, USA
+3Facility for Rare Isotope Beams, Michigan State University, MI 48824, USA
+(Dated: January 13, 2023)
+Emulators for low-energy nuclear physics can provide fast & accurate predictions of bound-state
+and scattering observables for applications that require repeated calculations with different param-
+eters, such as Bayesian uncertainty quantification. In this paper, we extend a scattering emulator
+based on the Kohn variational principle (KVP) to momentum space (including coupled channels)
+with arbitrary boundary conditions, which enable the mitigation of spurious singularities known as
+Kohn anomalies. We test it on a modern chiral nucleon-nucleon (NN) interaction, including emu-
+lation of the coupled channels. We provide comparisons between a Lippmann-Schwinger equation
+emulator and our KVP momentum-space emulator for a representative set of neutron-proton (np)
+scattering observables, and also introduce a quasi-spline-based approach for the KVP-based emula-
+tor. Our findings show that while there are some trade-offs between accuracy and speed, all three
+emulators perform well. Self-contained Jupyter notebooks that generate the results and figures in
+this paper are publicly available.
+I.
+INTRODUCTION
+Nucleon-nucleon (NN) scattering has long been used
+to fix parameters of microscopic Hamiltonians designed
+for ab initio few- and many-body calculations. But the
+uncertainty in most existing nuclear models has been un-
+derestimated because they have lacked two key ingredi-
+ents: a rigorous accounting of Hamiltonian uncertainty
+and a complete estimate of parameter uncertainty.
+In the case of chiral effective field theory (χEFT) [1–
+4], Hamiltonian uncertainty manifests as a truncation er-
+ror, which has been statistically modeled in Refs. [5–8].
+A holistic parameter estimation study would then both
+account for truncation errors in the likelihood, and esti-
+mate and propagate all plausible values of the low-energy
+constants (LECs) rather than finding a single parame-
+ter value maximizing the likelihood. Bayesian statisti-
+cal methods are particularly suitable for these tasks [9–
+13], but are computationally demanding, especially when
+generalizing to include few-body forces.
+Emulators—
+surrogate models that allow for fast & accurate (but ap-
+proximate) model predictions—have the potential to al-
+leviate some of these demands [14]. In this paper, we
+extend our recent explorations of emulators for NN scat-
+tering [15–17] to momentum-space wave functions and
+coupled channels, and test against a representative set of
+neutron-proton (np) scattering observables.
+The demand for emulators has led nuclear physics to
+the general field of parametric model order reduction
+(PMOR), where the goal is to extract the relevant in-
+formation from a model while reducing the computa-
+∗ garcia.823@osu.edu
+† drischler@ohio.edu
+‡ furnstahl.1@osu.edu
+§ melendez.27@osu.edu
+¶ zhangx@frib.msu.edu
+tional cost significantly.
+An efficient offline-online de-
+composition is crucial to construct an efficient emula-
+tor.
+In the offline stage, the emulator is trained with
+high-fidelity calculations1 for selected sets of parameters,
+also known as snapshots, while making predictions for
+any other set of parameters are performed in the online
+stage. The end result is a reduced-order model (ROM)
+that serves as an emulator. For general overviews of the
+literature on PMOR techniques and their applications,
+we refer the reader to Refs. [19, 20]. A pedagogical intro-
+duction to projection-based emulators for both scatter-
+ing and bound-state calculations, including interactive,
+open-source Python code, can be found in Refs. [21, 22].
+A particular snapshot-based ROM known as the re-
+duced basis method (RBM)2 has emerged as an efficient
+emulator for the prediction of both bound state and scat-
+tering observables [15, 23, 24]. The foundation of the first
+emulators for scattering is the Kohn variational princi-
+ple (KVP) [e.g., for the K matrix], whose snapshots are
+based on scattering solutions to the Schr¨odinger equa-
+tion [25, 26].
+It has been demonstrated for a variety
+of real and optical potentials that such emulators can be
+trained for two- and three-body3 scattering in coordinate
+space, then evaluated in the form of matrix inversions
+with low-dimensional matrices [15, 16, 27].
+Subsequently, an emulator of the Lippmann-Schwinger
+(LS) equation using the Newton variational principle
+(NVP) [28] was introduced in Ref. [17]. In contrast to the
+KVP emulator, the variational trial basis is composed of
+1 Following the terminology of Ref. [18], we will refer to the cal-
+culational machinery that generates high-fidelity solutions (e.g.,
+LS equation solver) as a simulator.
+2 The RBM has been rediscovered in the low-energy nuclear theory
+community as eigenvector continuation (EC). See Ref. [19] for
+more details.
+3 In Ref. [27], the offline training stage involves calculations in both
+momentum and coordinate space.
+arXiv:2301.05093v1 [nucl-th] 12 Jan 2023
+
+ID2
+TABLE I. Notation used in this work.
+Notation
+Description
+θ
+vector of parameters; θi are the parameters
+for the ith snapshot
+s, s′
+indices for the exit and entrance channels of
+the scattering process, e.g., 3S1 and 3D1
+t, t′
+indices for available channels (summation
+convention implied)
+ψs
+i
+wave function in the channel s used for train-
+ing and associated with the ith snapshot with
+θi [high-fidelity solution of Eq. (1)]
+�ψs
+snapshot-based trial wave function in the
+channel s (3) applied to the KVP func-
+tional (2)
+Lss′
+E
+a generic scattering matrix at energy E
+Lss′[ �ψ]
+a functional whose stationary point is an ap-
+proximation of the generic L-matrix; i.e.,
+L[ �ψ + δ �ψ] = Lss′
+E + O(δL2)
+βi
+to-be-determined coefficient of the ith snap-
+shot in the trial wave function with �
+i βi = 1
+∆�U ss′
+ij (θ)
+nb × nb kernel matrix defined in Eq. (5)
+scattering matrices (e.g., K matrices) rather than scat-
+tering wave functions. Both approaches were shown to
+quickly and accurately predict the np phase shifts from a
+chiral Hamiltonian across a range of parameter values. In
+this paper, we compare a momentum-space KVP-based
+emulator, including emulation of coupled channels and
+allowing for arbitrary boundary conditions, to the NVP
+emulator for a representative set of np observables. For a
+comparison of the KVP and NVP emulators in a Galerkin
+framework and a survey on other emulators see Ref. [21].
+The paper is organized as follows. In Sec. II, we review
+the underlying formalism of the KVP emulators and its
+extension to momentum space and coupled channels. We
+then show results for the momentum-space KVP emula-
+tor and compare them to the K matrix (NVP) emulator
+in Sec. III. We demonstrate that spurious singularities
+known as Kohn (or Schwartz) anomalies [29, 30] are mit-
+igated using methods from Ref. [16]. Section IV has a
+summary and outlook and additional details of the im-
+plementation are given in several appendices. The self-
+contained set of codes that generate all results and figures
+shown in this paper is publicly available [31].
+II.
+FORMALISM
+Our goal is to emulate the partial-wave Schr¨odinger
+equation for NN scattering at the center-of-mass energy
+E > 0
+�H(θ) |ψs⟩ ≡
+� �T + �V (θ)
+�
+|ψs⟩ = E |ψs⟩ ,
+(1)
+where the vector θ is composed of parameters used by
+the theoretical model to match results with experimen-
+tal observations (e.g., the LECs of χEFT). Building our
+snapshot-based MOR emulator begins by writing Eq. (1)
+in integral form.
+Here we choose the general (con-
+strained4) KVP, which is based on the functional [16, 32]
+Lss′[ �ψ] = �Lss′
+E − 2µk0
+det u ⟨ �ψs| �H(θ) − E| �ψs′⟩ ,
+(2)
+where �ψ is a trial scattering wave function, �Lss′
+E
+is a
+generic trial scattering matrix, u is a non-singular ma-
+trix [16, 32] used to parameterize the asymptotic bound-
+ary condition associated with �Lss′
+E (see Appendix A), and
+k0 = √2µE is the on-shell energy with µ being the re-
+duced mass.5 More details can be found in Ref. [16] and
+Appendix A. Table I summarizes the notation we use in
+this work. Note that we adopt the convention that the
+wave functions in a bra symbol ⟨·| in bra-ket notation are
+not complex conjugated [e.g., ⟨ �ψs| in Eq. (2)] [15, 16, 34].
+In Eq. (2), the superscripts s and s′ index the coupled
+channels (e.g., 3S1 and 3D1); for the uncoupled case this
+reduces to a single equation with s′ = s. Each combi-
+nation of (s′, s) will have their own, distinct emulator in
+our formulation. As an example, for a coupled-channel
+np interaction in Eq. (2), the (s′, s) pair could be one
+of 3S1–3S1, 3S1–3D1, 3D1–3S1, or 3D1–3D1, and for an
+uncoupled channel s′ = s could be 1S0.
+We use the
+np spin-triplet coupled channels as an exemplary case,
+but the general emulation procedure applies to general
+channel coupling (including spin-singlet spin-triplet np
+coupling [35]).
+The functional (2) yields Lss′[ �ψ] = Lss′
+E
+when �ψ is the
+exact wave function, and provides a stationary approxi-
+mation otherwise: Lss′[ �ψ + δ �ψ] = Lss′
+E + O(δL2). Rather
+than finding a wave function |ψ⟩ that satisfies Eq. (1),
+our task has now changed to finding a wave function that
+makes Eq. (2) stationary for a given choice of E.
+The key to creating an efficient PMOR emulator from
+Eq. (2) is to use a snapshot trial wave function,
+| �ψs⟩ ≡
+nb
+�
+i=1
+βi |ψs
+i ⟩ ,
+(3)
+where nb is the number of parameter vectors {θi}nb
+i=1 in
+the training set and {|ψs
+i ⟩}nb
+i=1 the associated high-fidelity
+solutions to Eq. (1), obtained by solving the LS equation
+directly (see also Sec. III). These solutions are determined
+once in the offline stage. The to-be-determined basis co-
+efficients ⃗β will not be the same for all the channels, re-
+sulting in independent emulators for each (s′, s) pair (see
+Appendix B for more details).
+For the np spin-triplet
+4 For a description of constrained and unconstrained emulators see
+Ref. [21]
+5 Throughout this paper we use boldface symbols to indicate vec-
+tors in parameter-space, arrows to indicate vectors in snapshot-
+space, natural units in which ℏ = c = 1, and follow the conven-
+tions for scattering matrices in Refs. [26, 33].
+
+3
+coupled channels, this will result in three distinct varia-
+tional principles being enforced: one for each of angular
+momentum s′ = s = j ± 1 and one for the off-diagonal
+component.
+The other off-diagonal component can be
+inferred through the unitarity of the S matrix.6
+Upon inserting the snapshot trial wave function (3)
+into the functional (2), the functional takes the form [15]
+Lss′[⃗β ] = βiLss′
+E,i − 1
+2βi∆�U ss′
+ij βj,
+(4)
+with the symmetric matrix
+∆�U ss′
+ij (θ) ≡ 2µk0
+det u
+�
+⟨ψs
+i | �H(θ) − E|ψs′
+j ⟩ + (i ↔ j)
+�
+= 2µk0
+det u
+�
+⟨ψs
+i |�V (θ) − �Vj|ψs′
+j ⟩ + (i ↔ j)
+�
+,
+(5)
+where, as in Eq. (2), s′ and s correspond to the entrance
+and exit channels. Equation (4) is a stationary approx-
+imation to the generic L-matrix at one energy, hence
+we build independent emulators for each value of an en-
+ergy grid. Equation (5) is obtained [15] by adding and
+subtracting �Vi ≡ �V (θi) and �Vj ≡ �V (θj) and applying
+Eq. (1). In this form, the constant terms in the poten-
+tials, such as a long-range Coulomb interaction (assuming
+the fine-structure constant is not varied), will cancel, and
+the matrix elements will only involve short-range physics.
+Emulating the scattering wave function [via Eq. (3)],
+and hence Lss′
+E ≈ Lss′[ �ψ] [via Eq. (4)], has now been re-
+duced to choosing an appropriate training set {θi} and
+then determining the values of βi that make Eq. (4) sta-
+tionary under the constraint that �
+i βi = 1. The latter
+is a consequence of maintaining a consistent asymptotic
+normalization for the scattering wave functions in Eq. (3)
+as required by the constrained KVP [15, 21]. A numeri-
+cally robust solution can be found by introducing a La-
+grange multiplier λ, and solving the matrix equation [16]
+�
+∆�U ss′ ⃗1
+⃗1 ⊺
+0
+� �⃗β⋆
+λ⋆
+�
+=
+�⃗Lss′
+E
+1
+�
+,
+(6)
+where ⃗1 is an nb × 1 vector of ones, ⃗Lss′
+E
+are the basis
+states used in the offline stage, and ⃗β⋆ is a vector of co-
+efficients of the trial wave function associated with the
+KVP’s stationary approximation. Since Eq. (6) is a lin-
+ear system, it will be a highly computationally efficient
+emulator for scattering systems if the number nb of basis
+functions is much smaller than the size of the high-fidelity
+wave function ψ.
+Thus far we have not specified whether the matrix el-
+ements ∆�U ss′
+ij
+are to be calculated in coordinate space
+or momentum space. The only difference between these
+6 For (complex-valued) optical potentials with two coupled chan-
+nels, one has four (instead of three) distinct variational principles
+because the S matrix is not unitary.
+implementations is the way we obtain the basis functions
+ψi used to construct the trial ansatz in Eq. (3), and thus
+the manner in which ∆�U ss′ is evaluated. To formulate a
+momentum-space wave function approach to MOR emu-
+lators for scattering, we initially solve for the K matrix
+and relate ψ to K before using Eq. (5). The scattering
+wave function in momentum space takes the form [36]
+ψst(k; k0) = 1
+k2 δ(k − k0)δst + 2
+π PKst(k, k0)/k0
+k2 − k2
+0
+,
+(7)
+which vanishes as k → ∞, but is singular at k = k0 =
+√2µE (the superscripts used for the K matrix in Eq. (7)
+are opposite Ref. [36]). Here, Kst is the reactance ma-
+trix (or just the K matrix), k0 the on-shell energy, P the
+Cauchy principal value, and the labeling st indicates the
+partial-wave or reaction channels.
+One can also write
+Eq. (5) in the momentum-space representation by insert-
+ing complete sets of states,7 resulting in
+∆�U ss′
+ij (θ) =
+¨ ∞
+0
+dk dp k2p2�
+ψts
+i (k)V tt′
+θ,j(k, p)ψt′s′
+j
+(p)
++ (i ↔ j)
+�
+,
+(8)
+with
+V tt′
+θ,j(k, p) ≡ 2µk0
+det u
+�
+V tt′(k, p; θ) − V tt′
+j
+(k, p)
+�
+,
+(9)
+where t and t′ are summed over the available channels
+and the dependence of ψ on k0 is left implicit. Moving
+forward, we will drop the channel superscripts on ∆�U.
+This is the general form of the momentum-space ∆ ˜U
+matrix. Note the ordering of the channel indices (t, s)
+in the left-hand wave function in Eq. (8), which follows
+from ψts(k) ≡ ⟨kt|ψs⟩ and the convention that ⟨ψ| = |ψ⟩⊺
+(without a complex conjugate), so that ψts(k) = ⟨ψs|kt⟩.
+Thus, if ψ has outgoing (ψ(+)), incoming (ψ(−)), or
+standing wave (ψ(0)) boundary conditions, then the same
+version of ψ(x) is used for both ψ(k) and ψ(p) in Eq. (8).
+No modification of Eq. (8) is needed in the case of optical
+potentials, where again the left-hand wave function is not
+conjugated relative to the right-hand wave function. For
+more details on how to build the general KVP emulator
+we refer the reader to Appendix C. Different boundary
+conditions will be used below to mitigate Kohn anomalies
+(see Sec. III B).
+The efficient evaluation of ∆�U across a range of θ val-
+ues is critical to the applicability of the emulator. If the
+Hamiltonian operators have an affine (i.e., factorizable)
+parameter dependence, denoted as
+�H(θ) =
+�
+n
+hn(θ) �Hn,
+(10)
+7 For example, for np scattering as in Sec. III, the complete set
+of states are relative-momentum partial-wave states with orbital
+angular momentum and spin coupled to total J and MJ.
+
+4
+then matrix elements of the Hn operators in a given basis
+only need to be calculated once in the offline stage rather
+than for every parameter set θi. Chiral NN interactions
+have the form of Eq. (10) and, when varying only the
+contact LECs, can even be cast into the form8
+�V (θ) = �V 0 + θ · �V 1,
+(11)
+so that Eq. (5) can then be written as
+∆�U(θ) = ∆�U 0 + θ · ∆ �U 1.
+(12)
+The matrices �V 0 and ∆�U 0 and vectors of matrices �V 1
+and ∆ �U 1, can now be pre-calculated during the emu-
+lator’s offline stage, allowing for considerable speed-up
+factors in the online stage where the value of ∆�U(θ) at
+any new parameter value is efficiently constructed.
+III.
+RESULTS
+In this section, we apply the KVP momentum-space
+emulator to calculate np scattering observables. We use
+the Reinert et al. semilocal momentum-space (SMS) reg-
+ularized chiral potential at N4LO+ with the momentum
+cutoff Λ = 450 MeV [37], which is a state-of-the-art chiral
+NN interaction. The parameters θ are composed of the
+NN contact LECs contributing to this potential.
+A.
+Emulator overview
+The snapshots used in the offline stage are the scatter-
+ing solutions given by Eq. (7). The K matrices used to
+calculate the second term in Eq. (7) are obtained from
+numerically solving the LS equation. The LS equation is
+reduced to a set of linear equations by approximating the
+integral as a sum over N quadrature points obtained from
+a Gauss–Legendre rule with corresponding weights (see
+Refs. [36, 38]). If the potential was calculated merely on
+the quadrature points, without appending the on-shell
+values, interpolation must be performed to obtain the
+(half-)on-shell potential so that one can (1) account for
+the singularity of the Green’s function when solving the
+LS equation [38], and (2) integrate the delta distribution
+in Eq (7) (explained in next paragraph). To generate the
+figures in this paper, we use a compound Gauss-Legendre
+quadrature mesh of N = 80 momentum points. For the
+observables, we use a lab energy range of 0 to 350 MeV
+with 350 points. For the partial waves plots, we use a fine
+8 Note that hn(θ) would include higher-order polynomials when
+also emulating the pion-nucleon coupling c2 (at N3LO) and axial
+coupling constant gA (already at LO). Nevertheless, the Hamil-
+tonian remains affine and thus the emulators discussed here are
+directly applicable.
+energy mesh of 3500 points over the same energy range
+previously mentioned.
+When performing the KVP emulation, we calculate
+Eq. (5) two different ways.
+The first is by inserting
+Eq. (7) into Eq. (5) and analytically integrating the delta
+distribution, which corresponds to appending the exact
+on-shell value of the potential. The remaining integrals
+are then solved numerically (see Appendix C). We re-
+fer to this method as the Standard method. The second
+is based on the global Gl¨ockle spline interpolation [39],
+which belongs to the family of quasi-spline methods that
+perform the mapping
+�
+k
+f(k)Sk(k0) ≈ f(k0),
+(13)
+for smooth functions f(k) sampled on a grid k that en-
+compasses k0 using the cubic spline polynomials Sk(k0)
+constructed in Ref. [39].
+This allows us to calculate
+Sk(k0) once in the offline stage and save the result for
+the online stage since it has no dependence on f(k) it-
+self. Using this method, we interpolate the solutions to
+the integrals that appear in Eq. (5) (i.e., k0 does not need
+to be appended to the mesh as opposed to the Standard
+method), thus decreasing the computational cost needed
+in the offline stage significantly at the expense of accu-
+racy. We compare the KVP emulator results using the
+Gl¨ockle and Standard method and compare those results
+to the NVP emulator described in Ref. [17].
+To reduce numerical errors in both the simulator and
+emulator, we compute snapshots {Ki} of the LS equation
+using non-interpolated potentials for partial waves that
+have a LEC-dependence and interpolated potentials for
+LEC-independent partial waves. When referring to in-
+terpolated potentials, we mean calculating the potential
+using only the momentum mesh and then using an in-
+terpolation method (such as the bivariate Gl¨ockle spline
+method) to interpolate the potential to k0.
+By non-
+interpolated, we mean that each k0 is appended to the
+momentum mesh and the potential evaluated at these
+points, which improves the accuracy of our potentials
+compared to interpolating the potential to k0. We chose
+to use non-interpolated potentials for the LEC-dependent
+partial waves since these are the only ones used to cal-
+culate Eq. (5) in the offline phase.
+The same LEC-
+independent partial waves are employed by the simulator
+and emulator. All potentials used for the emulators and
+simulator are pre-calculated for efficiency.
+The simulator used in this paper numerically solves
+the LS equation for each partial wave.
+The accuracy
+of our simulator was tested by comparing the simulator
+results to the analytical solution of a Gaussian separable
+potential, producing relative errors of ≈ 10−7 or better.
+Additionally, the simulator’s speed was roughly 4x slower
+when we doubled the mesh size from N = 80 to N = 160
+quadrature points.
+The accuracy of emulated observables depends on the
+size of the basis (see Sec. III C); here we use a basis size
+nb = 2na, where na is the number of LECs associated
+
+5
+with a given partial wave channel. The training points
+θi are randomly sampled within an interval of [−5, 5]
+using a Latin-hypercube for each partial wave, with the
+fitted coupling constants and appropriate units given in
+Ref. [37].
+The matrix ∆�U is increasingly ill-conditioned as the
+basis size nb increases. One can reduce numerical noise
+by (1) adding a regularization parameter (“nugget”) to
+the diagonal elements of the near-singular matrix [15], or
+(2) using a solver that performs some type of regulariza-
+tion. For the KVP emulator results in the figures, we use
+NumPy’s least-squares solver linalg.lstsq() [40] with
+a cut-off ratio for small singular values of 10−10 [16]. For
+the NVP emulator, we add a nugget of 10−10 to the di-
+agonal and use NumPy’s linalg.solve().
+The general KVP functional may not always provide
+a (unique) stationary approximation, giving rise to spu-
+rious singularities known as Kohn (or Schwartz) anoma-
+lies [29, 30]. The energies at which those anomalies occur
+depends on the training parameters θ used in the offline
+stage and the evaluation set used in the online stage.
+Reference [16] proposed detecting and mitigating these
+numerical instabilities by considering an array of KVPs
+with different boundary conditions (i.e., scattering ma-
+trices) within a partial wave and using the emulator so-
+lutions to obtain an estimated S matrix by a weighted
+sum of averages [see also Refs. [32, 41]].
+For our KVP emulator, the mitigation process involves
+first calculating Eq. (5) using the K matrix boundary
+condition. Once we have calculated ∆�U, the terms in
+Eq. (4) are rescaled to match the boundary conditions
+we want to emulate (here, L = K, K−1, and T). The
+anomalies are then detected by applying a consistency
+check to the (independent) emulated solutions of the dif-
+ferent boundary conditions. The emulator solutions that
+do not pass this check are discarded while those that
+pass are averaged to obtain an anomaly-free scattering
+matrix (here, the S matrix). All KVP emulator results
+in this paper are shown with anomaly mitigation unless
+otherwise stated. So far, such a mitigation protocol has
+not been implemented for the NVP emulator. However,
+one approach would be to use multiple emulators based
+on different variational principles [21] instead of multiple
+boundary conditions. See Appendix A for our implemen-
+tation and Ref. [16] for more information on emulation
+with arbitrary boundary conditions and ways to mitigate
+Kohn anomalies.
+B.
+Emulation of phase shifts
+We first apply the emulators to the uncoupled 1S0
+channel using Eq. (5) to calculate ∆�U (see Appendix C
+for explicit expressions).
+At N4LO+, this channel de-
+pends on na = 3 non-redundant LECs [37], and thus
+we choose our basis to be composed of nb = 6 training
+points. Figure 1 shows the phase shifts calculated using
+our simulator (black line) and the KVP emulator Stan-
+FIG. 1. Simulated (black solid line) and KVP emulated Stan-
+dard method (orange dots) 1S0 phase shifts for the N4LO+
+SMS potential with Λ = 450 MeV (top panel). The bottom
+panel shows the relative errors between the simulated and em-
+ulated phase shifts for the Gl¨ockle method (red dashed line),
+Standard method (blue solid line), and NVP emulator (green
+dotted line), respectively. The spike at Elab ≈ 270 MeV is
+due to the phase shift crossing zero.
+dard method prediction (orange dots) as a function of
+the laboratory energy in the top panel. The phase shifts
+associated with the training points are depicted by the
+light gray lines. In addition, the bottom panel shows the
+relative errors
+Rel. Error = 2
+����
+Simulator − Emulator
+Simulator + Emulator
+����
+(14)
+between the simulated and emulated phase shifts for
+the Gl¨ockle method (red dashed line), Standard method
+(blue solid line), and NVP emulator (blue dotted line).
+We find that our KVP emulator accurately reproduces
+the high-fidelity phase shifts over a large energy range
+for both methods, but the Standard method is much
+more accurate than the Gl¨ockle method.
+On average,
+the relative error for the Gl¨ockle method is on the order
+of ≈ 10−6 −10−5, while the Standard method has a rela-
+tive error on the order of ≈ 10−12 for the same basis size.
+The NVP emulator’s relative error is similar to the KVP
+Standard method, with an error of ≈ 10−13.
+We now turn to the coupled 3S1–3D1 channel. This
+channel depends on na = 6 non-redundant LECs [37]
+at N4LO+, which means that our basis will be com-
+posed of nb = 12 training points. Figure 2 shows the
+on-shell K matrix for the simulator calculation (black
+lines) and KVP emulator prediction (orange dots) as
+a function of the laboratory energy for each different
+partial-wave component. The errors are similar to the
+
+Basis
+- Simulator
+oooEmulator
+200
+[deg]
+100
+0
+-100
+100
+ Glockle
+Standard
+NVP
+4
+10
+Error
+0
+100
+200
+300
+Eiab
+[MeV]6
+FIG. 2. As in Fig. 1, but for the on-shell K matrix in the coupled 3S1–3D1 as a function of the laboratory energy. From left
+to right: pure D–wave, pure S–wave, and mixed S–D-wave component.
+1S0 channel, with the Standard method being much more
+accurate than the Gl¨ockle method, and the NVP emula-
+tor’s relative error being slightly better than the Stan-
+dard method. In all cases, we see a spike in the relative
+error at Elab ≈ 20 MeV where the K matrix is singular.
+The small spikes seen in the Standard method error are
+not Kohn anomalies, but can be attributed to a numerical
+instability of the principal value integral in the LS equa-
+tion. These spikes are mesh-dependent and appear when
+a k0 value is close to a momentum mesh point, thus caus-
+ing the denominator of the Green’s function to approach
+zero faster than the numerator. A way to decrease the
+relative error produced by these spikes is to not allow the
+k0 values to be close to momentum mesh points by mov-
+ing energies that are close to any momentum mesh point
+until the relative distance is greater than some threshold
+value; e.g., ε ≳ 10−2 MeV (see Appendix D for details).
+The oscillations that appear in the Gl¨ockle method’s rel-
+ative errors plots are potential-dependent, and increase
+in number, but decrease in separation, when increasing
+the mesh size.
+Overall, the emulators accurately predict the partial
+waves for the uncoupled 1S0 and coupled 3S1–3D1 chan-
+nels.
+When comparing the Gl¨ockle method emulation
+with the Standard method, we see that the relative error
+for the Standard method is much less than the Gl¨ockle
+method. For both partial waves shown, the NVP emula-
+tor is the one that most accurately reproduces its high-
+fidelity solution. Results for the other channels are sim-
+ilar to the ones presented here, with the only difference
+being that the relative error decreases as na gets smaller.
+This can be further explored with the Jupyter notebooks
+provided [31].
+C.
+Emulation of scattering observables
+Next, we examine the performance of the emulator for
+nuclear observables. As a demonstration, we use the SMS
+regularized chiral potential at N4LO+ for np scatter-
+ing with partial waves having total momentum quantum
+numbers j ⩽ jmax = 20. Overall, there are a total of 25
+parameters in θ that are being sampled using a Latin-
+hypercube design.
+As previously mentioned, the basis
+size is chosen as nb = 2na, where na is the number of
+LECs associated with the specific partial-wave, for a to-
+tal of 50 training points. Since these parameters are only
+present in the channels j ⩽ 4, the emulator only needs to
+be trained over these channels. The remaining channels
+do not change as the parameters are varied, therefore,
+they do not undergo a training process and need to be
+calculated only once by solving the LS equation directly.
+The emulation of observables is carried out by com-
+bining multiple emulators across different partial-wave
+channels. The total np cross section can be calculated
+using
+σtot(k0) =
+π
+2k2
+0
+jmax
+�
+j=0
+(2j + 1) Re{Tr[Sj(k0) − 14]},
+(15)
+where Sj = 14 − 2i(1 − iKj)−1Kj is the S matrix, Kj
+is the predicted on-shell K matrix, and Tr[·] denotes the
+trace. Both Sj and Kj are 4 × 4 matrices that contain
+both the triplet-triplet and the singlet-triplet channels.
+Figure 3 shows the simulator and emulator prediction
+for the total np cross section, which are calculated us-
+ing the fit values for the LECs determined in Ref. [37].
+The inset in Fig. 3 depicts the mean relative errors for
+all three emulators when randomly sampling 500 differ-
+ent combinations of np LECs (chosen within the same
+range as the training points), using these to calculate the
+emulated and simulated total cross section, and compar-
+ing the results.
+On average, the relative errors for all
+three emulators are similar to those for the partial-wave
+calculations discussed in Sec. III B. Although the mean
+relative errors for the Standard method and NVP emula-
+tors are very similar, the NVP emulator seems to be the
+one that most accurately reproduces its simulator.
+
+Basis
+1
+K
+K+
+Simulator
+(oy)M
+·o Emulator
+IOOOI
+Glockle
+2
+10°
+Standard
+NVP
+10
+Rel.
+10-16
+0
+100
+200
+300
+0
+100
+200
+300
+0
+100
+200
+300
+Eiab [MeV]
+Eiab[MeV]
+Eiab [MeV]7
+FIG. 3.
+Simulated (black solid line) and emulated (orange
+dots) np cross section with jmax = 20 for the N4LO+ SMS
+potential with Λ = 450 MeV as a function of the laboratory
+energy.
+The inset shows the relative mean errors between
+the emulator and the simulator using the Gl¨ockle, Standard
+method, and NVP emulator for 500 different sets of np LECs
+obtained from Latin-hypercube sampling. See the main text
+for details.
+As mentioned in Sec. III A and following Ref. [16],
+the Kohn anomalies found in the calculation were mit-
+igated by emulating with different boundary conditions
+and building the estimated S matrix. Figure 6 in Ap-
+pendix D shows a total cross section emulation result
+with one boundary condition, hence no anomaly mitiga-
+tion. From the figure, we see that anomalies contribute
+to the Standard method mean relative error at higher en-
+ergies with a magnitude of approximately 10−3. These
+spikes are reduced to approximately 10−9 with mitiga-
+tion. The Gl¨ockle method result exhibits anomaly con-
+tributions of order 10−3 at lower energies, which get re-
+duced to approximately 10−5–10−7 with mitigation. For
+additional information, see the discussion in Appendix D.
+Although the NVP emulator is subject to anomalies, they
+are not evident in the figures shown in this section, even
+though no mitigation strategy was applied. An example
+of noticeable anomaly contributions as large as 10−3 in
+the NVP emulation are seen in Fig. 7 in Appendix D.
+The remaining spikes in Fig. 3 (e.g., at Elab
+≈
+140 MeV) can be traced back to singularities in the on-
+shell K matrix for the 3S1–3D1 channel at those energies
+and are only seen for a few (specific) LEC values out
+of the 500 sampled (see also Fig. 2). The mesh-induced
+spikes seen in the Standard method relative error were
+also reduced in magnitude by preventing the on-shell k0
+value from being too close to a momentum mesh value
+(see Fig. 8 for result comparisons).
+We now turn our attention to spin-dependent observ-
+ables for non-identical particles. A detailed description
+of NN observables and their different conventions can be
+found in Refs. [35, 42–46]. In general, one can write the
+spin observables in terms of Saclay parameters, which are
+complex functions of the center-of-mass energy and an-
+gle θ. Here we only consider the differential cross section
+and analyzing power:
+dσ
+dΩ = 1
+2
+�
+|a|2 + |b|2 + |c|2 + |d|2 + |e|2 + |f|2�
+,
+(16)
+dσ
+dΩAy = Re(a∗ e + b∗ f),
+(17)
+where dσ/dΩ is the unpolarized differential cross sec-
+tion and Ay the analyzing power (also known as Pb).
+For identical particles, one has f = 0. More informa-
+tion on the description of the spin observables can be
+found in Refs. [44, 45]; see also Appendix D, which con-
+tains our emulation results for more spin observables.
+The Saclay parameters can be obtained from the spin-
+scattering M = M(θ, φ) matrix written in singlet-triplet
+space,
+M =
+�
+�
+�
+�
+M11
+M10e−iφ M1−1e−2iφ MST e−iφ
+M01eiφ
+M00
+M0−1e−iφ
+0
+M−11e2iφ M−10eiφ
+M−1−1
+MST eiφ
+MST eiφ
+0
+−MST e−iφ
+MSS
+�
+�
+�
+� ,
+(18)
+where the subscripts SS and ST represent the singlet-
+singlet and singlet-triplet channel,
+respectively [43].
+Equation (18) can be calculated using spherical harmon-
+ics and Clebsch-Gordan coefficients, and can be related
+to the Saclay parameters from the expressions:
+a = 1
+2(M11 + M00 − M1−1),
+(19)
+b = 1
+2(M11 + Mss − M1−1),
+(20)
+c = 1
+2(M11 − Mss − M1−1),
+(21)
+d = −
+1
+√
+2 sin θ(M01 + M01),
+(22)
+e = i
+2(M10 − M01),
+(23)
+f = −i
+√
+2MST .
+(24)
+The emulation process is performed similarly to the
+one for the total cross section, where multiple trained em-
+ulators are combined across different partial-wave chan-
+nels. Figures 4 and 5 show the simulator and emulator
+prediction for the differential cross section and analyzing
+power at three different energies calculated using the fit
+values for the LECs determined in Ref. [37]. The relative
+mean errors shown are obtained by randomly sampling
+500 different combinations of np LECs (the same LECs
+used for the sampled relative error calculation in Fig. 3)
+and comparing them against their respective simulator
+
+Standard
+Glockle
+NVP
+Error
+10-
+Mean Rel.
+10-7
+mb
+11
+10
+Otot
+-15
+10-
+0
+100
+200
+300
+Eiab[MeV]
+10
+01010101010101010
+Simulator
+Emulator
+000
+101
+0
+100
+200
+300
+Eiab[MeV]8
+FIG. 4. Simulated (solid lines) and emulated (dots) unpolar-
+ized differential cross section for the N4LO+ SMS potential
+with Λ = 450 MeV as a function of the center-of-mass angle
+at the three energies 60, 160, and 320 MeV (top panel). The
+bottom panel shows the mean relative errors between the em-
+ulators and their respective simulators for 500 different sets
+of np LECs obtained from Latin-hypercube sampling. The
+colors for the relative mean errors correspond to the energies
+in the top panel. The gray arrows point from the label asso-
+ciated with the emulator to its error. See the main text for
+details.
+calculation. On average, the spin observables emulator
+has a relative mean error on the order of ≈ 10−5 when
+employing the Gl¨ockle method and ≈ 10−14–10−11 when
+using the Standard method and NVP emulators, which
+are similar to the total cross section results. The results
+are similar to those obtained over the entire energy grid
+and for other observables (see Appendix. D).
+Table II details the angle-averaged relative errors be-
+tween the simulator and KVP emulators (based-10 log-
+arithm) for different spin observables with varying ba-
+sis size at a variety of energies. As can be seen, when
+training the emulator with basis size nb = na both the
+Standard and Gl¨ockle method emulators have large rela-
+tive errors of roughly 10−1 when compared to the high-
+fidelity model calculation. However, if we increase the
+basis size by doubling the parameters used per partial-
+wave, nb = 2na, the relative mean errors are signifi-
+cantly smaller, roughly 10−12–10−9 and 10−6–10−3, re-
+spectively.
+According to Ref. [47], the relative errors
+given by nb = 2na are below experimental uncertainties.
+When increasing the basis size to nb = 4na, the mean er-
+rors have mostly saturated and the improvement in accu-
+racy is insignificant compared to the basis size nb = 2na.
+Note that although only four energies are shown, these
+results are similar over the entire energy grid.
+FIG. 5. As in Fig. 4, but for the analyzing power Ay (also
+known as Pb). See the main text for details.
+The speed-up between the emulators and the simu-
+lator is highly implementation dependent (e.g., to-be-
+considered factors are the desired accuracy, idiosyncrasies
+of the solver, programming language, level of paralleliza-
+tion, hardware, etc.).
+The emulator speed-up will de-
+pend on the size of the quadrature mesh used by the
+simulator to obtain the high-fidelity solution. For repro-
+ducing the total cross section using the NVP emulator,
+Ref. [17] states an emulator speed-up factor of > 300x
+faster than the simulator in CPU time. When doubling
+the quadrature mesh size this factor becomes > 1000x.
+When comparing the KVP and NVP emulator speeds
+using one boundary condition (no anomaly checking) for
+the 1S0 uncoupled partial wave, the KVP emulator is
+slightly slower due to the Lagrange multiplier in Eq. (6)
+and numerical operations needed to solve Eq.(4). Mitiga-
+tion of Kohn anomalies (by emulating multiple boundary
+conditions) will further contribute to slowing down the
+KVP emulator, or any other emulator.
+IV.
+SUMMARY AND OUTLOOK
+We showed that the coordinate space KVP emulator
+for NN scattering [15, 16] can be extended to momen-
+tum space and coupled channels, and demonstrated its
+efficiency in accurately reproducing phase shifts and np
+observables using a modern chiral interaction at N4LO+.
+In addition, we provided two methods to implement the
+emulator, with the Gl¨ockle spline interpolation method
+having a faster offline stage, but less accurate online stage
+than the Standard method. By emulating (independent)
+scattering solutions associated with different asymptotic
+
+60 MeV
+160 MeV
+320 MeV
+15.0
+do/d2[mb/sr]
+10.0
+5.0
+0.0
+Glockle/NVP/Standard
+Error
+Mean Rel.
+10
+10-15
+0
+50
+100
+150
+Ocm [deg]60 MeV
+160 MeV
+320 MeV
+0.5
+0.2
+9
+0.0
+-0.2
+Glockle/NVP/Standard
+Error
+10
+Tean Rel.
+10
+10
+0
+50
+100
+150
+Ocm [deg]9
+TABLE II. Comparison of the angle-averaged relative errors (base-10 logarithm) between high-fidelity model and emulator
+for various angular observables with different basis size for 500 sets of np LECs using the N4LO+ SMS potential [37] with
+momentum cutoff Λ = 450 MeV (rounded to two significant figures). These results are similar over the entire energy mesh.
+Here, “Std.” refers to the Standard method emulator. See the main text for details.
+dσ/dΩ
+D
+Ay
+Ayy
+A
+Basis size
+E [MeV]
+Std.
+Gl¨ockle
+Std.
+Gl¨ockle
+Std.
+Gl¨ockle
+Std.
+Gl¨ockle
+Std.
+Gl¨ockle
+nb = na
+5
+−1.2
+−1.2
+−0.93
+−0.93
+−0.46
+−0.46
+−0.72
+−0.72
+−0.78
+−0.78
+100
+−0.73
+−0.73
+−0.47
+−0.47
+−0.12
+−0.12
+−0.20
+−0.20
+−0.28
+−0.28
+200
+−0.54
+−0.64
+−0.30
+−0.30
+−0.028
+−0.028
+−0.035
+−0.035
+−0.12
+−0.12
+300
+−0.49
+−0.49
+−0.24
+−0.24
+−0.066
+−0.066
+−0.037
+−0.037
+−0.043
+−0.043
+nb = 2na
+5
+−10
+−7.0
+−8.8
+−6.1
+−8.8
+−5.6
+−8.5
+−5.8
+−8.3
+−5.9
+100
+−12
+−6.3
+−11
+−5.1
+−10
+−4.9
+−10
+−4.9
+−11
+−5.3
+200
+−10
+−4.0
+−8.8
+−3.2
+−7.8
+−2.7
+−8.4
+−2.9
+−8.0
+−3.0
+300
+−12
+−4.9
+−11
+−4.0
+−11
+−3.9
+−9.9
+−3.8
+−11
+−3.9
+nb = 4na
+5
+−10
+−7.3
+−8.8
+−6.4
+−8.8
+−6.1
+−8.5
+−6.4
+−8.3
+−6.1
+100
+−13
+−6.5
+−12
+−5.3
+−11
+−5.1
+−11
+−5.0
+−11
+−5.4
+200
+−10
+−4.4
+−9.3
+−3.6
+−8.5
+−3.0
+−8.8
+−3.3
+−8.8
+−3.3
+300
+−12
+−5.1
+−11
+−4.0
+−10
+−4.1
+−10
+−3.8
+−11
+−4.0
+boundary conditions in each partial wave and weighting
+the results (e.g., for the S-matrix), spurious singularities
+known as Kohn anomalies were successfully mitigated for
+the KVP-based emulators [16].
+We also constructed an NVP-based emulator and as-
+sessed how well the three emulators reproduced their re-
+spective high-fidelity solution for the 1S0 and 3S1–3D1
+partial-waves, total and differential cross sections, and
+analyzing powers. While all emulators produced errors
+well below experimental errors [47], the KVP Standard
+method and NVP emulators most closely reproduced the
+simulator, while the KVP Gl¨ockle spline interpolation
+emulator was overall the least accurate. The KVP emula-
+tor was found to have a slower online stage than the NVP
+emulator because it has to evaluate a higher-dimensional
+matrix and perform overall more numerical operations.
+We stress, however, that the emulators’ speed-ups are
+highly implementation dependent and should be further
+investigated. Extensions of the NVP-based emulator for
+anomaly mitigation with minimal computational cost,
+similar to the KVP-based emulators, should also be in-
+vestigated [17].
+An alternative procedure for mitigat-
+ing anomalies would be constructing the estimated S
+matrix using solutions from emulator based on differ-
+ent variational principles, as opposed to emulating mul-
+tiple boundary conditions. Reference [21] provides fur-
+ther perspectives regarding different emulators (KVP-
+and NVP-based included) and efficient offline-online de-
+compositions.
+Although we considered here only χEFT NN potentials
+for np scattering, the constructed emulators are gener-
+ally applicable to two-body scattering, including pp scat-
+tering and nuclear reactions with complex-valued opti-
+cal potentials.
+To help implement these fast & accu-
+rate scattering emulators in Bayesian parameter estima-
+tions, we provide self-contained set of codes that gener-
+ate all results and figures shown in this paper [31]. Fur-
+thermore, we have written a pedagogical introduction to
+projection-based emulators [21] with interactive, open-
+source Python code [22] to facilitate implementations of
+fast & accurate emulators even further. However, taking
+full advantage of emulators for UQ in nuclear scattering
+and reaction calculations will require generalizations to
+higher-body scattering and non-affine potentials. Recent
+advances in this direction are already promising [27].
+ACKNOWLEDGMENTS
+We thank Evgeny Epelbaum for sharing a code that
+generates the SMS chiral potentials, Kyle Wendt for shar-
+ing a code that generates the spin obbservables, and
+Filomena Nunes for fruitful discussions. This work was
+supported in part by the National Science Foundation
+Award Nos.
+PHY-1913069 and PHY-2209442 and the
+NSF CSSI program under award number OAC-2004601
+(BAND Collaboration [48]), and the NUCLEI SciDAC
+Collaboration under U.S. Department of Energy MSU
+subcontract RC107839-OSU.
+This material is based
+upon work supported by the U.S. Department of Energy,
+Office of Science, Office of Nuclear Physics, under the
+FRIB Theory Alliance award DE-SC0013617.
+Appendix A: Mitigating Kohn anomalies
+We follow the method developed in Ref. [16] to detect
+and mitigate Kohn anomalies (see also Ref. [32]). The
+
+10
+estimated S matrix is calculated from the emulator so-
+lutions by using a weighted sum of averages. Letting L1
+and L2 be two independent KVP functional solutions,
+this weighted sum is computed by first calculating the
+relative residuals
+γrel(L1, L2) = max
+������
+S(L1)
+S(L2) − 1
+�����,
+�����
+S(L2)
+S(L1) − 1
+�����
+�
+,
+(A1)
+for all emulated KVP solutions without repetitions to
+avoid the trivial case where L1 = L2. Using a consistency
+check, γrel < ϵrel, with ϵrel = 10−1, we select the set
+of pairs P = {(L1, L2)} that satisfies this check. If at
+least one consistency check passes, the S matrix is now
+estimated by the weighted sum of averages
+[S](mixed)
+KVP
+=
+�
+(L1,L2)∈P
+ω(L1, L2)S(L1) + S(L2)
+2
+,
+(A2)
+ω(L1, L2) =
+γrel(L1, L2)−1
+�
+(L′
+1,L′
+2)∈P γrel(L′
+1, L′
+2)−1 .
+(A3)
+If no consistency check passes, one could change the ba-
+sis size to shift the position of the Kohn anomalies in the
+parameter space. However, we found that using Eq. (A2)
+was sufficient to mitigate Kohn anomalies in our appli-
+cations.
+We first calculate Eq. (5) using Eq. (7), then rescale
+Eq. (5) using the relations from Appendix B of Ref. [16],
+∆�U (u′) = C
+′−1(Li) C
+′−1(Lj) det u
+det u′ ∆�U (u),
+(A4)
+C′(L) = det u
+det u′
+u′
+11 − u′
+10K(L)
+u11 − u10K(L).
+(A5)
+Here, u and u′ are nonsingular matrices parameterizing
+the scattering boundary conditions; the K, K−1, and T
+scattering matrices, respectively, are given by
+uK =
+�
+1 0
+0 1
+�
+,
+uK−1 =
+�
+0 1
+1 0
+�
+,
+uT =
+�
+1 0
+i 1
+�
+.
+(A6)
+The u matrix parameterizes the initial boundary condi-
+tion associated with L, while the u′ parameterizes the
+final boundary condition associated with L′.
+The snapshots used in the emulator’s offline stage are
+transformed using the M¨obius transform [16]
+L′(L) = −u′
+01 + u′
+00K(L)
+u′
+11 − u′
+10K(L) .
+(A7)
+Once we obtain an emulator solution, we transform that
+solution back into its K matrix form using
+K(L) = u01 + u11L
+u00 + u10L.
+(A8)
+For the estimated S calculation, the KVP solution
+pairs (L1, L2) being evaluated are the K matrix solu-
+tions obtained from the different boundary conditions
+used [e.g., γrel(K(K), K(K−1)), γrel(K(K), K(T)), and
+γrel(K(K−1), K(T))]. See Ref. [16] for more details.
+Appendix B: Formalism details
+Here we provide clarifying remarks about how Eq. (4)
+arises in the coupled case.
+In particular, we focus on
+two questions about the specific manner in which the
+coefficients ⃗β enter into Eq. (4).
+Why can Lss′ be emulated separately for each ss′ pair
+rather than with one global set of coefficients for the cou-
+pled block?
+For uncoupled channels, each partial wave is inde-
+pendent of one another, thus they can be emulated
+individually using trial wave functions and coefficients
+that are specific to the channel under consideration.
+Without loss of generality, let us consider two uncoupled
+channels labeled as s = 0 and s = 1, and let ⃗β(0)
+and ⃗β(1) denote the independent sets of coefficients
+found by making each channel’s KVP stationary.
+To
+move toward the coupled regime, imagine adiabatically
+turning on the coupling between these two originally
+uncoupled channels.
+The coefficients for each channel
+should remain nearly fixed to their previously uncoupled
+values, but the coupling will introduce a new set of
+coefficients ⃗β(01) ̸= ⃗β(0) ̸= ⃗β(1) that must be determined.
+Hence, each independent channel in the coupled case
+will have its own set of coefficients. Attempting to force
+a global set of coefficients for a coupled system would be
+inconsistent with the treatment in the uncoupled case
+and also degrade accuracy in general. A more technical
+answer follows from the (Petrov-)Galerkin procedure
+described below.
+Should not each of |ψs′⟩ and ⟨ψs| have its own basis
+expansion with their own independent coefficients?
+No, there is only one set of coefficients that en-
+ter quadratically in Eq. (4).
+A way of understanding
+how the coefficients enter in Eq. (4) follows from the
+(Petrov-)Galerkin orthogonalization procedure (see also
+Ref. [21]). Rather than starting with a variational prin-
+ciple, the (Petrov-)Galerkin approach starts with the
+Schr¨odinger equation. Like the variational approach, it
+expands |ψs′⟩ as a linear combination of known functions,
+but determines the basis coefficients by enforcing orthog-
+onality against a set of test functions. For the diagonal
+channels, the test functions are chosen to have the same
+exit channel as the trial functions (standard Galerkin ap-
+proach). On the other hand, the test functions for the
+off-diagonal channels are chosen to have a different exit
+channel (s) than the trial functions (s′) (Petrov-Galerkin
+approach). The resulting set of linear equations is equiv-
+alent to those that follow from making the KVP station-
+ary for each combination of (s′, s) independently. Thus
+by following the (Petrov-)Galerkin procedure we can de-
+termine how the coefficients are to enter in Eq. (4).
+This discussion will follow closely that of Ref. [21],
+however using coupled-channel notation and more gen-
+eral boundary conditions consistent with the general
+
+11
+KVP. Starting from (the strong form of) the Schr¨odinger
+equation
+�H(θ) |ψs′⟩ = E |ψs′⟩ ,
+(B1)
+we can derive its weak form after multiplying by a test
+function ⟨ψs|
+⟨ψs| �H(θ) − E|ψs′⟩ = 0.
+(B2)
+This can be considered a Petrov-Galerkin approach be-
+cause s ̸= s′ in general. The boundary conditions can be
+made explicit via the relationship
+0 = ⟨ψs| �H(θ) − E|ψs′⟩
+= ⟨ψs| �H†(θ) − E|ψs′⟩ −
+�
+t
+W(rψts, rψts′; r)
+2µ
+�����
+∞
+r=0
+,
+(B3)
+where �H† denotes the operator acting to the left (via
+integration by parts) and where we have used ψts(r) =
+⟨rt|ψs⟩ = ⟨ψs|rt⟩ and defined the Wronskian
+W(φ, ψ; r) ≡ φ(r)ψ′(r) − φ′(r)ψ(r).
+(B4)
+The wave function rψ vanishes at the origin, so that only
+the limit as r → ∞ contributes. By adding Eqs. (B3)
+and (B2), we have
+⟨ψs| �H(θ) − E|ψs′⟩ + ⟨ψs| �H†(θ) − E|ψs′⟩
+=
+�
+t
+W(rψts, rψts′; r)
+2µ
+�����
+∞
+r=0
+.
+(B5)
+This is the weak form for general |ψs′⟩ and ⟨ψs|. We can
+arrive at the discrete form by inserting basis states |ψs
+i ⟩
+that satisfy the asymptotic boundary conditions
+ψst(r) −−−→
+r→∞ δst ¯φ(0)
+s (r) + Lst ¯φ(1)
+s (r) ,
+(B6)
+where
+�
+¯φ(0)
+ℓ (r)
+¯φ(1)
+ℓ (r)
+�
+∝
+�
+u00 u01
+u10 u11
+� �
+jℓ(qr)
+ηℓ(qr)
+�
+.
+(B7)
+With this substitution, we have, for i ∈ [1, nb],
+∆�U ss′
+ij βj = Lss′
+i
+�
+j
+βj − Ls′s
+j βj,
+(B8)
+where the expression for ∆�U ss′
+ij
+is given by Eq. (5). We
+must now implement the constraint �
+j βj = 1, which is
+performed here by a Lagrange multiplier λ mimicking a
+variational approach (see Ref. [19] for details):
+λ + ∆�U ss′
+ij βj = Lss′
+i
+�
+j
+βj − Ls′s
+j βj.
+(B9)
+The sum multiplying Lss′
+i
+can be evaluated using the
+constraint �
+j βj = 1, and we can make the redefinition
+λ′ ≡ λ + �
+j βjLs′s
+j
+without impacting the solution be-
+cause this term does not depend on i. Thus, we have
+λ′ − ⃗L(E) + ∆�U ⃗β⋆ = 0,
+(B10)
+which is exactly Eq. (6) found by making the KVP sta-
+tionary. This simplification can be understood by not-
+ing that if {⃗β⋆, λ⋆} satisfy Eq. (B9), then we know that
+{⃗β⋆, λ′
+⋆} is the unique solution to Eq. (B10). Therefore,
+we can solve Eq. (B10) to obtain ⃗β⋆ rather than Eq. (B9).
+In conclusion, using the Petrov-Galerkin projection of the
+homogeneous Schr¨odinger equation with trial and test
+bases of |ψs′
+i ⟩ and ⟨ψs
+i |, respectively, we were able to ob-
+tain the same coefficients as the KVP in Eq. (6), which
+yield the same on-shell Lss′ matrix when used in Eq. (4).
+Appendix C: KVP emulator construction details
+For single channel scattering over a k × p momentum
+grid using the K matrix (det u = 1), Eq. (8) becomes
+∆�Uij(θ) =
+∞
+¨
+0
+dk dp k2p2�
+ψi(k)Vθ,j(k, p)ψj(p) + (i ↔ j)
+�
+,
+(C1)
+with Vθ,j(k, p) defined as in Eq (9). We drop the super-
+scripts for the uncoupled case since s′ = s. Note that ψi
+is not complex conjugated. For the Gl¨ockle method, one
+would simply substitute Eq. (7) into Eq. (C1) and inter-
+polate the solutions to the integrals with the cubic spline
+polynomials Sk(k0). For the Standard method, the Dirac
+delta distribution is analytically integrated; thus we ob-
+tain the following expression for ∆�Uij
+∆�Uij(θ) = Vθ,j(k0, k0) + 2
+π (I1
+ij + I2
+ij) + 4
+π2 I3
+ij + (i ↔ j),
+(C2)
+with I1
+ij, I2
+ij, and I3
+ij defined as
+I1
+ij = P
+∞
+ˆ
+0
+dk k2
+k0
+Ki(k0, k)
+k2 − k2
+0
+Vθ,j(k, k0),
+(C3)
+I2
+ij = P
+∞
+ˆ
+0
+dp p2
+k0
+Vθ,j(k0, p)Kj(p, k0)
+p2 − k2
+0
+,
+(C4)
+I3
+ij = P
+∞
+¨
+0
+dk dp k2p2
+k2
+0
+Ki(k0, k)
+k2 − k2
+0
+Vθ,j(k, p)Kj(p, k0)
+p2 − k2
+0
+.
+(C5)
+
+12
+If V has an affine dependence on the parameters θ,
+applying Eq. (11) and Eq. (12) produces
+∆�U 0
+ij =
+∞
+¨
+0
+dk dp k2p2�
+ψi(k)V 0
+j (k, p)ψj(p) + (i ↔ j)
+�
+,
+(C6)
+∆ �U 1
+ij =
+∞
+¨
+0
+dk dp k2p2�
+ψi(k)V 1(k, p)ψj(p) + (i ↔ j)
+�
+,
+(C7)
+with
+V 0
+j (k, p) ≡ 2µk0
+�
+V 0(k, p) − Vj(k, p)
+�
+.
+(C8)
+For coupled-channel interactions (s′ ̸= s), the details
+of the emulation are more complex. In this case, we apply
+Eq. (4) to each individual channel in a partial-wave, but
+the real difference lies in how Eq. (5) is calculated. The
+usual way of solving for the phase shifts and mixing angle
+for the coupled channels involves building a 2 × 2 block
+matrix for the potential,
+V =
+�
+V 00 V 01
+V 10 V 11
+�
+.
+(C9)
+The same process can be applied to the emulator calcu-
+lation when calculating Eq. (5),
+∆�U =
+�
+∆�U 00 ∆�U 01
+∆�U 10 ∆�U 11
+�
+.
+(C10)
+Each of the four blocks in ∆�U has a separate functional,
+although there are contributions from the different wave
+functions and potentials (e.g., for the 3S1–3D1 partial
+wave ∆�U 00 depends on the 3S1–3S1, 3S1–3D1, and 3D1–
+3D1 wave functions and potentials).
+Additionally, Eq. (7) tells us that we can consider the
+momentum-space wave function for the individual chan-
+nels ψst. Using Eq. (8) with Eq. (9), the functionals for
+the individual channels in a coupled-channel calculation
+(using the 3S1–3D1 as an example) will be
+∆�U ss′
+ij
+=
+¨ ∞
+0
+dk dp k2p2�
+∆uss′
+ij + (i ↔ j)
+�
+,
+(C11)
+with
+∆u00
+ij = ψ00
+i (V 00
+θ,jψ00
+j + V 01
+θ,jψ10
+j )
++ ψ10
+i (V 10
+θ,jψ00
+j + V 11
+θ,jψ10
+j ),
+(C12)
+∆u01
+ij = ψ00
+i (V 00
+θ,jψ01
+j + V 01
+θ,jψ11
+j )
++ ψ10
+i (V 10
+θ,jψ01
+j + V 11
+θ,jψ11
+j ),
+(C13)
+∆u10
+ij = ψ01
+i (V 00
+θ,jψ00
+j + V 01
+θ,jψ10
+j )
++ ψ11
+i (V 10
+θ,jψ00
+j + V 11
+θ,jψ10
+j ),
+(C14)
+∆u11
+ij = ψ01
+i (V 00
+θ,jψ01
+j + V 01
+θ,jψ11
+j )
++ ψ11
+i (V 10
+θ,jψ01
+j + V 11
+θ,jψ11
+j ),
+(C15)
+where we have suppressed the arguments for compact-
+ness.
+Note that the weights βi in Eq. (4) are differ-
+ent for each channel (i.e., ∆�U 00, ∆�U 11, and ∆�U 01 =
+∆�U 10), and are determined independently of one an-
+other. Once Eqs. (C12) through (C15) are calculated,
+the steps for the uncoupled channel calculation are ap-
+plied to each ∆�U ss′
+ij
+to obtain the emulator prediction,
+in particular Eqs. (C2) through (C5), and the separation
+of ∆�U ss′(θ) into parameter-dependent and parameter-
+independent pieces as described by Eq. (12).
+Appendix D: Additional results
+FIG. 6.
+As in Fig. 3, but only emulating with the K ma-
+trix.
+The mesh-induced spikes have been removed for this
+calculation.
+Figure 6 shows the relative mean error for the total
+cross section using only the K matrix boundary condi-
+tion. Comparing to Fig. 3, where we apply the weighted
+sum (mixed) S approach, we see that for one bound-
+ary condition the relative mean error has Kohn anoma-
+lies (see Elab ≈ 270 MeV and ≈ 315 MeV for the stan-
+dard method and Elab ≈ 40 MeV and ≈ 130 MeV for
+the Gl¨ockle method) and a more spread-out error. From
+Fig. 8 and comparing to Fig. 3 and 6, we conclude that
+the mixed S approach is indeed successful in mitigating
+the Kohn anomalies.
+Figure 7 shows the relative mean error for the to-
+tal cross section with momentum cutoff 550 MeV. The
+weighted sum (mixed) S approach is used for the KVP
+emulator results. Here, the anomalies found in the NVP
+emulation are noticeable.
+Figure 8 shows the relative errors for the KVP emula-
+tors in the 1S0 channel. The figure on the left shows the
+
+Standard
+Glockle
+NVP
+Error
+Mean Rel.
+10-7
+mb
+11
+10
+Otot
+0
+100
+200
+300
+Eiab[MeV]
+10
+101001010101010101
+Simulator
+Emulator
+000
+101
+100
+0
+200
+300
+Eiab [MeV]13
+FIG. 7.
+As in Fig. 3, but for cutoff Λ = 550 MeV.
+relative error when emulating with the K−1 boundary
+condition and the one on the right shows the weighted
+sum (mixed) S errors. In the figure on the left we can
+see a spike around Elab ≈ 65 MeV, which disappears
+when using the weighted sum S approach.
+This is a
+clear example of the weighted sum S approach helping to
+mitigate these anomalies. Additionally, there are other
+smaller mesh-induced spikes (i.e., not anomalies) present
+throughout the energy grid in the figure on the left that
+are not in the figure on the right. These were mitigated
+by not allowing the k0 values to be close to any mo-
+mentum mesh points. See Sec. III for a more detailed
+description.
+Figures 9 through 12 show emulator results for the
+following spin observables:
+dσ
+dΩD = 1
+2
+�
+|a|2 + |b|2 − |c|2 − |d|2 + |e|2 + |f|2�
+, (D1)
+dσ
+dΩA = − Re(a∗ b − e∗ f) sin(α + θ
+2)
++ Re(c∗ d) sin(α − θ
+2)
+− Im(b∗ e + a∗ f) cos(α + θ
+2),
+(D2)
+dσ
+dΩAxx = Re(a∗ d) cos(θ) + Re(b∗ c) − Im(d∗ e) sin(θ),
+(D3)
+dσ
+dΩAyy = 1
+2
+�
+|a|2 + |b|2 − |c|2 − |d|2 + |e|2 + |f|2�
+, (D4)
+where D is the depolarization parameter, A is the spin-
+flip amplitude, Axx and Ayy are the spin-correlation am-
+plitudes, and α a relativistic spin rotating angle that van-
+ishes in the non-relativistic case [8]. For identical parti-
+cles, f = 0. The results and conclusions are similar to
+those described in Sec. III C.
+Figure 13 shows emulator results for the total cross
+section for the N4LO+ SMS potential with momentum
+cutoff 550 MeV. The results and conclusions are similar
+to the ones described in the text for the 450 MeV mo-
+mentum cutoff (see Sec. III C).
+Figures 14 and 15 shows emulator results for the dif-
+ferential cross section and analyzing power Ay for the
+N4LO+ SMS potential with momentum cutoff 550 MeV.
+The results and conclusions are similar to the ones de-
+scribed in the text for the 450 MeV momentum cutoff
+(see Sec. III C). These results and conclusions also ex-
+tend down to momentum cutoff 400 MeV. The spin ob-
+servables at 500 MeV show larger errors on order of 10−7
+for the NVP emulator at particular energies, which may
+come from Kohn anomalies at one or more of the sam-
+pled parameter sets (see Fig. 7); nevertheless, the errors
+are still well below experimental uncertainties [47].
+
+Standard
+Glockle
+NVP
+010101010101
+Error
+Mean Rel.
+10-
+mb
+1
+10
+Otot
+0
+100
+200
+300
+Eiab[MeV]
+10
+Simulator
+Emulator
+000
+101
+0
+100
+200
+300
+Eiab [MeV]14
+FIG. 8. Relative error of the 1S0 channel for a basis size of nb = 2na + 1 for the N4LO+ SMS potential with Λ = 450 MeV
+as a function of the laboratory energy.
+The left panel shows the relative error for an emulator using the K−1 boundary
+condition. There is a Kohn anomaly at Elab ≈ 65 MeV for both the Standard and Gl¨ockle emulators and mesh-induced spikes
+present throughout the energy grid. The right panel shows the relative error for the mixed S-matrix approach presented by
+Reference [16] with care taken to avoid the k0 values that correspond with a mesh point as described in Sec. III B. When
+comparing both graphs, the Kohn anomaly is no longer present and the mesh-induced spikes are much smaller in the right
+panel.
+
+Glockle
+Standard
+Mixed S
+emu.
+R
+Error
+10°
+Rel
+10-9
+10-12
+LA
+10-15
+0
+50
+200 0
+50
+100
+200
+100
+150
+150
+Eiab [MeV]
+Eiab [MeV]15
+FIG. 9. As in Fig. 4, but for the depolarization D.
+FIG. 10. As in Fig. 4, but for the spin-flip amplitude A.
+FIG. 11. As in Fig. 4, but for the spin-correlation amplitude
+Axx.
+FIG. 12. As in Fig. 4, but for the spin-correlation amplitude
+Ayy.
+
+60 MeV
+160 MeV
+320 MeV
+1.0
+0.5
+0.0
+-0.5
+Glockle/NVP/Standard
+Error
+Tean Rel.
+10
+0
+50
+100
+150
+cm [deg]60 MeV
+160 MeV
+320 MeV
+1.0
+0.5
+0.0
+-0.5
+Giockle/NVP/Standard
+Error
+Tean Rel.
+10
+10-
+10-15
+0
+50
+100
+150
+Ocm [deg]60 MeV
+160 MeV
+320 MeV
+0.5
+0.0
+-0.5
+-1.0
+Glockle/NVP/Standard
+Error
+10
+Tean Rel.
+10
+10-15
+0
+50
+100
+150
+Ocm [deg]60 MeV
+160 MeV
+320 MeV
+1.0
+0.5
+0.0
+-0.5
+-1.0
+Glockle/NVP/Standard
+Error
+10
+Rel.
+10
+Mean
+11
+10°
+0
+50
+100
+150
+Ocm [deg]16
+FIG. 13.
+As in Fig. 3, but for cutoff Λ = 550 MeV.
+FIG. 14. As in Fig. 3, but for cutoff Λ = 550 MeV.
+FIG. 15. As in Fig. 3, but for cutoff Λ = 550 MeV.
+
+Standard
+Glockle
+NVP
+ Error
+Mean Rel.
+10-
+mb
+Otot
+0
+100
+200
+300
+Eiab [MeV]
+10
+0101001010101010
+Simulator
+Emulator
+000
+101
+100
+0
+200
+300
+Eiab[MeV]60 MeV
+160 MeV
+320 MeV
+15.0
+do/d2[mb/sr]
+10.0
+5.0
+0.0
+Glockle/NVP/Standard
+Error
+Mean Rel.
+10
+10-15
+0
+50
+100
+150
+Ocm [deg]60 MeV
+160 MeV
+320 MeV
+0.5
+0.2
+9
+0.0
+-0.2
+Glockle/NVP/Standard
+Error
+3
+10°
+Tean Rel.
+10
+10-
+10-15
+0
+50
+100
+150
+Ocm [deg]17
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diff --git a/9dE4T4oBgHgl3EQfdwwo/content/tmp_files/load_file.txt b/9dE4T4oBgHgl3EQfdwwo/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..84440f638b3c6abdecee6a1a6c0c22ec3d56fda0
--- /dev/null
+++ b/9dE4T4oBgHgl3EQfdwwo/content/tmp_files/load_file.txt
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+page_content=' Garcia ,1, ∗ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Drischler ,2, 3, † R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
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+page_content=' Furnstahl ,1, ‡ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Melendez ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
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+page_content=' ¶ 1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The Ohio State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Columbus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
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+page_content=' Ohio University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Athens,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
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+page_content=' Michigan State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' MI 48824,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' USA (Dated: January 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 2023) Emulators for low-energy nuclear physics can provide fast & accurate predictions of bound-state and scattering observables for applications that require repeated calculations with different param- eters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' such as Bayesian uncertainty quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' In this paper, we extend a scattering emulator based on the Kohn variational principle (KVP) to momentum space (including coupled channels) with arbitrary boundary conditions, which enable the mitigation of spurious singularities known as Kohn anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' We test it on a modern chiral nucleon-nucleon (NN) interaction, including emu- lation of the coupled channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' We provide comparisons between a Lippmann-Schwinger equation emulator and our KVP momentum-space emulator for a representative set of neutron-proton (np) scattering observables, and also introduce a quasi-spline-based approach for the KVP-based emula- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Our findings show that while there are some trade-offs between accuracy and speed, all three emulators perform well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Self-contained Jupyter notebooks that generate the results and figures in this paper are publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' INTRODUCTION Nucleon-nucleon (NN) scattering has long been used to fix parameters of microscopic Hamiltonians designed for ab initio few- and many-body calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' But the uncertainty in most existing nuclear models has been un- derestimated because they have lacked two key ingredi- ents: a rigorous accounting of Hamiltonian uncertainty and a complete estimate of parameter uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' In the case of chiral effective field theory (χEFT) [1– 4], Hamiltonian uncertainty manifests as a truncation er- ror, which has been statistically modeled in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [5–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' A holistic parameter estimation study would then both account for truncation errors in the likelihood, and esti- mate and propagate all plausible values of the low-energy constants (LECs) rather than finding a single parame- ter value maximizing the likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Bayesian statisti- cal methods are particularly suitable for these tasks [9– 13], but are computationally demanding, especially when generalizing to include few-body forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Emulators— surrogate models that allow for fast & accurate (but ap- proximate) model predictions—have the potential to al- leviate some of these demands [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' In this paper, we extend our recent explorations of emulators for NN scat- tering [15–17] to momentum-space wave functions and coupled channels, and test against a representative set of neutron-proton (np) scattering observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The demand for emulators has led nuclear physics to the general field of parametric model order reduction (PMOR), where the goal is to extract the relevant in- formation from a model while reducing the computa- ∗ garcia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='823@osu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='edu † drischler@ohio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='edu ‡ furnstahl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='1@osu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='edu § melendez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='27@osu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='edu ¶ zhangx@frib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='msu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='edu tional cost significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' An efficient offline-online de- composition is crucial to construct an efficient emula- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' In the offline stage, the emulator is trained with high-fidelity calculations1 for selected sets of parameters, also known as snapshots, while making predictions for any other set of parameters are performed in the online stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The end result is a reduced-order model (ROM) that serves as an emulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' For general overviews of the literature on PMOR techniques and their applications, we refer the reader to Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' A pedagogical intro- duction to projection-based emulators for both scatter- ing and bound-state calculations, including interactive, open-source Python code, can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' A particular snapshot-based ROM known as the re- duced basis method (RBM)2 has emerged as an efficient emulator for the prediction of both bound state and scat- tering observables [15, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The foundation of the first emulators for scattering is the Kohn variational princi- ple (KVP) [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=', for the K matrix], whose snapshots are based on scattering solutions to the Schr¨odinger equa- tion [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' It has been demonstrated for a variety of real and optical potentials that such emulators can be trained for two- and three-body3 scattering in coordinate space, then evaluated in the form of matrix inversions with low-dimensional matrices [15, 16, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Subsequently, an emulator of the Lippmann-Schwinger (LS) equation using the Newton variational principle (NVP) [28] was introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' In contrast to the KVP emulator, the variational trial basis is composed of 1 Following the terminology of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [18], we will refer to the cal- culational machinery that generates high-fidelity solutions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=', LS equation solver) as a simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 2 The RBM has been rediscovered in the low-energy nuclear theory community as eigenvector continuation (EC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' See Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [19] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 3 In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [27], the offline training stage involves calculations in both momentum and coordinate space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='05093v1 [nucl-th] 12 Jan 2023 ID2 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Notation used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Notation Description θ vector of parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' θi are the parameters for the ith snapshot s, s′ indices for the exit and entrance channels of the scattering process, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=', 3S1 and 3D1 t, t′ indices for available channels (summation convention implied) ψs i wave function in the channel s used for train- ing and associated with the ith snapshot with θi [high-fidelity solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (1)] �ψs snapshot-based trial wave function in the channel s (3) applied to the KVP func- tional (2) Lss′ E a generic scattering matrix at energy E Lss′[ �ψ] a functional whose stationary point is an ap- proximation of the generic L-matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=', L[ �ψ + δ �ψ] = Lss′ E + O(δL2) βi to-be-determined coefficient of the ith snap- shot in the trial wave function with � i βi = 1 ∆�U ss′ ij (θ) nb × nb kernel matrix defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (5) scattering matrices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=', K matrices) rather than scat- tering wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Both approaches were shown to quickly and accurately predict the np phase shifts from a chiral Hamiltonian across a range of parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' In this paper, we compare a momentum-space KVP-based emulator, including emulation of coupled channels and allowing for arbitrary boundary conditions, to the NVP emulator for a representative set of np observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' For a comparison of the KVP and NVP emulators in a Galerkin framework and a survey on other emulators see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' II, we review the underlying formalism of the KVP emulators and its extension to momentum space and coupled channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' We then show results for the momentum-space KVP emula- tor and compare them to the K matrix (NVP) emulator in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' We demonstrate that spurious singularities known as Kohn (or Schwartz) anomalies [29, 30] are mit- igated using methods from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Section IV has a summary and outlook and additional details of the im- plementation are given in several appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The self- contained set of codes that generate all results and figures shown in this paper is publicly available [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' FORMALISM Our goal is to emulate the partial-wave Schr¨odinger equation for NN scattering at the center-of-mass energy E > 0 �H(θ) |ψs⟩ ≡ � �T + �V (θ) � |ψs⟩ = E |ψs⟩ , (1) where the vector θ is composed of parameters used by the theoretical model to match results with experimen- tal observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=', the LECs of χEFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Building our snapshot-based MOR emulator begins by writing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (1) in integral form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Here we choose the general (con- strained4) KVP, which is based on the functional [16, 32] Lss′[ �ψ] = �Lss′ E − 2µk0 det u ⟨ �ψs| �H(θ) − E| �ψs′⟩ , (2) where �ψ is a trial scattering wave function, �Lss′ E is a generic trial scattering matrix, u is a non-singular ma- trix [16, 32] used to parameterize the asymptotic bound- ary condition associated with �Lss′ E (see Appendix A), and k0 = √2µE is the on-shell energy with µ being the re- duced mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='5 More details can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [16] and Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Table I summarizes the notation we use in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Note that we adopt the convention that the wave functions in a bra symbol ⟨·| in bra-ket notation are not complex conjugated [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=', ⟨ �ψs| in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (2)] [15, 16, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (2), the superscripts s and s′ index the coupled channels (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=', 3S1 and 3D1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' for the uncoupled case this reduces to a single equation with s′ = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Each combi- nation of (s′, s) will have their own, distinct emulator in our formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' As an example, for a coupled-channel np interaction in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (2), the (s′, s) pair could be one of 3S1–3S1, 3S1–3D1, 3D1–3S1, or 3D1–3D1, and for an uncoupled channel s′ = s could be 1S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' We use the np spin-triplet coupled channels as an exemplary case, but the general emulation procedure applies to general channel coupling (including spin-singlet spin-triplet np coupling [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The functional (2) yields Lss′[ �ψ] = Lss′ E when �ψ is the exact wave function, and provides a stationary approxi- mation otherwise: Lss′[ �ψ + δ �ψ] = Lss′ E + O(δL2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Rather than finding a wave function |ψ⟩ that satisfies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (1), our task has now changed to finding a wave function that makes Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (2) stationary for a given choice of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The key to creating an efficient PMOR emulator from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (2) is to use a snapshot trial wave function, | �ψs⟩ ≡ nb � i=1 βi |ψs i ⟩ , (3) where nb is the number of parameter vectors {θi}nb i=1 in the training set and {|ψs i ⟩}nb i=1 the associated high-fidelity solutions to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (1), obtained by solving the LS equation directly (see also Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' These solutions are determined once in the offline stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The to-be-determined basis co- efficients ⃗β will not be the same for all the channels, re- sulting in independent emulators for each (s′, s) pair (see Appendix B for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' For the np spin-triplet 4 For a description of constrained and unconstrained emulators see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [21] 5 Throughout this paper we use boldface symbols to indicate vec- tors in parameter-space, arrows to indicate vectors in snapshot- space, natural units in which ℏ = c = 1, and follow the conven- tions for scattering matrices in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [26, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 3 coupled channels, this will result in three distinct varia- tional principles being enforced: one for each of angular momentum s′ = s = j ± 1 and one for the off-diagonal component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The other off-diagonal component can be inferred through the unitarity of the S matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='6 Upon inserting the snapshot trial wave function (3) into the functional (2), the functional takes the form [15] Lss′[⃗β ] = βiLss′ E,i − 1 2βi∆�U ss′ ij βj, (4) with the symmetric matrix ∆�U ss′ ij (θ) ≡ 2µk0 det u � ⟨ψs i | �H(θ) − E|ψs′ j ⟩ + (i ↔ j) � = 2µk0 det u � ⟨ψs i |�V (θ) − �Vj|ψs′ j ⟩ + (i ↔ j) � , (5) where, as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (2), s′ and s correspond to the entrance and exit channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Equation (4) is a stationary approx- imation to the generic L-matrix at one energy, hence we build independent emulators for each value of an en- ergy grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Equation (5) is obtained [15] by adding and subtracting �Vi ≡ �V (θi) and �Vj ≡ �V (θj) and applying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' In this form, the constant terms in the poten- tials, such as a long-range Coulomb interaction (assuming the fine-structure constant is not varied), will cancel, and the matrix elements will only involve short-range physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Emulating the scattering wave function [via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (3)], and hence Lss′ E ≈ Lss′[ �ψ] [via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (4)], has now been re- duced to choosing an appropriate training set {θi} and then determining the values of βi that make Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (4) sta- tionary under the constraint that � i βi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The latter is a consequence of maintaining a consistent asymptotic normalization for the scattering wave functions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (3) as required by the constrained KVP [15, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' A numeri- cally robust solution can be found by introducing a La- grange multiplier λ, and solving the matrix equation [16] � ∆�U ss′ ⃗1 ⃗1 ⊺ 0 � �⃗β⋆ λ⋆ � = �⃗Lss′ E 1 � , (6) where ⃗1 is an nb × 1 vector of ones, ⃗Lss′ E are the basis states used in the offline stage, and ⃗β⋆ is a vector of co- efficients of the trial wave function associated with the KVP’s stationary approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (6) is a lin- ear system, it will be a highly computationally efficient emulator for scattering systems if the number nb of basis functions is much smaller than the size of the high-fidelity wave function ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Thus far we have not specified whether the matrix el- ements ∆�U ss′ ij are to be calculated in coordinate space or momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The only difference between these 6 For (complex-valued) optical potentials with two coupled chan- nels, one has four (instead of three) distinct variational principles because the S matrix is not unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' implementations is the way we obtain the basis functions ψi used to construct the trial ansatz in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (3), and thus the manner in which ∆�U ss′ is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' To formulate a momentum-space wave function approach to MOR emu- lators for scattering, we initially solve for the K matrix and relate ψ to K before using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The scattering wave function in momentum space takes the form [36] ψst(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' k0) = 1 k2 δ(k − k0)δst + 2 π PKst(k, k0)/k0 k2 − k2 0 , (7) which vanishes as k → ∞, but is singular at k = k0 = √2µE (the superscripts used for the K matrix in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (7) are opposite Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Here, Kst is the reactance ma- trix (or just the K matrix), k0 the on-shell energy, P the Cauchy principal value, and the labeling st indicates the partial-wave or reaction channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' One can also write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (5) in the momentum-space representation by insert- ing complete sets of states,7 resulting in ∆�U ss′ ij (θ) = ¨ ∞ 0 dk dp k2p2� ψts i (k)V tt′ θ,j(k, p)ψt′s′ j (p) + (i ↔ j) � , (8) with V tt′ θ,j(k, p) ≡ 2µk0 det u � V tt′(k, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' θ) − V tt′ j (k, p) � , (9) where t and t′ are summed over the available channels and the dependence of ψ on k0 is left implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Moving forward, we will drop the channel superscripts on ∆�U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' This is the general form of the momentum-space ∆ ˜U matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Note the ordering of the channel indices (t, s) in the left-hand wave function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (8), which follows from ψts(k) ≡ ⟨kt|ψs⟩ and the convention that ⟨ψ| = |ψ⟩⊺ (without a complex conjugate), so that ψts(k) = ⟨ψs|kt⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Thus, if ψ has outgoing (ψ(+)), incoming (ψ(−)), or standing wave (ψ(0)) boundary conditions, then the same version of ψ(x) is used for both ψ(k) and ψ(p) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' No modification of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (8) is needed in the case of optical potentials, where again the left-hand wave function is not conjugated relative to the right-hand wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' For more details on how to build the general KVP emulator we refer the reader to Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Different boundary conditions will be used below to mitigate Kohn anomalies (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' III B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The efficient evaluation of ∆�U across a range of θ val- ues is critical to the applicability of the emulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' If the Hamiltonian operators have an affine (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=', factorizable) parameter dependence, denoted as �H(θ) = � n hn(θ) �Hn, (10) 7 For example, for np scattering as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' III, the complete set of states are relative-momentum partial-wave states with orbital angular momentum and spin coupled to total J and MJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 4 then matrix elements of the Hn operators in a given basis only need to be calculated once in the offline stage rather than for every parameter set θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Chiral NN interactions have the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (10) and, when varying only the contact LECs, can even be cast into the form8 �V (θ) = �V 0 + θ · �V 1, (11) so that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (5) can then be written as ∆�U(θ) = ∆�U 0 + θ · ∆ �U 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (12) The matrices �V 0 and ∆�U 0 and vectors of matrices �V 1 and ∆ �U 1, can now be pre-calculated during the emu- lator’s offline stage, allowing for considerable speed-up factors in the online stage where the value of ∆�U(θ) at any new parameter value is efficiently constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' RESULTS In this section, we apply the KVP momentum-space emulator to calculate np scattering observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' We use the Reinert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' semilocal momentum-space (SMS) reg- ularized chiral potential at N4LO+ with the momentum cutoff Λ = 450 MeV [37], which is a state-of-the-art chiral NN interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The parameters θ are composed of the NN contact LECs contributing to this potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Emulator overview The snapshots used in the offline stage are the scatter- ing solutions given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The K matrices used to calculate the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (7) are obtained from numerically solving the LS equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The LS equation is reduced to a set of linear equations by approximating the integral as a sum over N quadrature points obtained from a Gauss–Legendre rule with corresponding weights (see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [36, 38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' If the potential was calculated merely on the quadrature points, without appending the on-shell values, interpolation must be performed to obtain the (half-)on-shell potential so that one can (1) account for the singularity of the Green’s function when solving the LS equation [38], and (2) integrate the delta distribution in Eq (7) (explained in next paragraph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' To generate the figures in this paper, we use a compound Gauss-Legendre quadrature mesh of N = 80 momentum points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' For the observables, we use a lab energy range of 0 to 350 MeV with 350 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' For the partial waves plots, we use a fine 8 Note that hn(θ) would include higher-order polynomials when also emulating the pion-nucleon coupling c2 (at N3LO) and axial coupling constant gA (already at LO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Nevertheless, the Hamil- tonian remains affine and thus the emulators discussed here are directly applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' energy mesh of 3500 points over the same energy range previously mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' When performing the KVP emulation, we calculate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (5) two different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The first is by inserting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (7) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (5) and analytically integrating the delta distribution, which corresponds to appending the exact on-shell value of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The remaining integrals are then solved numerically (see Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' We re- fer to this method as the Standard method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The second is based on the global Gl¨ockle spline interpolation [39], which belongs to the family of quasi-spline methods that perform the mapping � k f(k)Sk(k0) ≈ f(k0), (13) for smooth functions f(k) sampled on a grid k that en- compasses k0 using the cubic spline polynomials Sk(k0) constructed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' This allows us to calculate Sk(k0) once in the offline stage and save the result for the online stage since it has no dependence on f(k) it- self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Using this method, we interpolate the solutions to the integrals that appear in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (5) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=', k0 does not need to be appended to the mesh as opposed to the Standard method), thus decreasing the computational cost needed in the offline stage significantly at the expense of accu- racy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' We compare the KVP emulator results using the Gl¨ockle and Standard method and compare those results to the NVP emulator described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' To reduce numerical errors in both the simulator and emulator, we compute snapshots {Ki} of the LS equation using non-interpolated potentials for partial waves that have a LEC-dependence and interpolated potentials for LEC-independent partial waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' When referring to in- terpolated potentials, we mean calculating the potential using only the momentum mesh and then using an in- terpolation method (such as the bivariate Gl¨ockle spline method) to interpolate the potential to k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' By non- interpolated, we mean that each k0 is appended to the momentum mesh and the potential evaluated at these points, which improves the accuracy of our potentials compared to interpolating the potential to k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' We chose to use non-interpolated potentials for the LEC-dependent partial waves since these are the only ones used to cal- culate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (5) in the offline phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The same LEC- independent partial waves are employed by the simulator and emulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' All potentials used for the emulators and simulator are pre-calculated for efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The simulator used in this paper numerically solves the LS equation for each partial wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The accuracy of our simulator was tested by comparing the simulator results to the analytical solution of a Gaussian separable potential, producing relative errors of ≈ 10−7 or better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Additionally, the simulator’s speed was roughly 4x slower when we doubled the mesh size from N = 80 to N = 160 quadrature points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The accuracy of emulated observables depends on the size of the basis (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' III C);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' here we use a basis size nb = 2na, where na is the number of LECs associated 5 with a given partial wave channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The training points θi are randomly sampled within an interval of [−5, 5] using a Latin-hypercube for each partial wave, with the fitted coupling constants and appropriate units given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The matrix ∆�U is increasingly ill-conditioned as the basis size nb increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' One can reduce numerical noise by (1) adding a regularization parameter (“nugget”) to the diagonal elements of the near-singular matrix [15], or (2) using a solver that performs some type of regulariza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' For the KVP emulator results in the figures, we use NumPy’s least-squares solver linalg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='lstsq() [40] with a cut-off ratio for small singular values of 10−10 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' For the NVP emulator, we add a nugget of 10−10 to the di- agonal and use NumPy’s linalg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='solve().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The general KVP functional may not always provide a (unique) stationary approximation, giving rise to spu- rious singularities known as Kohn (or Schwartz) anoma- lies [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The energies at which those anomalies occur depends on the training parameters θ used in the offline stage and the evaluation set used in the online stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Reference [16] proposed detecting and mitigating these numerical instabilities by considering an array of KVPs with different boundary conditions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=', scattering ma- trices) within a partial wave and using the emulator so- lutions to obtain an estimated S matrix by a weighted sum of averages [see also Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [32, 41]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' For our KVP emulator, the mitigation process involves first calculating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (5) using the K matrix boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Once we have calculated ∆�U, the terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (4) are rescaled to match the boundary conditions we want to emulate (here, L = K, K−1, and T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The anomalies are then detected by applying a consistency check to the (independent) emulated solutions of the dif- ferent boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The emulator solutions that do not pass this check are discarded while those that pass are averaged to obtain an anomaly-free scattering matrix (here, the S matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' All KVP emulator results in this paper are shown with anomaly mitigation unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' So far, such a mitigation protocol has not been implemented for the NVP emulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' However, one approach would be to use multiple emulators based on different variational principles [21] instead of multiple boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' See Appendix A for our implemen- tation and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [16] for more information on emulation with arbitrary boundary conditions and ways to mitigate Kohn anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Emulation of phase shifts We first apply the emulators to the uncoupled 1S0 channel using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (5) to calculate ∆�U (see Appendix C for explicit expressions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' At N4LO+, this channel de- pends on na = 3 non-redundant LECs [37], and thus we choose our basis to be composed of nb = 6 training points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Figure 1 shows the phase shifts calculated using our simulator (black line) and the KVP emulator Stan- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Simulated (black solid line) and KVP emulated Stan- dard method (orange dots) 1S0 phase shifts for the N4LO+ SMS potential with Λ = 450 MeV (top panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The bottom panel shows the relative errors between the simulated and em- ulated phase shifts for the Gl¨ockle method (red dashed line), Standard method (blue solid line), and NVP emulator (green dotted line), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The spike at Elab ≈ 270 MeV is due to the phase shift crossing zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' dard method prediction (orange dots) as a function of the laboratory energy in the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The phase shifts associated with the training points are depicted by the light gray lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' In addition, the bottom panel shows the relative errors Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Error = 2 ���� Simulator − Emulator Simulator + Emulator ���� (14) between the simulated and emulated phase shifts for the Gl¨ockle method (red dashed line), Standard method (blue solid line), and NVP emulator (blue dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' We find that our KVP emulator accurately reproduces the high-fidelity phase shifts over a large energy range for both methods, but the Standard method is much more accurate than the Gl¨ockle method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' On average, the relative error for the Gl¨ockle method is on the order of ≈ 10−6 −10−5, while the Standard method has a rela- tive error on the order of ≈ 10−12 for the same basis size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The NVP emulator’s relative error is similar to the KVP Standard method, with an error of ≈ 10−13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' We now turn to the coupled 3S1–3D1 channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' This channel depends on na = 6 non-redundant LECs [37] at N4LO+, which means that our basis will be com- posed of nb = 12 training points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Figure 2 shows the on-shell K matrix for the simulator calculation (black lines) and KVP emulator prediction (orange dots) as a function of the laboratory energy for each different partial-wave component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The errors are similar to the Basis Simulator oooEmulator 200 [deg] 100 0 100 100 Glockle Standard NVP 4 10 Error 0 100 200 300 Eiab [MeV]6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 1, but for the on-shell K matrix in the coupled 3S1–3D1 as a function of the laboratory energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' From left to right: pure D–wave, pure S–wave, and mixed S–D-wave component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 1S0 channel, with the Standard method being much more accurate than the Gl¨ockle method, and the NVP emula- tor’s relative error being slightly better than the Stan- dard method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' In all cases, we see a spike in the relative error at Elab ≈ 20 MeV where the K matrix is singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The small spikes seen in the Standard method error are not Kohn anomalies, but can be attributed to a numerical instability of the principal value integral in the LS equa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' These spikes are mesh-dependent and appear when a k0 value is close to a momentum mesh point, thus caus- ing the denominator of the Green’s function to approach zero faster than the numerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' A way to decrease the relative error produced by these spikes is to not allow the k0 values to be close to momentum mesh points by mov- ing energies that are close to any momentum mesh point until the relative distance is greater than some threshold value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=', ε ≳ 10−2 MeV (see Appendix D for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The oscillations that appear in the Gl¨ockle method’s rel- ative errors plots are potential-dependent, and increase in number, but decrease in separation, when increasing the mesh size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Overall, the emulators accurately predict the partial waves for the uncoupled 1S0 and coupled 3S1–3D1 chan- nels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' When comparing the Gl¨ockle method emulation with the Standard method, we see that the relative error for the Standard method is much less than the Gl¨ockle method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' For both partial waves shown, the NVP emula- tor is the one that most accurately reproduces its high- fidelity solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Results for the other channels are sim- ilar to the ones presented here, with the only difference being that the relative error decreases as na gets smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' This can be further explored with the Jupyter notebooks provided [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Emulation of scattering observables Next, we examine the performance of the emulator for nuclear observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' As a demonstration, we use the SMS regularized chiral potential at N4LO+ for np scatter- ing with partial waves having total momentum quantum numbers j ⩽ jmax = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Overall, there are a total of 25 parameters in θ that are being sampled using a Latin- hypercube design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' As previously mentioned, the basis size is chosen as nb = 2na, where na is the number of LECs associated with the specific partial-wave, for a to- tal of 50 training points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Since these parameters are only present in the channels j ⩽ 4, the emulator only needs to be trained over these channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The remaining channels do not change as the parameters are varied, therefore, they do not undergo a training process and need to be calculated only once by solving the LS equation directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The emulation of observables is carried out by com- bining multiple emulators across different partial-wave channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The total np cross section can be calculated using σtot(k0) = π 2k2 0 jmax � j=0 (2j + 1) Re{Tr[Sj(k0) − 14]}, (15) where Sj = 14 − 2i(1 − iKj)−1Kj is the S matrix, Kj is the predicted on-shell K matrix, and Tr[·] denotes the trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Both Sj and Kj are 4 × 4 matrices that contain both the triplet-triplet and the singlet-triplet channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Figure 3 shows the simulator and emulator prediction for the total np cross section, which are calculated us- ing the fit values for the LECs determined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The inset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 3 depicts the mean relative errors for all three emulators when randomly sampling 500 differ- ent combinations of np LECs (chosen within the same range as the training points), using these to calculate the emulated and simulated total cross section, and compar- ing the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' On average, the relative errors for all three emulators are similar to those for the partial-wave calculations discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Although the mean relative errors for the Standard method and NVP emula- tors are very similar, the NVP emulator seems to be the one that most accurately reproduces its simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Basis 1 K K+ Simulator (oy)M o Emulator IOOOI Glockle 2 10° Standard NVP 10 Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 10-16 0 100 200 300 0 100 200 300 0 100 200 300 Eiab [MeV] Eiab[MeV] Eiab [MeV]7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Simulated (black solid line) and emulated (orange dots) np cross section with jmax = 20 for the N4LO+ SMS potential with Λ = 450 MeV as a function of the laboratory energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The inset shows the relative mean errors between the emulator and the simulator using the Gl¨ockle, Standard method, and NVP emulator for 500 different sets of np LECs obtained from Latin-hypercube sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' See the main text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' As mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' III A and following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [16], the Kohn anomalies found in the calculation were mit- igated by emulating with different boundary conditions and building the estimated S matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Figure 6 in Ap- pendix D shows a total cross section emulation result with one boundary condition, hence no anomaly mitiga- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' From the figure, we see that anomalies contribute to the Standard method mean relative error at higher en- ergies with a magnitude of approximately 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' These spikes are reduced to approximately 10−9 with mitiga- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The Gl¨ockle method result exhibits anomaly con- tributions of order 10−3 at lower energies, which get re- duced to approximately 10−5–10−7 with mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' For additional information, see the discussion in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Although the NVP emulator is subject to anomalies, they are not evident in the figures shown in this section, even though no mitigation strategy was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' An example of noticeable anomaly contributions as large as 10−3 in the NVP emulation are seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 7 in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The remaining spikes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 3 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=', at Elab ≈ 140 MeV) can be traced back to singularities in the on- shell K matrix for the 3S1–3D1 channel at those energies and are only seen for a few (specific) LEC values out of the 500 sampled (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The mesh-induced spikes seen in the Standard method relative error were also reduced in magnitude by preventing the on-shell k0 value from being too close to a momentum mesh value (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 8 for result comparisons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' We now turn our attention to spin-dependent observ- ables for non-identical particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' A detailed description of NN observables and their different conventions can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [35, 42–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' In general, one can write the spin observables in terms of Saclay parameters, which are complex functions of the center-of-mass energy and an- gle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Here we only consider the differential cross section and analyzing power: dσ dΩ = 1 2 � |a|2 + |b|2 + |c|2 + |d|2 + |e|2 + |f|2� , (16) dσ dΩAy = Re(a∗ e + b∗ f), (17) where dσ/dΩ is the unpolarized differential cross sec- tion and Ay the analyzing power (also known as Pb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' For identical particles, one has f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' More informa- tion on the description of the spin observables can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [44, 45];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' see also Appendix D, which con- tains our emulation results for more spin observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The Saclay parameters can be obtained from the spin- scattering M = M(θ, φ) matrix written in singlet-triplet space, M = � � � � M11 M10e−iφ M1−1e−2iφ MST e−iφ M01eiφ M00 M0−1e−iφ 0 M−11e2iφ M−10eiφ M−1−1 MST eiφ MST eiφ 0 −MST e−iφ MSS � � � � , (18) where the subscripts SS and ST represent the singlet- singlet and singlet-triplet channel, respectively [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Equation (18) can be calculated using spherical harmon- ics and Clebsch-Gordan coefficients, and can be related to the Saclay parameters from the expressions: a = 1 2(M11 + M00 − M1−1), (19) b = 1 2(M11 + Mss − M1−1), (20) c = 1 2(M11 − Mss − M1−1), (21) d = − 1 √ 2 sin θ(M01 + M01), (22) e = i 2(M10 − M01), (23) f = −i √ 2MST .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (24) The emulation process is performed similarly to the one for the total cross section, where multiple trained em- ulators are combined across different partial-wave chan- nels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Figures 4 and 5 show the simulator and emulator prediction for the differential cross section and analyzing power at three different energies calculated using the fit values for the LECs determined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The relative mean errors shown are obtained by randomly sampling 500 different combinations of np LECs (the same LECs used for the sampled relative error calculation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 3) and comparing them against their respective simulator Standard Glockle NVP Error 10- Mean Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 10-7 mb 11 10 Otot 15 10- 0 100 200 300 Eiab[MeV] 10 01010101010101010 Simulator Emulator 000 101 0 100 200 300 Eiab[MeV]8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Simulated (solid lines) and emulated (dots) unpolar- ized differential cross section for the N4LO+ SMS potential with Λ = 450 MeV as a function of the center-of-mass angle at the three energies 60, 160, and 320 MeV (top panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The bottom panel shows the mean relative errors between the em- ulators and their respective simulators for 500 different sets of np LECs obtained from Latin-hypercube sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The colors for the relative mean errors correspond to the energies in the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The gray arrows point from the label asso- ciated with the emulator to its error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' See the main text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' On average, the spin observables emulator has a relative mean error on the order of ≈ 10−5 when employing the Gl¨ockle method and ≈ 10−14–10−11 when using the Standard method and NVP emulators, which are similar to the total cross section results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The results are similar to those obtained over the entire energy grid and for other observables (see Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Table II details the angle-averaged relative errors be- tween the simulator and KVP emulators (based-10 log- arithm) for different spin observables with varying ba- sis size at a variety of energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' As can be seen, when training the emulator with basis size nb = na both the Standard and Gl¨ockle method emulators have large rela- tive errors of roughly 10−1 when compared to the high- fidelity model calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' However, if we increase the basis size by doubling the parameters used per partial- wave, nb = 2na, the relative mean errors are signifi- cantly smaller, roughly 10−12–10−9 and 10−6–10−3, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' According to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [47], the relative errors given by nb = 2na are below experimental uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' When increasing the basis size to nb = 4na, the mean er- rors have mostly saturated and the improvement in accu- racy is insignificant compared to the basis size nb = 2na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Note that although only four energies are shown, these results are similar over the entire energy grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 4, but for the analyzing power Ay (also known as Pb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' See the main text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The speed-up between the emulators and the simu- lator is highly implementation dependent (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=', to-be- considered factors are the desired accuracy, idiosyncrasies of the solver, programming language, level of paralleliza- tion, hardware, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The emulator speed-up will de- pend on the size of the quadrature mesh used by the simulator to obtain the high-fidelity solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' For repro- ducing the total cross section using the NVP emulator, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [17] states an emulator speed-up factor of > 300x faster than the simulator in CPU time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' When doubling the quadrature mesh size this factor becomes > 1000x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' When comparing the KVP and NVP emulator speeds using one boundary condition (no anomaly checking) for the 1S0 uncoupled partial wave, the KVP emulator is slightly slower due to the Lagrange multiplier in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (6) and numerical operations needed to solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Mitiga- tion of Kohn anomalies (by emulating multiple boundary conditions) will further contribute to slowing down the KVP emulator, or any other emulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' SUMMARY AND OUTLOOK We showed that the coordinate space KVP emulator for NN scattering [15, 16] can be extended to momen- tum space and coupled channels, and demonstrated its efficiency in accurately reproducing phase shifts and np observables using a modern chiral interaction at N4LO+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' In addition, we provided two methods to implement the emulator, with the Gl¨ockle spline interpolation method having a faster offline stage, but less accurate online stage than the Standard method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' By emulating (independent) scattering solutions associated with different asymptotic 60 MeV 160 MeV 320 MeV 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='0 do/d2[mb/sr] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='0 Glockle/NVP/Standard Error Mean Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 10 10-15 0 50 100 150 Ocm [deg]60 MeV 160 MeV 320 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='2 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='2 Glockle/NVP/Standard Error 10 Tean Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 10 10 0 50 100 150 Ocm [deg]9 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Comparison of the angle-averaged relative errors (base-10 logarithm) between high-fidelity model and emulator for various angular observables with different basis size for 500 sets of np LECs using the N4LO+ SMS potential [37] with momentum cutoff Λ = 450 MeV (rounded to two significant figures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' These results are similar over the entire energy mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Here, “Std.” refers to the Standard method emulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' See the main text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' dσ/dΩ D Ay Ayy A Basis size E [MeV] Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Gl¨ockle Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Gl¨ockle Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Gl¨ockle Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Gl¨ockle Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Gl¨ockle nb = na 5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='93 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='93 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='72 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='72 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='78 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='78 100 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='47 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='47 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='28 200 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='54 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='028 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='028 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='035 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='035 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='12 300 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='49 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='49 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='066 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='066 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='037 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='037 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='043 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='043 nb = 2na 5 −10 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='0 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='8 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='1 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='8 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='6 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='5 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='8 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='3 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='9 100 −12 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='3 −11 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='1 −10 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='9 −10 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='9 −11 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='3 200 −10 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='0 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='8 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='2 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='8 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='7 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='4 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='9 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='0 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='0 300 −12 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='9 −11 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='0 −11 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='9 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='9 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='8 −11 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='9 nb = 4na 5 −10 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='3 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='8 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='4 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='8 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='1 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='5 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='4 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='3 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='1 100 −13 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='5 −12 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='3 −11 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='1 −11 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='0 −11 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='4 200 −10 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='4 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='3 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='6 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='5 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='0 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='8 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='3 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='8 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='3 300 −12 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='1 −11 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='0 −10 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='1 −10 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='8 −11 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='0 boundary conditions in each partial wave and weighting the results (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=', for the S-matrix), spurious singularities known as Kohn anomalies were successfully mitigated for the KVP-based emulators [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' We also constructed an NVP-based emulator and as- sessed how well the three emulators reproduced their re- spective high-fidelity solution for the 1S0 and 3S1–3D1 partial-waves, total and differential cross sections, and analyzing powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' While all emulators produced errors well below experimental errors [47], the KVP Standard method and NVP emulators most closely reproduced the simulator, while the KVP Gl¨ockle spline interpolation emulator was overall the least accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The KVP emula- tor was found to have a slower online stage than the NVP emulator because it has to evaluate a higher-dimensional matrix and perform overall more numerical operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' We stress, however, that the emulators’ speed-ups are highly implementation dependent and should be further investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Extensions of the NVP-based emulator for anomaly mitigation with minimal computational cost, similar to the KVP-based emulators, should also be in- vestigated [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' An alternative procedure for mitigat- ing anomalies would be constructing the estimated S matrix using solutions from emulator based on differ- ent variational principles, as opposed to emulating mul- tiple boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Reference [21] provides fur- ther perspectives regarding different emulators (KVP- and NVP-based included) and efficient offline-online de- compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Although we considered here only χEFT NN potentials for np scattering, the constructed emulators are gener- ally applicable to two-body scattering, including pp scat- tering and nuclear reactions with complex-valued opti- cal potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' To help implement these fast & accu- rate scattering emulators in Bayesian parameter estima- tions, we provide self-contained set of codes that gener- ate all results and figures shown in this paper [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Fur- thermore, we have written a pedagogical introduction to projection-based emulators [21] with interactive, open- source Python code [22] to facilitate implementations of fast & accurate emulators even further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' However, taking full advantage of emulators for UQ in nuclear scattering and reaction calculations will require generalizations to higher-body scattering and non-affine potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Recent advances in this direction are already promising [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' ACKNOWLEDGMENTS We thank Evgeny Epelbaum for sharing a code that generates the SMS chiral potentials, Kyle Wendt for shar- ing a code that generates the spin obbservables, and Filomena Nunes for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' This work was supported in part by the National Science Foundation Award Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' PHY-1913069 and PHY-2209442 and the NSF CSSI program under award number OAC-2004601 (BAND Collaboration [48]), and the NUCLEI SciDAC Collaboration under U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Department of Energy MSU subcontract RC107839-OSU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' This material is based upon work supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Department of Energy, Office of Science, Office of Nuclear Physics, under the FRIB Theory Alliance award DE-SC0013617.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Appendix A: Mitigating Kohn anomalies We follow the method developed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [16] to detect and mitigate Kohn anomalies (see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The 10 estimated S matrix is calculated from the emulator so- lutions by using a weighted sum of averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Letting L1 and L2 be two independent KVP functional solutions, this weighted sum is computed by first calculating the relative residuals γrel(L1, L2) = max ������ S(L1) S(L2) − 1 �����, ����� S(L2) S(L1) − 1 ����� � , (A1) for all emulated KVP solutions without repetitions to avoid the trivial case where L1 = L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Using a consistency check, γrel < ϵrel, with ϵrel = 10−1, we select the set of pairs P = {(L1, L2)} that satisfies this check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' If at least one consistency check passes, the S matrix is now estimated by the weighted sum of averages [S](mixed) KVP = � (L1,L2)∈P ω(L1, L2)S(L1) + S(L2) 2 , (A2) ω(L1, L2) = γrel(L1, L2)−1 � (L′ 1,L′ 2)∈P γrel(L′ 1, L′ 2)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (A3) If no consistency check passes, one could change the ba- sis size to shift the position of the Kohn anomalies in the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' However, we found that using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (A2) was sufficient to mitigate Kohn anomalies in our appli- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' We first calculate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (5) using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (7), then rescale Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (5) using the relations from Appendix B of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [16], ∆�U (u′) = C ′−1(Li) C ′−1(Lj) det u det u′ ∆�U (u), (A4) C′(L) = det u det u′ u′ 11 − u′ 10K(L) u11 − u10K(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (A5) Here, u and u′ are nonsingular matrices parameterizing the scattering boundary conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' the K, K−1, and T scattering matrices, respectively, are given by uK = � 1 0 0 1 � , uK−1 = � 0 1 1 0 � , uT = � 1 0 i 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (A6) The u matrix parameterizes the initial boundary condi- tion associated with L, while the u′ parameterizes the final boundary condition associated with L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The snapshots used in the emulator’s offline stage are transformed using the M¨obius transform [16] L′(L) = −u′ 01 + u′ 00K(L) u′ 11 − u′ 10K(L) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (A7) Once we obtain an emulator solution, we transform that solution back into its K matrix form using K(L) = u01 + u11L u00 + u10L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (A8) For the estimated S calculation, the KVP solution pairs (L1, L2) being evaluated are the K matrix solu- tions obtained from the different boundary conditions used [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=', γrel(K(K), K(K−1)), γrel(K(K), K(T)), and γrel(K(K−1), K(T))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' See Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [16] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Appendix B: Formalism details Here we provide clarifying remarks about how Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (4) arises in the coupled case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' In particular, we focus on two questions about the specific manner in which the coefficients ⃗β enter into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Why can Lss′ be emulated separately for each ss′ pair rather than with one global set of coefficients for the cou- pled block?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' For uncoupled channels, each partial wave is inde- pendent of one another, thus they can be emulated individually using trial wave functions and coefficients that are specific to the channel under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Without loss of generality, let us consider two uncoupled channels labeled as s = 0 and s = 1, and let ⃗β(0) and ⃗β(1) denote the independent sets of coefficients found by making each channel’s KVP stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' To move toward the coupled regime, imagine adiabatically turning on the coupling between these two originally uncoupled channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The coefficients for each channel should remain nearly fixed to their previously uncoupled values, but the coupling will introduce a new set of coefficients ⃗β(01) ̸= ⃗β(0) ̸= ⃗β(1) that must be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Hence, each independent channel in the coupled case will have its own set of coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Attempting to force a global set of coefficients for a coupled system would be inconsistent with the treatment in the uncoupled case and also degrade accuracy in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' A more technical answer follows from the (Petrov-)Galerkin procedure described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Should not each of |ψs′⟩ and ⟨ψs| have its own basis expansion with their own independent coefficients?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' No, there is only one set of coefficients that en- ter quadratically in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' A way of understanding how the coefficients enter in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (4) follows from the (Petrov-)Galerkin orthogonalization procedure (see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Rather than starting with a variational prin- ciple, the (Petrov-)Galerkin approach starts with the Schr¨odinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Like the variational approach, it expands |ψs′⟩ as a linear combination of known functions, but determines the basis coefficients by enforcing orthog- onality against a set of test functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' For the diagonal channels, the test functions are chosen to have the same exit channel as the trial functions (standard Galerkin ap- proach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' On the other hand, the test functions for the off-diagonal channels are chosen to have a different exit channel (s) than the trial functions (s′) (Petrov-Galerkin approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The resulting set of linear equations is equiv- alent to those that follow from making the KVP station- ary for each combination of (s′, s) independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Thus by following the (Petrov-)Galerkin procedure we can de- termine how the coefficients are to enter in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' This discussion will follow closely that of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [21], however using coupled-channel notation and more gen- eral boundary conditions consistent with the general 11 KVP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Starting from (the strong form of) the Schr¨odinger equation �H(θ) |ψs′⟩ = E |ψs′⟩ , (B1) we can derive its weak form after multiplying by a test function ⟨ψs| ⟨ψs| �H(θ) − E|ψs′⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (B2) This can be considered a Petrov-Galerkin approach be- cause s ̸= s′ in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The boundary conditions can be made explicit via the relationship 0 = ⟨ψs| �H(θ) − E|ψs′⟩ = ⟨ψs| �H†(θ) − E|ψs′⟩ − � t W(rψts, rψts′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' r) 2µ ����� ∞ r=0 , (B3) where �H† denotes the operator acting to the left (via integration by parts) and where we have used ψts(r) = ⟨rt|ψs⟩ = ⟨ψs|rt⟩ and defined the Wronskian W(φ, ψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' r) ≡ φ(r)ψ′(r) − φ′(r)ψ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (B4) The wave function rψ vanishes at the origin, so that only the limit as r → ∞ contributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' By adding Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (B3) and (B2), we have ⟨ψs| �H(θ) − E|ψs′⟩ + ⟨ψs| �H†(θ) − E|ψs′⟩ = � t W(rψts, rψts′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' r) 2µ ����� ∞ r=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (B5) This is the weak form for general |ψs′⟩ and ⟨ψs|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' We can arrive at the discrete form by inserting basis states |ψs i ⟩ that satisfy the asymptotic boundary conditions ψst(r) −−−→ r→∞ δst ¯φ(0) s (r) + Lst ¯φ(1) s (r) , (B6) where � ¯φ(0) ℓ (r) ¯φ(1) ℓ (r) � ∝ � u00 u01 u10 u11 � � jℓ(qr) ηℓ(qr) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (B7) With this substitution, we have, for i ∈ [1, nb], ∆�U ss′ ij βj = Lss′ i � j βj − Ls′s j βj, (B8) where the expression for ∆�U ss′ ij is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' We must now implement the constraint � j βj = 1, which is performed here by a Lagrange multiplier λ mimicking a variational approach (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [19] for details): λ + ∆�U ss′ ij βj = Lss′ i � j βj − Ls′s j βj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (B9) The sum multiplying Lss′ i can be evaluated using the constraint � j βj = 1, and we can make the redefinition λ′ ≡ λ + � j βjLs′s j without impacting the solution be- cause this term does not depend on i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Thus, we have λ′ − ⃗L(E) + ∆�U ⃗β⋆ = 0, (B10) which is exactly Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (6) found by making the KVP sta- tionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' This simplification can be understood by not- ing that if {⃗β⋆, λ⋆} satisfy Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (B9), then we know that {⃗β⋆, λ′ ⋆} is the unique solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (B10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Therefore, we can solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (B10) to obtain ⃗β⋆ rather than Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (B9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' In conclusion, using the Petrov-Galerkin projection of the homogeneous Schr¨odinger equation with trial and test bases of |ψs′ i ⟩ and ⟨ψs i |, respectively, we were able to ob- tain the same coefficients as the KVP in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (6), which yield the same on-shell Lss′ matrix when used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Appendix C: KVP emulator construction details For single channel scattering over a k × p momentum grid using the K matrix (det u = 1), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (8) becomes ∆�Uij(θ) = ∞ ¨ 0 dk dp k2p2� ψi(k)Vθ,j(k, p)ψj(p) + (i ↔ j) � , (C1) with Vθ,j(k, p) defined as in Eq (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' We drop the super- scripts for the uncoupled case since s′ = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Note that ψi is not complex conjugated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' For the Gl¨ockle method, one would simply substitute Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (7) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (C1) and inter- polate the solutions to the integrals with the cubic spline polynomials Sk(k0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' For the Standard method, the Dirac delta distribution is analytically integrated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' thus we ob- tain the following expression for ∆�Uij ∆�Uij(θ) = Vθ,j(k0, k0) + 2 π (I1 ij + I2 ij) + 4 π2 I3 ij + (i ↔ j), (C2) with I1 ij, I2 ij, and I3 ij defined as I1 ij = P ∞ ˆ 0 dk k2 k0 Ki(k0, k) k2 − k2 0 Vθ,j(k, k0), (C3) I2 ij = P ∞ ˆ 0 dp p2 k0 Vθ,j(k0, p)Kj(p, k0) p2 − k2 0 , (C4) I3 ij = P ∞ ¨ 0 dk dp k2p2 k2 0 Ki(k0, k) k2 − k2 0 Vθ,j(k, p)Kj(p, k0) p2 − k2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (C5) 12 If V has an affine dependence on the parameters θ, applying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (11) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (12) produces ∆�U 0 ij = ∞ ¨ 0 dk dp k2p2� ψi(k)V 0 j (k, p)ψj(p) + (i ↔ j) � , (C6) ∆ �U 1 ij = ∞ ¨ 0 dk dp k2p2� ψi(k)V 1(k, p)ψj(p) + (i ↔ j) � , (C7) with V 0 j (k, p) ≡ 2µk0 � V 0(k, p) − Vj(k, p) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (C8) For coupled-channel interactions (s′ ̸= s), the details of the emulation are more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' In this case, we apply Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (4) to each individual channel in a partial-wave, but the real difference lies in how Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (5) is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The usual way of solving for the phase shifts and mixing angle for the coupled channels involves building a 2 × 2 block matrix for the potential, V = � V 00 V 01 V 10 V 11 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (C9) The same process can be applied to the emulator calcu- lation when calculating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (5), ∆�U = � ∆�U 00 ∆�U 01 ∆�U 10 ∆�U 11 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (C10) Each of the four blocks in ∆�U has a separate functional, although there are contributions from the different wave functions and potentials (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=', for the 3S1–3D1 partial wave ∆�U 00 depends on the 3S1–3S1, 3S1–3D1, and 3D1– 3D1 wave functions and potentials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Additionally, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (7) tells us that we can consider the momentum-space wave function for the individual chan- nels ψst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (8) with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (9),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' the functionals for the individual channels in a coupled-channel calculation (using the 3S1–3D1 as an example) will be ∆�U ss′ ij = ¨ ∞ 0 dk dp k2p2� ∆uss′ ij + (i ↔ j) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (C11) with ∆u00 ij = ψ00 i (V 00 θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='jψ00 j + V 01 θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='jψ10 j ) + ψ10 i (V 10 θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='jψ00 j + V 11 θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='jψ10 j ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (C12) ∆u01 ij = ψ00 i (V 00 θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='jψ01 j + V 01 θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='jψ11 j ) + ψ10 i (V 10 θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='jψ01 j + V 11 θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='jψ11 j ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (C13) ∆u10 ij = ψ01 i (V 00 θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='jψ00 j + V 01 θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='jψ10 j ) + ψ11 i (V 10 θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='jψ00 j + V 11 θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='jψ10 j ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (C14) ∆u11 ij = ψ01 i (V 00 θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='jψ01 j + V 01 θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='jψ11 j ) + ψ11 i (V 10 θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='jψ01 j + V 11 θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='jψ11 j ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (C15) where we have suppressed the arguments for compact- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Note that the weights βi in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (4) are differ- ent for each channel (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=', ∆�U 00, ∆�U 11, and ∆�U 01 = ∆�U 10), and are determined independently of one an- other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Once Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (C12) through (C15) are calculated, the steps for the uncoupled channel calculation are ap- plied to each ∆�U ss′ ij to obtain the emulator prediction, in particular Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (C2) through (C5), and the separation of ∆�U ss′(θ) into parameter-dependent and parameter- independent pieces as described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Appendix D: Additional results FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 3, but only emulating with the K ma- trix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The mesh-induced spikes have been removed for this calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Figure 6 shows the relative mean error for the total cross section using only the K matrix boundary condi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Comparing to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 3, where we apply the weighted sum (mixed) S approach, we see that for one bound- ary condition the relative mean error has Kohn anoma- lies (see Elab ≈ 270 MeV and ≈ 315 MeV for the stan- dard method and Elab ≈ 40 MeV and ≈ 130 MeV for the Gl¨ockle method) and a more spread-out error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 8 and comparing to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 3 and 6, we conclude that the mixed S approach is indeed successful in mitigating the Kohn anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Figure 7 shows the relative mean error for the to- tal cross section with momentum cutoff 550 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The weighted sum (mixed) S approach is used for the KVP emulator results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Here, the anomalies found in the NVP emulation are noticeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Figure 8 shows the relative errors for the KVP emula- tors in the 1S0 channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The figure on the left shows the Standard Glockle NVP Error Mean Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 10-7 mb 11 10 Otot 0 100 200 300 Eiab[MeV] 10 101001010101010101 Simulator Emulator 000 101 100 0 200 300 Eiab [MeV]13 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 3, but for cutoff Λ = 550 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' relative error when emulating with the K−1 boundary condition and the one on the right shows the weighted sum (mixed) S errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' In the figure on the left we can see a spike around Elab ≈ 65 MeV, which disappears when using the weighted sum S approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' This is a clear example of the weighted sum S approach helping to mitigate these anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Additionally, there are other smaller mesh-induced spikes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=', not anomalies) present throughout the energy grid in the figure on the left that are not in the figure on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' These were mitigated by not allowing the k0 values to be close to any mo- mentum mesh points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' See Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' III for a more detailed description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Figures 9 through 12 show emulator results for the following spin observables: dσ dΩD = 1 2 � |a|2 + |b|2 − |c|2 − |d|2 + |e|2 + |f|2� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (D1) dσ dΩA = − Re(a∗ b − e∗ f) sin(α + θ 2) + Re(c∗ d) sin(α − θ 2) − Im(b∗ e + a∗ f) cos(α + θ 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (D2) dσ dΩAxx = Re(a∗ d) cos(θ) + Re(b∗ c) − Im(d∗ e) sin(θ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (D3) dσ dΩAyy = 1 2 � |a|2 + |b|2 − |c|2 − |d|2 + |e|2 + |f|2� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' (D4) where D is the depolarization parameter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' A is the spin- flip amplitude,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Axx and Ayy are the spin-correlation am- plitudes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' and α a relativistic spin rotating angle that van- ishes in the non-relativistic case [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' For identical parti- cles, f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The results and conclusions are similar to those described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' III C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Figure 13 shows emulator results for the total cross section for the N4LO+ SMS potential with momentum cutoff 550 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The results and conclusions are similar to the ones described in the text for the 450 MeV mo- mentum cutoff (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' III C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Figures 14 and 15 shows emulator results for the dif- ferential cross section and analyzing power Ay for the N4LO+ SMS potential with momentum cutoff 550 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The results and conclusions are similar to the ones de- scribed in the text for the 450 MeV momentum cutoff (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' III C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' These results and conclusions also ex- tend down to momentum cutoff 400 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The spin ob- servables at 500 MeV show larger errors on order of 10−7 for the NVP emulator at particular energies, which may come from Kohn anomalies at one or more of the sam- pled parameter sets (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' nevertheless, the errors are still well below experimental uncertainties [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Standard Glockle NVP 010101010101 Error Mean Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 10- mb 1 10 Otot 0 100 200 300 Eiab[MeV] 10 Simulator Emulator 000 101 0 100 200 300 Eiab [MeV]14 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Relative error of the 1S0 channel for a basis size of nb = 2na + 1 for the N4LO+ SMS potential with Λ = 450 MeV as a function of the laboratory energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The left panel shows the relative error for an emulator using the K−1 boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' There is a Kohn anomaly at Elab ≈ 65 MeV for both the Standard and Gl¨ockle emulators and mesh-induced spikes present throughout the energy grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' The right panel shows the relative error for the mixed S-matrix approach presented by Reference [16] with care taken to avoid the k0 values that correspond with a mesh point as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' When comparing both graphs, the Kohn anomaly is no longer present and the mesh-induced spikes are much smaller in the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Glockle Standard Mixed S emu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' R Error 10° Rel 10-9 10-12 LA 10-15 0 50 200 0 50 100 200 100 150 150 Eiab [MeV] Eiab [MeV]15 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 4, but for the depolarization D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 4, but for the spin-flip amplitude A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 4, but for the spin-correlation amplitude Axx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 4, but for the spin-correlation amplitude Ayy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 60 MeV 160 MeV 320 MeV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
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+page_content=' 10 0 50 100 150 cm [deg]60 MeV 160 MeV 320 MeV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
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+page_content='5 Giockle/NVP/Standard Error Tean Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 10 10- 10-15 0 50 100 150 Ocm [deg]60 MeV 160 MeV 320 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
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+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='0 Glockle/NVP/Standard Error 10 Tean Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 10 10-15 0 50 100 150 Ocm [deg]60 MeV 160 MeV 320 MeV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
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+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='0 Glockle/NVP/Standard Error 10 Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 10 Mean 11 10° 0 50 100 150 Ocm [deg]16 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 3, but for cutoff Λ = 550 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 3, but for cutoff Λ = 550 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 3, but for cutoff Λ = 550 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Standard Glockle NVP Error Mean Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 10- mb Otot 0 100 200 300 Eiab [MeV] 10 0101001010101010 Simulator Emulator 000 101 100 0 200 300 Eiab[MeV]60 MeV 160 MeV 320 MeV 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='0 do/d2[mb/sr] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='0 Glockle/NVP/Standard Error Mean Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 10 10-15 0 50 100 150 Ocm [deg]60 MeV 160 MeV 320 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='2 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
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+page_content=' 10 10- 10-15 0 50 100 150 Ocm [deg]17 [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Epelbaum, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Hammer, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Meißner, Mod- ern Theory of Nuclear Forces, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 81, 1773 (2009), arXiv:0811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='1338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [2] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Machleidt and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Entem, Chiral effective field theory and nuclear forces, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 503, 1 (2011), arXiv:1105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='2919.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [3] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Hammer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' K¨onig, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' van Kolck, Nuclear effective field theory: status and perspectives, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' 92, 025004 (2020), arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content='12122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' [4] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Epelbaum, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Krebs, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
+page_content=' Reinert, High- precision nuclear forces from chiral EFT: State-of-the- art, challenges and outlook, Front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfdwwo/content/2301.05093v1.pdf'}
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+Anisotropic Electron Heating in an Electron Cyclotron Resonance
+Thruster with Magnetic Nozzle
+J. Porto,1, 2 P.Q. Elias,1 and A. Ciardi2
+1)Physics - Instrumentation and Space Department, ONERA/DPHY, Université Paris Saclay
+F-91123 Palaiseau – France.
+2)Sorbonne Université, Observatoire de Paris, PSL Research University, LERMA, CNRS UMR 8112
+75005 Paris – France.
+(*Electronic mail: paul-quentin.elias@onera.fr)
+(*Electronic mail: jcportoh@gmail.com)
+(Dated: 30 January 2023)
+In a grid-less Electron Cyclotron Resonance (ECR) plasma thruster with a diverging magnetic nozzle, the magnitude
+of the ambipolar field accelerating the positive ions depends of the perpendicular energy gained by the electrons. This
+work investigates the heating of the electrons by electromagnetic waves, taking their bouncing motion into account in
+a confining well formed by the magnetic mirror force and the electrostatic potential of the thruster. An electromagnetic
+Particle-In-Cell (PIC) code is used to simulate the plasma in a magnetic field tube. The code’s Maxwell solver is based
+on a semi-Lagrangian scheme known as the Constrained Interpolation Profile (CIP) which enables larger time steps.
+The results show that anisotropic plasma heating takes place exclusively inside the coaxial chamber, along a Doppler-
+broadened zone. It is also shown that a trapped population of electrons with a larger perpendicular energy exists in the
+plume.
+I.
+INTRODUCTION
+Electric thrusters play a fundamental role in the field of
+space propulsion. Their main advantage lies in an efficient
+use of the propellant mass, and therefore a reduced consump-
+tion of propellant. Hall Effect Thrusters or Gridded Ion En-
+gines are examples of the most well-known and flight-proven
+technologies in the current propulsion market nowadays. Both
+technologies eject an ion beam which is subsequently neutral-
+ized to prevent the spacecraft from charging. Several compo-
+nents of these technologies, such as the acceleration grid or
+the neutralizer, are subject to erosion and wear and for this
+reason, meeting the challenging lifetime targets requires care-
+ful optimization and demanding testing1. The complexity of
+some of the components has driven the investigation of al-
+ternative concepts of propulsion devices that require neither
+a grid nor a neutralizer. The Electron Cyclotron Resonance
+(ECR) plasma thruster2,3 is one of these concepts and it is the
+subject of the present study.
+The ECR plasma thruster consists of a semi-open chamber
+where a quasi-neutral plasma is heated by electron cyclotron
+resonant microwaves at 2.45GHz, and accelerated by a mag-
+netic nozzle. This concept was first proposed in the 1960s in
+the works of Miller et al. 4 and Nagatomo 5, then further de-
+veloped by Sercel 6. These studies used a prototype with a
+wave-guide structure to couple the microwaves to the plasma.
+Their results showed that it was possible to achieve specific
+impulses and thrust values high enough to be of interest for
+space propulsion applications6. Nonetheless, the inefficiency,
+size and weight of the micro-wave sources and electromag-
+nets at that time led to a stagnation of the research on ECR
+thrusters for several years. Interest for this technology arose
+again recently with experimental works7,8. In particular, the
+use of coaxial microwave coupling structures and compact
+rare-earth permanent magnets were instrumental in designing
+compact sources (a schematic of the design is shown in Fig.
+1a).
+More experimental and theoretical efforts has since been
+made in order to get a deeper understanding of the phys-
+ical phenomena governing the plasma heating and acceler-
+ation in the thruster. Experimental characterizations of the
+plasma properties have been carried out using different mea-
+surement techniques such as Langmuir and Faraday probes,
+Laser Induced Fluorescence diagnostics, diamagnetic loops
+and thrust balances2,9–12. Unfortunately, most of the exper-
+imental studies so far have been limited to survey the plasma
+outside the thruster coaxial chamber.
+Recently, a resonant
+probe was developed to measure an electron density of about
+1×1011 cm−3 at the source exit plane, close to the coaxial
+chamber13. In the source, it is likely that the plasma density is
+higher (∼ 1×1012 cm−3) with electron temperatures of a few
+tens of eV.
+From a theoretical point of view, as a first step, global
+models describing the energy balance in the plasma source
+were proposed as a means to obtain the key parameters of
+the thruster14,15.
+While this approach yielded good agree-
+ment with measured electron temperature at high mass flow
+rate or high pressure, they failed at the lower mass-flow rate
+where the thruster achieves its best performance. Indeed, the
+assumptions of uniform electron temperature and isotropic
+Maxwellian electron distribution are too crude approxima-
+tions when collisionality decreases and the electron mean free
+path becomes much larger than the source length: in that range
+non-local effects become prevalent, as electrons undergo a
+bouncing motion along the magnetic field line. Those elec-
+trons which cross the ECR surface can gain energy depending
+on their phase in the gyromotion16, which leads to a strong
+anisotropy of the distribution function. An attempt to account
+for this stochastic heating in the plasma was made by consid-
+ering the electron heating as a random walk in phase space17.
+arXiv:2301.11411v1 [physics.plasm-ph] 26 Jan 2023
+
+2
+While this model provided a qualitative agreement with the
+measured ion energies, it could not account for the plasma
+feedback on the waves (assumed constant) and the collisions
+along the bouncing motion. Recently Sánchez-Villar et al. 18
+performed 2D axisymmetric simulations of the thruster with
+a hybrid model consisting of particle-in-cell (PIC) ions and a
+fluid model for the mass-less electrons. One of the main find-
+ings of this study was the identification of different regions in
+the source where the waves are either propagating or evanes-
+cent, with most of the power absorption taking place close to
+the inner conductor, near the ECR surface. By acting as a
+sink for the plasma, the inner conductor induces a decrease
+of the plasma density in its vicinity, enabling the propaga-
+tion of electromagnetic waves downstream of the ECR sur-
+face. These features lead to the formation of a hot electron
+beam close to the inner conductor, with a colder plasma in the
+bulk of the source. While these 2D results provided important
+insights on the operation of the thruster, some assumptions
+of the fluid model limit the validity of the results obtained
+from these simulations. The most important one being the
+assumption of isotropic electron temperature which excludes
+anisotropic heating in the directions parallel and perpendicu-
+lar to the magnetostatic field.
+This latter point is still an open question for this technology.
+Indeed, ECR heating is expected to lead to anisotropic heating
+of the electron translation modes. This difference affects the
+power losses near the source walls and the potential drop in
+the magnetic nozzle19. However, most of the electron temper-
+ature measurements performed in the thruster plume did not
+differentiate perpendicular and parallel electron temperature
+(with respect to the local magnetic field direction). A way to
+measure the electron temperature anisotropy is, for example,
+incoherent Thomson scattering20, but this type of measure-
+ment is not presently available in the ECR source. At any
+rate, this heating is intimately linked to the absorption of the
+electromagnetic waves in the coaxial source.
+Another issue is the non-local transport due to the bounc-
+ing magnetized electrons in the nozzle (the electron mean free
+path is greater than the source radius). In particular, the pro-
+duction and the heating of the electrons are not necessarily at
+the same location.
+While gaining a better understanding of these issues should
+firstly rely on experimental measurements, the challenges as-
+sociated with such an investigation are a strong incentive to
+use numerical models, even if simplified, to investigate the
+main physical processes at play in ECR thrusters.
+In par-
+ticular, such a model should be able to account for the self-
+consistent wave absorption and the anistropic heating, as well
+as the non-local effects and bouncing motion of the parti-
+cles. Electromagnetic kinetic models, such as Particle-In-Cell
+(PIC) or Vlasov methods, are natural candidates for this task.
+There are currently a few works using kinetic simulations of
+propulsion devices exploiting the ECR phenomenon, however
+the majority of these developments are concerned with grid-
+ded ion thrusters with ECR heating21–24, where the plasma
+acceleration is achieved by a grid-imposed electric field and
+not the plasma expansion in the magnetic nozzle, as in our
+design. Takao et al. 23 successfully modelled a gridded ion
+thruster where the ions are produced in an ECR source at 4.2
+GHz. The authors used a Particle-In-Cell (PIC) code consid-
+ering the microwave electric field as a temporal modulation
+of its initial amplitude obtained by simulating the microwave
+propagation without plasma. Therefore, in this approach the
+plasma feedback on the wave was considered negligible.
+The main purpose of the our study is to perform full-PIC
+electromagnetic simulations of the plasma in the thruster, tak-
+ing into account the plasma feedback on the wave propaga-
+tion, and to investigate the heating and confinement of the
+electrons.
+For this purpose it is necessary to simulate the
+microwave propagation and its interaction with the charged
+particles in the source and the nozzle region. However, due
+to the complexity and computational cost of simulating a full
+3D configuration (which should include the nozzle region),
+we restrict our investigation to the simplified case of an iso-
+lated magnetic flux tube. This approximation effectively re-
+stricts the phase space to 4 dimensions (1 dimension in space,
+3 dimensions in velocity space), and a 1D3V electromagnetic
+Particle-In-Cell can be used to model the ECR thruster.
+We show that the electron heating takes place over a
+broad region in the thruster source and leads to a signifi-
+cant anisotropy (ratio Te⊥/Te∥ ∈ [2.5,7.5]). The perpendicular
+electron temperature reaches a first maximum in the source
+and, surprisingly, has a second maximum in the downstream
+region. The explanation for these features lies in the confine-
+ment of electrons in the potential well formed by the com-
+bination of the diverging magnetic field and the electrostatic
+potential.
+II.
+NUMERICAL MODEL
+A.
+The Quasi-One-Dimensional Approach
+In the coaxial ECR thruster, an axially magnetized cylindri-
+cal permanent magnet creates a diverging static magnetic field
+BMS in the source and in the plume region12,14. This shape
+for the magnetostatic field was chosen to ensure a magnetic
+confinement at the close end of the coaxial chamber (called
+backplate in Fig. 1) while allowing the electrons to get accel-
+erated in the plume thanks to the divergence of the magnetic
+field lines.
+In fact, the ECR uses a diverging magnetic field whose
+magnitude decreases from approximately 100 mT at the back
+of the source to around 5 mT 10 cm downstream of the thruster
+exit plane. Under these conditions, assuming an electron tem-
+perature around Te ≃ 10eV, the Larmor radius of the elec-
+trons is between rL ≃ 0.07mm − 1.4mm. Thus electrons are
+strongly magnetized in the source and in the near-field plume
+region, while ions remain mostly unmagnetized. As a conse-
+quence, before the onset of plasma detachment, electrons and
+ions are bound to the magnetic field tube. Several mechanisms
+may account for the plasma detachment25 : collisions, stretch-
+ing of the magnetic field lines, electron demagnetization and
+plume instabilities. While it is out of the scope of this work to
+study the dominant mechanisms, several recent works have in-
+vestigated some of these effect in 2D PIC simulations26–28.In
+
+3
+(a)
+(b)
+FIG. 1: ECR thruster: (a) Schematic view of the coaxial
+source. The magnetic field lines are shown in red. The
+dashed surface corresponds to the flux tube (b) Schematic
+view flux tube used for the quasi-1D model. The exit plane of
+the coaxial source of length LS is represented by the dashed
+line. The end of the computational domain is reached at
+x = L. The axial magnetic profile and tube cross section
+along the axis are shown in red and blue, respectively.
+this work, we decided to rely on experimental evidence to de-
+fine the section of the nozzle where the plasma remains bound
+to the field lines. Recently, Little and Choueiri 29 have mapped
+the plasma potential in a magnetic nozzle to show that a good
+criterion for detachment is χp = rL/L∇B ≃ 0.1, where rL is the
+electron Larmor radius and L∇B = (∇B/B)−1 is the character-
+istic length scale of the magnetic field gradient. In the region
+of the nozzle where χp = rL/L∇B < 0.1, the plasma remains
+attached to the magnetic field. In our case, we considered a
+magnetic field with L∇B ≃ 5 − 10cm. Under this condition,
+we have χp < 0.1 up to L = 10cm downstream of the nozzle,
+and it is a reasonable assumption to consider that electrons do
+not detach from the magnetic field tube over this distance.
+As a consequence of this assumption, we decided to con-
+sider the creation and formation of the plasma enclosed in a
+magnetic field tube of length L = 10cm. More precisely, a
+portion of the thruster chamber and plume was represented
+by a quasi-1D model of a magnetic field tube with a varying
+cross-sectional area, as seen in Fig. 1b. There are several ex-
+amples of the use of quasi-1D models in the space propulsion
+field. Niewood and Martinez-Sanchez 30 used it to model a
+Magnetoplasmadynamic thruster, while De Giorgi and Fonta-
+narosa 31 studied a Vaporizing Liquid Microthruster with this
+approach. Recently, Saini and Ganesh 32 also used this ap-
+proach to model plasma expansion in a Radio-Frequency
+thruster. The moderate computational cost of a 1D3V model
+of the thruster facilitates the analysis of the plasma behavior in
+both the coaxial chamber and in the magnetic nozzle, and im-
+portantly, taking into account the nozzle is critical to resolve
+the bouncing motion of the electrons.
+The quasi-1D model assumes that the electrons and the ions
+are confined within a diverging magnetic flux tube, whose area
+is related to the axial magnetic field intensity through the con-
+servation of the magnetic flux:
+A(x)Bx(x) = A0B0
+(1)
+The model further assumes that the electromagnetic fields and
+all the plasma properties are constant across the section of the
+flux tube. For the ECR thruster under consideration12,14, the
+shape of the magnetic field lines close to the antenna is well
+approximated by an exponential function. For the sake of sim-
+plicity we approximated the magnetic field as:
+Bx(x) = B0 exp
+�
+− x
+LB
+�
+(2)
+In addition, we considered cylindrical symmetry for the static
+magnetic field around the field tube centerline.
+These assumptions mean that the particles guiding centers
+remain on the centerline. Since the plasma is assumed uni-
+form in the cross section, this approach does not allow the
+formation of a diamagnetic current and E ×B drifts.
+From now on, the term parallel and the subscript ∥ will
+refer to the direction parallel to the magnetostatic field lines.
+Similarly, perpendicular and the subscript ⊥ refer to the direc-
+tion perpendicular to the magnetostatic field lines. The source
+region, which corresponds to the coaxial cavity in Fig. 1b,
+was defined by 0 ≤ x ≤ LS, where LS is the coaxial source
+length. The plume region, which corresponds to the plasma
+expansion in vacuum, was defined by x ≥ LS.
+B.
+Particle-In-Cell Code overview
+The simulations were carried out with the Particle-In-Cell
+(PIC) code Rhei, which was developed to simulate low pres-
+sure cold plasmas and is adapted to parallel architectures. It
+
+4
+can be run with either a pure MPI or a hybrid MPI/OpenMp
+parallelization. The code integrates a Monte-Carlo Collision
+(MCC) module to simulate the collisions between the charged
+particles and a prescribed neutral background. At each it-
+eration, once the electrostatic and the electromagnetic fields
+were computed, the position of each macro particle labeled
+“p” was updated using dxp/dt = vp, and the velocity using
+Eq. 3. Each macro-particle represents W physical particles.
+The value of W used in the simulation is given in table I.
+ms
+dvp
+dt = qs
+�
+EESp +EEMp +vp ×
+�
+BMSp +BEMp
+��
+(3)
+In Eq. 3, qs is the charge of the particle, ms the mass, xp
+the position, and vp the velocity. Regarding the fields, they
+were computed at the location of the particle p using linear
+interpolation function, where EESp is electrostatic field from
+the charge distribution, BMSp is magnetostatic field from the
+permanent magnets and EEMp and BEMp are electromagnetic
+fields produced by the microwave source and by the plasma
+itself.
+The equations of motion were integrated using the leap-frog
+method and the Boris scheme to get the v × B rotation from
+the Lorentz force33. Details of the integration in the context
+of the quasi-1D model are provided in appendix A. Particle
+quantities were projected on a uniform grid using linear shape
+functions.
+The Rhei code development follows a test-driven approach
+to ensure the robustness and the maintainability of the code
+over time. Additionally, several test cases were run as a val-
+idation of the code. The first elementary test was the simu-
+lation of a magnetic bottle. The simulation domain, with a
+converging-diverging parabolic magnetic field, was uniformly
+loaded with a Maxwellian electron population. At the end of
+the simulation the electron distribution in velocity space v∥,v⊥
+was plotted to verify that the loss cone angle is coherent with
+the expected theoretical value arcsin
+��
+B0/BMax
+�
+. The sec-
+ond elementary test concerned the electromagnetic modes in
+a one-dimensional magnetized plasma. The simulation do-
+main was initialized with a uniform Maxwellian distribution
+of electrons and cold ions. The random fluctuations excited
+the modes of the plasma. The resulting dispersion curves were
+obtained by computing the discrete 2D Fourier transform of
+the electric fields during the simulation. This was compared
+to the expected theoretical description of the extraordinary and
+the ordinary wave. Finally, the third test case was the classi-
+cal capacitively coupled discharge in Helium, which verified
+in particular the collision module34.
+1.
+Collisions
+The Monte-Carlo Collision module used the Null Collision
+technique35 to speed-up the computation of the collisions by
+removing the velocity dependency of the total collision cross-
+section. Assuming Np collision processes defined by their re-
+spective cross sections σi(v),i = 1..Np, a null collision cross-
+section is defined as σ0(v) such that:
+σ0(v) = max
+v≥0
+� Np
+∑
+i=1
+σi(v)
+�
+−
+Np
+∑
+i=1
+σi(v)
+(4)
+A first test over all the particles of species s found the
+fraction of particles which undergo a collision with the back-
+ground. In that case the total cross section σT = ∑i=0 Npσi(v)
+(including the null collision process) did not depend on the
+velocity (thus avoiding a costly interpolation to get the cross
+section for all the particles). Then a second test among those
+selected particles computed all the collision cross sections
+for their given relative velocity and determined which cross
+section to use (including the null collision). When this test
+pointed to the null-collision cross section, then the particle
+did not experience an actual collision and was left unchanged.
+When the test pointed to another cross section, the the parti-
+cles experienced a collision.
+For the collisions of electrons with Xenon neutrals, we con-
+sidered a simplified set of three processes: elastic, ionization,
+and excitation. Excitation processes were lumped into a single
+process. Electron impact ionization and excitation were taken
+from the Morgan (Kinema Research & Software) database,
+while the total elastic scattering is from Ref. 36. For all elec-
+tronic processes, we assumed an isotropic scattering of the rel-
+ative velocity vector between the electron and the target dur-
+ing the collision. For the ionization collisions, the kinetic en-
+ergy of the projectile electron was equally split (after subtract-
+ing the threshold energy) between the secondaries. For the
+collisions of Xenon ions with Xenon neutrals, we considered
+isotropic scattering and backscattering34. Ion cross sections
+comes from Ref. 37. All electronic and ionic processes con-
+served momentum and total energy (kinetic plus internal). In
+order to start with a simplified description simulating weakly
+ionized plasmas, in which the collisions with the neutral par-
+ticles are the dominant process, Coulomb collisions were not
+considered in the code. Indeed, the electron-ion collision fre-
+quency νei, for Maxwellian electrons, is given by:
+νei = ωp
+Λei
+lnΛei
+(5)
+Here, ωp is the plasma frequency and lnΛ is the Coulomb log-
+arithm. For the typical simulation conditions in the thruster
+source, as it will be shown below, the maximum plasma den-
+sity was ne ∼ 1×1011 cm−3, the electron temperature was
+Te ∼ 10eV and the electron-neutral elastic collision frequency
+was νen ∼ 1×107 s−1.
+This gave lnΛ ∼ 12 − 15, ωp ∼
+1.8×1010 rads−1. Consequently, the maximum electron-ion
+collision frequency was νei ∼ 1×105 s−1, much less than the
+the electron-neutral collision frequency νen.
+The neutral gas in the thruster is injected at the backplate
+(see Fig. 1a) and expands in the source resulting in a decreas-
+ing density. Since there is no measurement of the neutral gas
+density profile in the thruster, and to avoid a costly particle
+simulation of the neutral particles, we modelled this expansion
+heuristically by assuming that the neutral background density
+
+5
+followed an exponential profile:
+nn(x) = nn0 exp
+�
+− x
+Ln
+�
+(6)
+where nn0 is the maximum density of neutrals found at the
+close end of the source, and Ln is the neutrals density char-
+acteristic length. The assumption of a time-independent neu-
+tral gas density profile means that the simulation did not con-
+serve the total mass, momentum and energy. In addition, it
+means that the neutral gas depletion due to ionizing collisions
+was not considered. However, both of those limitations are
+acceptable in the frame of this work which does not seek to
+compute the total thrust and energy balance but is concerned
+with the particle heating and trapping. To estimate the neutral
+depletion we note that the ion removal is driven by their ve-
+locity (at most 10kms−1), while the neutral removal is driven
+by their thermal speed, ∼ 200ms−1. From mass balance, the
+neutral inflow is balanced by the ion flux and the neutral out-
+flow. Using the characteristic speeds and the typical parame-
+ters for the gas density ng ∼ 1×1014 cm−3, and plasma den-
+sity nmax
+e
+∼ 3×1011 cm−3, gives a neutral depletion of at most
+10%, indicating that the assumption of a static background
+remained consistent with the assumed density profile.
+2.
+Fields solvers
+The Rhei code solved the Poisson equation to compute the
+electrostatic space potential Φ and the electric field (EES =
+Exx) using Eq.
+7 where the charge density is ρs.
+The
+solver implements a second order finite difference discretiza-
+tion and the resulting linear system is inverted using an itera-
+tive method (GMRES)38.
+∇2Φ(x,t) = −ρs(x,t)
+ε
+(7)
+In addition, an electromagnetic solver computed the fields
+produced by the microwave source and by the plasma itself:
+EEM = Eyy + Ezz and BEM = Byy + Bzz.
+This solver was
+based on the Constrained Interpolation Profile (CIP) method
+explained in detail in Ref. 39. This method considers not
+only the electromagnetic fields but also their spatial deriva-
+tives, therefore suppressing instabilities and providing lower
+numerical dispersion even when using coarse grids and large
+time steps40. The use of this method is a novel solution for a
+PIC code since most of the electromagnetic solvers are based
+on conventional approaches like the finite-difference time-
+domain method (FDTD). It was shown that it provides higher
+accuracy than the latter under the condition of identical cell
+size41.
+The CIP method is a semi-Lagrangian scheme that
+circumvents the Courant-Friedrichs-Lewy (CFL) stability
+condition42,43, i.e., (u∆t/∆x) < 1 where u is the magnitude
+of the velocity, ∆t is the time-step, and ∆x the length inter-
+val. This feature allows computations with CFL values ≥ 1.0,
+as can be seen in Ref.
+44 and 45 where the authors per-
+formed simulations using a CFL value of 2.6 in a Cartesian
+coordinate system. The gain in computational time, that is
+afforded by using high CFL values, is a key factor that en-
+ables the self-consistent kinetic simulations presented here to
+reach steady-state. In this paper, CFL values close to 3 were
+used for the simulations. As a check, simulations were also
+run with CFL=0.6 and compared to the results obtained with
+larger time steps. The results were identical to the one at larger
+time-steps, within small variations due to the noise inherent to
+the statistical nature of the PIC simulations.
+Finally, the CIP scheme does not necessarily maintain the
+divergence-free condition for the dynamic field BEM. How-
+ever, BEM is smaller than the magnetostatic field (which has
+divergence equal to zero by construction, see appendix A) by
+several orders of magnitude, over the whole computational do-
+main. Therefore, the resulting error on the total divergence
+was considered to be negligible.
+3.
+Boundary conditions
+As it was shown in Fig. 1b when describing the model, the
+domain goes from x = 0 at the left side which corresponds
+to the backplate and the microwave input, to the right-end at
+x = L, as discussed in II A.
+Electrostatic: At the right end of the computational do-
+main x = L, we imposed a Dirichlet boundary condition, with
+Φ(L) = 0, to simulate the presence of a grounded vacuum
+chamber wall. The dielectric backplate, at x = 0, is in con-
+tact with the plasma and therefore its surface voltage ΦBP is
+changed by the collection of charged particles. This can be
+modeled as a capacitor. Hence, the evolution of ΦBP is given
+by ∆ΦBP = ∆Q/(C∆t), where ∆Q is the charge deposited at
+the backplate at each time step, and C is an equivalent ca-
+pacitance under the assumption that the backplate is in con-
+tact with a grounded conductor.
+This capacitance is com-
+puted by considering that the backplate is a plane capacitor,
+its value is of a few picoFarads. Changing its magnitude mod-
+ifies the charging rate of the backplate and thus the transient
+phase of the computation. However, its does not affect the
+steady-state voltage of the backplate. This approach guaran-
+tees that at steady-state, the ion flux equates the electron flux
+on the backplate. In principle the steady-state value of the
+backplate potential is also affected by other processes such as
+secondary electron emission or charge migration. However,
+for this study, these processes were neglected.
+Electromagnetic: In the coaxial ECR thruster, the mi-
+crowaves are injected as Transverse Electro-Magnetic (TEM)
+mode. For this 1D simulation, the TEM mode can be seen
+as a linearly polarized wave, where the radial component of
+the electric field is along the transverse y axis, the azimuthal
+magnetic field defines the z axis and the wavevector direction
+is along the longitudinal x axis. Therefore, the microwaves
+were injected at the backplate as a propagating wave with a
+linear polarization along the y-axis. The incident wave was
+parametrized by its power per unit area Pin and its frequency
+fEM = ω/2π. The electric fields from the injected linearly
+polarized wave were computed as Ey = √µcPin sin(ωt) and
+Ez = 0.
+
+6
+The injected microwave input power per unit area Pin could
+be fixed, or it could be adapted to keep a roughly constant pre-
+defined number of particles Ntarget during the transient phase.
+This feature was intended to speed up the simulations by re-
+producing a faster plasma response to a given variation in the
+simulation’s parameters. The value of Pin can be regulated
+with an attenuation factor α ≤ 1 varying with the number of
+particles in the domain: α = exp(−Nparticles/Ntarget). A run
+performed without this regulation confirmed that it did not
+have an effect on the final steady state but only on the duration
+of the transient phase.
+Particles: We imposed a loss condition at both ends of the
+domain, for both ions and electrons. Particles crossing these
+boundaries are suppressed from the simulation. As a simpli-
+fying assumption, secondary emission processes on the back-
+plate were not considered in this first approach.
+4.
+Cross field diffusion loss model
+Electron cross-field diffusion is an important mechanism to
+model to get a more accurate representation of the discharge
+loss mechanisms. Previous works using PIC codes for elec-
+tric thrusters took it into account as wall losses by artificially
+increasing the collision rate or by using a profile of the cross
+field diffusion based on empirical evidence46,47. The electron
+balance equation is:
+∂ne(r,t)
+∂t
++∇⊥ ·neu⊥ +∇∥ ·neu∥ = kionne(r,t)
+(8)
+Where u⊥ and u∥ are the electron macroscopic velocity per-
+pendicular and parallel to the local magnetic field, respec-
+tively, and kion is the ionization rate.
+For our 1D3V simulations, the transport along the magnetic
+field is taken into account by the kinetic model. However, the
+perpendicular transport cannot be modeled with a 1D model.
+Therefore we simulated the particle losses into the coaxial
+chamber walls using a phenomenological, Monte Carlo loss
+model, as shown in Fig. 1b. The probability of an electron im-
+pacting the walls of the coaxial chamber was calculated from
+the diffusion equation of electrons across the magnetic field
+based on the assumption that their number density profile in
+the radial direction was independent of time and axial posi-
+tion. In a cylindrical coordinate system it can be expressed
+as the product ne(x,r,t) = ne0(x,t)g(r). The balance equation
+(for a constant diffusion coefficient D) integrated over the ra-
+dius of the flux tube rmax was then given by:
+∂ne0(x,t)
+∂t
++ ∂ne0(x,t)ux(x,t)
+∂x
+= −νLne0(x,t)+kionne0(x,t)
+(9)
+With the loss frequency given by:
+νL = −rmax
+g′(rmax)
+S
+D
+(10)
+Where rmax is the radius of the flux tube, and the weighted
+cross section S is given by:
+S =
+� rmax
+0
+rg(r)dr
+and
+(11)
+A first choice for the diffusion coefficient D would be a co-
+efficient based on classical diffusion obtained from theories on
+standard electron-neutral collisions. It can be seen in Eq. 12
+where τ = 1/ν is the collision period with the neutral back-
+ground. However, the electron mobility tends to be higher
+than the value predicted by this classical diffusion approach48.
+The cause of this discrepancy is an active area of research in
+the electric propulsion field49,50. As a consequence, we de-
+cided to use the Bohm coefficient, which is a phenomenologi-
+cal coefficient accounting for the anomalous cross-field diffu-
+sion.
+DBohm = 1
+16
+kBTe
+eB
+or
+Dclassical =
+ωcτ
+1+(ωcτ)2
+kBTe
+eB
+(12)
+The probability for a given particle to be lost between t and
+t + ∆t is given by pL = νL∆t. In our quasi-1D model, the
+flux conservation relates the magnetostatic field to the cross-
+sectional area of the magnetic field tube as shown in Eq. 1.
+In this work we assumed g(r) = J0(k0r/rmax), with k0 the first
+zero of the Bessel function. Then Eq. 10 has the form νL ∝
+k2
+0/r2
+maxD. Since D ∝ B−1 and in the model BS = Bπr2
+max is a
+constant, the loss probability does not depend on the position
+along the flux tube and is given by:
+pL(x) = 2
+3
+π
+16k2
+0
+�1
+2m⟨v(x)2⟩
+�
+dt
+eA0B0
+(13)
+Where ∆t is the time step, m⟨v(x)2⟩/2 is the electron’s mean
+kinetic energy, and A0 = A(0) is the cross-section of the mag-
+netic field tube at x = 0. The losses are computed at each
+time-step. For all electrons in the source (such as x ≤ LS, LS
+being the length of the coaxial chamber as shown in Fig. 1b),
+the probability pL is computed using equation 13. A random
+number x is drawn from a uniform distribution. If x ≤ pL, the
+electron and a neighboring ion are removed from the simula-
+tion.
+C.
+Simulation Setup
+The electron dynamics and the electromagnetic solver were
+updated every iteration. For these one-dimensional calcula-
+tions the real Xenon mass for the ions was used and to speed
+up the calculations a subcycling was used so the ion’s position
+and velocity that were updated every 10 time steps as given by
+∆tions in Table I. The collisions were also computed every 10
+time steps as given by ∆tcoll. The charged particle’s population
+was seeded using a uniform density distribution (N ∼ 103).
+The electron’s initial energy along each of the x-y-z axis was
+set to Te = 20eV, while ions were assumed cold Ti = 0.03eV.
+These values were intended to reproduce a non-equilibrium
+plasma at low density. The choice of the initial electron tem-
+perature Te = 20eV is somewhat arbitrary. Checks run with
+several energy values between 10 eV and 30 eV showed no
+impact of the initial electron energy on the final characteristic
+of the steady state. To sustain the plasma at the beginning,
+a plasma source located at 2 mm from the backplate injected
+electrons at 3×105 ms−1 and ions at 3×102 ms−1 during the
+
+7
+first 150 ns of the simulation. These velocities were specified
+along each of the x-y-z axis. Here the idea was to sustain the
+initial plasma long enough for the ionization to pick up and
+the plasma density to grow. The conditions for the simulation
+presented below are shown in table I. With this choice of mag-
+netic field profile, the resonance condition fEM = eB/2πm
+was met at x = 6.7mm.
+TABLE I: Simulation parameters for the electromagnetic full
+PIC simulations using the quasi-one-dimensional model.
+Parameter
+Description
+Value
+∆t
+Time step
+1.6 ps
+∆x
+Mesh spacing
+167 µm
+C
+CFL condition
+2.87
+fEM
+Microwave frequency
+2.45 GHz
+LS
+Coaxial chamber length
+20 mm
+xECR
+ECR surface position
+6.7 mm
+W
+Weight for the charged particles
+2×105
+LD
+Computational domain length
+100 mm
+nn0
+Maximum number density of neutrals
+8×1019m−3
+Ln
+Neutral density characteristic length
+1.0 cm
+AL
+Cross-sectional area for the loss module
+1cm2
+∆tions
+Time step to push the ions
+10∆t
+∆tcoll
+Time step for collisions
+10∆t
+The simulation was run until it reached a steady state, usu-
+ally after around 30µs which represents between 5 to 8 ion
+transit times. The definition of this steady state was done by
+following up the variation of the total number of particles in
+the domain, its mean kinetic energy, and the particle flux at
+the backplate and the plume since an equal number of ions
+and electrons must be impacting both surfaces, as shown in
+Fig. 2. At the end of the simulation, when the steady state
+was reached, the plasma properties were obtained by calcu-
+lating the time average for each parameter over several time
+steps. Overall, the wall time of the simulation was 44 hours,
+with 12 OpenMP threads.
+III.
+RESULTS
+Figure 3 shows the steady-state plasma potential distribu-
+tion over the whole computational domain. Except for slight
+random fluctuations on the instantaneous potential, no large
+scale fluctuations were observed. Time-averaging improved
+the signal to noise ratio but did not blur the shape of the pro-
+file. The backplate reached a positive steady state potential
+of around 70V. The peak of the plasma potential was 105V,
+and it was reached at around 3 mm, interestingly not at the
+ECR surface (indicated with a vertical dashed line). Indeed,
+the shapes of the plasma density and potential are driven by
+the ionization rate. For this simulation, the ionization rate
+was monotonically decreasing, because the background den-
+sity decrease was faster than the ionization rate increase due
+to the plasma heating. As a consequence, the maximum ion-
+ization was upstream of the ECR surface. This peak in the
+plasma potential formed a barrier. As a result, ions collected
+FIG. 2: Time evolution of simulation quantities. Top frame :
+total number of macro-particles (ions and electrons) in the
+simulation. Middle frame : Mean kinetic energy of the
+electrons. Bottom frame : particle fluxes at the boundaries
+(backplate and outlet) and particle source and sink terms in
+the whole computational domain. The volume loss gives the
+average number of particle lost per timestep due to the Bohm
+loss model. For the steady-state analysis, the particles
+quantities are sampled after t = 30µs
+on the backplate were necessarily created in a region where
+x ≤ 3mm, while ions collected downstream were created in
+a region where x ≥ 3mm and accelerated into the nozzle
+by the ambipolar electric field. Sheaths were formed at the
+backplate and the vacuum chamber wall. The plasma sheath
+width on the backplate was ∼ 0.1mm. At the downstream
+end x = L of the domain, as shown by Fig. 4, the plasma
+sheath began at around 90 mm. This size was consistent with
+a Debye length λD ∼ 1 − 5mm for a plasma density around
+1×108 cm−3. The electron and ion peak number density was
+1.12×1011 cm−3 at x = 1.5mm. Recall that the ECR condi-
+tion is met at x = 6.7mm.
+In Fig. 5 we plotted the electron’s mean kinetic energy in
+both the axial (e∥) and the perpendicular (e⊥) direction as a
+function of the axial position on the domain. First, we ob-
+served that the mean parallel kinetic energy remained nearly
+constant, around 4−5eV, over the whole simulation domain.
+
+e losses
+backplate
+ionization rate
+outlet
+Electrons
+Ions8
+FIG. 3: Plasma potential. The vertical dashed line indicates
+the ECR surface location. The horizontal dashed line shows
+the backplate potential ΦBP. The two colored zones delineate
+the regions where the plasma potential is above (∆Φ >> 0 or
+below (∆Φ < 0) the backplate potential.
+FIG. 4: Electron (solid) and ion (dashed) number densities.
+The dashed line indicates the ECR surface location.
+The perpendicular energy was higher than the parallel com-
+ponent, which underlined the anisotropic heating of the elec-
+trons in this thruster. More precisely, the mean perpendicular
+kinetic energy e⊥ reached a first peak at around x = 9mm and
+then decreased before reaching a global maximum of 25 eV at
+x = 45mm. After this point, e⊥ decreased until the end of the
+simulation domain. Over the whole simulation domain, the
+anisotropy ratio Te,⊥/Te,∥ was found to vary between 2.5 and
+7.5. Given that the ECR heating increases the perpendicular
+energy of the electrons, it was expected to see an anisotropic
+behavior depending on the direction parallel or perpendicular
+to the magnetic field lines. However, the second broad en-
+FIG. 5: Electron’s mean kinetic energies: e∥ longitudinal
+(blue line) and e⊥ perpendicular (red line) directions. The
+location of the ECR surface is shown by the dashed line.
+ergy peak in the downstream part of the magnetic nozzle was
+puzzling. To get a better understanding of these feature, it
+was necessary to evaluate the energy deposition by the elec-
+tromagnetic field.
+A.
+Electromagnetic Energy deposition in the source
+To understand how the field energy was transferred to the
+particles, we considered the energy balance equation, includ-
+ing the Poynting flux (its derivation is provided in appendix
+B).
+∂εEM +ε
+∂t
++∇·(Q+Π) = Scoll
+(14)
+In this equation, ε and εEM stands for the electron kinetic en-
+ergy density and the electromagnetic energy density, respec-
+tively. Q and Π are the kinetic energy flux and electromag-
+netic energy flux; Scoll, whose expression is given in Eq. B4,
+is the volume power loss term due to the collisions and the
+diffusion. This latter term account for the energy lost by elas-
+tic and inelastic collisions with the neutral background and by
+the particles removed by the loss model. Since we were inter-
+ested in the steady state regime, and considering that the field
+quantities depend on x only, this was further simplified to:
+1
+A
+∂
+∂xA(Qe +Qi +Π) = Se,coll +Si,coll
+(15)
+where we separated the time-averaged total energy flux into a
+kinetic contribution due to the electrons Qe , the ions Qi and
+the electromagnetic contribution Π. The kinetic energy flux of
+the electron was further separated into a flux of parallel energy
+Qe,∥ and perpendicular energy Qe,⊥ (see appendix B).
+To quantify the magnitude and direction of the energy ex-
+changes between the electromagnetic field and the particles,
+the different terms of Eq. 15 were evaluated. To do so, the
+particles and field quantities were sampled in the steady state
+phase (after t = 30µs, see fig. 2). First, particles were sorted
+in 120 spatial bins equally spaced along the axial direction. In
+
+△Φ<0
+△Φ>0ell
+el9
+each bin, the moments of particle distribution provided the to-
+tal energy flux Qe and Qi, as detailed in appendix B. Second,
+the cross product of the electric and magnetic field provided
+the axial component of the Poynting vector. This vector was
+time-averaged over a period corresponding to an integer num-
+ber of wave periods.
+FIG. 6: Energy source terms of eq. 15 along the axial
+direction. The location of the ECR surface is shown by the
+dashed line.
+FIG. 7: Perpendicular and parallel energy source terms for
+the electrons along the axial direction. The location of the
+ECR surface is shown by the vertical dashed line.
+The results, plotted in Fig.
+6, showed first that the en-
+ergy source terms are negligible in the plume region (x ≥
+20mm) compared to the source region (x < 20mm). In the
+plume region, the magnitude of the source terms is below
+1×104 Wm−3. For that magnitude, the signal is dominated
+by the statistical noise of the PIC simulation.
+This noise
+drowns the finer features of the source terms, especially for
+the electrons which are more prone to statistical noise. Never-
+theless, this underlines that most of the energy exchanges take
+place in the source region and cannot explain the secondary
+peak for mean perpendicular kinetic energy e⊥ observed in
+the plume region (Fig. 5).
+Second, the collision source term is negative over the whole
+source region. Its magnitude is maximum near the backplate,
+where the neutral and plasma densities are higher, and de-
+creases along the source axis. This behaviour is not unex-
+pected, since this term lumps together the contribution of in-
+elastic collisions and the diffusion model: these two processes
+are loss mechanisms for the plasma. Given that the plasma
+density and the neutral gas density decrease as we move away
+from the backplate, the collision frequency drops and the mag-
+nitude of the energy loss decreases.
+Third, the sign and magnitude of the source terms reveal
+different phenomena. For x ≤ 3mm, the terms linked to the
+Poynting vector are positive, while the source term due to the
+electron energy flux is negative. This indicates an energy con-
+version from the electron kinetic energy to the field energy. In
+parallel, the ion source term is positive, which points to a gain
+of energy in the sheath. For the 3mm < x ≤ 10mm range,
+the Poynting source term shows a negative peak, while the
+source term due to the electron energy flux is positive, with
+a peak centered on the ECR surface location. In that case,
+there is a transfer of energy from the field to the electrons.
+This latter feature corresponds to the ECR heating of the per-
+pendicular energy mode of the electrons. In fact, this appears
+when considering the parallel and perpendicular contributions
+to the source term in Fig. 7. The perpendicular source term
+dominates over the parallel source term, with a peak centered
+on the ECR surface. The axial extent of this peak shows that
+the perpendicular mode of the electrons is heated in a zone of
+about ∆xECR ≈ 6mm, i.e., from x = 3mm to x = 9mm. Con-
+sidering that the ECR condition is only met on a specific sur-
+face, the presence of an extended region may seem surprising.
+However, as will be discussed later, most of the electrons in
+the magnetic field tube are confined and undergo a bouncing
+motion in the potential well formed by the electrostatic field
+and the magnetic mirror force. These bouncing electrons can
+cross the resonance surface with a significant parallel velocity,
+thus one may expect a shift of the resonance condition due to
+the Doppler effect. The width ∆xECR can be compared with
+the expected value for a Doppler broadened resonance ∆xD in
+Eq. 1651.
+∆xD =
+�
+�
+�
+�
+2πv∥
+ωc
+BMS
+���
+∂BMS
+∂x
+���
+(16)
+Using
+the
+electrons’
+mean
+axial
+velocity
+v∥
+is
+1.1×106 ms−1 we obtain ∆xD = 4.7mm.
+However, as
+it will be shown later in Fig. 8, there is a high dispersion
+for the values of v∥. Therefore, we can expect a much larger
+Doppler broadening for the fastest electrons. The electrons’
+axial velocity can reach values up to 3.0×106 ms−1 around
+the ECR zone. With this velocity, we can compute a max-
+imum Doppler broadening of 7.8mm, which means that
+4.8mm < ∆xD < 7.8mm. The ECR heating zone obtained in
+the simulations is consistent with the one expected analyti-
+cally, indicating that the Doppler effect is a good candidate to
+explain the width of the heating zone observed in Fig. 6.
+
+1
+AQ
+Ao x
+1
+AQi
+Ao x
+1
+AII
+Ao
+X
+S
+coll0
+AQe.ll
+Ao x
+1 0
+AQe,1
+Ao x
+A(Qe. lI +Qe,↓)10
+Because of Doppler-broadening, ECR heating of the elec-
+trons can even occur when the ECR surface is outside the
+plasma source. Indeed, the fact that the plasma in the thruster
+can be sustained even with an ECR zone outside the coax-
+ial chamber was demonstrated experimentally by Vialis 52,
+where the location of the resonance surface was placed at x =
+−0.17mm and x = −0.77mm. Doppler broadening is a pos-
+sible explanation to this finding and this hypothesis was tested
+in simulations using the same parameters described in Table
+I but with an input microwave frequency of fEM = 2.9GHz.
+It was possible to sustain a discharge for this frequency even
+though the ECR condition was meet at x = −1.78mm (i.e.,
+upstream of the plasma source).
+In summary, the analysis of the power deposition shows that
+the energy transfer occurs mainly in the source region. The
+electrons absorb the wave energy over a Doppler-broadened
+volume. This energy goes preferentially to the perpendicu-
+lar energy mode and leads to an anisotropy ratio Te,⊥/Te,∥ ∈
+[2.5,7.5]. This energy deposition from the field to the perpen-
+dicular energy mode can explain the first peak in perpendicu-
+lar energy seen in Fig. 5. However, the source terms are neg-
+ligible in the plume region and thus cannot explain the broad
+peak observed in this region.
+B.
+Electron confinement in the magnetic nozzle
+To determine the factors driving the evolution of the mean
+electron energy, we considered the electron distribution in the
+nozzle region. In Fig. 8 we plotted the normalized electron
+distribution in the velocity space v∥,v⊥ plane. The distribution
+was plotted at different locations in the computational domain.
+The results in Fig. 8 show that as we move downstream into
+the nozzle, we see an increased electron population with high
+energies in the perpendicular direction. To understand this
+phenomenon, we must first get a more detailed description of
+the electron confinement, i.e., how they get trapped and under
+which conditions they can leave the thruster.
+Three loss pathways are identified for the electrons pro-
+duced in the source.
+1. Cross field losses : electrons can diffuse across the mag-
+netic field, due to anomalous transport, collision, etc.
+This is modeled by the phenomenological cross-field
+diffusion model presented in section II B 4.
+2. Losses at the downstream of the nozzle electrons which
+have a kinetic energy sufficient to overcome the confin-
+ing potential well are lost, alongside ions accelerated by
+this same potential drop.
+3. Electrons than can overcome both the repelling poten-
+tial of the plasma sheath at the close-end of the source
+and the mirror-force are collected on the dielectric plate
+and will contribute to its surface charge.
+While the first loss mechanism does not depend on the ki-
+netic energy of a single electron but rather on the mean kinetic
+energy at a given location, the two other mechanisms depend
+on the electron kinetic energy. An electron moving along the
+magnetic field will see both an electrostatic potential Φ (Fig.
+3) and the magnetostatic field B. Depending on its initial ki-
+netic and potential energies, its trajectory might have turning
+points (where v∥ = 0) within the domain, or out of the domain.
+In the first case, this electron is confined in the potential well.
+In the latter case, the electron is lost, either downstream (loss
+pathway 2) or at the backplate (loss pathway 3). Now the lim-
+iting case between confined / unconfined electrons is when the
+turning points are located at the boundaries. This will define
+the necessary conditions for the electrons confinement. Let
+us note ΦBP the potential of the backplate. Neglecting the
+plasma-wave interaction, and noting µ the magnetic moment,
+the equation for the total energy of an electron is:
+Etotal = 1
+2mv2
+∥ + µB−eΦ
+(17)
+From Eq. 17 we can say that the electron is oscillating in
+an effective potential given by Uef f = µB − eΦ, where µB
+represents the magnetic confinement as the electron moves
+towards the backplate while −eΦ is the electrostatic confine-
+ment given by the plasma potential. It can be seen in Fig. 9 for
+different arbitrary values of the magnetic moment taken from
+the results of the simulation. Its concave shape explains the
+confinement of the electrons in the ECR thruster inside this
+potential well.
+Let us now consider an electron moving from an arbitrary
+initial point to a turning point at a position x0 along the longi-
+tudinal direction, such that v∥(x0) = 0. The energy conserva-
+tion between any initial location and the turning point gives:
+v2
+∥ +v2
+⊥
+�
+1− Bx0
+B
+�
+= −2e
+m ∆Φ
+(18)
+Where ∆Φ = Φx0 − Φ. Now let us consider the loss path-
+ways 2 and 3 identified above.
+For an electron lost to the downstream boundary (pathway
+2), the confinement condition is obtained by setting x0 = L. In
+that case, given the divergence of the magnetic field, we have
+0 < 1−B(L)/B < 1 and ∆Φ < 0. Thus equation 18 describes
+an ellipse in the v∥ − v⊥ plane. Electrons in the ellipse have
+there turning points x0 ≤ L and remain confined by the elec-
+trostatic well. Electrons out of the ellipse can overcome this
+electrostatic confinement and are lost downstream.
+For an electron lost to the backplate (pathway 3), the con-
+finement condition is obtained by setting x0 = 0. Because the
+magnetic field is monotonically decreasing, 1 − B(0)/B < 0.
+In addition, Φ(0) = ΦBP. We define the loss cone angle as:
+sin(θ) =
+�
+B
+B(0)
+(19)
+Equation 18 can be recast as:
+v2
+⊥ = tan2(θ)
+�
+v2
+∥ + 2e
+m ∆Φ
+�
+(20)
+Depending on the sign of ∆Φ, three cases are possible:
+
+11
+(a)
+(b)
+(c)
+FIG. 8: Electrons distribution in the velocity space v∥,v⊥
+plane at different locations on the simulation domain. The
+number of electrons has been normalized by the total number
+of electrons for each case independently. For each case, we
+also plotted what we call the confinement boundaries
+described by an analytical model (Eq. 19 and 20). The dotted
+line is the magnetic confinement, the dashed line the
+electrostatic potential at the backplate, and the solid line is
+the electrostatic confinement on the plume. (a) x = 5 mm, (b)
+20 mm, (c) x = 80 mm.
+FIG. 9: Schematic view of the effective potential profile for
+arbitrary values of the magnetic moment.
+• ∆Φ = ΦBP − Φ = 0: The confinement of the electron
+at the backplate is given exclusively for the topology of
+the magnetostatic field according to the loss cone angle
+θ. Those electrons with values for v∥,v⊥ such as they
+are located in the loss cone and will therefore be lost at
+the backplate (Fig. 10a).
+• ∆Φ = ΦBP −Φ < 0: The backplate potential repels the
+negative charges and thus confines the electron. Conse-
+quently, a fraction of the electron in the loss cone will
+be reflected back and stay confined (Fig. 10b).
+• ∆Φ = ΦBP −Φ > 0: The backplate potential attracts the
+electrons. Thus, even electrons out of the loss cone are
+collected. Therefore, the confined electrons are those
+that meet two conditions: they are not on the loss cone
+for the magnetic field, and they are energetic enough
+in the perpendicular direction to avoid being lost at
+the backplate thanks to the electrostatic acceleration to-
+wards it (Fig. 10c).
+Fig. 3, displays the sign of ∆Φ in the nozzle. The combina-
+tion of the loss conditions at the backplate (pathway 3) and
+downstream (pathway 2) delimits a confinement volume in
+phase space where electrons are confined, as shown in Fig.
+10. Pitch-angle scattering, either caused by collisions with the
+neutral background or by the electromagnetic field, enables
+electrons to cross the confinement volume boundaries. De-
+pending on which boundary is crossed, electrons are lost at the
+backplate or at the downstream side of the nozzle. Indeed, it is
+important to recall that the electron deconfinement is mainly
+driven by these two phenomena in the source region. Since
+the neutral background density decreases exponentially, most
+of the collisions occur in the source. In addition, as shown
+above, wave absorption happens over a few millimeters in the
+source. As a consequence, the electron deconfinement rate is
+driven by the wave interaction and the collisions:
+• Interaction with the electromagnetic wave is akin to a
+scattering of the electron momentum53. After several
+passages through the ECR heating zone, the electron
+may gain enough energy to overcome the electrostatic
+barrier and escape into the plume.
+
+12
+(a)
+(b)
+(c)
+FIG. 10: Confinement boundaries in velocity space on the
+v∥,v⊥ plane where the orange colored zones describe the
+electrons being trapped in the ECR thruster. Solid line for the
+plume electrostatic confinement, and dashed (electrostatic)
+plus dotted (magnetic) lines for the backplate confinement.
+Where: (a) ∆Φ = 0, (b) ∆Φ < 0, and (c) ∆Φ > 0.
+• If the electron undergoes an elastic collision, it will ran-
+domly scatter its velocity vector. If the electron scat-
+tered momentum falls in the loss region defined by Eq.
+20 and shown in Fig. 10, the particle is lost at the back-
+plate.
+If we now go back to the results in Fig. 8 for the elec-
+tron distribution in the velocity space v∥,v⊥ plane, we notice
+that as we move downstream into the plume, the confinement
+boundaries change. There is a transition from a confinement
+boundary as the one in Fig. 10b (∆Φ < 0) to the one in Fig.
+10c (∆Φ > 0). This transition is a consequence of the fact
+that, as shown in Fig. 3 the plasma potential is greater than the
+backplate potential in the source Φ > ΦBP and lesser than ΦBP
+downstream. Thus, inside the coaxial chamber, the plasma po-
+tential is such that ∆Φ = ΦBP − Φ < 0 (Fig. 10b), and in the
+plume section it is such that ∆Φ = ΦBP − Φ > 0 (Fig. 10c).
+As a consequence, as we move downstream into the magnetic
+nozzle, the mean perpendicular kinetic energy can increase.
+However, this is not given by an additional heating phase but
+as a result of confining only a highly energetic electron pop-
+ulation in the perpendicular direction. Those electrons with a
+low perpendicular kinetic energy (i.e., below the dashed line)
+are lost at the backplate as previously described. It can be
+seen as a filtering process where only the hot electrons are
+confined, and this is what we see when plotting the electron
+perpendicular kinetic energy in Fig. 5. Further downstream
+of the magnetic nozzle, the magnitude of the potential well to
+the end of the nozzle decreases, while the magnitude of the at-
+tracting potential drop to the backplate increases. This results
+in a narrower distribution for the confined population and fi-
+nally a decrease in the mean perpendicular kinetic energy.
+IV.
+CONCLUSIONS
+We have performed electromagnetic full-PIC simulations
+of the ECR thruster using a 1D3V model that allowed us
+to shed light onto some of its working principles. The re-
+sults confirmed the expected anisotropic behavior for the elec-
+trons’ energies in the direction perpendicular and parallel to
+the magnetic field lines and a peak for the mean perpendic-
+ular energy near the resonance zone. The microwave energy
+injected at the backplate of the thruster propagates through
+the coaxial chamber while being absorbed by the electrons
+increasing their kinetic energy perpendicular to the magnetic
+field lines. The absorption takes place exclusively inside the
+coaxial chamber on a zone of 6 mm around the resonance
+condition.
+This zone is coherent with the predicted value
+from Doppler broadening. The width of this heating zone
+may explain why the thruster works even with a configura-
+tion in which the resonance condition is met outside the coax-
+ial chamber52. From a practical point of view, this feature
+improves the reliability of the thruster, since it means that the
+thruster can still operate even when the magnetostatic field de-
+creases, for example due to excessive heating of the magnets.
+The results also show, unexpectedly, that the electrons’
+mean perpendicular energy has a second peak in the plume
+due to the confinement of highly energetic electrons.
+The
+confinement is determined by the backplate’s potential, the
+magnetostatic field, and the potential drop on the plume. As
+a consequence, there is a population of trapped electrons
+with significant perpendicular kinetic energy in the down-
+stream region of the magnetic nozzle. The existence of dou-
+bly trapped electron population has been investigated using
+a kinetic model with a paraxial approximation similar to this
+work54. While this work was assuming the shape of the initial
+distribution function, it has also been seen in the case of an
+anistropic distribution that the perpendicular electron temper-
+
+= arcsin(V B,)
+0-2e△Φ
+V=
+m2e△Φ
+*
+Vi = tan(0)
+m13
+ature could increase in the diverging part of the nozzle55. We
+can speculate that these high temperature electrons trapped in
+the plume could be important to drive some instabilities ob-
+served in the plume that are thought to enhance cross-field
+transport of electrons and thus play a role in the detachment.
+In particular, Lower Hybrid drift instabilities have been re-
+cently observed in diverging magnetic nozzle. In these ex-
+periments, diamagnetic drift vD = ∇pe⊥ ×B/enB2 was iden-
+tified for the primary energy source for the instability56. The
+trapped electrons could enhance the radial gradient in perpen-
+dicular pressure and thus enhance the diamagnetic drift.
+Appendix A: Electron motion integration in the Quasi-1D
+model
+On one hand, the model assumes that the axial static mag-
+netic field in the flux tube depends on x only
+dBx
+dx = α(x)
+(A1)
+In the quasi-1D model, the field remains constant across the
+section of the tube. On the other hand, in the ECR thruster
+the electromagnetic part of the magnetic field (the part due
+to the propagation of the electromagnetic power injected in
+the source) remains negligible compared to the static part.
+Indeed, assuming a locally plane wave, the Poynting vector
+S = E × B/µ0. Since E ≃ cB and S ≃ 1Wcm−2, the order
+of magnitude for the electromagnetic component of the mag-
+netic field is B ≃ 1×10−2 mT. The static part of the field is
+between 10 mT and 10 mT, much greater than the electromag-
+netic component. Therefore, using the divergence equation
+for the static magnetic field and neglecting the electromag-
+netic component, it is possible to obtain the radial component
+of the magnetic field :
+Br(x,r) = −α(x)r
+2
+(A2)
+Assuming this value for the radial part of the magnetic field
+ensures that the divergence condition is automatically en-
+forced for the static part of the field. The electron guiding
+center is on the flux tube centerline. The Larmor radius of the
+electrons is given by:
+rL(x) = V⊥(x)
+ωc(x)
+(A3)
+Where V⊥ =
+�
+v2y +v2z, ωc(x) = eBx(x)/me. The gyromotion
+of the electron is shown in Fig. 11. Thus, knowing the parti-
+cle velocities it is possible to obtain the phase angle θ in its
+gyromotion, as given in Eq. A5.
+cos(θ) =
+y
+rL(x) = vz
+V⊥
+(A4)
+sin(θ) =
+z
+rL(x) = − vy
+V⊥
+(A5)
+Knowing the phase angle, sine and cosine, the By an Bz com-
+ponents can be deduced from eqs. A2 and A5.
+FIG. 11: Gyromotion in the plane normal to the axial static
+field Bx.
+Appendix B: Energy equation
+For the energy equation for the particles we consider the
+second order moment of the Vlasov equation. Multiplying the
+Vlasov equation for the electrons by
+� mev2/2, we obtain:
+∂
+∂t
+� mev2
+2
+fed3v+ ∂
+∂x ·
+� mev2
+2
+v fed3v
+(B1)
++qe
+� v2
+2 (E+v×B)· ∂ fe
+∂v d3v =
+� mev2
+2
+�∂ fe
+∂t
+�
+col
+d3v
+Where the right hand side lumps the contribution of colli-
+sions and the loss model detailed in section II B 1 and II B 4,
+respectively. Eq. B1 can be rewritten as:
+∂εe
+∂t +∇·Qe =−je ·E+ScollQe =
+� mev2
+2
+v fed3v (B2)
+εe =
+� mev2
+2
+fed3vr
+(B3)
+Se,coll =
+� mev2
+2
+�
+∂ fe
+∂t
+�
+coll d3v
+(B4)
+The same procedure can be applied to the ions:
+∂εi
+∂t +∇·Qi =−ji ·E+ScollQi =
+� Miv2
+2
+vfid3v (B5)
+εi =
+� Miv2
+2
+fid3v
+(B6)
+Si,coll =
+� Miv2
+2
+�
+∂ fi
+∂t
+�
+coll d3v
+(B7)
+Considering the electron population, the total heat flux can be
+written as:
+Qe = qe +Pe ·ue +neue(eK +EK)
+(B8)
+
+14
+Where the density is given by ne =
+� fed3v and the macro-
+scopic velocity by neue =
+� fevd3v. The random part of the
+velocity is c = v−ue The different terms are then:
+qe =
+� me
+2 c2cd3c
+(B9)
+Pe =
+�
+meccd3c
+(B10)
+EK = 1
+2meu2
+e
+(B11)
+eK =
+� me
+2 c2d3c
+(B12)
+Considering the one-dimensional approximation, we are con-
+sidering only the axial (parallel) part of the total heat flux.
+Thus, after averaging over a period of the incoming wave :
+< ... >= 1
+T
+� ...dt, we obtain:
+∂
+∂xA(x)
+�
+qe∥
+�
++A
+�
+Pe∥u∥ +Pe⊥u⊥
+�
++A(x)
+�
+ue∥ +neue;∥(eK +EK)
+�
+= −A(x)⟨je ·E⟩+A(x)
+�
+Se,coll
+�
+(B13)
+A similar expression can be written for the ions. if we make
+use of the Poynting theorem, we can relate the time averaged
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+µ0 :
+A(x)⟨(je +ji)·E⟩+ ∂
+∂x ⟨Π⟩ = 0
+(B14)
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf,len=912
+page_content='Anisotropic Electron Heating in an Electron Cyclotron Resonance Thruster with Magnetic Nozzle J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Porto,1, 2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Elias,1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Ciardi2 1)Physics - Instrumentation and Space Department, ONERA/DPHY, Université Paris Saclay F-91123 Palaiseau – France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 2)Sorbonne Université, Observatoire de Paris, PSL Research University, LERMA, CNRS UMR 8112 75005 Paris – France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' (*Electronic mail: paul-quentin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='elias@onera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='fr) (*Electronic mail: jcportoh@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='com) (Dated: 30 January 2023) In a grid-less Electron Cyclotron Resonance (ECR) plasma thruster with a diverging magnetic nozzle, the magnitude of the ambipolar field accelerating the positive ions depends of the perpendicular energy gained by the electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This work investigates the heating of the electrons by electromagnetic waves, taking their bouncing motion into account in a confining well formed by the magnetic mirror force and the electrostatic potential of the thruster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' An electromagnetic Particle-In-Cell (PIC) code is used to simulate the plasma in a magnetic field tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The code’s Maxwell solver is based on a semi-Lagrangian scheme known as the Constrained Interpolation Profile (CIP) which enables larger time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The results show that anisotropic plasma heating takes place exclusively inside the coaxial chamber, along a Doppler- broadened zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' It is also shown that a trapped population of electrons with a larger perpendicular energy exists in the plume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' INTRODUCTION Electric thrusters play a fundamental role in the field of space propulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Their main advantage lies in an efficient use of the propellant mass, and therefore a reduced consump- tion of propellant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Hall Effect Thrusters or Gridded Ion En- gines are examples of the most well-known and flight-proven technologies in the current propulsion market nowadays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Both technologies eject an ion beam which is subsequently neutral- ized to prevent the spacecraft from charging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Several compo- nents of these technologies, such as the acceleration grid or the neutralizer, are subject to erosion and wear and for this reason, meeting the challenging lifetime targets requires care- ful optimization and demanding testing1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The complexity of some of the components has driven the investigation of al- ternative concepts of propulsion devices that require neither a grid nor a neutralizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The Electron Cyclotron Resonance (ECR) plasma thruster2,3 is one of these concepts and it is the subject of the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The ECR plasma thruster consists of a semi-open chamber where a quasi-neutral plasma is heated by electron cyclotron resonant microwaves at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='45GHz, and accelerated by a mag- netic nozzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This concept was first proposed in the 1960s in the works of Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 4 and Nagatomo 5, then further de- veloped by Sercel 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' These studies used a prototype with a wave-guide structure to couple the microwaves to the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Their results showed that it was possible to achieve specific impulses and thrust values high enough to be of interest for space propulsion applications6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Nonetheless, the inefficiency, size and weight of the micro-wave sources and electromag- nets at that time led to a stagnation of the research on ECR thrusters for several years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Interest for this technology arose again recently with experimental works7,8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In particular, the use of coaxial microwave coupling structures and compact rare-earth permanent magnets were instrumental in designing compact sources (a schematic of the design is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' More experimental and theoretical efforts has since been made in order to get a deeper understanding of the phys- ical phenomena governing the plasma heating and acceler- ation in the thruster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Experimental characterizations of the plasma properties have been carried out using different mea- surement techniques such as Langmuir and Faraday probes, Laser Induced Fluorescence diagnostics, diamagnetic loops and thrust balances2,9–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Unfortunately, most of the exper- imental studies so far have been limited to survey the plasma outside the thruster coaxial chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Recently, a resonant probe was developed to measure an electron density of about 1×1011 cm−3 at the source exit plane, close to the coaxial chamber13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In the source, it is likely that the plasma density is higher (∼ 1×1012 cm−3) with electron temperatures of a few tens of eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' From a theoretical point of view, as a first step, global models describing the energy balance in the plasma source were proposed as a means to obtain the key parameters of the thruster14,15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' While this approach yielded good agree- ment with measured electron temperature at high mass flow rate or high pressure, they failed at the lower mass-flow rate where the thruster achieves its best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Indeed, the assumptions of uniform electron temperature and isotropic Maxwellian electron distribution are too crude approxima- tions when collisionality decreases and the electron mean free path becomes much larger than the source length: in that range non-local effects become prevalent, as electrons undergo a bouncing motion along the magnetic field line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Those elec- trons which cross the ECR surface can gain energy depending on their phase in the gyromotion16, which leads to a strong anisotropy of the distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' An attempt to account for this stochastic heating in the plasma was made by consid- ering the electron heating as a random walk in phase space17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='11411v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='plasm-ph] 26 Jan 2023 2 While this model provided a qualitative agreement with the measured ion energies, it could not account for the plasma feedback on the waves (assumed constant) and the collisions along the bouncing motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Recently Sánchez-Villar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 18 performed 2D axisymmetric simulations of the thruster with a hybrid model consisting of particle-in-cell (PIC) ions and a fluid model for the mass-less electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' One of the main find- ings of this study was the identification of different regions in the source where the waves are either propagating or evanes- cent, with most of the power absorption taking place close to the inner conductor, near the ECR surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' By acting as a sink for the plasma, the inner conductor induces a decrease of the plasma density in its vicinity, enabling the propaga- tion of electromagnetic waves downstream of the ECR sur- face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' These features lead to the formation of a hot electron beam close to the inner conductor, with a colder plasma in the bulk of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' While these 2D results provided important insights on the operation of the thruster, some assumptions of the fluid model limit the validity of the results obtained from these simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The most important one being the assumption of isotropic electron temperature which excludes anisotropic heating in the directions parallel and perpendicu- lar to the magnetostatic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This latter point is still an open question for this technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Indeed, ECR heating is expected to lead to anisotropic heating of the electron translation modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This difference affects the power losses near the source walls and the potential drop in the magnetic nozzle19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' However, most of the electron temper- ature measurements performed in the thruster plume did not differentiate perpendicular and parallel electron temperature (with respect to the local magnetic field direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' A way to measure the electron temperature anisotropy is, for example, incoherent Thomson scattering20, but this type of measure- ment is not presently available in the ECR source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' At any rate, this heating is intimately linked to the absorption of the electromagnetic waves in the coaxial source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Another issue is the non-local transport due to the bounc- ing magnetized electrons in the nozzle (the electron mean free path is greater than the source radius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In particular, the pro- duction and the heating of the electrons are not necessarily at the same location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' While gaining a better understanding of these issues should firstly rely on experimental measurements, the challenges as- sociated with such an investigation are a strong incentive to use numerical models, even if simplified, to investigate the main physical processes at play in ECR thrusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In par- ticular, such a model should be able to account for the self- consistent wave absorption and the anistropic heating, as well as the non-local effects and bouncing motion of the parti- cles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Electromagnetic kinetic models, such as Particle-In-Cell (PIC) or Vlasov methods, are natural candidates for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' There are currently a few works using kinetic simulations of propulsion devices exploiting the ECR phenomenon, however the majority of these developments are concerned with grid- ded ion thrusters with ECR heating21–24, where the plasma acceleration is achieved by a grid-imposed electric field and not the plasma expansion in the magnetic nozzle, as in our design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Takao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 23 successfully modelled a gridded ion thruster where the ions are produced in an ECR source at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='2 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The authors used a Particle-In-Cell (PIC) code consid- ering the microwave electric field as a temporal modulation of its initial amplitude obtained by simulating the microwave propagation without plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Therefore, in this approach the plasma feedback on the wave was considered negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The main purpose of the our study is to perform full-PIC electromagnetic simulations of the plasma in the thruster, tak- ing into account the plasma feedback on the wave propaga- tion, and to investigate the heating and confinement of the electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' For this purpose it is necessary to simulate the microwave propagation and its interaction with the charged particles in the source and the nozzle region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' However, due to the complexity and computational cost of simulating a full 3D configuration (which should include the nozzle region), we restrict our investigation to the simplified case of an iso- lated magnetic flux tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This approximation effectively re- stricts the phase space to 4 dimensions (1 dimension in space, 3 dimensions in velocity space), and a 1D3V electromagnetic Particle-In-Cell can be used to model the ECR thruster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' We show that the electron heating takes place over a broad region in the thruster source and leads to a signifi- cant anisotropy (ratio Te⊥/Te∥ ∈ [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='5,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The perpendicular electron temperature reaches a first maximum in the source and, surprisingly, has a second maximum in the downstream region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The explanation for these features lies in the confine- ment of electrons in the potential well formed by the com- bination of the diverging magnetic field and the electrostatic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' NUMERICAL MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The Quasi-One-Dimensional Approach In the coaxial ECR thruster, an axially magnetized cylindri- cal permanent magnet creates a diverging static magnetic field BMS in the source and in the plume region12,14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This shape for the magnetostatic field was chosen to ensure a magnetic confinement at the close end of the coaxial chamber (called backplate in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 1) while allowing the electrons to get accel- erated in the plume thanks to the divergence of the magnetic field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In fact, the ECR uses a diverging magnetic field whose magnitude decreases from approximately 100 mT at the back of the source to around 5 mT 10 cm downstream of the thruster exit plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Under these conditions, assuming an electron tem- perature around Te ≃ 10eV, the Larmor radius of the elec- trons is between rL ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='07mm − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='4mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Thus electrons are strongly magnetized in the source and in the near-field plume region, while ions remain mostly unmagnetized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' As a conse- quence, before the onset of plasma detachment, electrons and ions are bound to the magnetic field tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Several mechanisms may account for the plasma detachment25 : collisions, stretch- ing of the magnetic field lines, electron demagnetization and plume instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' While it is out of the scope of this work to study the dominant mechanisms, several recent works have in- vestigated some of these effect in 2D PIC simulations26–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='In 3 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 1: ECR thruster: (a) Schematic view of the coaxial source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The magnetic field lines are shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The dashed surface corresponds to the flux tube (b) Schematic view flux tube used for the quasi-1D model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The exit plane of the coaxial source of length LS is represented by the dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The end of the computational domain is reached at x = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The axial magnetic profile and tube cross section along the axis are shown in red and blue, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' this work, we decided to rely on experimental evidence to de- fine the section of the nozzle where the plasma remains bound to the field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Recently, Little and Choueiri 29 have mapped the plasma potential in a magnetic nozzle to show that a good criterion for detachment is χp = rL/L∇B ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='1, where rL is the electron Larmor radius and L∇B = (∇B/B)−1 is the character- istic length scale of the magnetic field gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In the region of the nozzle where χp = rL/L∇B < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='1, the plasma remains attached to the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In our case, we considered a magnetic field with L∇B ≃ 5 − 10cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Under this condition, we have χp < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='1 up to L = 10cm downstream of the nozzle, and it is a reasonable assumption to consider that electrons do not detach from the magnetic field tube over this distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' As a consequence of this assumption, we decided to con- sider the creation and formation of the plasma enclosed in a magnetic field tube of length L = 10cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' More precisely, a portion of the thruster chamber and plume was represented by a quasi-1D model of a magnetic field tube with a varying cross-sectional area, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' There are several ex- amples of the use of quasi-1D models in the space propulsion field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Niewood and Martinez-Sanchez 30 used it to model a Magnetoplasmadynamic thruster, while De Giorgi and Fonta- narosa 31 studied a Vaporizing Liquid Microthruster with this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Recently, Saini and Ganesh 32 also used this ap- proach to model plasma expansion in a Radio-Frequency thruster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The moderate computational cost of a 1D3V model of the thruster facilitates the analysis of the plasma behavior in both the coaxial chamber and in the magnetic nozzle, and im- portantly, taking into account the nozzle is critical to resolve the bouncing motion of the electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The quasi-1D model assumes that the electrons and the ions are confined within a diverging magnetic flux tube, whose area is related to the axial magnetic field intensity through the con- servation of the magnetic flux: A(x)Bx(x) = A0B0 (1) The model further assumes that the electromagnetic fields and all the plasma properties are constant across the section of the flux tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' For the ECR thruster under consideration12,14, the shape of the magnetic field lines close to the antenna is well approximated by an exponential function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' For the sake of sim- plicity we approximated the magnetic field as: Bx(x) = B0 exp � − x LB � (2) In addition, we considered cylindrical symmetry for the static magnetic field around the field tube centerline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' These assumptions mean that the particles guiding centers remain on the centerline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Since the plasma is assumed uni- form in the cross section, this approach does not allow the formation of a diamagnetic current and E ×B drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' From now on, the term parallel and the subscript ∥ will refer to the direction parallel to the magnetostatic field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Similarly, perpendicular and the subscript ⊥ refer to the direc- tion perpendicular to the magnetostatic field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The source region, which corresponds to the coaxial cavity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 1b, was defined by 0 ≤ x ≤ LS, where LS is the coaxial source length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The plume region, which corresponds to the plasma expansion in vacuum, was defined by x ≥ LS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Particle-In-Cell Code overview The simulations were carried out with the Particle-In-Cell (PIC) code Rhei, which was developed to simulate low pres- sure cold plasmas and is adapted to parallel architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' It 4 can be run with either a pure MPI or a hybrid MPI/OpenMp parallelization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The code integrates a Monte-Carlo Collision (MCC) module to simulate the collisions between the charged particles and a prescribed neutral background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' At each it- eration, once the electrostatic and the electromagnetic fields were computed, the position of each macro particle labeled “p” was updated using dxp/dt = vp, and the velocity using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Each macro-particle represents W physical particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The value of W used in the simulation is given in table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' ms dvp dt = qs � EESp +EEMp +vp × � BMSp +BEMp �� (3) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 3, qs is the charge of the particle, ms the mass, xp the position, and vp the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Regarding the fields, they were computed at the location of the particle p using linear interpolation function, where EESp is electrostatic field from the charge distribution, BMSp is magnetostatic field from the permanent magnets and EEMp and BEMp are electromagnetic fields produced by the microwave source and by the plasma itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The equations of motion were integrated using the leap-frog method and the Boris scheme to get the v × B rotation from the Lorentz force33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Details of the integration in the context of the quasi-1D model are provided in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Particle quantities were projected on a uniform grid using linear shape functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The Rhei code development follows a test-driven approach to ensure the robustness and the maintainability of the code over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Additionally, several test cases were run as a val- idation of the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The first elementary test was the simu- lation of a magnetic bottle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The simulation domain, with a converging-diverging parabolic magnetic field, was uniformly loaded with a Maxwellian electron population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' At the end of the simulation the electron distribution in velocity space v∥,v⊥ was plotted to verify that the loss cone angle is coherent with the expected theoretical value arcsin �� B0/BMax � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The sec- ond elementary test concerned the electromagnetic modes in a one-dimensional magnetized plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The simulation do- main was initialized with a uniform Maxwellian distribution of electrons and cold ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The random fluctuations excited the modes of the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The resulting dispersion curves were obtained by computing the discrete 2D Fourier transform of the electric fields during the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This was compared to the expected theoretical description of the extraordinary and the ordinary wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Finally, the third test case was the classi- cal capacitively coupled discharge in Helium, which verified in particular the collision module34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Collisions The Monte-Carlo Collision module used the Null Collision technique35 to speed-up the computation of the collisions by removing the velocity dependency of the total collision cross- section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Assuming Np collision processes defined by their re- spective cross sections σi(v),i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='.Np, a null collision cross- section is defined as σ0(v) such that: σ0(v) = max v≥0 � Np ∑ i=1 σi(v) � − Np ∑ i=1 σi(v) (4) A first test over all the particles of species s found the fraction of particles which undergo a collision with the back- ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In that case the total cross section σT = ∑i=0 Npσi(v) (including the null collision process) did not depend on the velocity (thus avoiding a costly interpolation to get the cross section for all the particles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Then a second test among those selected particles computed all the collision cross sections for their given relative velocity and determined which cross section to use (including the null collision).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' When this test pointed to the null-collision cross section, then the particle did not experience an actual collision and was left unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' When the test pointed to another cross section, the the parti- cles experienced a collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' For the collisions of electrons with Xenon neutrals, we con- sidered a simplified set of three processes: elastic, ionization, and excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Excitation processes were lumped into a single process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Electron impact ionization and excitation were taken from the Morgan (Kinema Research & Software) database, while the total elastic scattering is from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' For all elec- tronic processes, we assumed an isotropic scattering of the rel- ative velocity vector between the electron and the target dur- ing the collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' For the ionization collisions, the kinetic en- ergy of the projectile electron was equally split (after subtract- ing the threshold energy) between the secondaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' For the collisions of Xenon ions with Xenon neutrals, we considered isotropic scattering and backscattering34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Ion cross sections comes from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' All electronic and ionic processes con- served momentum and total energy (kinetic plus internal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In order to start with a simplified description simulating weakly ionized plasmas, in which the collisions with the neutral par- ticles are the dominant process, Coulomb collisions were not considered in the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Indeed, the electron-ion collision fre- quency νei, for Maxwellian electrons, is given by: νei = ωp Λei lnΛei (5) Here, ωp is the plasma frequency and lnΛ is the Coulomb log- arithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' For the typical simulation conditions in the thruster source, as it will be shown below, the maximum plasma den- sity was ne ∼ 1×1011 cm−3, the electron temperature was Te ∼ 10eV and the electron-neutral elastic collision frequency was νen ∼ 1×107 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This gave lnΛ ∼ 12 − 15, ωp ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='8×1010 rads−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Consequently, the maximum electron-ion collision frequency was νei ∼ 1×105 s−1, much less than the the electron-neutral collision frequency νen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The neutral gas in the thruster is injected at the backplate (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 1a) and expands in the source resulting in a decreas- ing density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Since there is no measurement of the neutral gas density profile in the thruster, and to avoid a costly particle simulation of the neutral particles, we modelled this expansion heuristically by assuming that the neutral background density 5 followed an exponential profile: nn(x) = nn0 exp � − x Ln � (6) where nn0 is the maximum density of neutrals found at the close end of the source, and Ln is the neutrals density char- acteristic length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The assumption of a time-independent neu- tral gas density profile means that the simulation did not con- serve the total mass, momentum and energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In addition, it means that the neutral gas depletion due to ionizing collisions was not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' However, both of those limitations are acceptable in the frame of this work which does not seek to compute the total thrust and energy balance but is concerned with the particle heating and trapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' To estimate the neutral depletion we note that the ion removal is driven by their ve- locity (at most 10kms−1), while the neutral removal is driven by their thermal speed, ∼ 200ms−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' From mass balance, the neutral inflow is balanced by the ion flux and the neutral out- flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Using the characteristic speeds and the typical parame- ters for the gas density ng ∼ 1×1014 cm−3, and plasma den- sity nmax e ∼ 3×1011 cm−3, gives a neutral depletion of at most 10%, indicating that the assumption of a static background remained consistent with the assumed density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Fields solvers The Rhei code solved the Poisson equation to compute the electrostatic space potential Φ and the electric field (EES = Exx) using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 7 where the charge density is ρs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The solver implements a second order finite difference discretiza- tion and the resulting linear system is inverted using an itera- tive method (GMRES)38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' ∇2Φ(x,t) = −ρs(x,t) ε (7) In addition, an electromagnetic solver computed the fields produced by the microwave source and by the plasma itself: EEM = Eyy + Ezz and BEM = Byy + Bzz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This solver was based on the Constrained Interpolation Profile (CIP) method explained in detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This method considers not only the electromagnetic fields but also their spatial deriva- tives, therefore suppressing instabilities and providing lower numerical dispersion even when using coarse grids and large time steps40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The use of this method is a novel solution for a PIC code since most of the electromagnetic solvers are based on conventional approaches like the finite-difference time- domain method (FDTD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' It was shown that it provides higher accuracy than the latter under the condition of identical cell size41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The CIP method is a semi-Lagrangian scheme that circumvents the Courant-Friedrichs-Lewy (CFL) stability condition42,43, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=', (u∆t/∆x) < 1 where u is the magnitude of the velocity, ∆t is the time-step, and ∆x the length inter- val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This feature allows computations with CFL values ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='0, as can be seen in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 44 and 45 where the authors per- formed simulations using a CFL value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='6 in a Cartesian coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The gain in computational time, that is afforded by using high CFL values, is a key factor that en- ables the self-consistent kinetic simulations presented here to reach steady-state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In this paper, CFL values close to 3 were used for the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' As a check, simulations were also run with CFL=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='6 and compared to the results obtained with larger time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The results were identical to the one at larger time-steps, within small variations due to the noise inherent to the statistical nature of the PIC simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Finally, the CIP scheme does not necessarily maintain the divergence-free condition for the dynamic field BEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' How- ever, BEM is smaller than the magnetostatic field (which has divergence equal to zero by construction, see appendix A) by several orders of magnitude, over the whole computational do- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Therefore, the resulting error on the total divergence was considered to be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Boundary conditions As it was shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 1b when describing the model, the domain goes from x = 0 at the left side which corresponds to the backplate and the microwave input, to the right-end at x = L, as discussed in II A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Electrostatic: At the right end of the computational do- main x = L, we imposed a Dirichlet boundary condition, with Φ(L) = 0, to simulate the presence of a grounded vacuum chamber wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The dielectric backplate, at x = 0, is in con- tact with the plasma and therefore its surface voltage ΦBP is changed by the collection of charged particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This can be modeled as a capacitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Hence, the evolution of ΦBP is given by ∆ΦBP = ∆Q/(C∆t), where ∆Q is the charge deposited at the backplate at each time step, and C is an equivalent ca- pacitance under the assumption that the backplate is in con- tact with a grounded conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This capacitance is com- puted by considering that the backplate is a plane capacitor, its value is of a few picoFarads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Changing its magnitude mod- ifies the charging rate of the backplate and thus the transient phase of the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' However, its does not affect the steady-state voltage of the backplate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This approach guaran- tees that at steady-state, the ion flux equates the electron flux on the backplate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In principle the steady-state value of the backplate potential is also affected by other processes such as secondary electron emission or charge migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' However, for this study, these processes were neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Electromagnetic: In the coaxial ECR thruster, the mi- crowaves are injected as Transverse Electro-Magnetic (TEM) mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' For this 1D simulation, the TEM mode can be seen as a linearly polarized wave, where the radial component of the electric field is along the transverse y axis, the azimuthal magnetic field defines the z axis and the wavevector direction is along the longitudinal x axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Therefore, the microwaves were injected at the backplate as a propagating wave with a linear polarization along the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The incident wave was parametrized by its power per unit area Pin and its frequency fEM = ω/2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The electric fields from the injected linearly polarized wave were computed as Ey = √µcPin sin(ωt) and Ez = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 6 The injected microwave input power per unit area Pin could be fixed, or it could be adapted to keep a roughly constant pre- defined number of particles Ntarget during the transient phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This feature was intended to speed up the simulations by re- producing a faster plasma response to a given variation in the simulation’s parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The value of Pin can be regulated with an attenuation factor α ≤ 1 varying with the number of particles in the domain: α = exp(−Nparticles/Ntarget).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' A run performed without this regulation confirmed that it did not have an effect on the final steady state but only on the duration of the transient phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Particles: We imposed a loss condition at both ends of the domain, for both ions and electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Particles crossing these boundaries are suppressed from the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' As a simpli- fying assumption, secondary emission processes on the back- plate were not considered in this first approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Cross field diffusion loss model Electron cross-field diffusion is an important mechanism to model to get a more accurate representation of the discharge loss mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Previous works using PIC codes for elec- tric thrusters took it into account as wall losses by artificially increasing the collision rate or by using a profile of the cross field diffusion based on empirical evidence46,47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The electron balance equation is: ∂ne(r,t) ∂t +∇⊥ ·neu⊥ +∇∥ ·neu∥ = kionne(r,t) (8) Where u⊥ and u∥ are the electron macroscopic velocity per- pendicular and parallel to the local magnetic field, respec- tively, and kion is the ionization rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' For our 1D3V simulations, the transport along the magnetic field is taken into account by the kinetic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' However, the perpendicular transport cannot be modeled with a 1D model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Therefore we simulated the particle losses into the coaxial chamber walls using a phenomenological, Monte Carlo loss model, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The probability of an electron im- pacting the walls of the coaxial chamber was calculated from the diffusion equation of electrons across the magnetic field based on the assumption that their number density profile in the radial direction was independent of time and axial posi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In a cylindrical coordinate system it can be expressed as the product ne(x,r,t) = ne0(x,t)g(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The balance equation (for a constant diffusion coefficient D) integrated over the ra- dius of the flux tube rmax was then given by: ∂ne0(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='t) ∂t + ∂ne0(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='t)ux(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='t) ∂x = −νLne0(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='t)+kionne0(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='t) (9) With the loss frequency given by: νL = −rmax g′(rmax) S D (10) Where rmax is the radius of the flux tube,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' and the weighted cross section S is given by: S = � rmax 0 rg(r)dr and (11) A first choice for the diffusion coefficient D would be a co- efficient based on classical diffusion obtained from theories on standard electron-neutral collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' It can be seen in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 12 where τ = 1/ν is the collision period with the neutral back- ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' However, the electron mobility tends to be higher than the value predicted by this classical diffusion approach48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The cause of this discrepancy is an active area of research in the electric propulsion field49,50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' As a consequence, we de- cided to use the Bohm coefficient, which is a phenomenologi- cal coefficient accounting for the anomalous cross-field diffu- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' DBohm = 1 16 kBTe eB or Dclassical = ωcτ 1+(ωcτ)2 kBTe eB (12) The probability for a given particle to be lost between t and t + ∆t is given by pL = νL∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In our quasi-1D model, the flux conservation relates the magnetostatic field to the cross- sectional area of the magnetic field tube as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In this work we assumed g(r) = J0(k0r/rmax), with k0 the first zero of the Bessel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 10 has the form νL ∝ k2 0/r2 maxD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Since D ∝ B−1 and in the model BS = Bπr2 max is a constant, the loss probability does not depend on the position along the flux tube and is given by: pL(x) = 2 3 π 16k2 0 �1 2m⟨v(x)2⟩ � dt eA0B0 (13) Where ∆t is the time step, m⟨v(x)2⟩/2 is the electron’s mean kinetic energy, and A0 = A(0) is the cross-section of the mag- netic field tube at x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The losses are computed at each time-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' For all electrons in the source (such as x ≤ LS, LS being the length of the coaxial chamber as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 1b), the probability pL is computed using equation 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' A random number x is drawn from a uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' If x ≤ pL, the electron and a neighboring ion are removed from the simula- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Simulation Setup The electron dynamics and the electromagnetic solver were updated every iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' For these one-dimensional calcula- tions the real Xenon mass for the ions was used and to speed up the calculations a subcycling was used so the ion’s position and velocity that were updated every 10 time steps as given by ∆tions in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The collisions were also computed every 10 time steps as given by ∆tcoll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The charged particle’s population was seeded using a uniform density distribution (N ∼ 103).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The electron’s initial energy along each of the x-y-z axis was set to Te = 20eV, while ions were assumed cold Ti = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='03eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' These values were intended to reproduce a non-equilibrium plasma at low density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The choice of the initial electron tem- perature Te = 20eV is somewhat arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Checks run with several energy values between 10 eV and 30 eV showed no impact of the initial electron energy on the final characteristic of the steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' To sustain the plasma at the beginning, a plasma source located at 2 mm from the backplate injected electrons at 3×105 ms−1 and ions at 3×102 ms−1 during the 7 first 150 ns of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' These velocities were specified along each of the x-y-z axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Here the idea was to sustain the initial plasma long enough for the ionization to pick up and the plasma density to grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The conditions for the simulation presented below are shown in table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' With this choice of mag- netic field profile, the resonance condition fEM = eB/2πm was met at x = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='7mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' TABLE I: Simulation parameters for the electromagnetic full PIC simulations using the quasi-one-dimensional model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Parameter Description Value ∆t Time step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='6 ps ∆x Mesh spacing 167 µm C CFL condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='87 fEM Microwave frequency 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='45 GHz LS Coaxial chamber length 20 mm xECR ECR surface position 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='7 mm W Weight for the charged particles 2×105 LD Computational domain length 100 mm nn0 Maximum number density of neutrals 8×1019m−3 Ln Neutral density characteristic length 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='0 cm AL Cross-sectional area for the loss module 1cm2 ∆tions Time step to push the ions 10∆t ∆tcoll Time step for collisions 10∆t The simulation was run until it reached a steady state, usu- ally after around 30µs which represents between 5 to 8 ion transit times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The definition of this steady state was done by following up the variation of the total number of particles in the domain, its mean kinetic energy, and the particle flux at the backplate and the plume since an equal number of ions and electrons must be impacting both surfaces, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' At the end of the simulation, when the steady state was reached, the plasma properties were obtained by calcu- lating the time average for each parameter over several time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Overall, the wall time of the simulation was 44 hours, with 12 OpenMP threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' RESULTS Figure 3 shows the steady-state plasma potential distribu- tion over the whole computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Except for slight random fluctuations on the instantaneous potential, no large scale fluctuations were observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Time-averaging improved the signal to noise ratio but did not blur the shape of the pro- file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The backplate reached a positive steady state potential of around 70V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The peak of the plasma potential was 105V, and it was reached at around 3 mm, interestingly not at the ECR surface (indicated with a vertical dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Indeed, the shapes of the plasma density and potential are driven by the ionization rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' For this simulation, the ionization rate was monotonically decreasing, because the background den- sity decrease was faster than the ionization rate increase due to the plasma heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' As a consequence, the maximum ion- ization was upstream of the ECR surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This peak in the plasma potential formed a barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' As a result, ions collected FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 2: Time evolution of simulation quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Top frame : total number of macro-particles (ions and electrons) in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Middle frame : Mean kinetic energy of the electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Bottom frame : particle fluxes at the boundaries (backplate and outlet) and particle source and sink terms in the whole computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The volume loss gives the average number of particle lost per timestep due to the Bohm loss model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' For the steady-state analysis, the particles quantities are sampled after t = 30µs on the backplate were necessarily created in a region where x ≤ 3mm, while ions collected downstream were created in a region where x ≥ 3mm and accelerated into the nozzle by the ambipolar electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Sheaths were formed at the backplate and the vacuum chamber wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The plasma sheath width on the backplate was ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='1mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' At the downstream end x = L of the domain, as shown by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 4, the plasma sheath began at around 90 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This size was consistent with a Debye length λD ∼ 1 − 5mm for a plasma density around 1×108 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The electron and ion peak number density was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='12×1011 cm−3 at x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='5mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Recall that the ECR condi- tion is met at x = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='7mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 5 we plotted the electron’s mean kinetic energy in both the axial (e∥) and the perpendicular (e⊥) direction as a function of the axial position on the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' First, we ob- served that the mean parallel kinetic energy remained nearly constant, around 4−5eV, over the whole simulation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' e losses backplate ionization rate outlet Electrons Ions8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 3: Plasma potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The vertical dashed line indicates the ECR surface location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The horizontal dashed line shows the backplate potential ΦBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The two colored zones delineate the regions where the plasma potential is above (∆Φ >> 0 or below (∆Φ < 0) the backplate potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 4: Electron (solid) and ion (dashed) number densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The dashed line indicates the ECR surface location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The perpendicular energy was higher than the parallel com- ponent, which underlined the anisotropic heating of the elec- trons in this thruster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' More precisely, the mean perpendicular kinetic energy e⊥ reached a first peak at around x = 9mm and then decreased before reaching a global maximum of 25 eV at x = 45mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' After this point, e⊥ decreased until the end of the simulation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Over the whole simulation domain, the anisotropy ratio Te,⊥/Te,∥ was found to vary between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='5 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Given that the ECR heating increases the perpendicular energy of the electrons, it was expected to see an anisotropic behavior depending on the direction parallel or perpendicular to the magnetic field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' However, the second broad en- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 5: Electron’s mean kinetic energies: e∥ longitudinal (blue line) and e⊥ perpendicular (red line) directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The location of the ECR surface is shown by the dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' ergy peak in the downstream part of the magnetic nozzle was puzzling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' To get a better understanding of these feature, it was necessary to evaluate the energy deposition by the elec- tromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Electromagnetic Energy deposition in the source To understand how the field energy was transferred to the particles, we considered the energy balance equation, includ- ing the Poynting flux (its derivation is provided in appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' ∂εEM +ε ∂t +∇·(Q+Π) = Scoll (14) In this equation, ε and εEM stands for the electron kinetic en- ergy density and the electromagnetic energy density, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Q and Π are the kinetic energy flux and electromag- netic energy flux;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Scoll, whose expression is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' B4, is the volume power loss term due to the collisions and the diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This latter term account for the energy lost by elas- tic and inelastic collisions with the neutral background and by the particles removed by the loss model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Since we were inter- ested in the steady state regime, and considering that the field quantities depend on x only, this was further simplified to: 1 A ∂ ∂xA(Qe +Qi +Π) = Se,coll +Si,coll (15) where we separated the time-averaged total energy flux into a kinetic contribution due to the electrons Qe , the ions Qi and the electromagnetic contribution Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The kinetic energy flux of the electron was further separated into a flux of parallel energy Qe,∥ and perpendicular energy Qe,⊥ (see appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' To quantify the magnitude and direction of the energy ex- changes between the electromagnetic field and the particles, the different terms of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 15 were evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' To do so, the particles and field quantities were sampled in the steady state phase (after t = 30µs, see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' First, particles were sorted in 120 spatial bins equally spaced along the axial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In △Φ<0 △Φ>0ell el9 each bin, the moments of particle distribution provided the to- tal energy flux Qe and Qi, as detailed in appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Second, the cross product of the electric and magnetic field provided the axial component of the Poynting vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This vector was time-averaged over a period corresponding to an integer num- ber of wave periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 6: Energy source terms of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 15 along the axial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The location of the ECR surface is shown by the dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 7: Perpendicular and parallel energy source terms for the electrons along the axial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The location of the ECR surface is shown by the vertical dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The results, plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 6, showed first that the en- ergy source terms are negligible in the plume region (x ≥ 20mm) compared to the source region (x < 20mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In the plume region, the magnitude of the source terms is below 1×104 Wm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' For that magnitude, the signal is dominated by the statistical noise of the PIC simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This noise drowns the finer features of the source terms, especially for the electrons which are more prone to statistical noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Never- theless, this underlines that most of the energy exchanges take place in the source region and cannot explain the secondary peak for mean perpendicular kinetic energy e⊥ observed in the plume region (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Second, the collision source term is negative over the whole source region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Its magnitude is maximum near the backplate, where the neutral and plasma densities are higher, and de- creases along the source axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This behaviour is not unex- pected, since this term lumps together the contribution of in- elastic collisions and the diffusion model: these two processes are loss mechanisms for the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Given that the plasma density and the neutral gas density decrease as we move away from the backplate, the collision frequency drops and the mag- nitude of the energy loss decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Third, the sign and magnitude of the source terms reveal different phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' For x ≤ 3mm, the terms linked to the Poynting vector are positive, while the source term due to the electron energy flux is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This indicates an energy con- version from the electron kinetic energy to the field energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In parallel, the ion source term is positive, which points to a gain of energy in the sheath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' For the 3mm < x ≤ 10mm range, the Poynting source term shows a negative peak, while the source term due to the electron energy flux is positive, with a peak centered on the ECR surface location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In that case, there is a transfer of energy from the field to the electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This latter feature corresponds to the ECR heating of the per- pendicular energy mode of the electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In fact, this appears when considering the parallel and perpendicular contributions to the source term in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The perpendicular source term dominates over the parallel source term, with a peak centered on the ECR surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The axial extent of this peak shows that the perpendicular mode of the electrons is heated in a zone of about ∆xECR ≈ 6mm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=', from x = 3mm to x = 9mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Con- sidering that the ECR condition is only met on a specific sur- face, the presence of an extended region may seem surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' However, as will be discussed later, most of the electrons in the magnetic field tube are confined and undergo a bouncing motion in the potential well formed by the electrostatic field and the magnetic mirror force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' These bouncing electrons can cross the resonance surface with a significant parallel velocity, thus one may expect a shift of the resonance condition due to the Doppler effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The width ∆xECR can be compared with the expected value for a Doppler broadened resonance ∆xD in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 1651.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' ∆xD = � � � � 2πv∥ ωc BMS ��� ∂BMS ∂x ��� (16) Using the electrons’ mean axial velocity v∥ is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='1×106 ms−1 we obtain ∆xD = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='7mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' However, as it will be shown later in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 8, there is a high dispersion for the values of v∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Therefore, we can expect a much larger Doppler broadening for the fastest electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The electrons’ axial velocity can reach values up to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='0×106 ms−1 around the ECR zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' With this velocity, we can compute a max- imum Doppler broadening of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='8mm, which means that 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='8mm < ∆xD < 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='8mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The ECR heating zone obtained in the simulations is consistent with the one expected analyti- cally, indicating that the Doppler effect is a good candidate to explain the width of the heating zone observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 1 AQ Ao x 1 AQi Ao x 1 AII Ao X S coll0 AQe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='ll Ao x 1 0 AQe,1 Ao x A(Qe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' lI +Qe,↓)10 Because of Doppler-broadening, ECR heating of the elec- trons can even occur when the ECR surface is outside the plasma source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Indeed, the fact that the plasma in the thruster can be sustained even with an ECR zone outside the coax- ial chamber was demonstrated experimentally by Vialis 52, where the location of the resonance surface was placed at x = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='17mm and x = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='77mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Doppler broadening is a pos- sible explanation to this finding and this hypothesis was tested in simulations using the same parameters described in Table I but with an input microwave frequency of fEM = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='9GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' It was possible to sustain a discharge for this frequency even though the ECR condition was meet at x = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='78mm (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=', upstream of the plasma source).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In summary, the analysis of the power deposition shows that the energy transfer occurs mainly in the source region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The electrons absorb the wave energy over a Doppler-broadened volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This energy goes preferentially to the perpendicu- lar energy mode and leads to an anisotropy ratio Te,⊥/Te,∥ ∈ [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='5,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This energy deposition from the field to the perpen- dicular energy mode can explain the first peak in perpendicu- lar energy seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' However, the source terms are neg- ligible in the plume region and thus cannot explain the broad peak observed in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Electron confinement in the magnetic nozzle To determine the factors driving the evolution of the mean electron energy, we considered the electron distribution in the nozzle region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 8 we plotted the normalized electron distribution in the velocity space v∥,v⊥ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The distribution was plotted at different locations in the computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 8 show that as we move downstream into the nozzle, we see an increased electron population with high energies in the perpendicular direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' To understand this phenomenon, we must first get a more detailed description of the electron confinement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=', how they get trapped and under which conditions they can leave the thruster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Three loss pathways are identified for the electrons pro- duced in the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Cross field losses : electrons can diffuse across the mag- netic field, due to anomalous transport, collision, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This is modeled by the phenomenological cross-field diffusion model presented in section II B 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Losses at the downstream of the nozzle electrons which have a kinetic energy sufficient to overcome the confin- ing potential well are lost, alongside ions accelerated by this same potential drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Electrons than can overcome both the repelling poten- tial of the plasma sheath at the close-end of the source and the mirror-force are collected on the dielectric plate and will contribute to its surface charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' While the first loss mechanism does not depend on the ki- netic energy of a single electron but rather on the mean kinetic energy at a given location, the two other mechanisms depend on the electron kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' An electron moving along the magnetic field will see both an electrostatic potential Φ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 3) and the magnetostatic field B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Depending on its initial ki- netic and potential energies, its trajectory might have turning points (where v∥ = 0) within the domain, or out of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In the first case, this electron is confined in the potential well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In the latter case, the electron is lost, either downstream (loss pathway 2) or at the backplate (loss pathway 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Now the lim- iting case between confined / unconfined electrons is when the turning points are located at the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This will define the necessary conditions for the electrons confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Let us note ΦBP the potential of the backplate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Neglecting the plasma-wave interaction, and noting µ the magnetic moment, the equation for the total energy of an electron is: Etotal = 1 2mv2 ∥ + µB−eΦ (17) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 17 we can say that the electron is oscillating in an effective potential given by Uef f = µB − eΦ, where µB represents the magnetic confinement as the electron moves towards the backplate while −eΦ is the electrostatic confine- ment given by the plasma potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' It can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 9 for different arbitrary values of the magnetic moment taken from the results of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Its concave shape explains the confinement of the electrons in the ECR thruster inside this potential well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Let us now consider an electron moving from an arbitrary initial point to a turning point at a position x0 along the longi- tudinal direction, such that v∥(x0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The energy conserva- tion between any initial location and the turning point gives: v2 ∥ +v2 ⊥ � 1− Bx0 B � = −2e m ∆Φ (18) Where ∆Φ = Φx0 − Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Now let us consider the loss path- ways 2 and 3 identified above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' For an electron lost to the downstream boundary (pathway 2), the confinement condition is obtained by setting x0 = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In that case, given the divergence of the magnetic field, we have 0 < 1−B(L)/B < 1 and ∆Φ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Thus equation 18 describes an ellipse in the v∥ − v⊥ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Electrons in the ellipse have there turning points x0 ≤ L and remain confined by the elec- trostatic well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Electrons out of the ellipse can overcome this electrostatic confinement and are lost downstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' For an electron lost to the backplate (pathway 3), the con- finement condition is obtained by setting x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Because the magnetic field is monotonically decreasing, 1 − B(0)/B < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In addition, Φ(0) = ΦBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' We define the loss cone angle as: sin(θ) = � B B(0) (19) Equation 18 can be recast as: v2 ⊥ = tan2(θ) � v2 ∥ + 2e m ∆Φ � (20) Depending on the sign of ∆Φ, three cases are possible: 11 (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 8: Electrons distribution in the velocity space v∥,v⊥ plane at different locations on the simulation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The number of electrons has been normalized by the total number of electrons for each case independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' For each case, we also plotted what we call the confinement boundaries described by an analytical model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 19 and 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The dotted line is the magnetic confinement, the dashed line the electrostatic potential at the backplate, and the solid line is the electrostatic confinement on the plume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' (a) x = 5 mm, (b) 20 mm, (c) x = 80 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 9: Schematic view of the effective potential profile for arbitrary values of the magnetic moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' ∆Φ = ΦBP − Φ = 0: The confinement of the electron at the backplate is given exclusively for the topology of the magnetostatic field according to the loss cone angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Those electrons with values for v∥,v⊥ such as they are located in the loss cone and will therefore be lost at the backplate (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 10a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' ∆Φ = ΦBP −Φ < 0: The backplate potential repels the negative charges and thus confines the electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Conse- quently, a fraction of the electron in the loss cone will be reflected back and stay confined (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 10b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' ∆Φ = ΦBP −Φ > 0: The backplate potential attracts the electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Thus, even electrons out of the loss cone are collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Therefore, the confined electrons are those that meet two conditions: they are not on the loss cone for the magnetic field, and they are energetic enough in the perpendicular direction to avoid being lost at the backplate thanks to the electrostatic acceleration to- wards it (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 10c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 3, displays the sign of ∆Φ in the nozzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The combina- tion of the loss conditions at the backplate (pathway 3) and downstream (pathway 2) delimits a confinement volume in phase space where electrons are confined, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Pitch-angle scattering, either caused by collisions with the neutral background or by the electromagnetic field, enables electrons to cross the confinement volume boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' De- pending on which boundary is crossed, electrons are lost at the backplate or at the downstream side of the nozzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Indeed, it is important to recall that the electron deconfinement is mainly driven by these two phenomena in the source region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Since the neutral background density decreases exponentially, most of the collisions occur in the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In addition, as shown above, wave absorption happens over a few millimeters in the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' As a consequence, the electron deconfinement rate is driven by the wave interaction and the collisions: Interaction with the electromagnetic wave is akin to a scattering of the electron momentum53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' After several passages through the ECR heating zone, the electron may gain enough energy to overcome the electrostatic barrier and escape into the plume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 12 (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 10: Confinement boundaries in velocity space on the v∥,v⊥ plane where the orange colored zones describe the electrons being trapped in the ECR thruster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Solid line for the plume electrostatic confinement, and dashed (electrostatic) plus dotted (magnetic) lines for the backplate confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Where: (a) ∆Φ = 0, (b) ∆Φ < 0, and (c) ∆Φ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' If the electron undergoes an elastic collision, it will ran- domly scatter its velocity vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' If the electron scat- tered momentum falls in the loss region defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 20 and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 10, the particle is lost at the back- plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' If we now go back to the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 8 for the elec- tron distribution in the velocity space v∥,v⊥ plane, we notice that as we move downstream into the plume, the confinement boundaries change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' There is a transition from a confinement boundary as the one in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 10b (∆Φ < 0) to the one in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 10c (∆Φ > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This transition is a consequence of the fact that, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 3 the plasma potential is greater than the backplate potential in the source Φ > ΦBP and lesser than ΦBP downstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Thus, inside the coaxial chamber, the plasma po- tential is such that ∆Φ = ΦBP − Φ < 0 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 10b), and in the plume section it is such that ∆Φ = ΦBP − Φ > 0 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 10c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' As a consequence, as we move downstream into the magnetic nozzle, the mean perpendicular kinetic energy can increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' However, this is not given by an additional heating phase but as a result of confining only a highly energetic electron pop- ulation in the perpendicular direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Those electrons with a low perpendicular kinetic energy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=', below the dashed line) are lost at the backplate as previously described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' It can be seen as a filtering process where only the hot electrons are confined, and this is what we see when plotting the electron perpendicular kinetic energy in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Further downstream of the magnetic nozzle, the magnitude of the potential well to the end of the nozzle decreases, while the magnitude of the at- tracting potential drop to the backplate increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This results in a narrower distribution for the confined population and fi- nally a decrease in the mean perpendicular kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' CONCLUSIONS We have performed electromagnetic full-PIC simulations of the ECR thruster using a 1D3V model that allowed us to shed light onto some of its working principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The re- sults confirmed the expected anisotropic behavior for the elec- trons’ energies in the direction perpendicular and parallel to the magnetic field lines and a peak for the mean perpendic- ular energy near the resonance zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The microwave energy injected at the backplate of the thruster propagates through the coaxial chamber while being absorbed by the electrons increasing their kinetic energy perpendicular to the magnetic field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The absorption takes place exclusively inside the coaxial chamber on a zone of 6 mm around the resonance condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' This zone is coherent with the predicted value from Doppler broadening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The width of this heating zone may explain why the thruster works even with a configura- tion in which the resonance condition is met outside the coax- ial chamber52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' From a practical point of view, this feature improves the reliability of the thruster, since it means that the thruster can still operate even when the magnetostatic field de- creases, for example due to excessive heating of the magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The results also show, unexpectedly, that the electrons’ mean perpendicular energy has a second peak in the plume due to the confinement of highly energetic electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The confinement is determined by the backplate’s potential, the magnetostatic field, and the potential drop on the plume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' As a consequence, there is a population of trapped electrons with significant perpendicular kinetic energy in the down- stream region of the magnetic nozzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The existence of dou- bly trapped electron population has been investigated using a kinetic model with a paraxial approximation similar to this work54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' While this work was assuming the shape of the initial distribution function, it has also been seen in the case of an anistropic distribution that the perpendicular electron temper- = arcsin(V B,) 0-2e△Φ V= m2e△Φ Vi = tan(0) m13 ature could increase in the diverging part of the nozzle55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' We can speculate that these high temperature electrons trapped in the plume could be important to drive some instabilities ob- served in the plume that are thought to enhance cross-field transport of electrons and thus play a role in the detachment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In particular, Lower Hybrid drift instabilities have been re- cently observed in diverging magnetic nozzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' In these ex- periments, diamagnetic drift vD = ∇pe⊥ ×B/enB2 was iden- tified for the primary energy source for the instability56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The trapped electrons could enhance the radial gradient in perpen- dicular pressure and thus enhance the diamagnetic drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Appendix A: Electron motion integration in the Quasi-1D model On one hand, the model assumes that the axial static mag- netic field in the flux tube depends on x only dBx dx = α(x) (A1) In the quasi-1D model, the field remains constant across the section of the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' On the other hand, in the ECR thruster the electromagnetic part of the magnetic field (the part due to the propagation of the electromagnetic power injected in the source) remains negligible compared to the static part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Indeed, assuming a locally plane wave, the Poynting vector S = E × B/µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Since E ≃ cB and S ≃ 1Wcm−2, the order of magnitude for the electromagnetic component of the mag- netic field is B ≃ 1×10−2 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The static part of the field is between 10 mT and 10 mT, much greater than the electromag- netic component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Therefore, using the divergence equation for the static magnetic field and neglecting the electromag- netic component, it is possible to obtain the radial component of the magnetic field : Br(x,r) = −α(x)r 2 (A2) Assuming this value for the radial part of the magnetic field ensures that the divergence condition is automatically en- forced for the static part of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The electron guiding center is on the flux tube centerline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The Larmor radius of the electrons is given by: rL(x) = V⊥(x) ωc(x) (A3) Where V⊥ = � v2y +v2z, ωc(x) = eBx(x)/me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The gyromotion of the electron is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Thus, knowing the parti- cle velocities it is possible to obtain the phase angle θ in its gyromotion, as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' cos(θ) = y rL(x) = vz V⊥ (A4) sin(θ) = z rL(x) = − vy V⊥ (A5) Knowing the phase angle, sine and cosine, the By an Bz com- ponents can be deduced from eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' A2 and A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 11: Gyromotion in the plane normal to the axial static field Bx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Appendix B: Energy equation For the energy equation for the particles we consider the second order moment of the Vlasov equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Multiplying the Vlasov equation for the electrons by � mev2/2, we obtain: ∂ ∂t � mev2 2 fed3v+ ∂ ∂x · � mev2 2 v fed3v (B1) +qe � v2 2 (E+v×B)· ∂ fe ∂v d3v = � mev2 2 �∂ fe ∂t � col d3v Where the right hand side lumps the contribution of colli- sions and the loss model detailed in section II B 1 and II B 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' B1 can be rewritten as: ∂εe ∂t +∇·Qe =−je ·E+ScollQe = � mev2 2 v fed3v (B2) εe = � mev2 2 fed3vr (B3) Se,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='coll = � mev2 2 � ∂ fe ∂t � coll d3v (B4) The same procedure can be applied to the ions: ∂εi ∂t +∇·Qi =−ji ·E+ScollQi = � Miv2 2 vfid3v (B5) εi = � Miv2 2 fid3v (B6) Si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='coll = � Miv2 2 � ∂ fi ∂t � coll d3v (B7) Considering the electron population,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' the total heat flux can be written as: Qe = qe +Pe ·ue +neue(eK +EK) (B8) 14 Where the density is given by ne = � fed3v and the macro- scopic velocity by neue = � fevd3v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' The random part of the velocity is c = v−ue The different terms are then: qe = � me 2 c2cd3c (B9) Pe = � meccd3c (B10) EK = 1 2meu2 e (B11) eK = � me 2 c2d3c (B12) Considering the one-dimensional approximation, we are con- sidering only the axial (parallel) part of the total heat flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Thus, after averaging over a period of the incoming wave : < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' >= 1 T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='dt, we obtain: ∂ ∂xA(x) � qe∥ � +A � Pe∥u∥ +Pe⊥u⊥ � +A(x) � ue∥ +neue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='∥(eK +EK) � = −A(x)⟨je ·E⟩+A(x) � Se,coll � (B13) A similar expression can be written for the ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' if we make use of the Poynting theorem, we can relate the time averaged joule term to the divergence of the pointing flux Π = E×B µ0 : A(x)⟨(je +ji)·E⟩+ ∂ ∂x ⟨Π⟩ = 0 (B14) If we drop the < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' > symbol for simplicity, recalling that all quantities are time-averaged, one obtains equation (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' 1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Holste, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Dietz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Scharmann, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Keil, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Henning, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Zschätzsch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Reitemeyer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Nauschütt, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Kiefer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Kunze, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Zorn, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Heiliger, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Joshi, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Probst, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Thüringer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
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+page_content=' Packan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
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+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
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+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
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+page_content=' Schippers, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Hannemann, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
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+page_content=' 2J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
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+page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
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+page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
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+page_content=' 10S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
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+page_content=' Jarrige, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Packan, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
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+page_content=' 11S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Correyero, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Merino, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Elias, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Jarrige, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Packan, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Ahedo, “Characterization of diamagnetism inside an ECR thruster with a diamag- netic loop,” Physics of Plasmas 26, 053511 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
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+page_content=' Jarrige, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Aanesland, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
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+page_content=' 13F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
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+page_content=' Jarrige, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content=' Désangles, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
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+page_content='1063/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
+page_content='0040175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFIT4oBgHgl3EQf-iyR/content/2301.11411v1.pdf'}
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+Multiplexed random-access optical memory in warm cesium vapor
+Leon Meßner,1, 2, a) Elizabeth Robertson,2, 3 Luisa Esguerra,2, 3 Kathy L¨udge,4 and Janik Wolters2, 3
+1)Institut f¨ur Physik, Humboldt-Universit¨at zu Berlin, Newtonstr. 15, 12489 Berlin, Germany.
+2)Deutsches Zentrum f¨ur Luft- und Raumfahrt e.V. (DLR), Institute of Optical Sensor Systems, Rutherfordstr. 2, 12489 Berlin,
+Germany.
+3)Technische Universit¨at Berlin, Institut f¨ur Optik und Atomare Physik, Str. des 17 Juni 135, 10623 Berlin,
+Germany
+4)Technische Universit¨at Ilmenau, Institut f¨ur Physik, Weimarer Straße 25, 98693 Ilmenau,
+Germany
+(Dated: 13 January 2023)
+The ability to store large amounts of photonic quantum states is regarded as substantial for future optical quantum
+computation and communication technologies. However, research for multiplexed quantum memories has been focused
+on systems that show good performance only after an elaborate preparation of the storage media. This makes it generally
+more difficult to apply outside a laboratory environment. In this work, we demonstrate a multiplexed random-access
+memory to store up to four optical pulses using electromagnetically induced transparency in warm cesium vapor. Using
+a Λ-System on the hyperfine transitions of the Cs D1 line, we achieve a mean internal storage efficiency of 36% and a
+1/e lifetime of 3.2 µs. In combination with future improvements, this work facilitates the implementation of multiplexed
+memories in future quantum communication and computation infrastructures.
+I.
+INTRODUCTION
+Quantum memories are considered to be a main component for the realization of many future second generation quantum
+technologies. Their potential use ranges from synchronizing inputs into various types of quantum systems1 to re-configurable
+optical reservoir computing2. They enable on-demand operation of otherwise probabilistic single-photon sources and quantum
+gates3, which will significantly enhance their rate of operation4. Moreover, they have been identified as an essential device
+required to realize a quantum repeater5, a key technology needed for long-distance quantum communication. When specifically
+considering the implementation of a global quantum communication network, satellite based quantum communication has been
+hallmarked as a most promising system if enhanced with a multiplexed quantum memory. It has been shown that a significant
+increase in communication rate is already achievable with around 1000 randomly accessible storage modes6,7. Consequently,
+the realization of suitable multiplexed quantum memories will be an important milestone in extending quantum communication
+over long distances.
+Quantum memories have been demonstrated using a variety of storage protocols in a number of single emitter and ensemble-
+based matter systems. These include solid state systems, ultra-cold atoms and warm atomic vapors1. Although the routine
+formation of subnanokelvin Bose-Einstein condensates in earth’s orbit has been demonstrated8 and is pursued in future projects9,
+reducing the technological requirements for space-borne quantum memories is a key step. This makes memories based on warm
+atomic vapors favored for applications, as they require no vacuum, laser cooling or strong magnetic fields.
+The memory used in this experiment utilizes the effect of electromagnetically induced transparency (EIT) on the Λ-system
+composed by the 62S1/2F=3, F=4 and the 62P1/2F=3 atomic hyperfine levels of an ensemble of cesium atoms to map optical
+excitations to a long-lived spin-wave, i.e. a coherence of the two hyperfine ground states of the atomic ensemble10,11. Due to its
+coherent ensemble origin, this spin-wave shows comparatively low dephasing and loss. Subsequent retrieval of the spin-wave
+excitation into the input optical mode can then be performed at some chosen time that is smaller than the spin-wave lifetime.
+Light storage for up to 1 s12 and single-photon operation13,14 have been demonstrated in separate experiments in warm atomic
+vapors.
+For a quantum memory to be most useful within a quantum communication system, the memory must be scaleable with
+the possibility to access individual storage modes in a way that is not significantly limited by the used technology. Various
+forms of quantum memory multiplexing have been studied in the past, including time bin15, orbital angular momentum16, and
+spatial17,18 multiplexing. Among these approaches, Ref.18 shows a clear foreseeable path to achieving the required number of
+1000 randomly accessible modes. However, the technological overhead of a cold atom setup complicates operating these outside
+of a laboratory. In this work we demonstrate a memory that combines the advantages of using warm vapor with a path towards
+scaleable multi-rail operation.
+a)Electronic mail: messner@physik.tu-berlin.de
+arXiv:2301.04885v1 [quant-ph] 12 Jan 2023
+
+2
+FIG. 1. Memory scheme. a) Scheme of the Cs D1 energy levels’ hyperfine structure that forms the Λ-system used. Signal and control fields
+are red detuned from resonance by ∆ and have angular frequencies ωs and ωc respectively. The transition F = 4 → F ′ = 4 is resonantly
+driven by the pump field with angular frequency ωp. b) Time traces from an experiment for writing a signal pulse onto an atomic spin-wave at
+time t0 = 0 and retrieving it at t0 + 0.4 µs. The bottom panel shows the operation performed on the memory (W: write, R: read).
+II.
+EXPERIMENT
+The multi-rail memory presented here uses an EIT memory scheme19 on the hyperfine ground state transitions of the Cs
+D1 line, as shown in Fig. 1(a). The control and signal lasers (14-pin butterfly external cavity diode laser (ECDL) modules by
+Sacher Lasertechnik) are set on the F=4 → F ′=3 and the F=3 → F ′=3 transitions respectively. To stabilize the frequency
+difference between signal and control laser to the Cs ground state hyperfine splitting, light from both lasers is superimposed
+on a fast photodiode (Electro Optics Technology, ET-3500FEXR) and their beat frequency is offset locked20 using a RedPitaya
+FPGA-board running the Linien21 locking software. We generate Gaussian signal and control pulses with a full width at half
+maximum (FWHM) of 25 ns and 43.75 ns respectively using an arbitrary function generator (AFG, Tek AFG31152). These
+pulses are modulated onto cw laser beams with electro-optic amplitude modulators (EOMs, Jenoptik AM905). The experiment
+is designed such that the signal and control pulses have linear and orthogonal polarization to each other to reduce the control
+light leaking into the detection path.
+The atomic storage medium is confined to a cylindrical, 25x75 mm anti-reflection coated cesium vapor cell filled with 5 torr
+N2 of buffer gas. It is kept at 60°C and shielded from ambient magnetic fields by a double-layered mu-metal housing. Spatial
+addressing of different rails within the cell is performed by one acousto-optic deflector (AOD) in front and one behind the vapor
+cell. The AODs (AA MT200-B100A0,5-800) have an aperture of 0.5x2 mm2 and a measured deflection of 0.2 mrad/MHz.
+Each of them is driven by the frequency sum of an arbitrary function generator (AFG, Tek AFG31152) and a local oscillator
+(Mini-Circuits, ZOS-300+) resulting in 200±50 MHz of carrier frequency. We refer to the position of memory rails by the
+AOD driving frequency used to deflect the beam to that position; the distance between rails is thus expressed as the difference
+in driving frequency. Changing the AOD driving frequency by 8 MHz changes the lateral position of the deflected beam by
+270 µm, equaling one signal beam radius.
+While the signal pulses enter the memory unmodified after their generation, the control laser pulses are amplified by a self-
+made tapered amplifier (TA, see22) and then spectrally filtered by a dielectric bandpass interference filter (IF) with a 1 nm
+FWHM. This results in 200 mW of coupled cw power. To increase the control laser’s on-off ratio, the TA diode’s driving current
+is only switched on for 120 ns, centered on the optical pulse, using a 4 A dc-coupled input follower driver with 2 ns rise/fall
+time. This reduces the unwanted interaction between control laser and atoms in times when no control pulse is generated.
+The beam paths of the cross-polarized signal and control lasers are overlapped at a polarizing beam splitter (PBS) on one side
+of the memory and then propagate collinearly through the cesium cell, and both AODs. At the position of the cell, control and
+signal beam have a 1/e2-level radius of 350 µm and 270 µm respectively. This yields an atomic transit time of ∆t = 3.7 µs for
+one signal beam radius when using a 2D diffusion model of ∆x =
+√
+4D∆t for the diffusion length and an assumed diffusion
+constant of D0 = 0.24 cm2s−123 at T0 = 0 K and P0 = 760 torr. After traversing the second AOD, the signal and control
+beams are split by a second PBS and are then individually coupled to fibers and detected by either a Si photodiode (Thorlabs
+DET10A2) or a Si avalanche photodiode (Menlo Systems APD210).
+Optical pumping of the Cs atoms into the 62S1/2F=3 state is performed by a third ECDL laser locked to the 62S1/2F=4 →
+
+a)
+b)
+5.
+SIGNAL
+62P1/ 2
+F'= 4
+F'= 3
+[arb.]
+CONTROL
+Intensity
+m
+m
+dm
+DETECTOR
+F= 4
+62S1/ 2
+9.2 GHz
+F= 3
+PUMP
+W
+R
+R
+- 0.5
+0.0
+0.5
+1.0
+Time [μs]3
+FIG. 2. Sketch of the experiment with A) laser sources, spectroscopy and optical pulse shaping, B) TA based pulse amplification, C) multi-rail
+storage system and D) CCD image of the four used rails, with the camera at the place of the Cs cell. HWP: half-wave plate, QWP: quarter-wave
+plate, DET: detector, (P)BS: (polarizing) beam splitter, L1;L2/L3: aspheric/cylindrical lens, AOD: acousto-optic deflector, EOM: electro-optic
+modulator, AFG: arbitrary function generator, IF: interference filter, OL: offset lock.
+62P1/2F=4 transition by saturated absorption spectroscopy. The pump light power is controlled via transmission through an
+electrically pulsed semiconductor optical amplifier (SOA) and illuminates each memory rail with 20 mW of optical power for
+900 ns prior to the memory experiment sequence.
+Figure 1(b) shows a typical time trace for a single-rail storage experiment and a sketch of the experimental setup can be seen
+in Fig. 2. Several features of storage within an EIT medium can be observed in the time trace. At t = 0 µs a signal pulse enters
+the atomic medium and is partly mapped to an atomic spin wave by the control laser field. The portion of that signal pulse that
+is transmitted through the atomic vapor is detected by the photodiode as leakage. After 0.4 µs the control laser field is switched
+on again and retrieves the spin-wave excitation back into the signal beams optical mode. A third pulse of the control laser at
+t = 0.8 µs serves to determine if all the excitation has been retrieved and also allows to estimate the signal noise induced by
+the control laser field. Not having a significant detection event during this last pulse, we conclude that nearly all the spin-wave
+excitation is mapped backed to optical and signal to noise ratio is not a limiting factor for this experiment with aforementioned
+laser pulses.
+III.
+RESULTS
+Prior to performing multi-rail storage, we first identify optimal operating conditions for the multi-rail memory by assessing
+the influence of rail separation on the interactions between two memory rails, and subsequently minimizing the cross-talk. For
+comparison of the single and multi-rail operation, we measure the 1/e lifetime and memory efficiency per rail.
+To assess the influence of rail separation on their interaction, multiple storage experiments at different rail separations are
+conducted. For effective operation of a memory, we require that operations on a given memory rail do not affect its neighbors.
+To determine the minimal separation that shows no cross-talk, we write into a rail fixed at 190 MHz, read from a neighboring rail,
+and then read from the 190 MHz rail again. This write/read/read sequence is depicted in the inset of Fig. 3. The rail separation
+is varied from 0 to 25 MHz at steps of 1 MHz, and the retrieval peak intensities after the first and second read are measured. The
+results are depicted in Fig. 3. Below 5 to 8 MHz of separation no excitation is left for the second retrieval pulse and both read
+pulses address the same ensemble of atoms. At a separation of 20 MHz the influence of the read operation on the neighboring
+rail is no longer visible.
+The AOD device used has a 100 MHz bandwidth and a 25% reduced diffraction efficiency at the edges of the frequency range;
+consequently we chose to limit this experiment to four memory rails spaced by 20 MHz. A CCD image of the four rails, taken
+at the position of the Cs cell, is shown in Fig. 2(D). A straightforward method to increase the number of rails, is to use AODs
+with a higher number of resolvable spots.
+The 1/e storage lifetime per rail is determined by performing storage experiments with increasing time delay between the
+memory write and read operation, for each rail. The delay was varied between 0.4 µs and 11.2 µs in steps of 400 ns, and for each
+
+A)
+duwnd
+EOM
+C)
+ signal
+HWP
+AOD
+L2
+AOD
+HWP
+PBS
+HwP
+L1
+QWP
+DET
+LASER
+to AWG
+BS
+PBS
+PBS
+to OL
+control
+HWP
+PBS
+LASER
+EOM
+D)
+B)
+BP
+HWP
+1nm HWP
+PBS
+20 MHz
+『』 TA-diode L2L3
+opt, isolator
+PD
+625 um4
+Peak 1
+Peak 2
+0
+5
+10
+15
+20
+25
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+Rail separation [MHz]
+Intensity [arb.]
+0.0
+0.5
+1.0
+Time [µs]
+W
+R
+R
+FIG. 3. Cross-talk estimation. Intensities detected in the first (Peak 1, orange rail) and second (Peak 2, green rail) read peak depending on the
+rail separation for the experiment sequence depicted in the inset. For a difference of 20 MHz in AOD driving frequency (rail separation), the
+influence between neighboring rails vanishes. The inset shows the used experiment sequence consisting of a write on the green rail at t = 0, a
+read on a neighboring orange rail at t = 0.4 µs and finally a read on the green rail at t = 0.8 µs.
+delay, we measure the retrieved peak intensities, averaged over 500 repetitions. Uncertainties are given by the standard deviation
+of the intensities. The intensities were fitted with an exponential function to extract the 1/e lifetime. Measured retrieval peak
+intensities and fit function are displayed in Fig. 4.
+The resulting lifetime values per rail are shown in Table I together with the achieved internal memory efficiencies ηmem at
+t = 0.
+Since the measured lifetimes are consistent with the estimate using the simple diffusion model presented earlier, it is reasonable
+to assume that diffusion is the most important lifetime-limiting process for the beam diameters chosen in this work.
+Independent investigation on a single rail setup also showed that spin polarization lifetimes at least on the order of several
+hundred microseconds are possible with larger beam diameters.
+The memory efficiencies are calculated by extrapolating the pulse energy of a retrieved pulse after t = 0 µs of storage from a
+retrieved pulse after t = 0.4 µs of storage using the memory lifetime. This is then divided by the energy of a normalization pulse
+to yield the efficiency. To obtain the normalization pulse, we set the signal laser frequency 2 GHz below the F=3 → F ′=3
+transition frequency, block the control beam and record the transmitted signal pulse. Under these conditions, we assume the
+pulses not to be absorbed by the atoms.
+Using the insights and results from the measurements on lifetime, efficiency and rail separation, we now explore the possibility
+of random-access operation in the memory setup. For this purpose an experimental sequence was designed that highlights
+important criteria for use as a random-access quantum memory. Figure 5 illustrates this experiment.
+The bottom panel depicts the operation performed on each specific rail and the top panel shows the intensity detected by
+the APD over a time span of about 5 µs. The experimental sequence contains 12 operations, either read (r) or write (w). We
+define three features which are necessary for use as a memory: a) that reading or writing to a rail should not affect its neighbors
+(interaction-free), b) rails which have not been written to should not return a retrieved pulse (empty state) and c) a read should
+leave the memory empty; a subsequent read pulse should yield no excitation (full retrieval).
+Rail (MHz)
+170
+190
+210
+230 Mean
+Lifetime (µs)
+4.3(5) 5.4(7) 3.3(3) 2.6(3) 3.2(2)
+Efficiency (%)
+32
+35
+39
+36
+36
+TABLE I. Measured 1/e-lifetime and retrieval efficiency for each rail and weighted mean.
+
+5
+FIG. 4. Measured retrieval amplitudes for storage times between 0.4 and 11.4 µs together with an exponential fit to the values. The inset shows
+the per rail 1/e storage lifetime deduced from the fit.
+0.00
+0.05
+0.10
+0.15
+0.20
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+Intensity [arb.]
+Operation #
+0
+1
+2
+3
+4
+230MHz
+210MHz
+190MHz
+170MHz
+W
+W
+R
+R
+W
+R
+W
+R
+R
+R
+W
+R
+Time [µs]
+FIG. 5. Storage experiment in the random-access memory using four rails with detected intensity in the signal path in the top and performed
+operations (read/write) in the bottom panel. A total of 12 operations are performed over a span of 5 µs.
+
+0.20
+1/e Lifetime [μus]
+0.15
+4
+Intensity [arb.]
+3
+王
+0.10
+2
+170
+190
+210
+230
+Rail [MHz]
+0.05
+0.00F
+0
+2
+4
+6
+8
+10
+12
+Time [μs]6
+Interaction-free operation is ensured by choosing an adequate rail separation and then confirmed by looking at the storage
+performance of a specific rail while there are operations performed on the neighboring rails. The rails at 230 MHz (blue) and
+190 MHz (green) can be used to show that operations are interaction free. Between the write (t = 1.6 µs) and read (t = 3.6 µs)
+operation on the 190 MHz rail, four operations are performed on the neighboring rails and there is no visible impact on the read
+peak shape or height. On the 230 MHz rail a pulse is written at t = 0 and then retrieved during the last operation on the memory
+at t = 4.4 µs. Taking into account the 2.6 µs lifetime of this rail, the high remaining intensity of the retrieval peak clearly shows
+that there is no significant detrimental influence from multi-rail operation.
+Reading an empty rail should not result in a significant amount of intensity. We verify this by reading the 210 MHz and
+170 MHz rail in a state that should not have excitation. In the 210 MHz rail a pulse is written to the memory at t = 0.4 µs
+and then this rail is immediately read twice. The second read operation at t = 0.8 µs yields negligible intensity compared to
+the first read operation at 0.6 µs. Additionally the same rail is read again at 3.2 µs to observe the amount of noise, which is
+found to be comparable to the read at 1.2 µs. The first operation on the 190 MHz rail at t = 2 µs is a read of a rail that has
+not been used before. This allows us to determine how well the memory was initialized by the pumping that is performed prior
+to the experimental sequence. Observing a larger intensity peak would point to insufficient polarization of the medium. As the
+observed peak is similar to the other reads of an empty rail mentioned above, we conclude that pumping is sufficient and reading
+an empty rail, regardless of its history, does not lead to the detection of a significant peak. In combination with the measurements
+on lifetime and rail interaction it follows that this setup allows random-access storage and retrieval of optical pulses for times
+comparable to the mean rail lifetime of 3.2 µs.
+IV.
+CONCLUSION
+We have presented a multiplexed optical random-access memory, realised within a single vapor cell at a temperature of 60°C.
+Using an EIT based storage scheme in a Λ-system on the cesium hyperfine transitions, we achieved a mean storage lifetime
+and internal efficiency of 3.2(2) µs and 36% respectively in multi-rail operation. According to the chosen rail separation of
+20 MHz, we performed random-access storage and retrieval in four parallel rails without observing reciprocal influence between
+the different rails.
+The time between successive operations was chosen to be 400 ns for the sake of simplifying experiment control. This time
+could be reduced considerably with the lower bound determined by the AODs switching time of 48 ns. Increasing the storage
+lifetime and number of addressable rails is possible by increasing the beam diameters and using AODs that have a higher
+time-bandwidth product respectively. This step will be important for applications in quantum communication and repeater
+networks. Reaching beyond the threshold number of 1000 individually addressable modes is possible by using 2-axis AODs and
+multiplexing into a two dimensional grid of parallel storage modes.
+FUNDING
+This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project number
+445183921. E.R. acknowledges funding through the Helmholtz Einstein International Berlin Research School in Data Science
+(HEIBRiDS).
+DISCLOSURES
+The authors declare no conflicts of interest.
+DATA AVAILABILITY
+The data presented in this paper is available from the authors upon reasonable request.
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+efficiency Raman memory by suppressing radiation trapping,” New Journal of Physics 19, 063034 (2017), arXiv:1610.03743.
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf,len=512
+page_content='Multiplexed random-access optical memory in warm cesium vapor Leon Meßner,1, 2, a) Elizabeth Robertson,2, 3 Luisa Esguerra,2, 3 Kathy L¨udge,4 and Janik Wolters2, 3 1)Institut f¨ur Physik, Humboldt-Universit¨at zu Berlin, Newtonstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 15, 12489 Berlin, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 2)Deutsches Zentrum f¨ur Luft- und Raumfahrt e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' (DLR), Institute of Optical Sensor Systems, Rutherfordstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 2, 12489 Berlin, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 3)Technische Universit¨at Berlin, Institut f¨ur Optik und Atomare Physik, Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' des 17 Juni 135, 10623 Berlin, Germany 4)Technische Universit¨at Ilmenau, Institut f¨ur Physik, Weimarer Straße 25, 98693 Ilmenau, Germany (Dated: 13 January 2023) The ability to store large amounts of photonic quantum states is regarded as substantial for future optical quantum computation and communication technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' However, research for multiplexed quantum memories has been focused on systems that show good performance only after an elaborate preparation of the storage media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' This makes it generally more difficult to apply outside a laboratory environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' In this work, we demonstrate a multiplexed random-access memory to store up to four optical pulses using electromagnetically induced transparency in warm cesium vapor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Using a Λ-System on the hyperfine transitions of the Cs D1 line, we achieve a mean internal storage efficiency of 36% and a 1/e lifetime of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='2 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' In combination with future improvements, this work facilitates the implementation of multiplexed memories in future quantum communication and computation infrastructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' INTRODUCTION Quantum memories are considered to be a main component for the realization of many future second generation quantum technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Their potential use ranges from synchronizing inputs into various types of quantum systems1 to re-configurable optical reservoir computing2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' They enable on-demand operation of otherwise probabilistic single-photon sources and quantum gates3, which will significantly enhance their rate of operation4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Moreover, they have been identified as an essential device required to realize a quantum repeater5, a key technology needed for long-distance quantum communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' When specifically considering the implementation of a global quantum communication network, satellite based quantum communication has been hallmarked as a most promising system if enhanced with a multiplexed quantum memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' It has been shown that a significant increase in communication rate is already achievable with around 1000 randomly accessible storage modes6,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Consequently, the realization of suitable multiplexed quantum memories will be an important milestone in extending quantum communication over long distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Quantum memories have been demonstrated using a variety of storage protocols in a number of single emitter and ensemble- based matter systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' These include solid state systems, ultra-cold atoms and warm atomic vapors1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Although the routine formation of subnanokelvin Bose-Einstein condensates in earth’s orbit has been demonstrated8 and is pursued in future projects9, reducing the technological requirements for space-borne quantum memories is a key step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' This makes memories based on warm atomic vapors favored for applications, as they require no vacuum, laser cooling or strong magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The memory used in this experiment utilizes the effect of electromagnetically induced transparency (EIT) on the Λ-system composed by the 62S1/2F=3, F=4 and the 62P1/2F=3 atomic hyperfine levels of an ensemble of cesium atoms to map optical excitations to a long-lived spin-wave, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' a coherence of the two hyperfine ground states of the atomic ensemble10,11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Due to its coherent ensemble origin, this spin-wave shows comparatively low dephasing and loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Subsequent retrieval of the spin-wave excitation into the input optical mode can then be performed at some chosen time that is smaller than the spin-wave lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Light storage for up to 1 s12 and single-photon operation13,14 have been demonstrated in separate experiments in warm atomic vapors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' For a quantum memory to be most useful within a quantum communication system, the memory must be scaleable with the possibility to access individual storage modes in a way that is not significantly limited by the used technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Various forms of quantum memory multiplexing have been studied in the past, including time bin15, orbital angular momentum16, and spatial17,18 multiplexing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Among these approaches, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='18 shows a clear foreseeable path to achieving the required number of 1000 randomly accessible modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' However, the technological overhead of a cold atom setup complicates operating these outside of a laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' In this work we demonstrate a memory that combines the advantages of using warm vapor with a path towards scaleable multi-rail operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' a)Electronic mail: messner@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='tu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='de arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='04885v1 [quant-ph] 12 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Memory scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' a) Scheme of the Cs D1 energy levels’ hyperfine structure that forms the Λ-system used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Signal and control fields are red detuned from resonance by ∆ and have angular frequencies ωs and ωc respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The transition F = 4 → F ′ = 4 is resonantly driven by the pump field with angular frequency ωp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' b) Time traces from an experiment for writing a signal pulse onto an atomic spin-wave at time t0 = 0 and retrieving it at t0 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='4 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The bottom panel shows the operation performed on the memory (W: write, R: read).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' EXPERIMENT The multi-rail memory presented here uses an EIT memory scheme19 on the hyperfine ground state transitions of the Cs D1 line, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The control and signal lasers (14-pin butterfly external cavity diode laser (ECDL) modules by Sacher Lasertechnik) are set on the F=4 → F ′=3 and the F=3 → F ′=3 transitions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' To stabilize the frequency difference between signal and control laser to the Cs ground state hyperfine splitting, light from both lasers is superimposed on a fast photodiode (Electro Optics Technology, ET-3500FEXR) and their beat frequency is offset locked20 using a RedPitaya FPGA-board running the Linien21 locking software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' We generate Gaussian signal and control pulses with a full width at half maximum (FWHM) of 25 ns and 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='75 ns respectively using an arbitrary function generator (AFG, Tek AFG31152).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' These pulses are modulated onto cw laser beams with electro-optic amplitude modulators (EOMs, Jenoptik AM905).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The experiment is designed such that the signal and control pulses have linear and orthogonal polarization to each other to reduce the control light leaking into the detection path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The atomic storage medium is confined to a cylindrical, 25x75 mm anti-reflection coated cesium vapor cell filled with 5 torr N2 of buffer gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' It is kept at 60°C and shielded from ambient magnetic fields by a double-layered mu-metal housing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Spatial addressing of different rails within the cell is performed by one acousto-optic deflector (AOD) in front and one behind the vapor cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The AODs (AA MT200-B100A0,5-800) have an aperture of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='5x2 mm2 and a measured deflection of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='2 mrad/MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Each of them is driven by the frequency sum of an arbitrary function generator (AFG, Tek AFG31152) and a local oscillator (Mini-Circuits, ZOS-300+) resulting in 200±50 MHz of carrier frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' We refer to the position of memory rails by the AOD driving frequency used to deflect the beam to that position;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' the distance between rails is thus expressed as the difference in driving frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Changing the AOD driving frequency by 8 MHz changes the lateral position of the deflected beam by 270 µm, equaling one signal beam radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' While the signal pulses enter the memory unmodified after their generation, the control laser pulses are amplified by a self- made tapered amplifier (TA, see22) and then spectrally filtered by a dielectric bandpass interference filter (IF) with a 1 nm FWHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' This results in 200 mW of coupled cw power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' To increase the control laser’s on-off ratio, the TA diode’s driving current is only switched on for 120 ns, centered on the optical pulse, using a 4 A dc-coupled input follower driver with 2 ns rise/fall time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' This reduces the unwanted interaction between control laser and atoms in times when no control pulse is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The beam paths of the cross-polarized signal and control lasers are overlapped at a polarizing beam splitter (PBS) on one side of the memory and then propagate collinearly through the cesium cell, and both AODs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' At the position of the cell, control and signal beam have a 1/e2-level radius of 350 µm and 270 µm respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' This yields an atomic transit time of ∆t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='7 µs for one signal beam radius when using a 2D diffusion model of ∆x = √ 4D∆t for the diffusion length and an assumed diffusion constant of D0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='24 cm2s−123 at T0 = 0 K and P0 = 760 torr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' After traversing the second AOD, the signal and control beams are split by a second PBS and are then individually coupled to fibers and detected by either a Si photodiode (Thorlabs DET10A2) or a Si avalanche photodiode (Menlo Systems APD210).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Optical pumping of the Cs atoms into the 62S1/2F=3 state is performed by a third ECDL laser locked to the 62S1/2F=4 → a) b) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=" SIGNAL 62P1/ 2 F'= 4 F'= 3 [arb." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='] CONTROL Intensity m m dm DETECTOR F= 4 62S1/ 2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='2 GHz F= 3 PUMP W R R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='0 Time [μs]3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Sketch of the experiment with A) laser sources, spectroscopy and optical pulse shaping, B) TA based pulse amplification, C) multi-rail storage system and D) CCD image of the four used rails, with the camera at the place of the Cs cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' HWP: half-wave plate, QWP: quarter-wave plate, DET: detector, (P)BS: (polarizing) beam splitter, L1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='L2/L3: aspheric/cylindrical lens, AOD: acousto-optic deflector, EOM: electro-optic modulator, AFG: arbitrary function generator, IF: interference filter, OL: offset lock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 62P1/2F=4 transition by saturated absorption spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The pump light power is controlled via transmission through an electrically pulsed semiconductor optical amplifier (SOA) and illuminates each memory rail with 20 mW of optical power for 900 ns prior to the memory experiment sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Figure 1(b) shows a typical time trace for a single-rail storage experiment and a sketch of the experimental setup can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Several features of storage within an EIT medium can be observed in the time trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' At t = 0 µs a signal pulse enters the atomic medium and is partly mapped to an atomic spin wave by the control laser field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The portion of that signal pulse that is transmitted through the atomic vapor is detected by the photodiode as leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' After 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='4 µs the control laser field is switched on again and retrieves the spin-wave excitation back into the signal beams optical mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' A third pulse of the control laser at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='8 µs serves to determine if all the excitation has been retrieved and also allows to estimate the signal noise induced by the control laser field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Not having a significant detection event during this last pulse, we conclude that nearly all the spin-wave excitation is mapped backed to optical and signal to noise ratio is not a limiting factor for this experiment with aforementioned laser pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' RESULTS Prior to performing multi-rail storage, we first identify optimal operating conditions for the multi-rail memory by assessing the influence of rail separation on the interactions between two memory rails, and subsequently minimizing the cross-talk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' For comparison of the single and multi-rail operation, we measure the 1/e lifetime and memory efficiency per rail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' To assess the influence of rail separation on their interaction, multiple storage experiments at different rail separations are conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' For effective operation of a memory, we require that operations on a given memory rail do not affect its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' To determine the minimal separation that shows no cross-talk, we write into a rail fixed at 190 MHz, read from a neighboring rail, and then read from the 190 MHz rail again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' This write/read/read sequence is depicted in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The rail separation is varied from 0 to 25 MHz at steps of 1 MHz, and the retrieval peak intensities after the first and second read are measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The results are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Below 5 to 8 MHz of separation no excitation is left for the second retrieval pulse and both read pulses address the same ensemble of atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' At a separation of 20 MHz the influence of the read operation on the neighboring rail is no longer visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The AOD device used has a 100 MHz bandwidth and a 25% reduced diffraction efficiency at the edges of the frequency range;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' consequently we chose to limit this experiment to four memory rails spaced by 20 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' A CCD image of the four rails, taken at the position of the Cs cell, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' A straightforward method to increase the number of rails, is to use AODs with a higher number of resolvable spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The 1/e storage lifetime per rail is determined by performing storage experiments with increasing time delay between the memory write and read operation, for each rail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The delay was varied between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='4 µs and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='2 µs in steps of 400 ns, and for each A) duwnd EOM C) signal HWP AOD L2 AOD HWP PBS HwP L1 QWP DET LASER to AWG BS PBS PBS to OL control HWP PBS LASER EOM D) B) BP HWP 1nm HWP PBS 20 MHz 『』 TA-diode L2L3 opt, isolator PD 625 um4 Peak 1 Peak 2 0 5 10 15 20 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='10 Rail separation [MHz] Intensity [arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='0 Time [µs] W R R FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Cross-talk estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Intensities detected in the first (Peak 1, orange rail) and second (Peak 2, green rail) read peak depending on the rail separation for the experiment sequence depicted in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' For a difference of 20 MHz in AOD driving frequency (rail separation), the influence between neighboring rails vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The inset shows the used experiment sequence consisting of a write on the green rail at t = 0, a read on a neighboring orange rail at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='4 µs and finally a read on the green rail at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='8 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' delay, we measure the retrieved peak intensities, averaged over 500 repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Uncertainties are given by the standard deviation of the intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The intensities were fitted with an exponential function to extract the 1/e lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Measured retrieval peak intensities and fit function are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The resulting lifetime values per rail are shown in Table I together with the achieved internal memory efficiencies ηmem at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Since the measured lifetimes are consistent with the estimate using the simple diffusion model presented earlier, it is reasonable to assume that diffusion is the most important lifetime-limiting process for the beam diameters chosen in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Independent investigation on a single rail setup also showed that spin polarization lifetimes at least on the order of several hundred microseconds are possible with larger beam diameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The memory efficiencies are calculated by extrapolating the pulse energy of a retrieved pulse after t = 0 µs of storage from a retrieved pulse after t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='4 µs of storage using the memory lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' This is then divided by the energy of a normalization pulse to yield the efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' To obtain the normalization pulse, we set the signal laser frequency 2 GHz below the F=3 → F ′=3 transition frequency, block the control beam and record the transmitted signal pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Under these conditions, we assume the pulses not to be absorbed by the atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Using the insights and results from the measurements on lifetime, efficiency and rail separation, we now explore the possibility of random-access operation in the memory setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' For this purpose an experimental sequence was designed that highlights important criteria for use as a random-access quantum memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Figure 5 illustrates this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The bottom panel depicts the operation performed on each specific rail and the top panel shows the intensity detected by the APD over a time span of about 5 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The experimental sequence contains 12 operations, either read (r) or write (w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' We define three features which are necessary for use as a memory: a) that reading or writing to a rail should not affect its neighbors (interaction-free), b) rails which have not been written to should not return a retrieved pulse (empty state) and c) a read should leave the memory empty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' a subsequent read pulse should yield no excitation (full retrieval).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Rail (MHz) 170 190 210 230 Mean Lifetime (µs) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='3(5) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='4(7) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='3(3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='6(3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='2(2) Efficiency (%) 32 35 39 36 36 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Measured 1/e-lifetime and retrieval efficiency for each rail and weighted mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Measured retrieval amplitudes for storage times between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='4 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='4 µs together with an exponential fit to the values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The inset shows the per rail 1/e storage lifetime deduced from the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='20 1 2 3 4 5 6 7 8 9 10 11 12 Intensity [arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='] Operation # 0 1 2 3 4 230MHz 210MHz 190MHz 170MHz W W R R W R W R R R W R Time [µs] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Storage experiment in the random-access memory using four rails with detected intensity in the signal path in the top and performed operations (read/write) in the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' A total of 12 operations are performed over a span of 5 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='20 1/e Lifetime [μus] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='15 4 Intensity [arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='] 3 王 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='10 2 170 190 210 230 Rail [MHz] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='00F 0 2 4 6 8 10 12 Time [μs]6 Interaction-free operation is ensured by choosing an adequate rail separation and then confirmed by looking at the storage performance of a specific rail while there are operations performed on the neighboring rails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The rails at 230 MHz (blue) and 190 MHz (green) can be used to show that operations are interaction free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Between the write (t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='6 µs) and read (t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='6 µs) operation on the 190 MHz rail, four operations are performed on the neighboring rails and there is no visible impact on the read peak shape or height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' On the 230 MHz rail a pulse is written at t = 0 and then retrieved during the last operation on the memory at t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='4 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Taking into account the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='6 µs lifetime of this rail, the high remaining intensity of the retrieval peak clearly shows that there is no significant detrimental influence from multi-rail operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Reading an empty rail should not result in a significant amount of intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' We verify this by reading the 210 MHz and 170 MHz rail in a state that should not have excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' In the 210 MHz rail a pulse is written to the memory at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='4 µs and then this rail is immediately read twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The second read operation at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='8 µs yields negligible intensity compared to the first read operation at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='6 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Additionally the same rail is read again at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='2 µs to observe the amount of noise, which is found to be comparable to the read at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='2 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The first operation on the 190 MHz rail at t = 2 µs is a read of a rail that has not been used before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' This allows us to determine how well the memory was initialized by the pumping that is performed prior to the experimental sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Observing a larger intensity peak would point to insufficient polarization of the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' As the observed peak is similar to the other reads of an empty rail mentioned above, we conclude that pumping is sufficient and reading an empty rail, regardless of its history, does not lead to the detection of a significant peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' In combination with the measurements on lifetime and rail interaction it follows that this setup allows random-access storage and retrieval of optical pulses for times comparable to the mean rail lifetime of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='2 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' CONCLUSION We have presented a multiplexed optical random-access memory, realised within a single vapor cell at a temperature of 60°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Using an EIT based storage scheme in a Λ-system on the cesium hyperfine transitions, we achieved a mean storage lifetime and internal efficiency of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='2(2) µs and 36% respectively in multi-rail operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' According to the chosen rail separation of 20 MHz, we performed random-access storage and retrieval in four parallel rails without observing reciprocal influence between the different rails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The time between successive operations was chosen to be 400 ns for the sake of simplifying experiment control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' This time could be reduced considerably with the lower bound determined by the AODs switching time of 48 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Increasing the storage lifetime and number of addressable rails is possible by increasing the beam diameters and using AODs that have a higher time-bandwidth product respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' This step will be important for applications in quantum communication and repeater networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Reaching beyond the threshold number of 1000 individually addressable modes is possible by using 2-axis AODs and multiplexing into a two dimensional grid of parallel storage modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' FUNDING This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project number 445183921.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' acknowledges funding through the Helmholtz Einstein International Berlin Research School in Data Science (HEIBRiDS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' DISCLOSURES The authors declare no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' DATA AVAILABILITY The data presented in this paper is available from the authors upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
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+Draft version January 30, 2023
+Typeset using LATEX twocolumn style in AASTeX63
+Strong Variability in AzV 493, an Extreme Oe-Type Star in the SMC
+M. S. Oey,1 N. Castro,2 M. Renzo,3 I. Vargas-Salazar,1 M. W. Suffak,4 M. Ratajczak,5 J. D. Monnier,1
+M. K. Szymanski,5 G. D. Phillips,1 N. Calvet,1 A. Chiti,6, 7 G. Micheva,1, 8 K. C. Rasmussen,1, 9 and
+R. H. D. Townsend10
+1Astronomy Department, University of Michigan, 1085 South University Ave., Ann Arbor, MI, 48109, USA
+2Leibniz-Institut f¨ur Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482, Potsdam, Germany
+3Center for Computational Astrophysics, Flatiron Institute, 162 5th Ave, New York, NY 10010, USA
+4Department of Physics and Astronomy, Western University, London, ON N6A 3K7, Canada
+5Astronomical Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland
+6Department of Astronomy & Astrophysics, University of Chicago, 5640 S Ellis Avenue, Chicago, IL 60637, USA
+7Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA
+8Present address: Leibniz-Institut f¨ur Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482, Potsdam, Germany
+9Present address: Astronomy Department, University of Washington, Box 351580, Seattle, WA 98195, USA
+10Astronomy Department, University of Wisconsin, Madison, WI 53706, USA
+(Accepted January 23, 2023; to appear in the Astrophysical Journal)
+ABSTRACT
+We present 18 years of OGLE photometry together with spectra obtained over 12 years, revealing that
+the early Oe star AzV 493 shows strong photometric (∆I < 1.2 mag) and spectroscopic variability with
+a dominant, 14.6-year pattern and ∼40-day oscillations. We estimate stellar parameters Teff = 42000
+K, log L/L⊙ = 5.83 ± 0.15, M/M⊙ = 50 ± 9, and v sin i = 370 ± 40 km s−1. Direct spectroscopic
+evidence shows episodes of both gas ejection and infall. There is no X-ray detection, and it is likely
+a runaway star. AzV 493 may have an unseen companion on a highly eccentric (e > 0.93) orbit.
+We propose that close interaction at periastron excites ejection of the decretion disk, whose variable
+emission-line spectrum suggests separate inner and outer components, with an optically thick outer
+component obscuring both the stellar photosphere and the emission-line spectrum of the inner disk at
+early phases in the
+photometric cycle. It is plausible that AzV 493’s mass and rotation have been
+enhanced by binary interaction followed by the core-collapse supernova explosion of the companion,
+which now could be either a black hole or neutron star. This system in the Small Magellanic Cloud
+can potentially shed light on OBe decretion disk formation and evolution, massive binary evolution,
+and compact binary progenitors.
+Keywords: early-type stars — Oe stars — Be stars — high-mass X-ray binary stars — circumstellar
+disks — stellar pulsations — interacting binary stars — compact objects — runaway stars
+— variable stars — Small Magellanic Cloud
+1. INTRODUCTION
+Binary interactions are now understood to be a funda-
+mental component of massive star evolution, and they
+are the progenitors of a wide variety of energetic phe-
+nomena including high-mass X-ray binaries (HMXBs),
+ultra-luminous X-ray sources (ULXs), stripped-envelope
+core-collapse supernovae (SNe), and gravitational wave
+events. A consensus is emerging that classical OBe stars
+appear to originate from close massive binary systems,
+wherein they have spun up through mass and angular
+momentum transfer from their mass donors (e.g, Pols
+et al. 1991; Vinciguerra et al. 2020; Bodensteiner et al.
+2020, see also Rivinius et al. 2013 for a review). When
+donor stars subsequently explode as supernovae, result-
+ing post-explosion bound binaries are more likely to
+be eccentric, since they result from tight binaries (e.g.,
+Brandt & Podsiadlowski 1995; Tauris & Takens 1998;
+Renzo et al. 2019). Thus, a substantial subset of classi-
+cal OBe stars are likely to have eccentric orbits. In this
+paper, we present photometric and spectrocopic time-
+series data showing that the star AzV 493 exhibits dra-
+matic variability and may be an eccentric binary system.
+AzV 493 (Azzopardi et al. 1975) or [M2002]SMC-
+77616 (Massey 2002) was identified as an extreme, clas-
+arXiv:2301.11433v1 [astro-ph.SR] 26 Jan 2023
+
+2
+sical Oe star by Golden-Marx et al. (2016).
+In that
+work, it was found to be the earliest classical Oe star
+in our sample of field OB stars in the Small Magel-
+lanic Cloud (SMC), based on a spectrum obtained in
+2009 that shows double-peaked emission, not only in
+the Balmer lines, but also in He i and He ii λ4686, the
+latter feature being rarely observed in other Oe stars
+(Conti & Leep 1974). Specifically, it is classified as an
+Ope star, indicating that the He i absorption lines show
+infilled emission (Sota et al. 2011).
+As an extreme object, AzV 493 offers unique oppor-
+tunities to study massive binary evolution and decre-
+tion disk formation, structure, and dynamics. Section 2
+presents the unusual light curve and periodicity, and
+Section 3 presents our multi-epoch spectroscopy with
+resulting derived stellar parameters and individual spec-
+tral features. We then present two possible models for
+the AzV 493 system in Sections 4 and 5, one based on
+ejection of an optically thick disk near periastron; and
+another based on disk growth and disruption. Section 6
+discusses the likely binary origin of the system, and Sec-
+tion 7 summarizes our findings.
+2. PHOTOMETRIC LIGHT CURVE
+2.1. Long-term light curve
+The I and V -band light curves of AzV 493 from the
+OGLE Project (Udalski et al. 2008, 2015) are presented
+in Figure 1. The I-band shows a short eruption with
+the peak of the light curve on
+MJD 52212, followed
+by an abrupt decline of approximately 1.2 mag, to a
+minimum on MJD 52303 in early 2002. After this, the
+star eventually recovers its original luminosity. Another
+photometric minimum is seen in 2016 on MJD 57626,
+followed by the same brightening pattern.
+The gray
+symbols in Figure 1 show the I-band photometry from
+the 2016 cycle overplotted on the data from 2002 cy-
+cle. This shows that the minimum luminosity and subse-
+quent increase are quantitatively identical, although the
+photometry immediately preceding the minimum differs.
+Cross-correlating these segments yields a long-cycle pe-
+riod of 5311 days (14.55 years). There is no evidence of
+a similar eruption preceding the minimum in the 2016
+cycle on the same 91-day timescale, although the pho-
+tometry is incomplete in this range.
+After the minimum, the brightness increases and then
+starts to gradually decrease again, over a period of sev-
+eral years. Approximately in 2008, AzV 493 appears to
+go into a multiple outburst event. After this, the light
+curve drastically changes, showing a multi-mode pulsa-
+tion behavior that evolves with time (Section 2.2). The
+pulsation ends with another 0.2 – 0.3 mag drop, followed
+by a steady increase, repeating the light curve cycle that
+started in 2002, 14.55 years before.
+2.2. Photometric Oscillations
+Figure 2 shows short-term variability on the order of
+30 – 45 days. We quantify the evolution of these oscil-
+lations seen in the I-band light curve using Generalized
+Lomb-Scargle periodograms (Zechmeister & K¨urster
+2009) for the six contiguous OGLE datasets from 2010 –
+2016 (Figure 1). The individual fits to these six ranges
+are shown in Appendix A. Comparison of the periods
+shown in Figure 3 with the light curve (Figure 1) shows
+that they qualitatively appear to correlate with stellar
+brightness.
+The OGLE survey provides V -band magnitudes for
+a subset of the survey epochs, which are shown in red
+in Figure 1. Figure 4 displays the color-magnitude dia-
+gram (CMD) in V vs V − I for those days where both
+bands were observed.
+Figure 4a compares AzV 493’s
+color variations with data for the remainder of the RI-
+OTS4 sample stars (Lamb et al. 2016). The latter corre-
+spond to single-epoch photometry from the OGLE cat-
+alog of Poleski et al. (2012). Those stars classified as
+OBe stars by Lamb et al. (2016) are marked in the plot.
+The blue plume of non-emission-line stars is clearly sepa-
+rated from the cloud of OBe stars at redder colors in the
+CMD, a phenomenon already known from different pho-
+tometric bands (e.g., Bonanos et al. 2010; Castro et al.
+2018). The color variation of AzV 493 spans almost the
+entire range of V − I colors covered by the emission-line
+stars.
+Figure 4b shows a zoom in the CMD with the path of
+AzV 493 traced out. The star appears red during the
+broad peak of the light curve around 2006 (Figure 1),
+and then moves to bluer colors reaching the bluest V −I
+color during the pulsation phase.
+Approximately in
+2017, when the light curve is brightening after the mini-
+mum, AzV 493 shows redder colors again, moving to the
+original position observed in 2005 with V − I ∼ 0.18.
+Similar, semi-periodic variability with timescales on
+the order of weeks to months is seen in many other OBe
+stars, and their origin is unknown (e.g., Labadie-Bartz
+et al. 2017).
+Proposed explanations include forms of
+non-radial pulsations of the star and transitory or orbit-
+ing density enhancements in the disk, which may be the
+most likely scenario. The associated cyclical variation in
+the CMD (Figure 4) is also consistent with some kind of
+stellar radial pulsation. This is supported by the corre-
+lation between period and luminosity (cf. Figures 3 and
+4). In that case, the relatively long period implies that
+they could be an induced gravity mode or pulsational
+instability. However, there are many other possible ex-
+
+3
+Figure 1. AzV 493 OGLE light curves in I (black) and V (red) bands. The last segment of the I-band curve is overplotted
+(light grey dots) on the beginning of the dataset phase 14.6 years (5311 days) earlier. V − I is shown in the lower panel. The
+dashed lines mark the epochs for the observed spectra, assigned alphabetically in chronological sequence. The green shaded
+regions show consecutive 2656-day segments starting with the light curve maximum in 2001.
+Figure 2. Zoom on light curve (top) showing ∼ 40-day oscillations, and color variation (bottom).
+
+Time [year]
+2002
+2004
+2006
+2008
+2010
+2012
+2014
+2016
+2018
+2020
+2022
+D
+H
+13.65
+E
+F
+13.80
+B
+C
+G
+13.95
+@ 14.10
+nitude
+14.25
+Magr
+14.40
+14.55
+14.70-
+I band
+14.85
+V band
+I band shifted -14 years
+C
+0.2
+:
+0.1
+io
+0.0
+-0.1 -
+53000
+54000
+55000
+56000
+57000
+58000
+59000
+MJD [days]Time [year]
+2010.4
+2010.6
+2010.8
+2011.0
+2011.2
+2011.4
+2011.6
+2011.8
+2012.0
+2012.2
+I band
+14.24-
+V band
+14.32
+.
+3
+14.40
+.
+.
+.
+.
+::
+.
+.
+:
+i
+14.64
+.
+.
+14.72
+14.80
+0.025
+0.000
+.
+-0.050
+.
+-0.075
+C
+-0.100
+55300
+55400
+55500
+55600
+55700
+55800
+55900
+56000
+MjD[days]4
+Figure 3.
+Fitted periods for the six contiguous OGLE
+datasets between ∼ 2010 – 2016, as a function of time.
+planations, perhaps including interactions with another
+star in a close orbit. We note that de Wit et al. (2006,
+see also Rivinius et al. 2013) reported similar loop-like
+excursions in the CMD of other OBe stars, and ascribed
+the anti-clockwise variation to the formation and dissi-
+pation of the circumstellar decretion disks in those ob-
+jects.
+2.3. Light curve period
+It is possible that the multiple-outburst event in 2008
+– 2009 may represent another periastron. Figure 1 shows
+the 5311-day cycle initiated at the light-curve peak at
+MJD 52212 instead of at the minima. We see that the
+mid-cycle occurs during this multiple-outburst event, al-
+though due to the OGLE observing cadence, it is unclear
+whether it occurs near the end or near the middle. In
+Section 3 below, we show that the spectrum obtained
+around this time, Epoch A (Figure 1), shows an un-
+usually strong emission-line spectrum, consistent with
+maximum disk activation and flaring. However, the light
+curve does not repeat the cycle minimum seen in 2002
+and 2016, and OBe stars are known to show temporary
+outbursts of activity (e.g., Labadie-Bartz et al. 2017;
+Baade et al. 2018).
+Thus, it is not clear whether 2008 – 2009 corresponds
+to the mid-cycle or not. The light curve does not repeat
+regularly in detail, and we caution that the period, if the
+system is a binary, is uncertain. Assuming that there is
+indeed a fundamental physical period, the same phases
+may not all generate the same observational signatures,
+which may depend on other factors such as disk orienta-
+tion and/or varying physical processes. In what follows,
+we adopt a system period of 5311 (2656) days, or 14.55
+(7.28) years, where the values in parentheses allow for
+the possibility that the period may be half of the long
+cycle.
+3. SPECTROSCOPY
+Spectroscopic observations of AzV 493 were obtained
+in the course of the RIOTS4 spectroscopic survey of
+field OB stars in the SMC (Lamb et al. 2016), and
+follow-up radial velocity monitoring of the SMC Wing
+region (Vargas-Salazar et al. 2023, in preparation). The
+observations were carried out using the Magellan tele-
+scopes at Las Campanas, Chile. Three different spectro-
+graphs were used: IMACS (Bigelow & Dressler 2003),
+MIKE (Bernstein et al. 2003) and M2FS (Mateo et al.
+2012). Table 1 gives details of our spectroscopic observa-
+tions, including the modified Julian day (MJD), signal-
+to-noise, spectral resolution, spectral range, phase in the
+light curve cycle, radial velocity, Hβ peak separation
+(Section 3.2), and instrument used. Figure 5 displays
+the 11 spectra in chronological sequence, labeled A – K
+as shown.
+IMACS was operated by default in multi-slit mode
+with the f/4 camera and 1200 lines/mm grating, which
+provides a resolving power of R ∼ 3000 and a wave-
+length coverage spanning ∼3800 – 5200 ˚A. One obser-
+vation (Epoch I) was observed with the f/2 camera, re-
+sulting in lower resolution (Table 1). The reduction was
+performed using the cosmos pipeline1. MIKE data were
+obtained using a 1′′ slit width for a spectral resolution
+of R ∼ 28000, covering the wavelength range ∼3600
+– 10000 ˚A. The spectra were processed with the the
+Carnegie Python (CarPy2) pipeline software (Kelson
+et al. 2000; Kelson 2003), except for Epoch B, which was
+extracted using IRAF3. M2FS data were observed us-
+ing a custom filter yielding ∼4080 – 4470 ˚A wavelength
+coverage at R ∼ 28000. The data were processed follow-
+ing the standard steps in fiber spectroscopic reduction
+using IRAF/PyRAF tasks implemented within python
+and designed for this instrument (see Walker et al. 2015).
+Figure 5 shows strong variability in the spectrum of
+AzV 493. The weaker epochs show a typical OBe spec-
+trum, with only Hβ showing double-peaked emission,
+and Hγ and Hδ absorption features showing evidence
+of infill; whereas Epochs A, B, and K show stronger
+emission-line spectra, with Hγ and He i often in emis-
+sion. Epoch F shows strong, high-order Balmer emis-
+sion and inverse P-Cygni features. These epochs will be
+discussed in Sections 3.3 – 3.4.
+1 http://code.obs.carnegiescience.edu/cosmos.
+2 http://code.obs.carnegiescience.edu/mike
+3 IRAF was distributed by the National Optical Astronomy Obser-
+vatory, which was managed by the Association of Universities for
+Research in Astronomy (AURA) under a cooperative agreement
+with the National Science Foundation.
+
+44
+42
+40
+[days]
+38
+Period
+36
+34
+32
+30
+55500 55750 56000 56250 56500 56750 57000 57250
+MJD [days]5
+Figure 4. Color-magnitude diagram (CMD) based on available V - and I-band OGLE photometry (see Figure 1). The variation
+of AzV 493 in the CMD is colored according to the MJD, and compared to single-epoch OGLE photometry (Poleski et al. 2012)
+for the RIOTS4 OB-star sample (Lamb et al. 2016) (grey dots). Objects classified as OBe by Lamb et al. (2016) are highlighted
+with circles. The right panel is a zoom of the same data around the track of AzV 493.
+Table 1. Spectroscopic Observations of AzV 493
+Epoch
+Date [UTC]
+MJD
+S/N
+R
+Wavelength
+Phasea
+RV
+∆v(Hβ)b
+Instrument
+Range [˚A]
+(km s−1)
+(km s−1)
+A
+2009-08-26T01:43:36.0
+55069.071944
+140
+3000
+3825–5422
+0.538 (0.076)
+152 ± 200
+279
+IMACS
+B
+2015-01-14T02:12:03.0
+57036.091701
+120
+28000
+3362–9397
+0.908 (0.817)
+192 ± 18
+(213)c
+MIKE
+C
+2016-06-15T07:47:54.3
+57554.324935
+130
+3000
+3879–5479
+0.006 (0.012)
+171 ± 60
+346
+IMACS
+D
+2016-09-08T01:42:08.0
+57639.070926
+60
+28000
+4079–4466
+0.022 (0.044)
+217 ± 50
+· · ·
+M2FS
+E
+2016-09-11T02:49:33.0
+57642.117743
+90
+28000
+4080–4465
+0.022 (0.045)
+239 ± 46
+· · ·
+M2FS
+F
+2016-09-22T05:36:51.0
+57653.233924
+150
+28000
+3538–9397
+0.024 (0.049)
+192 ± 29
+334
+MIKE
+G
+2016-12-04T04:09:41.5
+57726.173397
+110
+3000
+3862–5458
+0.038 (0.076)
+243 ± 38
+319
+IMACS
+H
+2017-06-05T06:35:11.2
+57909.274435
+50
+3000
+3871–5471
+0.073 (0.145)
+235 ± 54
+322
+IMACS
+I
+2017-06-07T08:08:18.9
+57911.339108
+130
+1300
+3900–8000
+0.073 (0.146)
+231 ± 83
+295
+IMACSd
+J
+2017-07-10T09:05:00.5
+57944.378478
+190
+3000
+3854–5468
+0.079 (0.159)
+181 ± 39
+303
+IMACS
+K
+2021-09-25T07:38:18.0
+59482.318264
+210
+28000
+3362–9397
+0.369 (0.738)
+183 ± 17
+289
+MIKE
+aPhase relative to the light curve peak at MJD 52212 (54868), adopting a period of 5311 (2655.5) days.
+b Hβ peak separation obtained by fitting two gaussians with fixed width of 2 ˚A (∼ 120 km s−1).
+c Epoch B does not show a double-peaked profile (see Figure 7 and Section 3.4); the value for ∆v(Hβ) assumes that two
+components exist, as they do for other epochs.
+dEpoch I was observed with the f/2 camera while the other IMACS observations were obtained with the f/4 camera.
+
+13.0
+O
+13.5
+O
+O
+14.0
+O
+> 14.5
+O
+O
+O
+0
+15.0
+8
+O
+0
+O
+0
+15.5
+O
+16.0
+-0.3
+-0.2
+-0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+V-I14.0
+57600
+14.1
+57000
+14.2
+56400
+14.3
+MJD [days]
+55800
+> 14.4
+55200
+Q
+14.5
+00
+54600
+14.6
+00
+54000
+14.7
+O
+53400
+14.8
+-0.10
+-0.05
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+V-I6
+Figure 5. AzV 493 multi-epoch spectroscopic observations sorted by MJD and normalized to the continuum. Epoch I is low
+resolution (Table 1).
+
+M
+H
+normalized flux
+G
+F
+E
+D
+C
+B
+A
+1
+He lI
+He llI
+He ll
+4600
+4000
+4200
+4400
+4800
+5000
+wavelength [A]7
+3.1. Stellar fundamental parameters
+The photospheric He ii lines λ4200, λ4541, and λ5411
+lines at all epochs confirm the early O spectral type
+assigned by Golden-Marx et al. (2016). To improve S/N
+in the He ii λ4541 absorption line, we combine epochs
+C, G, H and J, which are all IMACS spectra obtained
+in 2016 – 2017.
+We use this composite spectrum to
+estimate the projected rotational velocity (υ sin i) using
+the iacob-broad code (Sim´on-D´ıaz & Herrero 2014,
+2007). We obtain υ sin i = 370±40 km s−1. As discussed
+in Section 4, the angle of inclination i is likely high,
+based on the amount of obscuration from the disk, and
+so the rotational velocity might be ≲ 450 km s−1.
+The combined spectrum was modelled using the stellar
+atmosphere code fastwind (Santolaya-Rey et al. 1997;
+Puls et al. 2005; Rivero Gonz´alez et al. 2012), using
+the same technique and stellar grid described in Cas-
+tro et al. (2018).
+The cores of the Balmer lines are
+omitted from the fit to ameliorate contamination from
+disk emission. Our best model yields effective tempera-
+ture Teff = 42000 K and surface gravity log g = 3.4 dex,
+which reproduce the main He i and He ii lines (Figure 6).
+Since He i photospheric features are not detected, this
+Teff may be a lower limit. The derived temperature is
+consistent with an O3-5 spectral type (Martins & Pala-
+cios 2021), matching the early O-type classification of
+AzV 493 (Lamb et al. 2016). However, we caution that
+the wings of the Balmer lines, which are the main spec-
+troscopic anchors for deriving the surface gravity, may
+be affected by the circumstellar emission, resulting in an
+underestimate of log g, as found for OBe stars by Castro
+et al. (2018).
+The stellar luminosity was calculated using the optical
+and IR photometry for AzV 493 (Massey 2002; Skrut-
+skie et al. 2006), adopting a distance to the SMC of
+62.1 kpc (Graczyk et al. 2014) and the synthetic fast-
+wind spectral energy distribution (SED) derived above.
+We explored the extinction curves published by Fitz-
+patrick & Massa (2007) until the observed photometry
+was reproduced by the fastwind synthetic SED. We
+obtain a luminosity log L/L⊙ = 5.83 ± 0.15 and radius
+R⋆/R⊙ = 15±3, in agreement with the expected values
+for an early O-type star of luminosity class III – V (e.g.
+Martins et al. 2005). We compare the position of AzV
+493 in the Hertzsprung–Russell diagram with the rotat-
+ing evolutionary tracks by Brott et al. (2011) for SMC
+metallicity. Based on the Teff and L/L⊙ and their re-
+spective uncertainties, we estimate that the stellar mass
+is M/M⊙ = 50±9. If the observed luminosity is overes-
+timated by the inferred log g, or includes a contribution
+from a non-compact binary companion and/or the disk
+continuum, then the stellar mass may be somewhat over-
+estimated; for reference, a factor of two overestimate in
+luminosity implies M/M⊙ ∼ 40.
+3.2. Hβ emission-line profile
+Variability in the emission lines is a common charac-
+teristic of the Be phenomenon (e.g., Rivinius et al. 2013;
+Richardson et al. 2021). One effect is the violet-to-red
+(V/R) variations, which are cycles that can last weeks
+or decades. The V/R variations describe changes in the
+dominant peak strength for double-peaked emission lines
+observed in some stars. These cycles are attributed to
+variation in the morphology and density of the circum-
+stellar disks (Poeckert 1982; Okazaki 1991).
+Figure 7 shows Hβ profiles in the spectroscopic epochs
+where this line is available, and Gaussian models used to
+disentangle the V and R components. The two peaks are
+clearly resolved in all our observations of Hβ, except for
+Epoch B, which instead shows a P-Cygni profile (Fig-
+ures 5, 7; see Section 3.4). Table 1 gives the peak separa-
+tions ∆Hβ fitted in Figure 7. The V peak is usually more
+prominent than R. There may be a long-timescale V/R
+cycle, but further spectroscopic monitoring is needed to
+determine whether V/R indeed oscillates, or whether
+there is any trend in ∆Hβ with phase.
+3.3.
+Epochs A and K: Evidence of disk evolution
+Epoch A is observed at a phase of 0.54 (0.08), soon
+after the apparent eruption event in 2009 (Figure 1, Ta-
+ble 1). This spectrum shows the strongest helium line
+emission (Figure 5), although we have no other spec-
+troscopic observations within several years of this data
+point. Only photospheric He ii is seen in absorption in
+this spectrum; the H i and He i lines are all in emission
+or filled in. Moreover, He ii λ4686 is also in emission,
+which prompted Golden-Marx et al. (2016), to identify
+this spectrum as the hottest-known observation of the
+OBe phenomenon. Nebular He ii is only generated by
+the very hottest O stars (e.g., Martins & Palacios 2021).
+All of the emission lines in Epoch A are double peaked.
+Hβ and Hγ show larger peak separations than the He i
+and He ii emission lines. For a Keplerian disk, this would
+imply that the higher-temperature species is dominated
+by larger radii than the Hβ and Hγ emission. Figure 5
+shows that the emission is slightly redshifted relative to
+the photospheric Balmer absorption.
+Epoch K, observed at phase 0.37 (0.74) (Figure 5; Ta-
+ble 1) shows the opposite relation between ionization
+and disk radius. Here, the He i lines have larger peak
+separations than Hβ, implying that the hotter species
+dominates at smaller radii, unlike Epoch A. We also
+see that the Hβ and Hγ line profiles show high-velocity
+wings that are not observed at other epochs, consistent
+
+8
+Figure 6. Spectroscopic analysis of the composite IMACS spectrum from epochs C, G, H and J (black; cf. Fig. 5). The
+best fastwind (Santolaya-Rey et al. 1997; Puls et al. 2005; Rivero Gonz´alez et al. 2012) stellar atmosphere synthetic model is
+overplotted (red). The main transitions used in the analysis are marked.
+Figure 7. Hβ emission-line profiles from our spectra of AzV 493. The best-fit photospheric model (Figure 6) is subtracted,
+after which the violet and red components are fitted by two Gaussian profiles having fixed widths of 2 ˚A. The figure shows
+the data overplotted by these summed fitted Gaussians. The resulting peak values are shown by the vertical lines, and their
+separations are given in Table 1. Epoch I has low spectral resolution and is not included in this figure.
+with high-velocity gas at smaller orbital radii. Epoch K
+is similar in emission-line strength
+to Epoch A and
+shows He i in emission, but He ii λ4686 is in absorption
+in this observation, as it is in all the other observations
+of this line.
+3.4. Epochs B and F: Gas Outflow and Infall
+Epoch B shows P-Cygni emission-line profiles in Hβ
+and Hγ (Figures 5, 7), suggesting an outflow episode.
+This is also the only spectrum obtained during the pe-
+riod where the strong pulsations dominate the flux (Fig-
+ure 1), and it is observed at the latest phase, 0.91 (0.82).
+Figure 13 shows that the observation coincides with a
+local minimum in the light curve. Thus the P-Cygni fea-
+tures could suggest that the pulsations may be directly
+linked to mass ejection, since it coincides with the star
+reaching its smallest radius.
+The spectrum of Epoch F is dramatically different
+from most of the other spectra (Figure 5).
+It shows
+strong, asymmetric Balmer and He i emission that show
+remarkable, inverse P-Cygni line profiles, with red-
+shifted absorption and blue-shifted emission. Figure 8
+
+1.3
+He lI
+He lI
+He ll
+He l
+Hel
+Hel
+Hel
+1.2
+1.1
+1
+1.0
+0.9
+0.8-
+0.7
+0.6.
+4000
+4100
+4200
+4300
+4400
+4500
+4600
+4700
+1.3
+Hel
+He llI
+1.2
+≤ 1.1
+1.0
+0.9
+0.8
+0.7
+0.6
+4800
+4900
+5000
+5100
+5200
+5300
+5400
+5500
+wavelength [A]1.3
+3
+1.3
+1.3
+B
+C
+A
+1.2
+1.2
+1.2
+1.1
+1.1
+1.1
+1.1
+1.0
+1.0
+1.0
+1.0
+0.9
+0.9
+0.9
+0.9
+0.8
+0.8
+0.8
+0.8
+-500
+500
+-500
+0
+500
+-500
+0
+500
+-500
+0
+500
+0
+velocity - 200 [kms-1]
+1.3
+1.3
+1.3
+1.3
+G
+H
+K
+1.2
+1.2
+1.2
+1.2
+1.1
+1.1
+1.1
+1.0
+1.0
+1.0
+1.0
+0.9
+0.9
+0.9
+0.9
+0.8
+0.8
+0.8
+0.8
+-500
+0
+500
+-500
+0
+500
+-500
+0
+500
+-500
+0
+5009
+1.8
+1.6
+1.4
+1.2
+1
+-2000
+-1000
+0
+Velocity (km/s)
+1000
+2000
+Hα
+1.2
+1.15
+1.1
+1.05
+1
+.95
+_._ _ _ _ _
+___._
+_ _ _ _
+__. _ _ _ _ _
+.__ _ _ _ _
+..__ _ _ _ _
+_.___.
+-3000
+-2000
+-1000
+0
+1000
+Velocity (km/s)
+2000
+3000
+Hβ
+1.2
+1.15
+1.1
+1.05
+1
+.95
+-3000
+-2000
+-1000
+0
+Velocity (km/s)
+1000
+2000
+3000
+Hγ
+1.2
+1.1
+1
+. 9
+-7500
+-5000
+-2500
+0
+Velocity (km/s)
+2500
+5000
+Hδ
+He I 4026
+1.15
+1.1
+1.05
+1
+.95
+-3000
+-2000
+-1000
+0
+1000
+Velocity (km/s)
+2000
+3000
+He I 4471
+Figure 8. Epoch F line profiles for Balmer and He i emission lines, as shown, centered at the systemic velocity obtained from
+the He ii absorption. This Magellan/MIKE observation was obtained on 2016 September 22 (Table 1).
+shows the line profiles relative to the systemic velocity of
+the He ii photospheric lines. Such observations are usu-
+ally interpreted as infall of matter (e.g., Hartmann et al.
+2016), which appears to imply a re-absorption of decre-
+tion disk material. The free-fall velocity at the stellar
+surface for our adopted stellar parameters (Section 3.1)
+is ∼ 800 km s−1, which is consistent with the red edge
+of the absorption trough seen in Hδ and He i λ4471.
+The Balmer emission-line intensities do not follow the
+Balmer decrement and are almost uniform (Figures 6
+and 8), indicating optically thick emission. This sug-
+gests that the infalling material is also likely dense, and
+thus has high emissivity.
+Although Epochs D and E are taken only 14 and 11
+days before Epoch F, respectively, Epochs D and E show
+most lines in absorption with no sign of these features.
+Similarly, Epoch G is obtained only 73 days after Epoch
+F, and also shows primarily an absorption spectrum.
+Thus, this infall episode corresponds to a short-lived
+event, which we fortuitously captured with this MIKE
+observation. In the spectra observed before and after
+Epoch F, the Balmer emission, which presumably origi-
+nates from the disk, does not seem substantially different
+in intensity. This suggests that the reabsorbed material
+corresponds to a negligible fraction of the disk mass.
+The timing of Epoch F is at a very early phase, 0.024
+(0.05), only 27 days after the light curve minimum on
+
+10
+MJD 57626. There is no significant feature in the pho-
+tometry near the time of Epoch F, and the light curve is
+gradually brightening during this phase. This similarly
+implies that the continuum luminosity is dominated by
+the star and/or disk sources unrelated to the P-Cygni
+event.
+4. DISK EJECTION SCENARIO
+The distinctive shape of the light curve seen in 2002
+– 2004, and again in 2016 – 2018, showing a strong
+drop in brightness followed by gradual increase (Fig-
+ure 1), is seen in some other emission-line stars (Riv-
+inius et al. 2013). We suggest that this may be due to
+the repeated ejection of an optically thick circumstel-
+lar decretion disk, perhaps related to interaction with
+a binary companion.
+The exact reproduction of this
+part of the light curve across two cycles, starting with
+a 1.2-magnitude drop in brightness, suggests a geomet-
+ric extinction effect caused by an optically thick disk.
+This event’s pattern in photometry and Hβ line profile
+is consistent with a disk ejection outburst, similar to,
+e.g., HD 38708 (Labadie-Bartz et al. 2017).
+Assuming that an optically thick disk is indeed ex-
+pelled to generate the deep light-curve mimima (I ∼
+14.85) in 2002 and 2016, we can estimate the geometric
+obscuration by considering the maximum flux following
+these minima, which peaks around I ∼ 14.0. The differ-
+ence of 0.85 mag corresponds to reduction in flux by a
+factor of ∼ 0.46, or over half, assuming that all of this
+difference is due to obscuration. This suggests not only
+a fairly high angle of inclination, but also a thick, or in
+particular, a geometrically flared disk, which is consis-
+tent with spectroscopic evidence (Section 3.3).
+In this model, most of the emission lines originate from
+an inner disk region that experiences variable obscura-
+tion to our line of sight from a thicker outer disk or
+torus. The weaker spectroscopic epochs in Figure 5 with
+the typical OBe spectrum are the most obscured, while
+Epochs A, B, and K are less obscured and therefore show
+stronger emission-line spectra. Epoch C is observed in
+2016 at a phase of 0.01 (0.01), and thus near the same
+phase as the light curve peak in late 2001 (2009) (Fig-
+ure 1; Table 1). However, as noted above (Section 2.1),
+although the light curve repeats the disk ejection pat-
+tern, there is no evidence of a corresponding peak pre-
+ceding this sequence on the same timescale as that in
+2002.
+The Epoch C Hβ profile (Figure 7) is consis-
+tent with the optically thick disk already having formed.
+Epochs D and E, observed immediately after this min-
+imum, are similarly unremarkable, although they cover
+a much shorter spectral range. Since we see that a pu-
+tative disk ejection apparently occurred in 2016, it may
+be that the system has precessed such that an associ-
+ated photometric outburst is obscured by the ejection
+process.
+The emission lines in Epoch A are dominated by
+higher temperature species at larger radii, whereas
+Epoch K shows the opposite effect (Section 3.3).
+Epoch A is consistent with very dense, optically thick
+disks that have extended vertical flaring, as shown in
+models by, e.g., Sigut et al. (2009), where the emission,
+including from harder radiation, is dominated by this
+outer region. In contrast, the disk geometry at Epoch K
+is dominated by high-density gas near the center and
+no flaring, thus differing significantly from Epoch A.
+Epoch A is observed at a phase of 0.54 (0.08), and
+Epoch K shows the system at a phase of 0.37 (0.74;
+Table 1, Figure 1). This suggests that the disk changes
+between having a large, flared outer region at Epoch A
+that contributes significantly to the emission, and a con-
+figuration where flaring is insignificant and emission is
+dominated by a dense central region at Epoch K, per-
+haps also reaccreting material onto the star. The exis-
+tence of two different components dominated by inner
+and outer regions, respectively, could also be due to disk
+tearing, resulting in an inner disk and outer, expanding
+annulus with different inclinations (Suffak et al. 2022;
+Marr et al. 2022).
+The decreasing Hβ peak separations seen from Epoch
+C (346 km s−1) to Epoch J (303 km s−1) and to Epoch
+K (289 km s−1; Table 1) suggest that the emission is
+weighted toward increasing radii over this period, which
+is consistent with the inner disk dissipating or forming
+an annular disk with an expanding inner radius. How-
+ever, this scenario does not explain the strong line emis-
+sion in Epochs A and K (Figure 5), which have the min-
+imum Hβ peak separations. If the inner radius is indeed
+expanding, then the emitting region either must become
+dense, or the disk must precess to lower inclination an-
+gles to reveal stronger line emission. The latter could
+also contribute to a model in which the decreasing peak
+separation is due to decreasing obscuration of the disk,
+allowing emission at larger radii to dominate. This is
+consistent with the system’s increasing brightness over
+this period (Figure 1). The extinction may result from
+the outer component, or optically thick torus or flare in
+the disk which either precesses or dissipates. However,
+we caution that such a fast precession rate may not be
+feasible. Moreover, if the long-term photometric cycle is
+due to precession, the light curve should be symmetric
+around the minima, whereas the observed strong, sud-
+den drops (Figure 1) are difficult to explain with such a
+model.
+
+11
+The outflow and inflow episodes described in Sec-
+tion 3.4 apparently are not significant in mass relative to
+the entire disk. If the minima of the 14-year light curve
+indeed correspond to the bulk of disk ejection, followed
+by gradual disk dissipation, then the mass ejection as-
+sociated with the P-Cygni features in Epoch B are not
+likely to be a dominant source of disk material. How-
+ever, we note that pulsations have been suggested to be
+important in replenishing the disk in other OBe systems
+(e.g., Baade et al. 2016, 2018).
+The timing of Epoch F is 27 days after the light curve
+minimum on MJD 57626. Although there are 3 other
+intermediate spectroscopic epochs between the putative
+disk ejection and Epoch F, this still takes place dur-
+ing what we assume is the heavily obscured phase in
+the light curve. The lack of any photometric event near
+the appearance of inverse P-Cygni features in epoch F
+suggests that the reabsorbed material is an insignificant
+portion of the disk material. The disk is therefore sub-
+stantial and can plausibly provide material that may fall
+back to the star. This is consistent with the optically
+thick conditions indicated by the Balmer decrement in
+Epoch F.
+Thus, this model is driven by repeated ejection of a
+flared, optically thick disk whose outer region gradually
+dissipates, revealing the inner, line-emitting region. A
+flared disk is most clearly implied by the ionization and
+emission-line peak separation in Epoch A (Section 3.3),
+and is also consistent with a maximum geometric ob-
+scuration that may be > 50% implied by this model.
+The spectroscopic variation could also be caused by
+disk tearing or precession of the system. The decreas-
+ing trend in Hβ peak separations with increasing flux
+suggests that more light from larger radii can be seen
+(Section 3.2). Additionally, the high-amplitude, semi-
+regular pulsations with the ∼month-long period become
+visible at low extinction (Figure 1). Other photometric
+and spectral variations may be due to contributions from
+the inner disk’s radial expansion, reabsorption, or evap-
+oration/ionization, and possible geometric distortion or
+warping of the disk system.
+5. DISK GROWTH SCENARIO
+However, some observations seem inconsistent with a
+disk ejection model. For example, the system is bluest
+when faintest (Figure 1), contrary to expectations for
+reddening.
+As noted above, the strong emission-line
+spectra at Epochs A and K seem inconsistent with a
+dissipating inner disk scenario implied by the trend in
+∆Hβ. If the long-period cycle is attributed to disk pre-
+cession, it would require an additional mechanism to
+explain the assymmetric light curve, and also a third,
+external massive star that is not seen, to torque the
+disk. Thus, alternative models for the AzV 493 system
+should also be considered.
+Some other Be stars such as δ Sco (Suffak et al. 2020)
+and ω CMa (Ghoreyshi et al. 2018) show long-term pho-
+tometric variability in which the increasing flux is due to
+contributions from a growing disk, while the minima rep-
+resent episodes of disk destruction by the secondary at
+periastron. Such a model is therefore opposite to the one
+presented above. In this alternative scenario, the light
+curve minima of AzV 493 in 2002 and 2016 (Figure 1)
+correspond to episodes with the lowest disk contribu-
+tion. The disk then grows and brightens, recovering its
+full size around 2005. In this case, the decreasing trend
+in Hβ peak separation with increasing flux is simply due
+to the disk growth itself. This scenario is consistent with
+the blue color at the light curve minimum in 2016 (Fig-
+ure 1), and the weak emission-line spectra near the 2016
+minimum (epochs C – J; Figure 5).
+If the disk is responsible for the factor of 2.2 increase in
+flux, then the equivalent width (EW) of stellar absorp-
+tion features should decrease proportionately. Figure 9
+shows the EW of He ii λ4200 and λ4540 as a function of
+V and I magnitude. A slight trend is indeed apparent,
+although not as large as a factor of two in amplitude.
+These lines are in the B-band, and thus not in the range
+of our photometry. Figure 1 shows that the amplitude
+of the photometric variations may be smaller at bluer
+wavelengths, although with the given V -band sampling
+it is not entirely clear. It may be challenging for the
+alternative model to produce and maintain the viscous
+disk necessary to generate continuum luminosities that
+compete with those of the star, given the harsh circum-
+stellar environment of an extreme, early-type O-star.
+The extinction-dominated model is supported by the
+lack of correlation between the strength of the emission-
+line spectrum and photometric flux from the sys-
+tem. There is no significant variation between spectral
+Epochs C – J (Figure 5), which should correspond to the
+period of strong disk growth in this model, whereas the
+obscuration-dominated model implies dissipation (Sec-
+tion 3.3). The one exception showing spectral variation,
+Epoch F, has P-Cygni emission and stronger emission-
+line features, yet it is photometrically unremarkable
+(Section 3.4).
+Another issue is that the photometric
+minimum corresponds to the bluest color (Figure 1),
+which is more consistent with the alternative model.
+However, the star itself may be changing substantially in
+magnitude and color. Blueing is also caused by scatter-
+ing in high-extinction conditions, as seen in the UXOR
+class of Herbig Ae stars (Natta & Whitney 2000).
+
+12
+Figure 9.
+Equivalent width of He ii λ4200 (red) and λ4540
+(blue) as a function of V (bottom) and I (top) magnitude.
+A constant value of 0.5 ˚A is shown for reference.
+The overall shape of the light curve for AzV 493
+is rather different from those of δ Sco and ω CMa,
+which show extended minima with more top-hat-like
+light curves (Ghoreyshi et al. 2018; Suffak et al. 2020).
+In contrast, AzV 493 shows sharp minima (Figure 1),
+implying very rapid disk destruction and immediate,
+regular regrowth in the alternative model. It seems hard
+to explain such sudden dissipation of a several-AU dense,
+viscous disk by a neutron star or black hole (Section 6)
+during the brief periastron passage. Moreover, the exact
+reproduction of the photometric cycle’s initial segment
+(Section 2.1) is unusual and may be harder to explain
+with a disk-growth model.
+Overall, the fundamental nature of the light curve and
+disk evolution remain unclear. Tailored modeling of this
+system and further multimode observational monitoring
+is needed to clarify the relationship between the decre-
+tion disk and interaction with a secondary star.
+6. AN EXTREME INTERACTING BINARY
+The fast surface rotation for this evolved O star is
+a natural signature of accretion during a mass transfer
+event (e.g., Packet 1981; Cantiello et al. 2007; Renzo
+& G¨otberg 2021), consistent with an interacting binary
+scenario.
+If the disk is induced by a periastron pas-
+sage of an undetected companion, then this may imply
+a long, 14.6 (7.3)-year period, and hence a large and
+highly eccentric orbit. For the AzV 493 stellar parame-
+ters obtained in Section 3.1, a neutron star companion
+of mass 1.4 M⊙ would require e ∼ 0.95 (0.93) and apas-
+tron of ∼ 43 (27) AU for a typical OBe star periastron
+distance of 40R⋆. These orbital parameters are similar
+to those of the Be star δ Sco (e.g., Che et al. 2012).
+The unseen companion could also be a somewhat more
+massive main-sequence or stripped star, or a black hole.
+The eccentricity may be lower, but if a binary compan-
+ion is responsible for disk ejection, then periastron must
+be small and the eccentricity high. The nominal peri-
+astron value used here would likely be an upper limit,
+since δ Sco showed no disk ejection at periastron (Che
+et al. 2012).
+6.1. Neutron star or black hole?
+Thus, if a binary companion excites disk ejection or
+is otherwise responsible for the observed properties of
+AzV 493, then it is probably an eccentric system, and
+the most likely explanation for such an orbit is that the
+companion has already experienced core collapse, receiv-
+ing a strong kick. Large natal kicks are routinely invoked
+in core-collapse events that form neutron stars (e.g., Ar-
+zoumanian et al. 2002; Podsiadlowski et al. 2004; Ver-
+bunt et al. 2017; Janka 2017). Natal kicks during black
+hole formation are still highly debated (e.g., Dray et al.
+2005; Janka 2013; Mandel 2016; Repetto et al. 2017;
+Atri et al. 2019; Renzo et al. 2019; Callister et al. 2020),
+but not excluded. Assuming a large 450 km s−1 kick,
+Brandt & Podsiadlowski (1995) found a broad correla-
+tion between eccentricity and orbital period of binaries
+surviving the first core-collapse. This is in agreement
+with the high e and long period we find for AzV 493.
+The present-day mass of AzV 493 can be used to con-
+strain the nature of a putative compact object. Assum-
+ing a flat distribution in initial mass ratio, the average
+initial binary mass ratio q = M2/M1 ≃ 0.5 (e.g., Moe &
+Di Stefano 2017). Without any accretion during mass
+transfer, the present-day mass of AzV 493, M2 ≃ 50 M⊙,
+would suggest M1 ≃ 100 M⊙, which at SMC metallicity
+implies that the compact object should be a black hole
+(e.g., Sukhbold et al. 2016; Couch et al. 2020; Zapartas
+et al. 2021). In this case, however, the rapid rotation of
+AzV 493 would need to be primordial.
+Instead, it is more likely that mass transfer has oc-
+curred, in which case M1 is likely to be quite different,
+depending on the mass transfer efficiency. A mass trans-
+fer phase during the donor’s main sequence (Case A) is
+expected to be slower and more conservative, possibly
+causing significant mass growth of the accretor with-
+out extreme chemical pollution. This scenario has been
+invoked to explain the formation of low-mass compact
+objects in very young regions (Belczynski et al. 2008),
+and in particular, the origin of very massive companions
+(van der Meij et al. 2021), such as we have for AzV 493.
+In this case, the zero-age-main-sequence (ZAMS) mass
+of M1 ∼ 30 − 40 M⊙ for the adopted q, also accounting
+for the final donor core mass.
+However, mass trans-
+fer is far more likely to occur after the donor main se-
+
+1.2
+1.0
+0.8
+[y]
+O
+EW
+0.6
+0.4
+:
+0.2
+0.0
+14.0
+14.2
+14.4
+14.6
+14.8
+I[mag]
+1.2
+1.0
+0.8
+EW
+0.6
+0.4
+8
+0.2
+0.0
+14.35
+14.40
+14.45
+14.50
+14.55
+14.60
+14.65
+V [mag]13
+quence (Case B), due to the star’s expansion (e.g., van
+den Heuvel 1969). It then takes place rapidly, on the
+thermal or He core-burning nuclear timescale (Klencki
+et al. 2022), and system mass loss is far more likely,
+implying a higher ZAMS mass for M1.
+Although post-SN outcomes are stochastic, black hole
+production is expected to dominate for Z⊙ progenitors
+with initial masses ≳ 20 M⊙. This nominal threshold
+ZAMS mass is expected to decrease for lower metallicity
+(e.g., Zhang et al. 2008; O’Connor & Ott 2011; Sukhbold
+et al. 2016), which in principle enhances the likelihood
+that the compact object should be a black hole. The
+high eccentricity in AzV 493 strongly suggests that a SN
+occurred. While this implies that the companion is more
+likely to be a neutron star, black holes can form from
+fall-back if the SN is insufficient to unbind the ejecta,
+which is more likely to happen at low metallicity (e.g.,
+Zhang et al. 2008). There are multiple mechanisms to
+produce core-collapse black holes, and if mass-loss oc-
+curs, a SN and/or kick to the system may result (e.g.,
+Janka 2013).
+We note that M1 ∼ 20 − 40 M⊙ is a
+range that has been extensively simulated and where
+explodability and fallback are uncertain (e.g., O’Connor
+& Ott 2011; Sukhbold et al. 2016; Janka 2013; Zhang
+et al. 2008). Establishing that a neutron star or black
+hole resulted from this ZAMS range, with some kind
+of kick, would provide an important empirical reference
+for theoretical models of the explosion and the binary
+interactions preceding it.
+Follow-up observations at subsequent periastra could
+more firmly establish whether AzV 493 has a compan-
+ion, and whether it is a black hole vs a neutron star. A
+74.33 ksec Chandra/HRC observation on 2012 February
+12 (MJD 55969) of a field including AzV 493 (ObsID
+14054) shows no detection. Given the tiny orbital inter-
+val during which the two stars interact, no significant ac-
+cretion onto the compact object is expected, explaining
+why the system is not a known high-mass X-ray binary.
+However, well-timed X-ray observations near periastron
+may be able to catch a brief flare event.
+6.2. Radial Velocities
+We also measure the radial velocity (RV) for the ob-
+tained spectra to search for evidence of a companion.
+This is challenging, since AzV 493 is a luminous, fast-
+rotating, early-type O-star, with few photospheric fea-
+tures, several of which are often in emission. We carried
+out cross-correlations against the FASTWIND model
+spectra for the entire observed spectral range using the
+iSpec code (Blanco-Cuaresma et al. 2014), as well as de-
+terminations based on cross-correlations against PoWR
+model spectra (Hainich et al. 2019) for only the He ii
+lines (λ4200, λ4540 lines, and λ4686), which are the
+only clean features appearing in all epochs.
+The lat-
+ter are carried out with the Markov Chain Monte Carlo
+code of Becker et al. (2015), and since they yield better
+results, we adopt these RV measurements (Table 1).
+We find that the mean systemic radial velocity is
+202 ± 9 km s−1, weighted inversely by the errors. We
+caution that the quoted standard error on this value
+underestimates the uncertainty if there is true variation.
+Given the difficulty of these measurements, with median
+error on individual epochs of 46 km s−1, it is difficult to
+evaluate any variability (Figure 10). There is possible
+evidence for very short-term RV variations; however, the
+data are ambiguous.
+We compute RV models for a possible periastron sug-
+gested in Section 2.3 at MJD 57523, which is near the
+second minimum in the light curve (Figure 1). For this
+7.3-year period, and the above, nominal periastron dis-
+tance of 40R⋆, the eccentricity e ∼ 0.93 and apastron
+∼ 28 AU. For this scenario, Figure 10 demonstrates that
+the RV signature of a neutron-star companion at perias-
+tron is very brief, on the order of 0.01 in orbital phase,
+and moreover, the observational uncertainties are larger
+than the expected amplitude. This is the case even for
+e = 0.99. Thus, our existing RV measurements do not
+strongly constrain whether MJD 57523 corresponds to
+a periastron, nor the existence and properties of a com-
+panion,
+6.3. Proper Motion
+A post-SN bound system can be expected to have been
+accelerated from its original rest frame. Relative to the
+blue stars from Massey (2002) within a 5′ radius, the
+Gaia EDR3 (Gaia Collaboration et al. 2021) residual
+proper motions of AzV 493 show two potential velocity
+vectors. Figure 11 provides the velocity histograms of
+these local field stars, showing strong bimodality in the
+R.A. components. These define two possible local ve-
+locity fields implying R.A. and Dec residual velocity for
+AzV 493 of either (vα, vδ) = (53 ± 11, 3 ± 12) km s−1;
+or (vα, vδ) = (−11 ± 11, 12 ± 13) km s−1. These yield
+total projected transverse velocities of 54±11 km s−1 or
+16 ± 12 km s−1.
+Figure 12 shows a wide-field view of the surround-
+ing environment, with the two possible proper motion
+vectors superposed.
+We see that the nearest massive
+star-forming region is the N84 complex (Henize 1956)
+about 15′ − 20′ or ∼ 300 pc to the west. If the velocity
+measurements are correct, the faster, east-bound veloc-
+ity is consistent with AzV 493 originating in N84 and
+traveling for ≳ 5 Myr. The lifetime itself of a 50 M⊙
+star with v sin i ∼ 500 km s−1 is about 5 Myr (Brott
+
+14
+Figure 10. Left: Heliocentric radial velocities measured from He ii photospheric absorption vs MJD for all epochs. Epoch A
+has only one available line of low quality, and hence has a very large uncertainty. The vertical dashed lines show the possible
+periastra at MJD 54686 and 57523. Right: Zoom for the same data showing RV models for eccentricities of 0.93 (dashed lines)
+and 0.99 (solid lines), assuming a periastron occurs at MJD 57523; and for inclination angles of 90◦ (black) and 45◦ (blue), for
+the 50 M⊙ primary and assuming a 3 M⊙ secondary. If a periastron is closer to the light curve minimum at MJD 57626, the
+models would shift to 103 days later.
+Figure 11. Distribution of Gaia proper motion velocities in R.A. (left) and Dec (right) for stars from Massey (2002) within 5′
+of AzV 493. The bimodal R.A. distribution defines two kinematic groups. The first group has 13 stars with median velocity
+(vα, vδ) = (254 ± 7, −378 ± 9) km s−1 and the second has 10 stars with (vα, vδ) = (318 ± 6, −386 ± 11) km s−1. The one star
+between the two groups in vα is included in both. The median velocities for these groups are shown with the vertical green and
+blue lines, together with the velocity of AzV 493 (red).
+et al. 2011), and for a SN ejection, its travel time would
+only be the post-SN lifetime. However, since the star
+presumably acquired its total mass and spin later in
+life, the system may have been ejected earlier by dy-
+namical processes as a tight, non-compact binary. If so,
+it would have been reaccelerated by the SN explosion,
+therefore implying that it may be a two-step ejection
+(Pflamm-Altenburg & Kroupa 2010). Supernova accel-
+erations are typically weaker than dynamical ejections
+(e.g., Renzo et al. 2019), and so the dominant velocity
+component could still be due to a dynamical ejection
+from N84. A dynamically active past in a dense stellar
+environment of N84 may also help to explain the eccen-
+tricity (e.g., Sim´on-D´ıaz et al. 2015), although it would
+seem unlikely that the system could maintain its highly
+eccentric configuration for 5 Myr. On the other hand,
+we note that the inferred runaway velocity, orbital ec-
+centricity, and period are still consistent with being due
+
+AzV 493
+350
+ (km/sec)
+300
+E
+250
+-
+B
+D
+200
+--
+K
+RV
+A
+Heliocentric
+150
+100
+-
+50
+0
+50
+--
+55000
+56000
+57000
+58000
+59000
+60000
+Date (MJD)Example RV Curves
+260
+e=0.99
+e=0.94
+240
+Primary RV (km/s)
+220
+200
+180
+160
+140
+-200
+0
+200
+400
+600
+Days around Periastron = MJD 5752377616 field v RA plot
+77616 field v DEC plot
+6
+8
+7
+5
+6
+4
+(stars)
+3
+4
+N
+N
+2
+2
+1
+1
+0
+0
+200
+250
+300
+350
+-450
+-400
+-350
+-300
+Velocity (km/s)
+Velocity (km/s)15
+Figure 12. Location of AzV 493 in the SMC field, with the green and blue proper motion vectors corresponding to the two
+field velocities indicated with the same color coding in Figure 11, superposed on an Hα images from Smith et al. (2005). The
+nearest massive star-forming region is the N84 complex (Henize 1956), indicated. For the adopted SMC distance, 10′ = 181 pc.
+solely to SN acceleration (e.g., Brandt & Podsiadlowski
+1995). Thus, in order to explain both the long travel
+time and high eccentricity, the most plausible scenario
+may be the two-step ejection.
+There is also a small possibility that the slow, alterna-
+tive proper motion vector (Figure 12) is correct. How-
+ever, this would mean that the AzV 493 system formed
+in isolation since there is no corresponding young clus-
+ter whence it could have originated (Figure 11). Vargas-
+Salazar et al. (2020) find that < 5% of OB stars, if any,
+formed in the field, and this is especially unlikely for
+AzV 493, given its high mass.
+We caution that the ve-
+locity errors do not include unknown systematic errors,
+and so these measurements need to be confirmed. Thus,
+although AzV 493 indeed appears to be a runaway star,
+this does not provide especially useful information to
+constrain its binary interaction history.
+6.4. Similar systems
+A comprehensive study by Marr et al. (2022) shows
+that the B8 Vpe star Pleione (HD 23862) has a light
+curve with a similar long-term pattern of slow growth
+with sudden drops, and similar variations in the Balmer
+emission-line profiles. It is a triple system with a close
+companion on a 218-day orbit (Katahira et al. 1996;
+Nemravov´a et al. 2010). Marr et al. (2022) suggest that
+the photometric drops correspond to the decretion disk
+tearing into two components, where one remains aligned
+with the star’s equatorial plane and the other is mis-
+aligned due to tidal torque from the close companion.
+Pleione’s long-term photometric cycle is 34 years, simi-
+lar in magnitude to that of AzV 493. Nemravov´a et al.
+(2010) find that the close companion is on an eccentric
+orbit with e > 0.7.
+AzV 493’s initial peak brightness and subsequent drop
+in 2001 (Figure 1) qualitatively resemble the photomet-
+ric pattern characteristic of heartbeat stars. These are a
+rare class of interacting binary systems with high eccen-
+tricities such that the periastron passage tidally induces
+regular photometric outbursts. However, the observed
+pattern in AzV 493 cannot be induced by this type of
+tidal interaction; preliminary simulations using new ca-
+pabilities in the GYRE stellar oscillation code (Sun et al.
+2023) suggest that the combined amplitude and width of
+the periastron pulse cannot be reproduced by eccentric
+tidal models. Nevertheless, given that AzV 493 seems
+likely to be a massive eccentric binary system, massive
+heartbeat stars thus share some similarities with this
+
+53.5±10.9 km/s
+73°00'00"
+16.0±12.1 km/s
+10'00"
+Dec (J2000)
+N84
+20'00"
+30'00"
+20°00'00"
+19°00'00"
+18°00'00"
+RA (12000)16
+object if a companion indeed interacts with the primary
+and/or its disk.
+Examples include the non-Be binary
+system ι Ori (O9 III + B1 III/IV), which has orbital
+period 29 d and eccentricity e = 0.764, as determined by
+Pablo et al. (2017). They find that the two components
+have masses of 23.2 and 13.4 M⊙, respectively, generat-
+ing tidally excited oscillations with periods on the order
+of ∼ 1 day. MACHO 80.7443.1718 is another heartbeat
+system with two stars of type B0 Iae and O9.5 V and
+masses of 35 and 16 M⊙, respectively (Jayasinghe et al.
+2021).
+The B0.5 Ve star δ Sco is has a B2 V star companion
+in an eccentric (e = 0.94) orbit with period 10.7 years
+(e.g., Tango et al. 2009; Tycner et al. 2011). The two
+components have masses of 13.9 M⊙ and 6 M⊙ (Che
+et al. 2012). This system shows a long-term photomet-
+ric cycle somewhat similar to that of AzV 493, although
+much more poorly defined. There is no obvious link be-
+tween the disk properties and binary interaction (Suffak
+et al. 2020; Che et al. 2012), but the long-term pho-
+tometry has a timescale similar to that of the orbital
+period.
+H 1145–619 is a Be X-ray binary whose primary is a
+B0.2e III star estimated to be 18.5 M⊙ (Alfonso-Garz´on
+et al. 2017), and the secondary is an X-ray pulsar. As
+shown by Alfonso-Garz´on et al. (2017), H 1145–619 has
+a light curve with a ∼ 10-year cycle together with un-
+explained multiple modes of much shorter periods (∼ 1
+year), qualitatively similar to what we see for AzV 493,
+which has a long cycle of 14.6 (7.3) years and short oscil-
+lations of ∼ 40 days. While it remains unclear whether
+the light curves of H 1145–619 and AzV 493 have fun-
+damental similarities, both stars are massive OBe stars.
+If they are related, the fact that H 1145–619 has a con-
+firmed compact binary companion may suggest that the
+unusual variability of AzV 493 may have a similar origin.
+These objects provide a context for AzV 493 that sup-
+ports this object being a member of this broad class of
+binary, massive OBe systems with high eccentricities.
+At 50 M⊙, AzV 493 is more massive than any of these
+similar objects.
+It is also one of the earliest O stars
+in the SMC, since there is no photospheric He i. Thus,
+AzV 493 may be the most extreme such object known,
+in terms of its mass and effective temperature. Its vari-
+ability amplitudes are also among the largest known.
+We note that, based on only the Epoch A spectrum
+(Figure 5), Golden-Marx et al. (2016) suggested that
+AzV 493 is a normal, but extremely early, classical Oe
+star. Given the strong spectroscopic and photometric
+variability, the nature of this spectrum may be some-
+what different than inferred in that work, and the origin
+of the strong line emission seen in this particular spec-
+trum is unclear (Section 3.3). Still, its status as a post-
+SN binary where the observed star was likely spun up
+by mass transfer from the compact object progenitor, is
+consistent with the origin of classical OBe stars. Indeed,
+given that most of the massive OBe stars are post-SN
+systems (e.g., Dallas & Oey 2022; Dorigo Jones et al.
+2020), we can expect that more of them are likely to be
+high-eccentricity, compact-object binaries.
+6.5. Alternative Companion Scenarios
+We now consider alternative scenarios for a putative
+binary component. First, such a companion might be
+an unexploded former donor in an interacting binary. In
+this case, it could be a stripped star (e.g., Schootemeijer
+et al. 2018; G¨otberg et al. 2017), which can be elusive to
+detect. Wang et al. (2021) identified hot, stripped star
+companions to Be stars based on FUV spectral cross-
+correlations; however, the extremely hot temperature of
+AzV 493, which is commensurate with the hottest O
+stars, poses a serious challenge for this method. If the
+observed star has previously experienced accretion from
+binary mass transfer, then its surface might be He- and
+N-enriched (e.g., Blaauw 1993; Renzo & G¨otberg 2021),
+although whether this occurs depends on the accretion
+efficiency and mixing processes in the accretor’s enve-
+lope. Since early O stars have few metal lines, it is again
+difficult to evaluate any enrichment, especially in a fast
+rotator like AzV 493. There is no immediate evidence
+for any unusual abundances in this star. Moreover, a
+non-degenerate companion does not naturally explain
+the high observed eccentricity, which would then have
+to be primordial, avoiding tidal dissipation, or of dy-
+namical origin.
+Alternatively, the high rotation rate and variability of
+AzV 493 might be caused by a non-standard internal
+structure of the star because of a merger.
+These are
+common among massive stars, occurring in 22+26
+−9 % of
+isolated massive binaries (Renzo et al. 2019), with an
+even higher rate if accounting for the presence of further
+companions (e.g. Toonen et al. 2020). For example, η
+Car has been suggested to originate from a merger in
+a hierarchical triple system, resulting in a present-day
+eccentric binary (e.g., Hirai et al. 2021). However, η Car
+is a luminous blue variable star and has other substantial
+differences from AzV 493.
+Yet another possibility is that AzV 493 might be a
+triple system with a third, also invisible, star on a
+shorter-period orbit.
+This speculative scenario might
+help to explain how the strong, 40-day pulsations are
+maintained (Section 2.2). It also might help explain the
+apparently sporadic ejection and accretion events seen
+in Epochs B and F (Section 3.4). Such a system would
+
+17
+be unstable, but the brief interaction phase with the sec-
+ondary may enhance its longevity. We note that the sys-
+tem is unlikely to be a triple in which the third star has
+an even larger orbit than the secondary. Although high
+orbital eccentricities can be produced by Kozai-Lidov
+cycles in such a system, this high-eccentricity phase of
+the cycle is short in duration. Thus, such extreme eccen-
+tricity may require a triple or higher-order multiple-star
+interaction in the system’s birth cluster, and may be
+linked to a dynamical ejection of AzV 493 into the field.
+Overall, however, it is challenging to explain AzV 493 in
+terms of a triple-star scenario. Unfortunately, RV mon-
+itoring is complicated due to the technical difficulty and
+possible presence of varying stellar pulsations, so it will
+be hard to evaluate whether the system consists of more
+than two stars.
+7. SUMMARY
+We present 18 years of OGLE Project photometric
+data and spectroscopic data over 12 years, revealing the
+remarkable variability of AzV 493. This is perhaps the
+earliest known classical Oe star, with Teff = 42000 K,
+log L/L⊙ = 5.83 ± 0.15, and R⋆/R⊙ = 15 ± 3. These
+parameters imply a mass of 50 ± 9 M⊙.
+The domi-
+nant photometric pattern is reproduced after 14.6 years.
+There are also large, semi-regular ∼ 40-day pulsations of
+unknown origin, as well as other structure in the light
+curve. It is not a known HMXB. The observed v sin i
+= 370± 40 km s−1, with a high inferred sin i, suggesting
+a rotational velocity of 400 − 450 km s−1. The system
+is ∼ 300 pc from the nearest massive star-forming com-
+plex and its proper motion shows that it is likely a run-
+away star from that region, with a transverse velocity
+of 54 ± 11 km s−1, possibly having experienced two-step
+acceleration.
+Altogether, the data suggest that this object is likely
+an eccentric, interacting binary system with an unde-
+tected compact companion.
+If so, the orbital period
+could correspond to the 14.6 (7.3)-year period, imply-
+ing a high eccentricity of at least e ∼ 0.95 (0.93) and
+apastron ∼ 43 (28) AU. If this binary scenario is cor-
+rect, AzV 493 would be among the most extreme sys-
+tems known, in terms of its early spectral type, high
+mass, and extreme eccentricity.
+In our favored model, an optically thick decretion disk
+is regularly ejected, likely by a periastron encounter. A
+two-component disk system forms, with the outer re-
+gion responsible for the 0.85-magnitude drop in I-band
+flux, while the inner disk is the origin of most of the
+observed emission-line spectrum. The spectra appear to
+show varying relative contributions from the inner and
+outer regions, consistent with the optically thick outer
+region dissipating over the cycle. The outer region may
+correspond to a flared disk, torus, or possibly, a separate
+inclined annulus formed by tearing from the inner disk.
+We see direct spectroscopic evidence for episodes of both
+matter ejection and infalling reabsorption of dense disk
+material onto the star. The lack of exact regularity of
+photometric and spectroscopic variations in the cycle
+implies that the geometry and/or mechanics of the disk
+ejection may vary. An alternative, opposite model seen
+in some Be stars, in which the brightness increases due
+to contribution from growing disk emission (e.g., Suf-
+fak et al. 2020; Ghoreyshi et al. 2018), should also be
+considered.
+If AzV 493 indeed has a highly eccentric orbit, it would
+suggest that the system experienced a strong SN kick,
+implying that the unseen companion is a neutron star
+or black hole. The high v sin i also suggests that mass
+transfer occurred before this event.
+For conservative,
+Case A mass transfer, the progenitor donor’s ZAMS
+mass would be 30 − 40 M⊙ for a typical q ∼ 0.5, and
+larger for non-conservative Case B mass transfer. This
+mass range is well within that suggested by models to
+produce black holes, although the occurrence of strong
+natal kicks in cases of black hole formation is less clear.
+Alternatively, the donor could be a stripped star; how-
+ever, this scenario cannot explain the extreme eccentric-
+ity, which would have to be dynamical or primordial.
+The system could also be a merger, but the eruptions
+and long-term pulsations seem less consistent with this
+scenario.
+AzV 493 could possibly be a triple system,
+which might explain how the strong photometric oscil-
+lations are maintained (Section 6.5).
+Establishing the existence and nature of the unseen
+companion(s) can provide important constraints on bi-
+nary evolution, core explodability, and the origin of
+compact binaries. AzV 493 may offer an opportunity
+to directly observe the relationship between the binary
+companion’s dynamical interaction and the disk ejec-
+tion. Since many classical OBe stars are massive, post-
+SN objects, it suggests a likely link between OBe stars
+and massive, eccentric systems. Further study of this
+fascinating object can more definitively confirm its sta-
+tus and exploit the opportunities it offers to learn about
+massive binary evolution and disk ejection.
+
+18
+ACKNOWLEDGMENTS
+We benefited from useful discussions with many peo-
+ple, including Jon Bjorkman, Paul Crowther, Julian
+Deman, Jim Fuller, Jay Gallagher, Carol Jones, Max
+Moe, Megan Reiter, Steve Shore, and Drew Weisser-
+man.
+Many thanks to Juliette Becker for the use of
+her code, and to Traci Johnson, Mario Mateo, and the
+M2FS Team for help with observing runs.
+We also
+thank the anonymous referees for valuable comments
+that greatly improved this paper. This work was sup-
+ported by NSF grant AST-1514838 to M.S.O. and by the
+University of Michigan. N. Castro acknowledges funding
+from the Deutsche Forschungsgemeinschaft (DFG), CA
+2551/1-1; M.R. is supported by EUH2020 OPTICON
+RadioNet Pilot grant No.
+101004719; and R.H.D.T.
+is supported by NASA grant 80NSSC20K0515.
+This
+research made use of Astropy, a community-developed
+core Python package for Astronomy (Astropy Collabo-
+ration et al. 2013). M.S.O. acknowledges MDRS, LLC,
+for pandemic hospitality.
+Facilities: Magellan, OGLE, Gaia
+
+19
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+APPENDIX
+A. GENERALIZED LOMB-SCARGLE PERIODOGRAMS
+Figure 13 shows the individual generalized Lomb-Scargle periodograms (Zechmeister & K¨urster 2009) and ancillary
+information for the six, roughly contiguous, OGLE datasets during ∼ 2010 – 2016 (Section 2.2).
+
+22
+Figure 13. Top panels show the generalized Lomb-Scargle periodogram for light curves shown in the middle-left panels. The
+fitted light curves are shown in the middle-right panels, with each cycle superposed according to color from the middle-left
+panel. Residuals are shown in the bottom panels, as a function of MJD and phase, as shown. The middle and bottom panels
+have the same x-axes. The fitted period is shown in the top panel as the inverse of the frequency f. The observation time of
+spectroscopic epoch B is shown by the vertical dashed line in the plots for the fifth dataset.
+
+Period [day]
+100
+20
+(ZK)
+1/f=37.3±0.2[day]
+Power
+0.5
+0.0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+Frequency f[day-1]
+A
+14.4
+14.6
+Residuals
+0.1
+0.0
+55350
+55400
+55450
+55500
+55550
+0
+5
+10
+15
+20
+25
+30
+35
+MJD [day]
+PhasePeriod [day]
+100
+20
+(ZK)
+1/f=31.8±0.1[day]
+Power
+0.5
+0.0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+Frequency f[day-1]
+14.4
+[mag]
+14.6
+0.05
+Residuals
+08
+00
+0.00
+.
+0.05
+:
+55700
+55750
+5580055850
+55900
+55950
+0
+5
+10
+15
+20
+25
+30
+MJD [day]
+Phase23
+Figure 13. (Continued)
+
+Period [day]
+100
+20
+(ZK)
+1/f=30.8±0.3[day]
+0.5
+Power
+0.0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+Frequency f [day-1]
+AA
+ 14.5
+ma
+:
+14.6
+14.7
+Residuals
+0.05
+.
+:
+.
+..
+0.00
+.
+.
+.
+.
+0.05
+:
+56100
+56150
+56200
+56250
+56300
+0
+5
+10
+15
+20
+25
+30
+MJD [day]
+PhasePeriod [day]
+100
+20
+(ZK)
+1/f = 34.5±1.2 [day]
+0.2
+Power
+0.0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+Frequency f[day-1]
+14.4
+14.5
+Residuals
+0.1
+0.0
+0.1
+56500
+56550
+56600
+56650
+0
+5
+10
+15
+20
+25
+30
+MJD [day]
+Phase24
+Figure 13. (Continued)
+
+Period [day]
+100
+20
+10
+(ZK)
+0.5
+1/f=43.6±0.8[day]
+Power
+0.0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+0.10
+Frequency f[day-1]
+iB
+[ma
+6
+14.6
+0.1
+Residuals
+159
+!
+0.0
+b60
+iB
+0.1
+iB
+56800
+56850
+56900
+56950
+57000
+57050
+10
+0
+20
+30
+40
+MJD [day]
+PhasePeriod [day]
+100
+20
+(ZK)
+0.4
+1/f= 42.3±1.2[day]
+Power
+0.2
+0.0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+Frequency f[day-1]
+[ma
+14.6
+Residuals
+0.1
+"
+0.0
+.
+000
+:
+U.
+57200
+57250
+57300
+57350
+57400
+0
+10
+20
+30
+40
+MJD [day]
+Phase
\ No newline at end of file
diff --git a/FtFJT4oBgHgl3EQfDSx3/content/tmp_files/load_file.txt b/FtFJT4oBgHgl3EQfDSx3/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..87a0100cee4ceeecb2f0d0f29731335dde234a9d
--- /dev/null
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Townsend10 1Astronomy Department, University of Michigan, 1085 South University Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Ann Arbor, MI, 48109, USA 2Leibniz-Institut f¨ur Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482, Potsdam, Germany 3Center for Computational Astrophysics, Flatiron Institute, 162 5th Ave, New York, NY 10010, USA 4Department of Physics and Astronomy, Western University, London, ON N6A 3K7, Canada 5Astronomical Observatory, University of Warsaw, Al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Ujazdowskie 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 00-478 Warszawa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Poland 6Department of Astronomy & Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' University of Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 5640 S Ellis Avenue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' IL 60637,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' USA 7Kavli Institute for Cosmological Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' University of Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' IL 60637,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' USA 8Present address: Leibniz-Institut f¨ur Astrophysik Potsdam (AIP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' An der Sternwarte 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 14482,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Potsdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Germany 9Present address: Astronomy Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' University of Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Box 351580,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Seattle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' WA 98195,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' USA 10Astronomy Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' University of Wisconsin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Madison,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' WI 53706,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' USA (Accepted January 23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' to appear in the Astrophysical Journal) ABSTRACT We present 18 years of OGLE photometry together with spectra obtained over 12 years, revealing that the early Oe star AzV 493 shows strong photometric (∆I < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='2 mag) and spectroscopic variability with a dominant, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='6-year pattern and ∼40-day oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' We estimate stellar parameters Teff = 42000 K, log L/L⊙ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='15, M/M⊙ = 50 ± 9, and v sin i = 370 ± 40 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Direct spectroscopic evidence shows episodes of both gas ejection and infall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' There is no X-ray detection, and it is likely a runaway star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' AzV 493 may have an unseen companion on a highly eccentric (e > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='93) orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' We propose that close interaction at periastron excites ejection of the decretion disk, whose variable emission-line spectrum suggests separate inner and outer components, with an optically thick outer component obscuring both the stellar photosphere and the emission-line spectrum of the inner disk at early phases in the photometric cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' It is plausible that AzV 493’s mass and rotation have been enhanced by binary interaction followed by the core-collapse supernova explosion of the companion, which now could be either a black hole or neutron star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' This system in the Small Magellanic Cloud can potentially shed light on OBe decretion disk formation and evolution, massive binary evolution, and compact binary progenitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Keywords: early-type stars — Oe stars — Be stars — high-mass X-ray binary stars — circumstellar disks — stellar pulsations — interacting binary stars — compact objects — runaway stars — variable stars — Small Magellanic Cloud 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' INTRODUCTION Binary interactions are now understood to be a funda- mental component of massive star evolution, and they are the progenitors of a wide variety of energetic phe- nomena including high-mass X-ray binaries (HMXBs), ultra-luminous X-ray sources (ULXs), stripped-envelope core-collapse supernovae (SNe), and gravitational wave events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' A consensus is emerging that classical OBe stars appear to originate from close massive binary systems, wherein they have spun up through mass and angular momentum transfer from their mass donors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g, Pols et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Vinciguerra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Bodensteiner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2020, see also Rivinius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2013 for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' When donor stars subsequently explode as supernovae, result- ing post-explosion bound binaries are more likely to be eccentric, since they result from tight binaries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Brandt & Podsiadlowski 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Tauris & Takens 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Renzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Thus, a substantial subset of classi- cal OBe stars are likely to have eccentric orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' In this paper, we present photometric and spectrocopic time- series data showing that the star AzV 493 exhibits dra- matic variability and may be an eccentric binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' AzV 493 (Azzopardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 1975) or [M2002]SMC- 77616 (Massey 2002) was identified as an extreme, clas- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='11433v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='SR] 26 Jan 2023 2 sical Oe star by Golden-Marx et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' In that work, it was found to be the earliest classical Oe star in our sample of field OB stars in the Small Magel- lanic Cloud (SMC), based on a spectrum obtained in 2009 that shows double-peaked emission, not only in the Balmer lines, but also in He i and He ii λ4686, the latter feature being rarely observed in other Oe stars (Conti & Leep 1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Specifically, it is classified as an Ope star, indicating that the He i absorption lines show infilled emission (Sota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' As an extreme object, AzV 493 offers unique oppor- tunities to study massive binary evolution and decre- tion disk formation, structure, and dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Section 2 presents the unusual light curve and periodicity, and Section 3 presents our multi-epoch spectroscopy with resulting derived stellar parameters and individual spec- tral features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' We then present two possible models for the AzV 493 system in Sections 4 and 5, one based on ejection of an optically thick disk near periastron;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' and another based on disk growth and disruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Section 6 discusses the likely binary origin of the system, and Sec- tion 7 summarizes our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' PHOTOMETRIC LIGHT CURVE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Long-term light curve The I and V -band light curves of AzV 493 from the OGLE Project (Udalski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2008, 2015) are presented in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The I-band shows a short eruption with the peak of the light curve on MJD 52212, followed by an abrupt decline of approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='2 mag, to a minimum on MJD 52303 in early 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' After this, the star eventually recovers its original luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Another photometric minimum is seen in 2016 on MJD 57626, followed by the same brightening pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The gray symbols in Figure 1 show the I-band photometry from the 2016 cycle overplotted on the data from 2002 cy- cle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' This shows that the minimum luminosity and subse- quent increase are quantitatively identical, although the photometry immediately preceding the minimum differs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Cross-correlating these segments yields a long-cycle pe- riod of 5311 days (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='55 years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' There is no evidence of a similar eruption preceding the minimum in the 2016 cycle on the same 91-day timescale, although the pho- tometry is incomplete in this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' After the minimum, the brightness increases and then starts to gradually decrease again, over a period of sev- eral years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Approximately in 2008, AzV 493 appears to go into a multiple outburst event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' After this, the light curve drastically changes, showing a multi-mode pulsa- tion behavior that evolves with time (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The pulsation ends with another 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='2 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='3 mag drop, followed by a steady increase, repeating the light curve cycle that started in 2002, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='55 years before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Photometric Oscillations Figure 2 shows short-term variability on the order of 30 – 45 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' We quantify the evolution of these oscil- lations seen in the I-band light curve using Generalized Lomb-Scargle periodograms (Zechmeister & K¨urster 2009) for the six contiguous OGLE datasets from 2010 – 2016 (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The individual fits to these six ranges are shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Comparison of the periods shown in Figure 3 with the light curve (Figure 1) shows that they qualitatively appear to correlate with stellar brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The OGLE survey provides V -band magnitudes for a subset of the survey epochs, which are shown in red in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Figure 4 displays the color-magnitude dia- gram (CMD) in V vs V − I for those days where both bands were observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Figure 4a compares AzV 493’s color variations with data for the remainder of the RI- OTS4 sample stars (Lamb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The latter corre- spond to single-epoch photometry from the OGLE cat- alog of Poleski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Those stars classified as OBe stars by Lamb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' (2016) are marked in the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The blue plume of non-emission-line stars is clearly sepa- rated from the cloud of OBe stars at redder colors in the CMD, a phenomenon already known from different pho- tometric bands (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Bonanos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The color variation of AzV 493 spans almost the entire range of V − I colors covered by the emission-line stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Figure 4b shows a zoom in the CMD with the path of AzV 493 traced out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The star appears red during the broad peak of the light curve around 2006 (Figure 1), and then moves to bluer colors reaching the bluest V −I color during the pulsation phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Approximately in 2017, when the light curve is brightening after the mini- mum, AzV 493 shows redder colors again, moving to the original position observed in 2005 with V − I ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Similar, semi-periodic variability with timescales on the order of weeks to months is seen in many other OBe stars, and their origin is unknown (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Labadie-Bartz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Proposed explanations include forms of non-radial pulsations of the star and transitory or orbit- ing density enhancements in the disk, which may be the most likely scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The associated cyclical variation in the CMD (Figure 4) is also consistent with some kind of stellar radial pulsation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' This is supported by the corre- lation between period and luminosity (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Figures 3 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' In that case, the relatively long period implies that they could be an induced gravity mode or pulsational instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' However, there are many other possible ex- 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' AzV 493 OGLE light curves in I (black) and V (red) bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The last segment of the I-band curve is overplotted (light grey dots) on the beginning of the dataset phase 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='6 years (5311 days) earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' V − I is shown in the lower panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The dashed lines mark the epochs for the observed spectra, assigned alphabetically in chronological sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The green shaded regions show consecutive 2656-day segments starting with the light curve maximum in 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Zoom on light curve (top) showing ∼ 40-day oscillations, and color variation (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Time [year] 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020 2022 D H 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='65 E F 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='80 B C G 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='95 @ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='10 nitude 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='25 Magr 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='40 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='55 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='70- I band 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='85 V band I band shifted -14 years C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='1 io 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='1 - 53000 54000 55000 56000 57000 58000 59000 MJD [days]Time [year] 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='4 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='6 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='8 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='0 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='2 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='4 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='6 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='8 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='0 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='2 I band 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='24- V band 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='32 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 3 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='40 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' :: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' : i 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='64 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='72 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='050 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='075 C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='100 55300 55400 55500 55600 55700 55800 55900 56000 MjD[days]4 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Fitted periods for the six contiguous OGLE datasets between ∼ 2010 – 2016, as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' planations, perhaps including interactions with another star in a close orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' We note that de Wit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' (2006, see also Rivinius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2013) reported similar loop-like excursions in the CMD of other OBe stars, and ascribed the anti-clockwise variation to the formation and dissi- pation of the circumstellar decretion disks in those ob- jects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Light curve period It is possible that the multiple-outburst event in 2008 – 2009 may represent another periastron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Figure 1 shows the 5311-day cycle initiated at the light-curve peak at MJD 52212 instead of at the minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' We see that the mid-cycle occurs during this multiple-outburst event, al- though due to the OGLE observing cadence, it is unclear whether it occurs near the end or near the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' In Section 3 below, we show that the spectrum obtained around this time, Epoch A (Figure 1), shows an un- usually strong emission-line spectrum, consistent with maximum disk activation and flaring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' However, the light curve does not repeat the cycle minimum seen in 2002 and 2016, and OBe stars are known to show temporary outbursts of activity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Labadie-Bartz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Baade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Thus, it is not clear whether 2008 – 2009 corresponds to the mid-cycle or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The light curve does not repeat regularly in detail, and we caution that the period, if the system is a binary, is uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Assuming that there is indeed a fundamental physical period, the same phases may not all generate the same observational signatures, which may depend on other factors such as disk orienta- tion and/or varying physical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' In what follows, we adopt a system period of 5311 (2656) days, or 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='55 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='28) years, where the values in parentheses allow for the possibility that the period may be half of the long cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' SPECTROSCOPY Spectroscopic observations of AzV 493 were obtained in the course of the RIOTS4 spectroscopic survey of field OB stars in the SMC (Lamb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2016), and follow-up radial velocity monitoring of the SMC Wing region (Vargas-Salazar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2023, in preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The observations were carried out using the Magellan tele- scopes at Las Campanas, Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Three different spectro- graphs were used: IMACS (Bigelow & Dressler 2003), MIKE (Bernstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2003) and M2FS (Mateo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Table 1 gives details of our spectroscopic observa- tions, including the modified Julian day (MJD), signal- to-noise, spectral resolution, spectral range, phase in the light curve cycle, radial velocity, Hβ peak separation (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='2), and instrument used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Figure 5 displays the 11 spectra in chronological sequence, labeled A – K as shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' IMACS was operated by default in multi-slit mode with the f/4 camera and 1200 lines/mm grating, which provides a resolving power of R ∼ 3000 and a wave- length coverage spanning ∼3800 – 5200 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' One obser- vation (Epoch I) was observed with the f/2 camera, re- sulting in lower resolution (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The reduction was performed using the cosmos pipeline1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' MIKE data were obtained using a 1′′ slit width for a spectral resolution of R ∼ 28000, covering the wavelength range ∼3600 – 10000 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The spectra were processed with the the Carnegie Python (CarPy2) pipeline software (Kelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Kelson 2003), except for Epoch B, which was extracted using IRAF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' M2FS data were observed us- ing a custom filter yielding ∼4080 – 4470 ˚A wavelength coverage at R ∼ 28000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The data were processed follow- ing the standard steps in fiber spectroscopic reduction using IRAF/PyRAF tasks implemented within python and designed for this instrument (see Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Figure 5 shows strong variability in the spectrum of AzV 493.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The weaker epochs show a typical OBe spec- trum, with only Hβ showing double-peaked emission, and Hγ and Hδ absorption features showing evidence of infill;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' whereas Epochs A, B, and K show stronger emission-line spectra, with Hγ and He i often in emis- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Epoch F shows strong, high-order Balmer emis- sion and inverse P-Cygni features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' These epochs will be discussed in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='3 – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 1 http://code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='carnegiescience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='edu/cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2 http://code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='carnegiescience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='edu/mike 3 IRAF was distributed by the National Optical Astronomy Obser- vatory, which was managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 44 42 40 [days] 38 Period 36 34 32 30 55500 55750 56000 56250 56500 56750 57000 57250 MJD [days]5 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Color-magnitude diagram (CMD) based on available V - and I-band OGLE photometry (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The variation of AzV 493 in the CMD is colored according to the MJD, and compared to single-epoch OGLE photometry (Poleski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2012) for the RIOTS4 OB-star sample (Lamb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2016) (grey dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Objects classified as OBe by Lamb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' (2016) are highlighted with circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The right panel is a zoom of the same data around the track of AzV 493.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Spectroscopic Observations of AzV 493 Epoch Date [UTC] MJD S/N R Wavelength Phasea RV ∆v(Hβ)b Instrument Range [˚A] (km s−1) (km s−1) A 2009-08-26T01:43:36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='0 55069.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='071944 140 3000 3825–5422 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='538 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='076) 152 ± 200 279 IMACS B 2015-01-14T02:12:03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='0 57036.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='091701 120 28000 3362–9397 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='908 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='817) 192 ± 18 (213)c MIKE C 2016-06-15T07:47:54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='3 57554.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='324935 130 3000 3879–5479 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='006 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='012) 171 ± 60 346 IMACS D 2016-09-08T01:42:08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='0 57639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='070926 60 28000 4079–4466 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='022 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='044) 217 ± 50 · · M2FS E 2016-09-11T02:49:33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='0 57642.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='117743 90 28000 4080–4465 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='022 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='045) 239 ± 46 · · M2FS F 2016-09-22T05:36:51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='0 57653.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='233924 150 28000 3538–9397 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='024 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='049) 192 ± 29 334 MIKE G 2016-12-04T04:09:41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='5 57726.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='173397 110 3000 3862–5458 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='038 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='076) 243 ± 38 319 IMACS H 2017-06-05T06:35:11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='2 57909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='274435 50 3000 3871–5471 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='073 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='145) 235 ± 54 322 IMACS I 2017-06-07T08:08:18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='9 57911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='339108 130 1300 3900–8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='073 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='146) 231 ± 83 295 IMACSd J 2017-07-10T09:05:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='5 57944.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='378478 190 3000 3854–5468 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='079 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='159) 181 ± 39 303 IMACS K 2021-09-25T07:38:18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='0 59482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='318264 210 28000 3362–9397 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='369 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='738) 183 ± 17 289 MIKE aPhase relative to the light curve peak at MJD 52212 (54868), adopting a period of 5311 (2655.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='5) days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' b Hβ peak separation obtained by fitting two gaussians with fixed width of 2 ˚A (∼ 120 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' c Epoch B does not show a double-peaked profile (see Figure 7 and Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' the value for ∆v(Hβ) assumes that two components exist, as they do for other epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' dEpoch I was observed with the f/2 camera while the other IMACS observations were obtained with the f/4 camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='0 O 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='5 O O 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='0 O > 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='5 O O O 0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='0 8 O 0 O 0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='5 O 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
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+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='4 V-I14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='0 57600 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='1 57000 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='2 56400 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='3 MJD [days] 55800 > 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='4 55200 Q 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='5 00 54600 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='6 00 54000 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='7 O 53400 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='25 V-I6 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' AzV 493 multi-epoch spectroscopic observations sorted by MJD and normalized to the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Epoch I is low resolution (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' M H normalized flux G F E D C B A 1 He lI He llI He ll 4600 4000 4200 4400 4800 5000 wavelength [A]7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Stellar fundamental parameters The photospheric He ii lines λ4200, λ4541, and λ5411 lines at all epochs confirm the early O spectral type assigned by Golden-Marx et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' To improve S/N in the He ii λ4541 absorption line, we combine epochs C, G, H and J, which are all IMACS spectra obtained in 2016 – 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' We use this composite spectrum to estimate the projected rotational velocity (υ sin i) using the iacob-broad code (Sim´on-D´ıaz & Herrero 2014, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' We obtain υ sin i = 370±40 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' As discussed in Section 4, the angle of inclination i is likely high, based on the amount of obscuration from the disk, and so the rotational velocity might be ≲ 450 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The combined spectrum was modelled using the stellar atmosphere code fastwind (Santolaya-Rey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Puls et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Rivero Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2012), using the same technique and stellar grid described in Cas- tro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The cores of the Balmer lines are omitted from the fit to ameliorate contamination from disk emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Our best model yields effective tempera- ture Teff = 42000 K and surface gravity log g = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='4 dex, which reproduce the main He i and He ii lines (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Since He i photospheric features are not detected, this Teff may be a lower limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The derived temperature is consistent with an O3-5 spectral type (Martins & Pala- cios 2021), matching the early O-type classification of AzV 493 (Lamb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' However, we caution that the wings of the Balmer lines, which are the main spec- troscopic anchors for deriving the surface gravity, may be affected by the circumstellar emission, resulting in an underestimate of log g, as found for OBe stars by Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The stellar luminosity was calculated using the optical and IR photometry for AzV 493 (Massey 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Skrut- skie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2006), adopting a distance to the SMC of 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='1 kpc (Graczyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2014) and the synthetic fast- wind spectral energy distribution (SED) derived above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' We explored the extinction curves published by Fitz- patrick & Massa (2007) until the observed photometry was reproduced by the fastwind synthetic SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' We obtain a luminosity log L/L⊙ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='15 and radius R⋆/R⊙ = 15±3, in agreement with the expected values for an early O-type star of luminosity class III – V (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Martins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' We compare the position of AzV 493 in the Hertzsprung–Russell diagram with the rotat- ing evolutionary tracks by Brott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' (2011) for SMC metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Based on the Teff and L/L⊙ and their re- spective uncertainties, we estimate that the stellar mass is M/M⊙ = 50±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' If the observed luminosity is overes- timated by the inferred log g, or includes a contribution from a non-compact binary companion and/or the disk continuum, then the stellar mass may be somewhat over- estimated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' for reference, a factor of two overestimate in luminosity implies M/M⊙ ∼ 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Hβ emission-line profile Variability in the emission lines is a common charac- teristic of the Be phenomenon (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Rivinius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' One effect is the violet-to-red (V/R) variations, which are cycles that can last weeks or decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The V/R variations describe changes in the dominant peak strength for double-peaked emission lines observed in some stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' These cycles are attributed to variation in the morphology and density of the circum- stellar disks (Poeckert 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Okazaki 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Figure 7 shows Hβ profiles in the spectroscopic epochs where this line is available, and Gaussian models used to disentangle the V and R components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The two peaks are clearly resolved in all our observations of Hβ, except for Epoch B, which instead shows a P-Cygni profile (Fig- ures 5, 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Table 1 gives the peak separa- tions ∆Hβ fitted in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The V peak is usually more prominent than R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' There may be a long-timescale V/R cycle, but further spectroscopic monitoring is needed to determine whether V/R indeed oscillates, or whether there is any trend in ∆Hβ with phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Epochs A and K: Evidence of disk evolution Epoch A is observed at a phase of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='54 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='08), soon after the apparent eruption event in 2009 (Figure 1, Ta- ble 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' This spectrum shows the strongest helium line emission (Figure 5), although we have no other spec- troscopic observations within several years of this data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Only photospheric He ii is seen in absorption in this spectrum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' the H i and He i lines are all in emission or filled in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Moreover, He ii λ4686 is also in emission, which prompted Golden-Marx et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' (2016), to identify this spectrum as the hottest-known observation of the OBe phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Nebular He ii is only generated by the very hottest O stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Martins & Palacios 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' All of the emission lines in Epoch A are double peaked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Hβ and Hγ show larger peak separations than the He i and He ii emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' For a Keplerian disk, this would imply that the higher-temperature species is dominated by larger radii than the Hβ and Hγ emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Figure 5 shows that the emission is slightly redshifted relative to the photospheric Balmer absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Epoch K, observed at phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='37 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='74) (Figure 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Ta- ble 1) shows the opposite relation between ionization and disk radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Here, the He i lines have larger peak separations than Hβ, implying that the hotter species dominates at smaller radii, unlike Epoch A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' We also see that the Hβ and Hγ line profiles show high-velocity wings that are not observed at other epochs, consistent 8 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Spectroscopic analysis of the composite IMACS spectrum from epochs C, G, H and J (black;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The best fastwind (Santolaya-Rey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Puls et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Rivero Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2012) stellar atmosphere synthetic model is overplotted (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The main transitions used in the analysis are marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Hβ emission-line profiles from our spectra of AzV 493.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The best-fit photospheric model (Figure 6) is subtracted, after which the violet and red components are fitted by two Gaussian profiles having fixed widths of 2 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The figure shows the data overplotted by these summed fitted Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The resulting peak values are shown by the vertical lines, and their separations are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Epoch I has low spectral resolution and is not included in this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' with high-velocity gas at smaller orbital radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Epoch K is similar in emission-line strength to Epoch A and shows He i in emission, but He ii λ4686 is in absorption in this observation, as it is in all the other observations of this line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Epochs B and F: Gas Outflow and Infall Epoch B shows P-Cygni emission-line profiles in Hβ and Hγ (Figures 5, 7), suggesting an outflow episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' This is also the only spectrum obtained during the pe- riod where the strong pulsations dominate the flux (Fig- ure 1), and it is observed at the latest phase, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='91 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='82).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Figure 13 shows that the observation coincides with a local minimum in the light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Thus the P-Cygni fea- tures could suggest that the pulsations may be directly linked to mass ejection, since it coincides with the star reaching its smallest radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The spectrum of Epoch F is dramatically different from most of the other spectra (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' It shows strong, asymmetric Balmer and He i emission that show remarkable, inverse P-Cygni line profiles, with red- shifted absorption and blue-shifted emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Figure 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='3 He lI He lI He ll He l Hel Hel Hel 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='1 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='8- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 4000 4100 4200 4300 4400 4500 4600 4700 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
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+page_content='_ _ _ _ _ ___.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='_ _ _ _ _ __.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
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+page_content='___.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 3000 2000 1000 0 1000 Velocity (km/s) 2000 3000 Hβ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
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+page_content='95 3000 2000 1000 0 Velocity (km/s) 1000 2000 3000 Hγ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
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+page_content=' 9 7500 5000 2500 0 Velocity (km/s) 2500 5000 Hδ He I 4026 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
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+page_content='95 3000 2000 1000 0 1000 Velocity (km/s) 2000 3000 He I 4471 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Epoch F line profiles for Balmer and He i emission lines, as shown, centered at the systemic velocity obtained from the He ii absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' This Magellan/MIKE observation was obtained on 2016 September 22 (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' shows the line profiles relative to the systemic velocity of the He ii photospheric lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Such observations are usu- ally interpreted as infall of matter (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Hartmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2016), which appears to imply a re-absorption of decre- tion disk material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The free-fall velocity at the stellar surface for our adopted stellar parameters (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='1) is ∼ 800 km s−1, which is consistent with the red edge of the absorption trough seen in Hδ and He i λ4471.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The Balmer emission-line intensities do not follow the Balmer decrement and are almost uniform (Figures 6 and 8), indicating optically thick emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' This sug- gests that the infalling material is also likely dense, and thus has high emissivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Although Epochs D and E are taken only 14 and 11 days before Epoch F, respectively, Epochs D and E show most lines in absorption with no sign of these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Similarly, Epoch G is obtained only 73 days after Epoch F, and also shows primarily an absorption spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Thus, this infall episode corresponds to a short-lived event, which we fortuitously captured with this MIKE observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' In the spectra observed before and after Epoch F, the Balmer emission, which presumably origi- nates from the disk, does not seem substantially different in intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' This suggests that the reabsorbed material corresponds to a negligible fraction of the disk mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The timing of Epoch F is at a very early phase, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='024 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='05), only 27 days after the light curve minimum on 10 MJD 57626.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' There is no significant feature in the pho- tometry near the time of Epoch F, and the light curve is gradually brightening during this phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' This similarly implies that the continuum luminosity is dominated by the star and/or disk sources unrelated to the P-Cygni event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' DISK EJECTION SCENARIO The distinctive shape of the light curve seen in 2002 – 2004, and again in 2016 – 2018, showing a strong drop in brightness followed by gradual increase (Fig- ure 1), is seen in some other emission-line stars (Riv- inius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' We suggest that this may be due to the repeated ejection of an optically thick circumstel- lar decretion disk, perhaps related to interaction with a binary companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The exact reproduction of this part of the light curve across two cycles, starting with a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='2-magnitude drop in brightness, suggests a geomet- ric extinction effect caused by an optically thick disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' This event’s pattern in photometry and Hβ line profile is consistent with a disk ejection outburst, similar to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', HD 38708 (Labadie-Bartz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Assuming that an optically thick disk is indeed ex- pelled to generate the deep light-curve mimima (I ∼ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='85) in 2002 and 2016, we can estimate the geometric obscuration by considering the maximum flux following these minima, which peaks around I ∼ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The differ- ence of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='85 mag corresponds to reduction in flux by a factor of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='46, or over half, assuming that all of this difference is due to obscuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' This suggests not only a fairly high angle of inclination, but also a thick, or in particular, a geometrically flared disk, which is consis- tent with spectroscopic evidence (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' In this model, most of the emission lines originate from an inner disk region that experiences variable obscura- tion to our line of sight from a thicker outer disk or torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The weaker spectroscopic epochs in Figure 5 with the typical OBe spectrum are the most obscured, while Epochs A, B, and K are less obscured and therefore show stronger emission-line spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Epoch C is observed in 2016 at a phase of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='01), and thus near the same phase as the light curve peak in late 2001 (2009) (Fig- ure 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' However, as noted above (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='1), although the light curve repeats the disk ejection pat- tern, there is no evidence of a corresponding peak pre- ceding this sequence on the same timescale as that in 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The Epoch C Hβ profile (Figure 7) is consis- tent with the optically thick disk already having formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Epochs D and E, observed immediately after this min- imum, are similarly unremarkable, although they cover a much shorter spectral range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Since we see that a pu- tative disk ejection apparently occurred in 2016, it may be that the system has precessed such that an associ- ated photometric outburst is obscured by the ejection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The emission lines in Epoch A are dominated by higher temperature species at larger radii, whereas Epoch K shows the opposite effect (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Epoch A is consistent with very dense, optically thick disks that have extended vertical flaring, as shown in models by, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Sigut et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' (2009), where the emission, including from harder radiation, is dominated by this outer region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' In contrast, the disk geometry at Epoch K is dominated by high-density gas near the center and no flaring, thus differing significantly from Epoch A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Epoch A is observed at a phase of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='54 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='08), and Epoch K shows the system at a phase of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='37 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='74;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Table 1, Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' This suggests that the disk changes between having a large, flared outer region at Epoch A that contributes significantly to the emission, and a con- figuration where flaring is insignificant and emission is dominated by a dense central region at Epoch K, per- haps also reaccreting material onto the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The exis- tence of two different components dominated by inner and outer regions, respectively, could also be due to disk tearing, resulting in an inner disk and outer, expanding annulus with different inclinations (Suffak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Marr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The decreasing Hβ peak separations seen from Epoch C (346 km s−1) to Epoch J (303 km s−1) and to Epoch K (289 km s−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Table 1) suggest that the emission is weighted toward increasing radii over this period, which is consistent with the inner disk dissipating or forming an annular disk with an expanding inner radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' How- ever, this scenario does not explain the strong line emis- sion in Epochs A and K (Figure 5), which have the min- imum Hβ peak separations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' If the inner radius is indeed expanding, then the emitting region either must become dense, or the disk must precess to lower inclination an- gles to reveal stronger line emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The latter could also contribute to a model in which the decreasing peak separation is due to decreasing obscuration of the disk, allowing emission at larger radii to dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' This is consistent with the system’s increasing brightness over this period (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The extinction may result from the outer component, or optically thick torus or flare in the disk which either precesses or dissipates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' However, we caution that such a fast precession rate may not be feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Moreover, if the long-term photometric cycle is due to precession, the light curve should be symmetric around the minima, whereas the observed strong, sud- den drops (Figure 1) are difficult to explain with such a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 11 The outflow and inflow episodes described in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='4 apparently are not significant in mass relative to the entire disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' If the minima of the 14-year light curve indeed correspond to the bulk of disk ejection, followed by gradual disk dissipation, then the mass ejection as- sociated with the P-Cygni features in Epoch B are not likely to be a dominant source of disk material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' How- ever, we note that pulsations have been suggested to be important in replenishing the disk in other OBe systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Baade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2016, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The timing of Epoch F is 27 days after the light curve minimum on MJD 57626.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Although there are 3 other intermediate spectroscopic epochs between the putative disk ejection and Epoch F, this still takes place dur- ing what we assume is the heavily obscured phase in the light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The lack of any photometric event near the appearance of inverse P-Cygni features in epoch F suggests that the reabsorbed material is an insignificant portion of the disk material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The disk is therefore sub- stantial and can plausibly provide material that may fall back to the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' This is consistent with the optically thick conditions indicated by the Balmer decrement in Epoch F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Thus, this model is driven by repeated ejection of a flared, optically thick disk whose outer region gradually dissipates, revealing the inner, line-emitting region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' A flared disk is most clearly implied by the ionization and emission-line peak separation in Epoch A (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='3), and is also consistent with a maximum geometric ob- scuration that may be > 50% implied by this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The spectroscopic variation could also be caused by disk tearing or precession of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The decreas- ing trend in Hβ peak separations with increasing flux suggests that more light from larger radii can be seen (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Additionally, the high-amplitude, semi- regular pulsations with the ∼month-long period become visible at low extinction (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Other photometric and spectral variations may be due to contributions from the inner disk’s radial expansion, reabsorption, or evap- oration/ionization, and possible geometric distortion or warping of the disk system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' DISK GROWTH SCENARIO However, some observations seem inconsistent with a disk ejection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' For example, the system is bluest when faintest (Figure 1), contrary to expectations for reddening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' As noted above, the strong emission-line spectra at Epochs A and K seem inconsistent with a dissipating inner disk scenario implied by the trend in ∆Hβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' If the long-period cycle is attributed to disk pre- cession, it would require an additional mechanism to explain the assymmetric light curve, and also a third, external massive star that is not seen, to torque the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Thus, alternative models for the AzV 493 system should also be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Some other Be stars such as δ Sco (Suffak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2020) and ω CMa (Ghoreyshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2018) show long-term pho- tometric variability in which the increasing flux is due to contributions from a growing disk, while the minima rep- resent episodes of disk destruction by the secondary at periastron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Such a model is therefore opposite to the one presented above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' In this alternative scenario, the light curve minima of AzV 493 in 2002 and 2016 (Figure 1) correspond to episodes with the lowest disk contribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The disk then grows and brightens, recovering its full size around 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' In this case, the decreasing trend in Hβ peak separation with increasing flux is simply due to the disk growth itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' This scenario is consistent with the blue color at the light curve minimum in 2016 (Fig- ure 1), and the weak emission-line spectra near the 2016 minimum (epochs C – J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' If the disk is responsible for the factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='2 increase in flux, then the equivalent width (EW) of stellar absorp- tion features should decrease proportionately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Figure 9 shows the EW of He ii λ4200 and λ4540 as a function of V and I magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' A slight trend is indeed apparent, although not as large as a factor of two in amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' These lines are in the B-band, and thus not in the range of our photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Figure 1 shows that the amplitude of the photometric variations may be smaller at bluer wavelengths, although with the given V -band sampling it is not entirely clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' It may be challenging for the alternative model to produce and maintain the viscous disk necessary to generate continuum luminosities that compete with those of the star, given the harsh circum- stellar environment of an extreme, early-type O-star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The extinction-dominated model is supported by the lack of correlation between the strength of the emission- line spectrum and photometric flux from the sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' There is no significant variation between spectral Epochs C – J (Figure 5), which should correspond to the period of strong disk growth in this model, whereas the obscuration-dominated model implies dissipation (Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The one exception showing spectral variation, Epoch F, has P-Cygni emission and stronger emission- line features, yet it is photometrically unremarkable (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Another issue is that the photometric minimum corresponds to the bluest color (Figure 1), which is more consistent with the alternative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' However, the star itself may be changing substantially in magnitude and color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Blueing is also caused by scatter- ing in high-extinction conditions, as seen in the UXOR class of Herbig Ae stars (Natta & Whitney 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 12 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Equivalent width of He ii λ4200 (red) and λ4540 (blue) as a function of V (bottom) and I (top) magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' A constant value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='5 ˚A is shown for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The overall shape of the light curve for AzV 493 is rather different from those of δ Sco and ω CMa, which show extended minima with more top-hat-like light curves (Ghoreyshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Suffak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' In contrast, AzV 493 shows sharp minima (Figure 1), implying very rapid disk destruction and immediate, regular regrowth in the alternative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' It seems hard to explain such sudden dissipation of a several-AU dense, viscous disk by a neutron star or black hole (Section 6) during the brief periastron passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Moreover, the exact reproduction of the photometric cycle’s initial segment (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='1) is unusual and may be harder to explain with a disk-growth model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Overall, the fundamental nature of the light curve and disk evolution remain unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Tailored modeling of this system and further multimode observational monitoring is needed to clarify the relationship between the decre- tion disk and interaction with a secondary star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' AN EXTREME INTERACTING BINARY The fast surface rotation for this evolved O star is a natural signature of accretion during a mass transfer event (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Packet 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Cantiello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Renzo & G¨otberg 2021), consistent with an interacting binary scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' If the disk is induced by a periastron pas- sage of an undetected companion, then this may imply a long, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='6 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='3)-year period, and hence a large and highly eccentric orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' For the AzV 493 stellar parame- ters obtained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='1, a neutron star companion of mass 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='4 M⊙ would require e ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='95 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='93) and apas- tron of ∼ 43 (27) AU for a typical OBe star periastron distance of 40R⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' These orbital parameters are similar to those of the Be star δ Sco (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Che et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The unseen companion could also be a somewhat more massive main-sequence or stripped star, or a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The eccentricity may be lower, but if a binary compan- ion is responsible for disk ejection, then periastron must be small and the eccentricity high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The nominal peri- astron value used here would likely be an upper limit, since δ Sco showed no disk ejection at periastron (Che et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Neutron star or black hole?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Thus, if a binary companion excites disk ejection or is otherwise responsible for the observed properties of AzV 493, then it is probably an eccentric system, and the most likely explanation for such an orbit is that the companion has already experienced core collapse, receiv- ing a strong kick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Large natal kicks are routinely invoked in core-collapse events that form neutron stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Ar- zoumanian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Podsiadlowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Ver- bunt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Janka 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Natal kicks during black hole formation are still highly debated (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Dray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Janka 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Mandel 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Repetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Atri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Renzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Callister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2020), but not excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Assuming a large 450 km s−1 kick, Brandt & Podsiadlowski (1995) found a broad correla- tion between eccentricity and orbital period of binaries surviving the first core-collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' This is in agreement with the high e and long period we find for AzV 493.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The present-day mass of AzV 493 can be used to con- strain the nature of a putative compact object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Assum- ing a flat distribution in initial mass ratio, the average initial binary mass ratio q = M2/M1 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='5 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Moe & Di Stefano 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Without any accretion during mass transfer, the present-day mass of AzV 493, M2 ≃ 50 M⊙, would suggest M1 ≃ 100 M⊙, which at SMC metallicity implies that the compact object should be a black hole (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Couch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Zapartas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' In this case, however, the rapid rotation of AzV 493 would need to be primordial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Instead, it is more likely that mass transfer has oc- curred, in which case M1 is likely to be quite different, depending on the mass transfer efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' A mass trans- fer phase during the donor’s main sequence (Case A) is expected to be slower and more conservative, possibly causing significant mass growth of the accretor with- out extreme chemical pollution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' This scenario has been invoked to explain the formation of low-mass compact objects in very young regions (Belczynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2008), and in particular, the origin of very massive companions (van der Meij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2021), such as we have for AzV 493.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' In this case, the zero-age-main-sequence (ZAMS) mass of M1 ∼ 30 − 40 M⊙ for the adopted q, also accounting for the final donor core mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' However, mass trans- fer is far more likely to occur after the donor main se- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='8 [y] O EW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='4 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
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+page_content='6 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='8 I[mag] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
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+page_content='8 EW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
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+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
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+page_content='40 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
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+page_content='50 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='55 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='60 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='65 V [mag]13 quence (Case B), due to the star’s expansion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', van den Heuvel 1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' It then takes place rapidly, on the thermal or He core-burning nuclear timescale (Klencki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2022), and system mass loss is far more likely, implying a higher ZAMS mass for M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Although post-SN outcomes are stochastic, black hole production is expected to dominate for Z⊙ progenitors with initial masses ≳ 20 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' This nominal threshold ZAMS mass is expected to decrease for lower metallicity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' O’Connor & Ott 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2016), which in principle enhances the likelihood that the compact object should be a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The high eccentricity in AzV 493 strongly suggests that a SN occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' While this implies that the companion is more likely to be a neutron star, black holes can form from fall-back if the SN is insufficient to unbind the ejecta, which is more likely to happen at low metallicity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' There are multiple mechanisms to produce core-collapse black holes, and if mass-loss oc- curs, a SN and/or kick to the system may result (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Janka 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' We note that M1 ∼ 20 − 40 M⊙ is a range that has been extensively simulated and where explodability and fallback are uncertain (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', O’Connor & Ott 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Janka 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Establishing that a neutron star or black hole resulted from this ZAMS range, with some kind of kick, would provide an important empirical reference for theoretical models of the explosion and the binary interactions preceding it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Follow-up observations at subsequent periastra could more firmly establish whether AzV 493 has a compan- ion, and whether it is a black hole vs a neutron star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' A 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='33 ksec Chandra/HRC observation on 2012 February 12 (MJD 55969) of a field including AzV 493 (ObsID 14054) shows no detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Given the tiny orbital inter- val during which the two stars interact, no significant ac- cretion onto the compact object is expected, explaining why the system is not a known high-mass X-ray binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' However, well-timed X-ray observations near periastron may be able to catch a brief flare event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Radial Velocities We also measure the radial velocity (RV) for the ob- tained spectra to search for evidence of a companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' This is challenging, since AzV 493 is a luminous, fast- rotating, early-type O-star, with few photospheric fea- tures, several of which are often in emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' We carried out cross-correlations against the FASTWIND model spectra for the entire observed spectral range using the iSpec code (Blanco-Cuaresma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2014), as well as de- terminations based on cross-correlations against PoWR model spectra (Hainich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2019) for only the He ii lines (λ4200, λ4540 lines, and λ4686), which are the only clean features appearing in all epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The lat- ter are carried out with the Markov Chain Monte Carlo code of Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' (2015), and since they yield better results, we adopt these RV measurements (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' We find that the mean systemic radial velocity is 202 ± 9 km s−1, weighted inversely by the errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' We caution that the quoted standard error on this value underestimates the uncertainty if there is true variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Given the difficulty of these measurements, with median error on individual epochs of 46 km s−1, it is difficult to evaluate any variability (Figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' There is possible evidence for very short-term RV variations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' however, the data are ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' We compute RV models for a possible periastron sug- gested in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='3 at MJD 57523, which is near the second minimum in the light curve (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' For this 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='3-year period, and the above, nominal periastron dis- tance of 40R⋆, the eccentricity e ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='93 and apastron ∼ 28 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' For this scenario, Figure 10 demonstrates that the RV signature of a neutron-star companion at perias- tron is very brief, on the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='01 in orbital phase, and moreover, the observational uncertainties are larger than the expected amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' This is the case even for e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Thus, our existing RV measurements do not strongly constrain whether MJD 57523 corresponds to a periastron, nor the existence and properties of a com- panion, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Proper Motion A post-SN bound system can be expected to have been accelerated from its original rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Relative to the blue stars from Massey (2002) within a 5′ radius, the Gaia EDR3 (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2021) residual proper motions of AzV 493 show two potential velocity vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Figure 11 provides the velocity histograms of these local field stars, showing strong bimodality in the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' These define two possible local ve- locity fields implying R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' and Dec residual velocity for AzV 493 of either (vα, vδ) = (53 ± 11, 3 ± 12) km s−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' or (vα, vδ) = (−11 ± 11, 12 ± 13) km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' These yield total projected transverse velocities of 54±11 km s−1 or 16 ± 12 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Figure 12 shows a wide-field view of the surround- ing environment, with the two possible proper motion vectors superposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' We see that the nearest massive star-forming region is the N84 complex (Henize 1956) about 15′ − 20′ or ∼ 300 pc to the west.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' If the velocity measurements are correct, the faster, east-bound veloc- ity is consistent with AzV 493 originating in N84 and traveling for ≳ 5 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The lifetime itself of a 50 M⊙ star with v sin i ∼ 500 km s−1 is about 5 Myr (Brott 14 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Left: Heliocentric radial velocities measured from He ii photospheric absorption vs MJD for all epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Epoch A has only one available line of low quality, and hence has a very large uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The vertical dashed lines show the possible periastra at MJD 54686 and 57523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Right: Zoom for the same data showing RV models for eccentricities of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='93 (dashed lines) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='99 (solid lines), assuming a periastron occurs at MJD 57523;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' and for inclination angles of 90◦ (black) and 45◦ (blue), for the 50 M⊙ primary and assuming a 3 M⊙ secondary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' If a periastron is closer to the light curve minimum at MJD 57626, the models would shift to 103 days later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Distribution of Gaia proper motion velocities in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' (left) and Dec (right) for stars from Massey (2002) within 5′ of AzV 493.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The bimodal R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' distribution defines two kinematic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The first group has 13 stars with median velocity (vα, vδ) = (254 ± 7, −378 ± 9) km s−1 and the second has 10 stars with (vα, vδ) = (318 ± 6, −386 ± 11) km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The one star between the two groups in vα is included in both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The median velocities for these groups are shown with the vertical green and blue lines, together with the velocity of AzV 493 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2011), and for a SN ejection, its travel time would only be the post-SN lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' However, since the star presumably acquired its total mass and spin later in life, the system may have been ejected earlier by dy- namical processes as a tight, non-compact binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' If so, it would have been reaccelerated by the SN explosion, therefore implying that it may be a two-step ejection (Pflamm-Altenburg & Kroupa 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Supernova accel- erations are typically weaker than dynamical ejections (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Renzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2019), and so the dominant velocity component could still be due to a dynamical ejection from N84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' A dynamically active past in a dense stellar environment of N84 may also help to explain the eccen- tricity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Sim´on-D´ıaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2015), although it would seem unlikely that the system could maintain its highly eccentric configuration for 5 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' On the other hand, we note that the inferred runaway velocity, orbital ec- centricity, and period are still consistent with being due AzV 493 350 (km/sec) 300 E 250 B D 200 -- K RV A Heliocentric 150 100 50 0 50 -- 55000 56000 57000 58000 59000 60000 Date (MJD)Example RV Curves 260 e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='99 e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='94 240 Primary RV (km/s) 220 200 180 160 140 200 0 200 400 600 Days around Periastron = MJD 5752377616 field v RA plot 77616 field v DEC plot 6 8 7 5 6 4 (stars) 3 4 N N 2 2 1 1 0 0 200 250 300 350 450 400 350 300 Velocity (km/s) Velocity (km/s)15 Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Location of AzV 493 in the SMC field, with the green and blue proper motion vectors corresponding to the two field velocities indicated with the same color coding in Figure 11, superposed on an Hα images from Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The nearest massive star-forming region is the N84 complex (Henize 1956), indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' For the adopted SMC distance, 10′ = 181 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' solely to SN acceleration (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Brandt & Podsiadlowski 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Thus, in order to explain both the long travel time and high eccentricity, the most plausible scenario may be the two-step ejection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' There is also a small possibility that the slow, alterna- tive proper motion vector (Figure 12) is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' How- ever, this would mean that the AzV 493 system formed in isolation since there is no corresponding young clus- ter whence it could have originated (Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Vargas- Salazar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' (2020) find that < 5% of OB stars, if any, formed in the field, and this is especially unlikely for AzV 493, given its high mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' We caution that the ve- locity errors do not include unknown systematic errors, and so these measurements need to be confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Thus, although AzV 493 indeed appears to be a runaway star, this does not provide especially useful information to constrain its binary interaction history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Similar systems A comprehensive study by Marr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' (2022) shows that the B8 Vpe star Pleione (HD 23862) has a light curve with a similar long-term pattern of slow growth with sudden drops, and similar variations in the Balmer emission-line profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' It is a triple system with a close companion on a 218-day orbit (Katahira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Nemravov´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Marr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' (2022) suggest that the photometric drops correspond to the decretion disk tearing into two components, where one remains aligned with the star’s equatorial plane and the other is mis- aligned due to tidal torque from the close companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Pleione’s long-term photometric cycle is 34 years, simi- lar in magnitude to that of AzV 493.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Nemravov´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' (2010) find that the close companion is on an eccentric orbit with e > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' AzV 493’s initial peak brightness and subsequent drop in 2001 (Figure 1) qualitatively resemble the photomet- ric pattern characteristic of heartbeat stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' These are a rare class of interacting binary systems with high eccen- tricities such that the periastron passage tidally induces regular photometric outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' However, the observed pattern in AzV 493 cannot be induced by this type of tidal interaction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' preliminary simulations using new ca- pabilities in the GYRE stellar oscillation code (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2023) suggest that the combined amplitude and width of the periastron pulse cannot be reproduced by eccentric tidal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Nevertheless, given that AzV 493 seems likely to be a massive eccentric binary system, massive heartbeat stars thus share some similarities with this 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='5±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='9 km/s 73°00\'00" 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='0±12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='1 km/s 10\'00" Dec (J2000) N84 20\'00" 30\'00" 20°00\'00" 19°00\'00" 18°00\'00" RA (12000)16 object if a companion indeed interacts with the primary and/or its disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Examples include the non-Be binary system ι Ori (O9 III + B1 III/IV), which has orbital period 29 d and eccentricity e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='764, as determined by Pablo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' They find that the two components have masses of 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='2 and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='4 M⊙, respectively, generat- ing tidally excited oscillations with periods on the order of ∼ 1 day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' MACHO 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='7443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='1718 is another heartbeat system with two stars of type B0 Iae and O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='5 V and masses of 35 and 16 M⊙, respectively (Jayasinghe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='5 Ve star δ Sco is has a B2 V star companion in an eccentric (e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='94) orbit with period 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='7 years (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Tango et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Tycner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The two components have masses of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='9 M⊙ and 6 M⊙ (Che et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' This system shows a long-term photomet- ric cycle somewhat similar to that of AzV 493, although much more poorly defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' There is no obvious link be- tween the disk properties and binary interaction (Suffak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Che et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2012), but the long-term pho- tometry has a timescale similar to that of the orbital period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' H 1145–619 is a Be X-ray binary whose primary is a B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='2e III star estimated to be 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='5 M⊙ (Alfonso-Garz´on et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2017), and the secondary is an X-ray pulsar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' As shown by Alfonso-Garz´on et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' (2017), H 1145–619 has a light curve with a ∼ 10-year cycle together with un- explained multiple modes of much shorter periods (∼ 1 year), qualitatively similar to what we see for AzV 493, which has a long cycle of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='6 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='3) years and short oscil- lations of ∼ 40 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' While it remains unclear whether the light curves of H 1145–619 and AzV 493 have fun- damental similarities, both stars are massive OBe stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' If they are related, the fact that H 1145–619 has a con- firmed compact binary companion may suggest that the unusual variability of AzV 493 may have a similar origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' These objects provide a context for AzV 493 that sup- ports this object being a member of this broad class of binary, massive OBe systems with high eccentricities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' At 50 M⊙, AzV 493 is more massive than any of these similar objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' It is also one of the earliest O stars in the SMC, since there is no photospheric He i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Thus, AzV 493 may be the most extreme such object known, in terms of its mass and effective temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Its vari- ability amplitudes are also among the largest known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' We note that, based on only the Epoch A spectrum (Figure 5), Golden-Marx et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' (2016) suggested that AzV 493 is a normal, but extremely early, classical Oe star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Given the strong spectroscopic and photometric variability, the nature of this spectrum may be some- what different than inferred in that work, and the origin of the strong line emission seen in this particular spec- trum is unclear (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Still, its status as a post- SN binary where the observed star was likely spun up by mass transfer from the compact object progenitor, is consistent with the origin of classical OBe stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Indeed, given that most of the massive OBe stars are post-SN systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Dallas & Oey 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Dorigo Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2020), we can expect that more of them are likely to be high-eccentricity, compact-object binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Alternative Companion Scenarios We now consider alternative scenarios for a putative binary component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' First, such a companion might be an unexploded former donor in an interacting binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' In this case, it could be a stripped star (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Schootemeijer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' G¨otberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2017), which can be elusive to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' (2021) identified hot, stripped star companions to Be stars based on FUV spectral cross- correlations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' however, the extremely hot temperature of AzV 493, which is commensurate with the hottest O stars, poses a serious challenge for this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' If the observed star has previously experienced accretion from binary mass transfer, then its surface might be He- and N-enriched (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Blaauw 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Renzo & G¨otberg 2021), although whether this occurs depends on the accretion efficiency and mixing processes in the accretor’s enve- lope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Since early O stars have few metal lines, it is again difficult to evaluate any enrichment, especially in a fast rotator like AzV 493.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' There is no immediate evidence for any unusual abundances in this star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Moreover, a non-degenerate companion does not naturally explain the high observed eccentricity, which would then have to be primordial, avoiding tidal dissipation, or of dy- namical origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Alternatively, the high rotation rate and variability of AzV 493 might be caused by a non-standard internal structure of the star because of a merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' These are common among massive stars, occurring in 22+26 −9 % of isolated massive binaries (Renzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2019), with an even higher rate if accounting for the presence of further companions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Toonen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' For example, η Car has been suggested to originate from a merger in a hierarchical triple system, resulting in a present-day eccentric binary (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Hirai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' However, η Car is a luminous blue variable star and has other substantial differences from AzV 493.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Yet another possibility is that AzV 493 might be a triple system with a third, also invisible, star on a shorter-period orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' This speculative scenario might help to explain how the strong, 40-day pulsations are maintained (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' It also might help explain the apparently sporadic ejection and accretion events seen in Epochs B and F (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Such a system would 17 be unstable, but the brief interaction phase with the sec- ondary may enhance its longevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' We note that the sys- tem is unlikely to be a triple in which the third star has an even larger orbit than the secondary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Although high orbital eccentricities can be produced by Kozai-Lidov cycles in such a system, this high-eccentricity phase of the cycle is short in duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Thus, such extreme eccen- tricity may require a triple or higher-order multiple-star interaction in the system’s birth cluster, and may be linked to a dynamical ejection of AzV 493 into the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Overall, however, it is challenging to explain AzV 493 in terms of a triple-star scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Unfortunately, RV mon- itoring is complicated due to the technical difficulty and possible presence of varying stellar pulsations, so it will be hard to evaluate whether the system consists of more than two stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' SUMMARY We present 18 years of OGLE Project photometric data and spectroscopic data over 12 years, revealing the remarkable variability of AzV 493.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' This is perhaps the earliest known classical Oe star, with Teff = 42000 K, log L/L⊙ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='15, and R⋆/R⊙ = 15 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' These parameters imply a mass of 50 ± 9 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The domi- nant photometric pattern is reproduced after 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='6 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' There are also large, semi-regular ∼ 40-day pulsations of unknown origin, as well as other structure in the light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' It is not a known HMXB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The observed v sin i = 370± 40 km s−1, with a high inferred sin i, suggesting a rotational velocity of 400 − 450 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The system is ∼ 300 pc from the nearest massive star-forming com- plex and its proper motion shows that it is likely a run- away star from that region, with a transverse velocity of 54 ± 11 km s−1, possibly having experienced two-step acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Altogether, the data suggest that this object is likely an eccentric, interacting binary system with an unde- tected compact companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' If so, the orbital period could correspond to the 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='6 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='3)-year period, imply- ing a high eccentricity of at least e ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='95 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='93) and apastron ∼ 43 (28) AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' If this binary scenario is cor- rect, AzV 493 would be among the most extreme sys- tems known, in terms of its early spectral type, high mass, and extreme eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' In our favored model, an optically thick decretion disk is regularly ejected, likely by a periastron encounter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' A two-component disk system forms, with the outer re- gion responsible for the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='85-magnitude drop in I-band flux, while the inner disk is the origin of most of the observed emission-line spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The spectra appear to show varying relative contributions from the inner and outer regions, consistent with the optically thick outer region dissipating over the cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The outer region may correspond to a flared disk, torus, or possibly, a separate inclined annulus formed by tearing from the inner disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' We see direct spectroscopic evidence for episodes of both matter ejection and infalling reabsorption of dense disk material onto the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The lack of exact regularity of photometric and spectroscopic variations in the cycle implies that the geometry and/or mechanics of the disk ejection may vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' An alternative, opposite model seen in some Be stars, in which the brightness increases due to contribution from growing disk emission (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Suf- fak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Ghoreyshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 2018), should also be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' If AzV 493 indeed has a highly eccentric orbit, it would suggest that the system experienced a strong SN kick, implying that the unseen companion is a neutron star or black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The high v sin i also suggests that mass transfer occurred before this event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' For conservative, Case A mass transfer, the progenitor donor’s ZAMS mass would be 30 − 40 M⊙ for a typical q ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='5, and larger for non-conservative Case B mass transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' This mass range is well within that suggested by models to produce black holes, although the occurrence of strong natal kicks in cases of black hole formation is less clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Alternatively, the donor could be a stripped star;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' how- ever, this scenario cannot explain the extreme eccentric- ity, which would have to be dynamical or primordial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' The system could also be a merger, but the eruptions and long-term pulsations seem less consistent with this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' AzV 493 could possibly be a triple system, which might explain how the strong photometric oscil- lations are maintained (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Establishing the existence and nature of the unseen companion(s) can provide important constraints on bi- nary evolution, core explodability, and the origin of compact binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' AzV 493 may offer an opportunity to directly observe the relationship between the binary companion’s dynamical interaction and the disk ejec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Since many classical OBe stars are massive, post- SN objects, it suggests a likely link between OBe stars and massive, eccentric systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Further study of this fascinating object can more definitively confirm its sta- tus and exploit the opportunities it offers to learn about massive binary evolution and disk ejection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 18 ACKNOWLEDGMENTS We benefited from useful discussions with many peo- ple, including Jon Bjorkman, Paul Crowther, Julian Deman, Jim Fuller, Jay Gallagher, Carol Jones, Max Moe, Megan Reiter, Steve Shore, and Drew Weisser- man.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Many thanks to Juliette Becker for the use of her code, and to Traci Johnson, Mario Mateo, and the M2FS Team for help with observing runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' We also thank the anonymous referees for valuable comments that greatly improved this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' This work was sup- ported by NSF grant AST-1514838 to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' and by the University of Michigan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' Castro acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG), CA 2551/1-1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' is supported by EUH2020 OPTICON RadioNet Pilot grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' 101004719;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
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+page_content=', Vigna-G´omez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
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+page_content=' 2020, MNRAS, 498, 4705 Walker, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
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+page_content=', Mateo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=', Olszewski, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
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+page_content=' 2015, ApJ, 808, 108 Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
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+page_content=' GENERALIZED LOMB-SCARGLE PERIODOGRAMS Figure 13 shows the individual generalized Lomb-Scargle periodograms (Zechmeister & K¨urster 2009) and ancillary information for the six, roughly contiguous, OGLE datasets during ∼ 2010 – 2016 (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
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+page_content=' Top panels show the generalized Lomb-Scargle periodogram for light curves shown in the middle-left panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFJT4oBgHgl3EQfDSx3/content/2301.11433v1.pdf'}
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diff --git a/FtFLT4oBgHgl3EQfGS-3/content/tmp_files/2301.11991v1.pdf.txt b/FtFLT4oBgHgl3EQfGS-3/content/tmp_files/2301.11991v1.pdf.txt
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+Real-time non-perturbative dynamics of jet production:
+quantum entanglement and vacuum modification
+Adrien Florio,1, ∗ David Frenklakh,2, † Kazuki Ikeda,2, 3, ‡ Dmitri Kharzeev,1, 2, 3, §
+Vladimir Korepin,4, ¶ Shuzhe Shi,2, ∗∗ and Kwangmin Yu5, ††
+1Department of Physics, Brookhaven National Laboratory, Upton, New York 11973-5000, USA
+2Center for Nuclear Theory, Department of Physics and Astronomy,
+Stony Brook University, Stony Brook, New York 11794-3800, USA
+3Co-design Center for Quantum Advantage, Department of Physics and Astronomy,
+Stony Brook University, Stony Brook, New York 11794-3800, USA
+4C.N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, New York, 11794-3840, USA
+5Computational Science Initiative, Brookhaven National Laboratory, Upton, New York 11973-5000, USA
+The production of jets should allow to test the real-time response of the QCD vacuum disturbed
+by the propagation of high-momentum color charges. Addressing this problem theoretically requires
+a real-time, non-perturbative method. As a step in developing such an approach, we report here on
+fully quantum simulations of a massive Schwinger model coupled to external sources representing
+quark and antiquark jets as produced in e+e− annihilation. It is well known that the Schwinger
+model [QED in (1 + 1) dimensions] shares many common properties with QCD, including confine-
+ment, chiral symmetry breaking and the existence of vacuum fermion condensate. This allows us to
+study, for the first time, the modification of the vacuum chiral condensate by the propagating jets,
+and the quantum entanglement between the fragmenting jets. Our results indicate strong entangle-
+ment between the fragmentation products of the two jets at rapidity separations ∆η ≤ 2 that can
+potentially be studied in experiment.
+Introduction:
+The discovery of jets played a crucial
+role in establishing Quantum Chromodynamics (QCD)
+as the theory of strong interactions, see [1, 2] for reviews.
+The production of the initial high momentum partons is
+a short-distance process that can be described in pertur-
+bative QCD due to asymptotic freedom. However, as the
+initial partons keep radiating gluons and quark-antiquark
+pairs as described by QCD evolution equations, the char-
+acteristic virtuality decreases, and non-perturbative phe-
+nomena should come into play.
+In particular, one expects that the propagating color
+charges will disturb the non-perturbative QCD vacuum,
+and the corresponding real-time response should contain
+valuable information about the vacuum structure. More-
+over, the initial partons should be entangled by the pro-
+duction process, but whether any trace of this entangle-
+ment can be found in fragmenting jets is not clear. The
+answers to these questions lie outside of the realm of per-
+turbative QCD, and finding them requires a real-time,
+non-perturbative method.
+Such an approach is enabled by the advent of quantum
+simulations. Unfortunately, the case of real (3+1) dimen-
+sional QCD is still out of reach for the existing quantum
+hardware, as well as for real-time simulations on classical
+computers. However one can start developing real-time
+non-perturbative methods using simpler models in lower
+number of space-time dimensions.
+In this respect QED in (1 + 1) dimensions (the
+Schwinger model [3]) holds a special place: just like QCD,
+it possesses confinement, chiral symmetry breaking, and
+fermion condensate [4]. In the massless fermion limit, the
+theory is exactly solvable by bosonization, and admits a
+dual description in terms of a free massive scalar theory.
+In 1974, Casher, Kogut, and Susskind [5] proposed to
+model quark-antiquark production in e+e− annihilation
+by coupling Schwinger model to external sources propa-
+gating along the light cone.
+An explicit analytical solution of this model has been
+found in [6, 7], where this setup was also used to de-
+scribe jet quenching in heavy ion collisions by introducing
+in-medium scattering of the sources, and the anomalous
+enhancement of soft photon production in jet fragmenta-
+tion [8] observed by the DELPHI Collaboration [9].
+A more realistic extension of this approach is based
+on a massive Schwinger model, which in the bosonized
+description is dual to an interacting meson theory. In this
+case, the model is no longer analytically solvable, and so
+a numerical approach is necessary. The first study of this
+setup was carried out in [10] using a numerical classical-
+statistical approach. Coupling the Schwinger model to an
+external Yukawa theory has also been used to mimic the
+propagation of jets through a thermal environment [11].
+Various other aspects of the Schwinger model have also
+been addressed using quantum simulations, see [12–17]
+for examples and [18] for a recent review of quantum
+simulations.
+In this work, using the massive Schwinger model cou-
+pled to external sources, we perform the first fully quan-
+tum simulation of jet production. In particular, we focus
+on real-time, non-perturbative effects that have not been
+studied before: the modification of the vacuum structure
+and the entanglement between the produced jets.
+The model:
+We use the massive Schwinger model
+Hamiltonian in temporal gauge A0 = 0 in the presence
+arXiv:2301.11991v1 [hep-ph] 27 Jan 2023
+
+2
+of an external current jµ
+ext describing the produced jets:
+HC = HC
+S + HC
+ext ,
+(1)
+HC
+S =
+�
+dx
+�1
+2E2 + ¯ψ(−iγ1∂1 + gγ1A1 + m)ψ
+�
+, (2)
+HC
+ext =
+�
+dx j1
+extA1 ,
+(3)
+where Aµ is the U(1) gauge potential, E = − ˙A1 is
+the corresponding electric field, ψ is a two-component
+fermionic field, m is the fermion mass, and γµ are two-
+dimensional γ-matrices satisfying Clifford algebra; we use
+ηµν = diag(1, −1) as our metric.
+The superscript C
+stands for “continuum”.
+The effect on the theory of the interaction with the
+external source Hext is to modify Gauss law to
+∂1E − j0 = j0
+ext .
+(4)
+with j0 = g ¯ψγ0ψ. In other words, the theory is gauge
+invariant up to the presence of the external charge j0
+ext;
+the external current is a “defect” of the U(1) gauge trans-
+formation.
+To mimic production of a pair of jets in e+e− annihila-
+tion, we choose the external current to represent charges
+of opposite sign flying apart along the light cone:
+j0
+ext(x, t) = g[δ(∆x − ∆t) − δ(∆x + ∆t)]θ(∆t) ,
+j1
+ext(x, t) = g[δ(∆x − ∆t) + δ(∆x + ∆t)]θ(∆t) ,
+(5)
+where (t0, x0) is the time and position of a point where
+the jet pair is produced, and ∆x ≡ x−x0 and ∆t ≡ t−t0
+are the space and time distance from this position.
+Note that in principle one could replace the external
+probe charges by “hard” dynamical fermions, which can,
+for instance, be produced by short lived pulses of electric
+fields. This has been done in [10], where it was found
+that, at least within the semiclassics, the use of exter-
+nal charges is a very good approximation to a pair of
+dynamical relativistic “hard” fermions. This motivates
+us to restrict ourselves to the simpler case of external
+currents.
+Our goal is to study the modification of the vacuum
+due to the presence of the external sources (5). To this
+end, we evolve the ground state of the massive Schwinger
+model with the time-dependent Hamiltonian (1). In or-
+der to solve this problem, we need to discretize space-
+time and approximate the theory by a finite-dimensional
+Hilbert space.
+Lattice model: We begin by discretizing space in a lat-
+tice of N points with lattice spacing a. We choose to
+work with staggered fermions χn [19, 20]. We use a non-
+compact formulation for the U(1) gauge fields, and in-
+troduce a lattice electric field operator Ln = E(an)/g, a
+lattice vector potential φn = ag A1(an), and a link opera-
+tor Un = e−iagA1(an). We further impose open-boundary
+conditions (OBC) χN+1 = LN = 0 on the fermion and
+gauge fields. Using the Dirac matrices γ0 = σz, γ1 = i σy,
+the Hamiltonian is
+HL(t) = HL
+S + HL
+ext(t) ,
+(6)
+HL
+S = − i
+2a
+N−1
+�
+n=1
+�
+U †
+nχ†
+nχn+1 − Unχ†
+n+1χn
+�
++ ag2
+2
+N−1
+�
+n=1
+L2
+n + m
+N
+�
+n=1
+(−1)nχ†
+nχn ,
+(7)
+HL
+ext(t) = 1
+g
+N−1
+�
+n=1
+j1
+ext(a n, t)φn ,
+(8)
+where the superscript L stands for “lattice”. Even in the
+presence of point charges, Gauss law is well defined when
+integrated over a lattice spacing and reads
+Ln − Ln−1 − Qn = 1
+g
+� (n+1/2)a
+(n−1/2)a
+dx j0
+ext(x, t) ,
+(9)
+with Qn = χ†
+nχn (1 − (−1)n) the lattice charge density
+operator. For the rest of this work, we insert the sources
+at the center of our lattice, x0 = a
+� N+1
+2
+�
+, at time t0
+a = 1.
+Before proceeding with the time evolution, we take
+advantage of the fact that the gauge fields are non-
+dynamical in (1+1) dimensions to express them in terms
+of fermionic operators through Gauss law. This has the
+advantage of drastically reducing the size of the discrete
+Hilbert space needed down to 2N, at the cost of intro-
+ducing non-localities. The former turns out to outweigh
+the latter for the method we use (direct diagonalization,
+or “exact diagonalization” of the Hamiltonian), see also
+the Supplementary Material.
+We then use the remaining freedom to perform a space-
+only dependent gauge transformation to set all gauge
+links to unity. The explicit gauge transformation which
+achieves this result is Ω1 = 1, Ωn = �n−1
+i=1 U †
+i [17]. Note
+that the existence of such a transformation is a pecu-
+liarity of (1 + 1) dimensions and is related to the fact
+that the gauge field is not dynamical. We then rewrite
+Ln = Ldyn,n + Lext,n and solve Gauss law (9) as follows:
+Ldyn,n =
+n
+�
+i=1
+Qi ,
+(10)
+Lext,n(t) = −θ
+�
+t − t0 −
+���x − x0 + a
+2
+���
+�
+.
+(11)
+The non-locality is contained in the dynamical gauge field
+and the external sources create a chain of electric fluxes
+between them.
+The Hamiltonian (6) is now directly suitable for di-
+agonalization.
+However, having in mind future quan-
+tum computing applications, we have used an equivalent
+form in terms of Pauli matrices X, Y, Z, or “spin” degrees
+
+3
+t/a
+A B
+2-
+3-
+4-
+5-
+6-
+7-
+8-
+9-
+10-
+source
+fermion
+anti-fermion
+electric charge
+dynamical
+electric field
+0
+5
+10
+15
+Eele,t - Eele,0
+a g2 / 2
+external only
+0.0
+0.5
+1.0
+1.5
+νt - ν0
+0.4
+0.6
+0.8
+1.0
+SEE
+0
+2
+4
+6
+8
+10
+0.0
+0.1
+t / a
+Qt
+FIG. 1.
+(Left) Time evolution of the local charge density (vertical bars) and of the electric field (arrows), with vacuum
+expectation values subtracted. Black(white) even(odd)-sites correspond to (anti)fermions. The position of the external sources
+is shown above each configuration. From top to bottom, the rows are for time values (in units of lattice spacing a) t/a = 2−
+to 10−, where n− ≡ n − ε with ε being an arbitrarily small positive number. (Right) (from top to bottom) Time evolution
+of electric energy, scalar fermion density, entanglement entropy, and electric charge. Dotted lines in the first panel show the
+electric energy generated by the external sources.
+of freedom. We employ the Jordan–Wigner transforma-
+tion [21]
+χn = Xn − iYn
+2
+n−1
+�
+j=1
+(−iZj),
+χ†
+n = Xn + iYn
+2
+n−1
+�
+j=1
+(iZj),
+(12)
+to obtain
+HL(t) = 1
+4a
+N−1
+�
+n=1
+(XnXn+1 + YnYn+1) + m
+2
+N
+�
+n=1
+(−1)nZn
++ ag2
+2
+N−1
+�
+n=1
+(Ldyn,n + Lext,n(t))2 .
+(13)
+Our simulations then proceed as follows.
+We start
+by finding the ground state |Ψ0⟩ of the usual massive
+Schwinger model HL(0).
+We then compute the state
+|Ψt⟩ = T e−i
+� t
+0 HL(t′)dt′ |Ψ0⟩ corresponding to the evolu-
+tion under the time-dependent Hamiltonian HL(t), with
+T being the time-ordering operator. The system is ef-
+fectively “quenched” at
+t
+a = t0
+a = 1, when the external
+sources are introduced. We then compute different time-
+dependent expectation values ⟨O⟩t ≡ ⟨Ψt| O |Ψt⟩ where
+O are the operators corresponding to observables of in-
+terest.
+Vacuum modification and quantum entanglement be-
+tween the jets: We measure the local electric charge den-
+sity, the total electric charge, the scalar fermion density
+⟨ ¯ψψ⟩, the local electric field strength, and the electric
+field energy, that are given respectively by
+qn,t ≡ ⟨ψ†(a n)ψ(a n)⟩t = ⟨Zn⟩t + (−1)n
+2a
+,
+(14)
+Qt ≡
+�
+⟨ψ†(x)ψ(x)⟩t dx = a
+N
+�
+n=1
+qn,t,
+(15)
+νn,t ≡ ⟨ ¯ψ(a n)ψ(a n)⟩t = (−1)n⟨Zn⟩t
+2a
+,
+(16)
+νt ≡
+�
+⟨ ¯ψ(x)ψ(x)⟩t dx = a
+N
+�
+n=1
+νn,t,
+(17)
+Πn,t ≡ ⟨E(a n)⟩t = g ⟨Ln⟩t,
+(18)
+Eele,t ≡ 1
+2
+�
+⟨E2(x)⟩t dx = a g2
+2
+N−1
+�
+n=1
+⟨L2
+n⟩t.
+(19)
+We also compute the entanglement entropy between the
+left- and the right-hand sides of the chain
+SEE(t) = −TrA(ρt,A log ρt,A),
+(20)
+with A = {1, · · · , N/2} and B = {N/2 + 1, · · · , N}. The
+operator ρt,A = TrBρt is the partial trace of the time
+dependent density matrix ρt ≡ |Ψt⟩ ⟨Ψt| over B [see il-
+lustration in Fig. 1(left)].
+In Fig. 1, we show the time evolution of local and
+global observables respectively, for parameters N = 20,
+m = 0.25/a, and g = 0.5/a. In the left panel, we show
+the full time evolution of our quantum state.
+We ob-
+serve that both the gauge fields and the fermion fields
+
+4
+are excited by the external sources, and their effects are
+constrained within the light cone spanned by them. In
+the right panel, we observe a step-like increase in electric
+field energy. The growth of νt − ν0 shown in Fig. 1 indi-
+cates destruction of the (negative) vacuum chiral conden-
+sate ν0 by the propagating jets [22]. This destruction is
+due to the pair production from the vacuum that also re-
+sults in the screening of the electric energy which appears
+smaller than the contribution from external sources.
+Since we can access the entire quantum state, we are
+able to compute also for the first time the entanglement
+entropy between the jets. The growth of this entangle-
+ment entropy (third panel) results from the pair creation.
+Lastly, as a consistency check, we also show in the lower
+panel the total electric charge, which remains zero, as
+expected.
+Observing quantum entanglement between the jets:
+With an eye towards possible experimental studies of
+quantum entanglement between the produced jets, we
+measure the two-point correlation of scalar fermion den-
+sity operators with the vacuum expectation value sub-
+tracted,
+⟨∆νN/2+ℓ ∆νN/2+1−ℓ⟩,
+(21)
+where ∆νn ≡ νn − ⟨νn⟩vac.
+The motivation behind this study is the following.
+In the bosonization dictionary of the massive Schwinger
+model, the correlation between the scalar fermion densi-
+ties translates into the correlation among the boson pairs
+(and higher order correlations). Therefore we hope that
+this correlation function may be used to infer informa-
+tion about quantum entanglement between the pion pairs
+produced in jet fragmentation. A concrete proposal of an
+observable correlation between pion pairs produced in jet
+fragmentation has been put forward in [23].
+To isolate the effect of entanglement between the jets,
+we measure the correlation function for the cases of cor-
+related and uncorrelated sources of fermion-antifermion
+pairs. Because the entanglement should stem from the
+correlation between the sources, the case of uncorrelated
+sources provides the classical baseline for the correlation
+functions.
+…
+…
+1
+2
+l=3
+(a) correlated:
+(b) left:
+(c) right:
+FIG. 2. Illustration of correlated and uncorrelated measure-
+ments of two point correlation functions. The uncorrelated
+setup is obtained as an uncorrelated linear superposition of
+jets created by a single (anti)fermion source moving to the
+(left)right.
+0
+1
+2
+3
+4
+l = 3
+5
+7
+9
+4
+6
+8
+correlated
+0
+2
+4
+6
+8
+10
+0.0
+0.1
+0.2
+0.3
+0.4
+t / a
+uncorrelated
+102 × 〈ΔνN/2+l ΔνN/2+1-l〉
+0.0
+0.5
+1.0
+1.5
+0
+1
+2
+3
+4
+ηs
+uncorrelated
+correlated
+t = 10 a
+102 × 〈Δν-ηs Δν+ηs〉
+FIG. 3.
+Time evolution of two-point correlation functions
+with various separations.
+The upper(lower) panel is for a
+correlated(uncorrelated) setup. The large difference between
+the two cases is a signature of quantum entanglement in the
+produced pairs. (Insert) Spatial-rapidity dependence of the
+two-point correlation at the end of the evolution.
+Our method of preparation of two uncorrelated quan-
+tum systems is illustrated in Fig. 2 (b, c).
+In one of
+these systems, there is only an antifermion source mov-
+ing to the left while the fermion source sits still at the
+origin.
+We denote the quantum state of such a sys-
+tem as |ψL⟩.
+We then define its counterpart, |ψR⟩,
+corresponding to the setup of Fig. 2(c), with fermion
+source moving to the right and the antifermion source
+fixed at the origin.
+The uncorrelated state is defined
+as the superposition of left and right state with a ran-
+dom phase, |ψuncorr⟩ =
+1
+√
+2 |ψL⟩ + eiϕ
+√
+2 |ψR⟩, and the ex-
+pectation value of any observable is obtained by aver-
+aging over this random phase, ⟨⟨ψuncorr|O|ψuncorr⟩⟩ ≡
+�
+⟨ψuncorr|O|ψuncorr⟩ dϕ
+2π = ⟨ψL|O|ψL⟩
+2
++ ⟨ψR|O|ψR⟩
+2
+.
+The correlation function (21) is designed to measure
+the points that are symmetric with respect to the jet
+production vertex. We measure the two-point correlation
+function with different separation distances as functions
+of time, and the results are presented in Fig. 3. We find
+that the correlation functions measured for the correlated
+state are an order of magnitude greater than those for the
+uncorrelated state. Note that it is non-zero in the latter
+case because of the classical correlation between the par-
+ticle production in left- and right-moving jets which is
+similar to the correlation that would be induced by the
+propagation of sound along the jets’ axes.
+Meanwhile, for the quantum correlated state, we ob-
+serve the propagation of a similar pattern for odd ℓ’s and
+
+5
+similarly for even ℓ’s, which is driven by the correlated
+moving sources. After a sufficiently large time, we take
+a snapshot and present the space dependence of the cor-
+relation functions in Fig. 3 (insert), where we have con-
+verted the site separation to spatial rapidity separation,
+ηs ≡ arctanh z
+t = arctanh (ℓ−1/2)a
+t
+.
+One can clearly see a big difference between the strong
+quantum correlation for the quantum state and the near
+absence of correlations for the uncorrelated baseline.
+This difference is especially pronounced for moderate ra-
+pidity separations ∆ηs = 2ηs ≤ 2. Using the approxi-
+mate equality of space-time and momentum space rapidi-
+ties in jet fragmentation, this suggests that one should
+look for quantum entanglement among the pions pro-
+duced in the fragmentation of the two jets at rapidity
+separation ∆η ≤ 2. An observation of correlations among
+these pion pairs would constitute a direct signature of en-
+tanglement between the jets.
+Specifically, it would be interesting to study the quan-
+tum correlations between the “handedness” of the pion
+pairs produced in the fragmentation of the quark and
+antiquark jets [23].
+Some hints of such correlations
+had been reported in the data from DELPHI Collabo-
+ration [24].
+To summarize, we have performed a real-time, non-
+perturbative study of jet fragmentation using a massive
+Schwinger model with external sources. Strong distortion
+of the vacuum chiral condensate by the propagating jets
+has been observed. We have also found strong quantum
+entanglement between the fragmenting jets for rapidity
+separation ∆η ≤ 2. We hope that this result will moti-
+vate dedicated experimental studies. Our work also paves
+the way for quantum simulations of jet fragmentation us-
+ing quantum hardware; we plan to address this problem
+in the near future.
+ACKNOWLEDGEMENT
+We thank Jo˜ao Barata, Fangcheng He, Yuta Kikuchi,
+Semeon Valgushev, Tzu-Chieh Wei, and Ismail Zahed
+for useful discussions and communications.
+This work
+was supported by the U.S. Department of Energy, Of-
+fice of Science, National Quantum Information Science
+Research Centers, Co-design Center for Quantum Ad-
+vantage (C2QA) under Contract No.DE-SC0012704 (AF,
+KI, VK), and the U.S. Department of Energy, Office of
+Science, Office of Nuclear Physics, Grants Nos.
+DE-
+FG88ER41450 (DF, DK, SS) and DE-SC0012704 (AF,
+DK, KY). This research used resources of the National
+Energy Research Scientific Computing Center, a DOE
+Office of Science User Facility supported by the Office of
+Science of the U.S. Department of Energy under Contract
+No. DE-AC02-05CH11231 using NERSC award NERSC
+DDR-ERCAP0022229.
+∗ aflorio@bnl.gov
+† david.frenklakh@stonybrook.edu
+‡ kazuki.ikeda@stonybrook.edu
+§ dmitri.kharzeev@stonybrook.edu
+¶ vladimir.korepin@stonybrook.edu
+∗∗ shuzhe.shi@stonybrook.edu
+†† kyu@bnl.gov
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+arXiv:hep-ph/0412013.
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+Morris, R. C. Pooser, M. Sanz, E. Solano, P. Lougovski,
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+Schwinger model dynamics using quantum computers,”
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+J. Unmuth-Yockey, “Tensor network formulation of the
+massless Schwinger model with staggered fermions,”
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+F. V. Pepe, and E. Ercolessi, “Real Time Dynamics and
+Confinement in the Zn Schwinger-Weyl lattice model
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+for 1+1 QED,” Quantum 4 (2020) 281,
+arXiv:1909.04821 [quant-ph].
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+dynamics from a digital quantum simulation,” Phys.
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+[hep-ph].
+[17] K. Ikeda, D. E. Kharzeev, and Y. Kikuchi, “Real-time
+dynamics of Chern-Simons fluctuations near a critical
+point,” Phys. Rev. D 103 no. 7, (2021) L071502,
+arXiv:2012.02926 [hep-ph].
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+Energy Physics,” arXiv:2204.03381 [quant-ph].
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+(1977) 3031–3039.
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+principle,” Z. Phys. 47 (1928) 631–651.
+[22] For the case of static sources, partial destruction of the
+chiral condensate in Schwinger model was studied in [?
+].
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+QCD vacuum in quark fragmentation,” Phys. Lett. B
+366 (1996) 311–315, arXiv:hep-ph/9506412.
+[24] “A Measurement of Quark Spin Correlations in
+Hadronic Z Decays,”.
+[25] M. Bruno, The energy scale of the 3-flavour Lambda
+parameter. PhD thesis, Humboldt-Universit¨at zu Berlin,
+Mathematisch-Naturwissenschaftliche Fakult¨at, 2016.
+
+7
+Supplementary Material
+In the main text, we study the evolution of the Schwinger model Hamiltonian in the presence of external charges
+moving on the light-cone. In this supplemental material, we show that despite the relatively modest lattice sizes,
+the volume dependence and effect of open-boundary conditions are well under control for the quantities and set of
+parameters we studied.
+0.0
+0.5
+1.0
+1.5
+νt - ν0
+N=6, Λ=3
+N=20
+16
+12
+8
+0
+2
+4
+6
+8
+10
+12
+Eele,t - Eele,0
+a g2 / 2
+0
+2
+4
+6
+8
+10
+0.4
+0.6
+0.8
+1.0
+t / a
+SEE,t
+-0.32
+-0.30
+-0.28
+-0.26
+-0.24
+-0.22
+νn [1/a]
+m=0.25/a
+g=0.5/a
+0
+5
+10
+15
+20
+0.05
+0.10
+0.15
+0.20
+0.25
+n
+Eele,n [1/a]
+-0.47
+-0.46
+-0.45
+-0.44
+-0.43
+νn [1/a]
+m=1/a
+g=1/a
+periodic, dynamical
+open, Gauss' law
+0
+2
+4
+6
+8
+10
+1.7
+1.8
+1.9
+2.0
+2.1
+2.2
+n
+Eele,n [10-3/a]
+FIG. 4.
+(Left) Time evolution of total electric field energy, mass creation, and entanglement entropy for periodic boundary
+condition with dynamical gauge field with N = 6 and Λ = 3 (black dotted) versus open boundary condition with gauge field
+fixed by the Gauss’ law with lattice size from 8(red) to 20 (purple). (Middle) Comparison of local electric field energy and
+chiral condensate. Black dotted lines are determined by the bulk values. In both left and middle panels, parameters are set to
+be N = 20, m = 0.25/a, and g = 0.5/a. (Right) Same as middle but with parameters with parameters N = 10, m = 1/a, and
+g = 1/a. Red dots correspond to open boundary condition with gauge field fixed by the Gauss’ law, whereas black lines are for
+periodic boundary condition with dynamical gauge field.
+In the left-hand side of Fig. 4, plain colored lines show the time evolution of the chiral condensate, electric field
+energy and entanglement entropy for different lattice sizes. The maximal time until which a simulation is meaningful
+is set by half the lattice site plus one unit of time, as after this the point sources exit the system. As illustrated by
+the agreement of the different curves, finite size effects are minimal.
+We also assess the effect of using open-boundary conditions. We expect that the introduction of a physical boundary
+to have the same effect as the introduction of a defect. Excitations localize on the boundary and affect the system in a
+“boundary zone” of order the correlation length of the system, see for instance [25]. We can see in the middle panel of
+Fig. 4 that this is indeed what happens. We show in the upper(lower) panel the value of the chiral condensate(electric
+energy density) as a function of lattice sites in the ground state. In both cases, we can clearly observe a boundary
+zone extending over approximately 4-5 lattice sites.
+It also matches the naive estimate of the correlation length
+ξ ∼ 1
+m = 4a.
+To further crosscheck our results, we also decided to implement simulations with periodic-boundary conditions,
+χN+1 = χ1 and χ†
+N+1 = χ†
+1, and to keep the gauge field as independent operators. The Hamiltonian reads
+HPBC = 1
+8a
+N
+�
+n=1
+�
+(Un + U †
+n) ⊗ (XnXn+1 + YnYn+1) + i(Un − U †
+n) ⊗ (XnYn+1 − YnXn+1)
+�
++ m
+2
+N
+�
+n=1
+(−1)nZn + a g2
+2
+N
+�
+n=1
+L2
+n + 1
+g
+N
+�
+n=1
+j1
+ext(xn)φn ,
+(22)
+where XN+1 ≡ (−1)
+N
+2 X1
+�N−1
+m=2 Zm, and likewise for YN+1. We implement the electric-field operator and the link
+
+8
+operator as
+Ln =
+Λ
+�
+ϵ=−Λ
+ϵ |ϵ⟩n ⟨ϵ|n ,
+(23)
+Un = |Λ⟩n ⟨−Λ|n +
+Λ−1
+�
+ϵ=−Λ
+|ϵ⟩n ⟨ϵ + 1|n ,
+(24)
+where Λ is a cutoff [15], the eigenbasis |ϵ⟩n of electric field operator Ln.
+The size of the discrete Hilbert space for a truncation Λ is (2Λ + 1)N 2N, namely it is (2Λ + 1)N times larger than
+in the case of open boundary conditions after integrating out the gauge fields through Gauss law. This also means
+that only smaller lattices can be simulated in this set-up.
+We show results of the chiral condensate and electric field energy for N = 6 and Λ = 3 as black dotted lines in the
+left-hand side of Fig. 4. No deviations from the open-boundary conditions can be seen.
+We also investigated the space-dependence of observables. In particular, we expect the bulk value of the open-
+boundary conditions to equal the periodic boundary condition average. Unfortunately, we could not directly verify
+this for the parameters used in the main text as the lattice size required are not achievable not integrating out gauge
+fields. As an alternative, we verified it for a larger mass and larger coupling ma = ga = 1 such that the boundary
+zone is smaller. The results are shown in the right-hand side panel of Fig. 4. Again, the two lattice sites affected by
+the boundary is in agreement with naive expectations. And as expected, the bulk value of the open-boundary system
+matches the value of the periodic one.
+
diff --git a/FtFLT4oBgHgl3EQfGS-3/content/tmp_files/load_file.txt b/FtFLT4oBgHgl3EQfGS-3/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..20342d68a3f38923359a8e2b7c262c134094d190
--- /dev/null
+++ b/FtFLT4oBgHgl3EQfGS-3/content/tmp_files/load_file.txt
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+page_content=' ∗ David Frenklakh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
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+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
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+page_content=' Upton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
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+page_content=' Stony Brook,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' New York 11794-3800,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
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+page_content=' Stony Brook,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' New York 11794-3800,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
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+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, New York, 11794-3840, USA 5Computational Science Initiative, Brookhaven National Laboratory, Upton, New York 11973-5000, USA The production of jets should allow to test the real-time response of the QCD vacuum disturbed by the propagation of high-momentum color charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Addressing this problem theoretically requires a real-time, non-perturbative method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' As a step in developing such an approach, we report here on fully quantum simulations of a massive Schwinger model coupled to external sources representing quark and antiquark jets as produced in e+e− annihilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' It is well known that the Schwinger model [QED in (1 + 1) dimensions] shares many common properties with QCD, including confine- ment, chiral symmetry breaking and the existence of vacuum fermion condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' This allows us to study, for the first time, the modification of the vacuum chiral condensate by the propagating jets, and the quantum entanglement between the fragmenting jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Our results indicate strong entangle- ment between the fragmentation products of the two jets at rapidity separations ∆η ≤ 2 that can potentially be studied in experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Introduction: The discovery of jets played a crucial role in establishing Quantum Chromodynamics (QCD) as the theory of strong interactions, see [1, 2] for reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The production of the initial high momentum partons is a short-distance process that can be described in pertur- bative QCD due to asymptotic freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' However, as the initial partons keep radiating gluons and quark-antiquark pairs as described by QCD evolution equations, the char- acteristic virtuality decreases, and non-perturbative phe- nomena should come into play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In particular, one expects that the propagating color charges will disturb the non-perturbative QCD vacuum, and the corresponding real-time response should contain valuable information about the vacuum structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' More- over, the initial partons should be entangled by the pro- duction process, but whether any trace of this entangle- ment can be found in fragmenting jets is not clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The answers to these questions lie outside of the realm of per- turbative QCD, and finding them requires a real-time, non-perturbative method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Such an approach is enabled by the advent of quantum simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Unfortunately, the case of real (3+1) dimen- sional QCD is still out of reach for the existing quantum hardware, as well as for real-time simulations on classical computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' However one can start developing real-time non-perturbative methods using simpler models in lower number of space-time dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In this respect QED in (1 + 1) dimensions (the Schwinger model [3]) holds a special place: just like QCD, it possesses confinement, chiral symmetry breaking, and fermion condensate [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In the massless fermion limit, the theory is exactly solvable by bosonization, and admits a dual description in terms of a free massive scalar theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In 1974, Casher, Kogut, and Susskind [5] proposed to model quark-antiquark production in e+e− annihilation by coupling Schwinger model to external sources propa- gating along the light cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' An explicit analytical solution of this model has been found in [6, 7], where this setup was also used to de- scribe jet quenching in heavy ion collisions by introducing in-medium scattering of the sources, and the anomalous enhancement of soft photon production in jet fragmenta- tion [8] observed by the DELPHI Collaboration [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' A more realistic extension of this approach is based on a massive Schwinger model, which in the bosonized description is dual to an interacting meson theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In this case, the model is no longer analytically solvable, and so a numerical approach is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The first study of this setup was carried out in [10] using a numerical classical- statistical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Coupling the Schwinger model to an external Yukawa theory has also been used to mimic the propagation of jets through a thermal environment [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Various other aspects of the Schwinger model have also been addressed using quantum simulations, see [12–17] for examples and [18] for a recent review of quantum simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In this work, using the massive Schwinger model cou- pled to external sources, we perform the first fully quan- tum simulation of jet production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In particular, we focus on real-time, non-perturbative effects that have not been studied before: the modification of the vacuum structure and the entanglement between the produced jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The model: We use the massive Schwinger model Hamiltonian in temporal gauge A0 = 0 in the presence arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='11991v1 [hep-ph] 27 Jan 2023 2 of an external current jµ ext describing the produced jets: HC = HC S + HC ext , (1) HC S = � dx �1 2E2 + ¯ψ(−iγ1∂1 + gγ1A1 + m)ψ � , (2) HC ext = � dx j1 extA1 , (3) where Aµ is the U(1) gauge potential, E = − ˙A1 is the corresponding electric field, ψ is a two-component fermionic field, m is the fermion mass, and γµ are two- dimensional γ-matrices satisfying Clifford algebra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' we use ηµν = diag(1, −1) as our metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The superscript C stands for “continuum”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The effect on the theory of the interaction with the external source Hext is to modify Gauss law to ∂1E − j0 = j0 ext .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' (4) with j0 = g ¯ψγ0ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In other words, the theory is gauge invariant up to the presence of the external charge j0 ext;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' the external current is a “defect” of the U(1) gauge trans- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' To mimic production of a pair of jets in e+e− annihila- tion, we choose the external current to represent charges of opposite sign flying apart along the light cone: j0 ext(x, t) = g[δ(∆x − ∆t) − δ(∆x + ∆t)]θ(∆t) , j1 ext(x, t) = g[δ(∆x − ∆t) + δ(∆x + ∆t)]θ(∆t) , (5) where (t0, x0) is the time and position of a point where the jet pair is produced, and ∆x ≡ x−x0 and ∆t ≡ t−t0 are the space and time distance from this position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Note that in principle one could replace the external probe charges by “hard” dynamical fermions, which can, for instance, be produced by short lived pulses of electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' This has been done in [10], where it was found that, at least within the semiclassics, the use of exter- nal charges is a very good approximation to a pair of dynamical relativistic “hard” fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' This motivates us to restrict ourselves to the simpler case of external currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Our goal is to study the modification of the vacuum due to the presence of the external sources (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' To this end, we evolve the ground state of the massive Schwinger model with the time-dependent Hamiltonian (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In or- der to solve this problem, we need to discretize space- time and approximate the theory by a finite-dimensional Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Lattice model: We begin by discretizing space in a lat- tice of N points with lattice spacing a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We choose to work with staggered fermions χn [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We use a non- compact formulation for the U(1) gauge fields, and in- troduce a lattice electric field operator Ln = E(an)/g, a lattice vector potential φn = ag A1(an), and a link opera- tor Un = e−iagA1(an).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We further impose open-boundary conditions (OBC) χN+1 = LN = 0 on the fermion and gauge fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Using the Dirac matrices γ0 = σz, γ1 = i σy, the Hamiltonian is HL(t) = HL S + HL ext(t) , (6) HL S = − i 2a N−1 � n=1 � U † nχ† nχn+1 − Unχ† n+1χn � + ag2 2 N−1 � n=1 L2 n + m N � n=1 (−1)nχ† nχn , (7) HL ext(t) = 1 g N−1 � n=1 j1 ext(a n, t)φn , (8) where the superscript L stands for “lattice”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Even in the presence of point charges, Gauss law is well defined when integrated over a lattice spacing and reads Ln − Ln−1 − Qn = 1 g � (n+1/2)a (n−1/2)a dx j0 ext(x, t) , (9) with Qn = χ† nχn (1 − (−1)n) the lattice charge density operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' For the rest of this work, we insert the sources at the center of our lattice, x0 = a � N+1 2 � , at time t0 a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Before proceeding with the time evolution, we take advantage of the fact that the gauge fields are non- dynamical in (1+1) dimensions to express them in terms of fermionic operators through Gauss law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' This has the advantage of drastically reducing the size of the discrete Hilbert space needed down to 2N, at the cost of intro- ducing non-localities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The former turns out to outweigh the latter for the method we use (direct diagonalization, or “exact diagonalization” of the Hamiltonian), see also the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We then use the remaining freedom to perform a space- only dependent gauge transformation to set all gauge links to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The explicit gauge transformation which achieves this result is Ω1 = 1, Ωn = �n−1 i=1 U † i [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Note that the existence of such a transformation is a pecu- liarity of (1 + 1) dimensions and is related to the fact that the gauge field is not dynamical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We then rewrite Ln = Ldyn,n + Lext,n and solve Gauss law (9) as follows: Ldyn,n = n � i=1 Qi , (10) Lext,n(t) = −θ � t − t0 − ���x − x0 + a 2 ��� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' (11) The non-locality is contained in the dynamical gauge field and the external sources create a chain of electric fluxes between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The Hamiltonian (6) is now directly suitable for di- agonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' However, having in mind future quan- tum computing applications, we have used an equivalent form in terms of Pauli matrices X, Y, Z, or “spin” degrees 3 t/a A B 2- 3- 4- 5- 6- 7- 8- 9- 10- source fermion anti-fermion electric charge dynamical electric field 0 5 10 15 Eele,t - Eele,0 a g2 / 2 external only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='5 νt - ν0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='0 SEE 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='1 t / a Qt FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' (Left) Time evolution of the local charge density (vertical bars) and of the electric field (arrows), with vacuum expectation values subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Black(white) even(odd)-sites correspond to (anti)fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The position of the external sources is shown above each configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' From top to bottom, the rows are for time values (in units of lattice spacing a) t/a = 2− to 10−, where n− ≡ n − ε with ε being an arbitrarily small positive number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' (Right) (from top to bottom) Time evolution of electric energy, scalar fermion density, entanglement entropy, and electric charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Dotted lines in the first panel show the electric energy generated by the external sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We employ the Jordan–Wigner transforma- tion [21] χn = Xn − iYn 2 n−1 � j=1 (−iZj), χ† n = Xn + iYn 2 n−1 � j=1 (iZj), (12) to obtain HL(t) = 1 4a N−1 � n=1 (XnXn+1 + YnYn+1) + m 2 N � n=1 (−1)nZn + ag2 2 N−1 � n=1 (Ldyn,n + Lext,n(t))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' (13) Our simulations then proceed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We start by finding the ground state |Ψ0⟩ of the usual massive Schwinger model HL(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We then compute the state |Ψt⟩ = T e−i � t 0 HL(t′)dt′ |Ψ0⟩ corresponding to the evolu- tion under the time-dependent Hamiltonian HL(t), with T being the time-ordering operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The system is ef- fectively “quenched” at t a = t0 a = 1, when the external sources are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We then compute different time- dependent expectation values ⟨O⟩t ≡ ⟨Ψt| O |Ψt⟩ where O are the operators corresponding to observables of in- terest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Vacuum modification and quantum entanglement be- tween the jets: We measure the local electric charge den- sity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' the total electric charge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' the scalar fermion density ⟨ ¯ψψ⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' the local electric field strength,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' and the electric field energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' that are given respectively by qn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='t ≡ ⟨ψ†(a n)ψ(a n)⟩t = ⟨Zn⟩t + (−1)n 2a ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' (14) Qt ≡ � ⟨ψ†(x)ψ(x)⟩t dx = a N � n=1 qn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' (15) νn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='t ≡ ⟨ ¯ψ(a n)ψ(a n)⟩t = (−1)n⟨Zn⟩t 2a ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' (16) νt ≡ � ⟨ ¯ψ(x)ψ(x)⟩t dx = a N � n=1 νn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' (17) Πn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='t ≡ ⟨E(a n)⟩t = g ⟨Ln⟩t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' (18) Eele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='t ≡ 1 2 � ⟨E2(x)⟩t dx = a g2 2 N−1 � n=1 ⟨L2 n⟩t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' (19) We also compute the entanglement entropy between the left- and the right-hand sides of the chain SEE(t) = −TrA(ρt,A log ρt,A), (20) with A = {1, · · · , N/2} and B = {N/2 + 1, · · · , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The operator ρt,A = TrBρt is the partial trace of the time dependent density matrix ρt ≡ |Ψt⟩ ⟨Ψt| over B [see il- lustration in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 1(left)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 1, we show the time evolution of local and global observables respectively, for parameters N = 20, m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='25/a, and g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='5/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In the left panel, we show the full time evolution of our quantum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We ob- serve that both the gauge fields and the fermion fields 4 are excited by the external sources, and their effects are constrained within the light cone spanned by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In the right panel, we observe a step-like increase in electric field energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The growth of νt − ν0 shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 1 indi- cates destruction of the (negative) vacuum chiral conden- sate ν0 by the propagating jets [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' This destruction is due to the pair production from the vacuum that also re- sults in the screening of the electric energy which appears smaller than the contribution from external sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Since we can access the entire quantum state, we are able to compute also for the first time the entanglement entropy between the jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The growth of this entangle- ment entropy (third panel) results from the pair creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Lastly, as a consistency check, we also show in the lower panel the total electric charge, which remains zero, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Observing quantum entanglement between the jets: With an eye towards possible experimental studies of quantum entanglement between the produced jets, we measure the two-point correlation of scalar fermion den- sity operators with the vacuum expectation value sub- tracted, ⟨∆νN/2+ℓ ∆νN/2+1−ℓ⟩, (21) where ∆νn ≡ νn − ⟨νn⟩vac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The motivation behind this study is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In the bosonization dictionary of the massive Schwinger model, the correlation between the scalar fermion densi- ties translates into the correlation among the boson pairs (and higher order correlations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Therefore we hope that this correlation function may be used to infer informa- tion about quantum entanglement between the pion pairs produced in jet fragmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' A concrete proposal of an observable correlation between pion pairs produced in jet fragmentation has been put forward in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' To isolate the effect of entanglement between the jets, we measure the correlation function for the cases of cor- related and uncorrelated sources of fermion-antifermion pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Because the entanglement should stem from the correlation between the sources, the case of uncorrelated sources provides the classical baseline for the correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' … … 1 2 l=3 (a) correlated: (b) left: (c) right: FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Illustration of correlated and uncorrelated measure- ments of two point correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The uncorrelated setup is obtained as an uncorrelated linear superposition of jets created by a single (anti)fermion source moving to the (left)right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 0 1 2 3 4 l = 3 5 7 9 4 6 8 correlated 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='4 t / a uncorrelated 102 × 〈ΔνN/2+l ΔνN/2+1-l〉 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='5 0 1 2 3 4 ηs uncorrelated correlated t = 10 a 102 × 〈Δν-ηs Δν+ηs〉 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Time evolution of two-point correlation functions with various separations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The upper(lower) panel is for a correlated(uncorrelated) setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The large difference between the two cases is a signature of quantum entanglement in the produced pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' (Insert) Spatial-rapidity dependence of the two-point correlation at the end of the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Our method of preparation of two uncorrelated quan- tum systems is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 2 (b, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In one of these systems, there is only an antifermion source mov- ing to the left while the fermion source sits still at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We denote the quantum state of such a sys- tem as |ψL⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We then define its counterpart, |ψR⟩, corresponding to the setup of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 2(c), with fermion source moving to the right and the antifermion source fixed at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The uncorrelated state is defined as the superposition of left and right state with a ran- dom phase, |ψuncorr⟩ = 1 √ 2 |ψL⟩ + eiϕ √ 2 |ψR⟩, and the ex- pectation value of any observable is obtained by aver- aging over this random phase, ⟨⟨ψuncorr|O|ψuncorr⟩⟩ ≡ � ⟨ψuncorr|O|ψuncorr⟩ dϕ 2π = ⟨ψL|O|ψL⟩ 2 + ⟨ψR|O|ψR⟩ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The correlation function (21) is designed to measure the points that are symmetric with respect to the jet production vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We measure the two-point correlation function with different separation distances as functions of time, and the results are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We find that the correlation functions measured for the correlated state are an order of magnitude greater than those for the uncorrelated state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Note that it is non-zero in the latter case because of the classical correlation between the par- ticle production in left- and right-moving jets which is similar to the correlation that would be induced by the propagation of sound along the jets’ axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Meanwhile, for the quantum correlated state, we ob- serve the propagation of a similar pattern for odd ℓ’s and 5 similarly for even ℓ’s, which is driven by the correlated moving sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' After a sufficiently large time, we take a snapshot and present the space dependence of the cor- relation functions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 3 (insert), where we have con- verted the site separation to spatial rapidity separation, ηs ≡ arctanh z t = arctanh (ℓ−1/2)a t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' One can clearly see a big difference between the strong quantum correlation for the quantum state and the near absence of correlations for the uncorrelated baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' This difference is especially pronounced for moderate ra- pidity separations ∆ηs = 2ηs ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Using the approxi- mate equality of space-time and momentum space rapidi- ties in jet fragmentation, this suggests that one should look for quantum entanglement among the pions pro- duced in the fragmentation of the two jets at rapidity separation ∆η ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' An observation of correlations among these pion pairs would constitute a direct signature of en- tanglement between the jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Specifically, it would be interesting to study the quan- tum correlations between the “handedness” of the pion pairs produced in the fragmentation of the quark and antiquark jets [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Some hints of such correlations had been reported in the data from DELPHI Collabo- ration [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' To summarize, we have performed a real-time, non- perturbative study of jet fragmentation using a massive Schwinger model with external sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Strong distortion of the vacuum chiral condensate by the propagating jets has been observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We have also found strong quantum entanglement between the fragmenting jets for rapidity separation ∆η ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We hope that this result will moti- vate dedicated experimental studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Our work also paves the way for quantum simulations of jet fragmentation us- ing quantum hardware;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' we plan to address this problem in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' ACKNOWLEDGEMENT We thank Jo˜ao Barata, Fangcheng He, Yuta Kikuchi, Semeon Valgushev, Tzu-Chieh Wei, and Ismail Zahed for useful discussions and communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' This work was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Department of Energy, Of- fice of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Ad- vantage (C2QA) under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='DE-SC0012704 (AF, KI, VK), and the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Department of Energy, Office of Science, Office of Nuclear Physics, Grants Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' DE- FG88ER41450 (DF, DK, SS) and DE-SC0012704 (AF, DK, KY).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Department of Energy under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' DE-AC02-05CH11231 using NERSC award NERSC DDR-ERCAP0022229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' ∗ aflorio@bnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='gov † david.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
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+page_content=' [24] “A Measurement of Quark Spin Correlations in Hadronic Z Decays,”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
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+page_content=' Bruno, The energy scale of the 3-flavour Lambda parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' PhD thesis, Humboldt-Universit¨at zu Berlin, Mathematisch-Naturwissenschaftliche Fakult¨at, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 7 Supplementary Material In the main text, we study the evolution of the Schwinger model Hamiltonian in the presence of external charges moving on the light-cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In this supplemental material, we show that despite the relatively modest lattice sizes, the volume dependence and effect of open-boundary conditions are well under control for the quantities and set of parameters we studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='5 νt - ν0 N=6, Λ=3 N=20 16 12 8 0 2 4 6 8 10 12 Eele,t - Eele,0 a g2 / 2 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
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+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='0 t / a SEE,t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
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+page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='22 νn [1/a] m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='25/a g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='5/a 0 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='25 n Eele,n [1/a] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content="43 νn [1/a] m=1/a g=1/a periodic, dynamical open, Gauss' law 0 2 4 6 8 10 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='2 n Eele,n [10-3/a] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' (Left) Time evolution of total electric field energy, mass creation, and entanglement entropy for periodic boundary condition with dynamical gauge field with N = 6 and Λ = 3 (black dotted) versus open boundary condition with gauge field fixed by the Gauss’ law with lattice size from 8(red) to 20 (purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' (Middle) Comparison of local electric field energy and chiral condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Black dotted lines are determined by the bulk values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In both left and middle panels, parameters are set to be N = 20, m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='25/a, and g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='5/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' (Right) Same as middle but with parameters with parameters N = 10, m = 1/a, and g = 1/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Red dots correspond to open boundary condition with gauge field fixed by the Gauss’ law, whereas black lines are for periodic boundary condition with dynamical gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In the left-hand side of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 4, plain colored lines show the time evolution of the chiral condensate, electric field energy and entanglement entropy for different lattice sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The maximal time until which a simulation is meaningful is set by half the lattice site plus one unit of time, as after this the point sources exit the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' As illustrated by the agreement of the different curves, finite size effects are minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We also assess the effect of using open-boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We expect that the introduction of a physical boundary to have the same effect as the introduction of a defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Excitations localize on the boundary and affect the system in a “boundary zone” of order the correlation length of the system, see for instance [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We can see in the middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 4 that this is indeed what happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We show in the upper(lower) panel the value of the chiral condensate(electric energy density) as a function of lattice sites in the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In both cases, we can clearly observe a boundary zone extending over approximately 4-5 lattice sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' It also matches the naive estimate of the correlation length ξ ∼ 1 m = 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' To further crosscheck our results, we also decided to implement simulations with periodic-boundary conditions, χN+1 = χ1 and χ† N+1 = χ† 1, and to keep the gauge field as independent operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The Hamiltonian reads HPBC = 1 8a N � n=1 � (Un + U † n) ⊗ (XnXn+1 + YnYn+1) + i(Un − U † n) ⊗ (XnYn+1 − YnXn+1) � + m 2 N � n=1 (−1)nZn + a g2 2 N � n=1 L2 n + 1 g N � n=1 j1 ext(xn)φn , (22) where XN+1 ≡ (−1) N 2 X1 �N−1 m=2 Zm, and likewise for YN+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We implement the electric-field operator and the link 8 operator as Ln = Λ � ϵ=−Λ ϵ |ϵ⟩n ⟨ϵ|n , (23) Un = |Λ⟩n ⟨−Λ|n + Λ−1 � ϵ=−Λ |ϵ⟩n ⟨ϵ + 1|n , (24) where Λ is a cutoff [15], the eigenbasis |ϵ⟩n of electric field operator Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The size of the discrete Hilbert space for a truncation Λ is (2Λ + 1)N 2N, namely it is (2Λ + 1)N times larger than in the case of open boundary conditions after integrating out the gauge fields through Gauss law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' This also means that only smaller lattices can be simulated in this set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We show results of the chiral condensate and electric field energy for N = 6 and Λ = 3 as black dotted lines in the left-hand side of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' No deviations from the open-boundary conditions can be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We also investigated the space-dependence of observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In particular, we expect the bulk value of the open- boundary conditions to equal the periodic boundary condition average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Unfortunately, we could not directly verify this for the parameters used in the main text as the lattice size required are not achievable not integrating out gauge fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' As an alternative, we verified it for a larger mass and larger coupling ma = ga = 1 such that the boundary zone is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The results are shown in the right-hand side panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Again, the two lattice sites affected by the boundary is in agreement with naive expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' And as expected, the bulk value of the open-boundary system matches the value of the periodic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
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+IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
+1
+IMKGA-SM: Interpretable Multimodal
+Knowledge Graph Answer Prediction via
+Sequence Modeling
+Yilin Wen, Biao Luo, Senior Member, IEEE, and Yuqian Zhao
+Abstract—Multimodal knowledge graph link prediction aims to improve the accuracy and efficiency of link prediction tasks for
+multimodal data. However, for complex multimodal information and sparse training data, it is usually difficult to achieve interpretability
+and high accuracy simultaneously for most methods. To address this difficulty, a new model is developed in this paper, namely
+Interpretable Multimodal Knowledge Graph Answer Prediction via Sequence Modeling (IMKGA-SM). First, a multi-modal fine-grained
+fusion method is proposed, and Vgg16 and Optical Character Recognition (OCR) techniques are adopted to effectively extract text
+information from images and images. Then, the knowledge graph link prediction task is modelled as an offline reinforcement learning
+Markov decision model, which is then abstracted into a unified sequence framework. An interactive perception-based reward
+expectation mechanism and a special causal masking mechanism are designed, which “converts” the query into an inference path.
+Then, an autoregressive dynamic gradient adjustment mechanism is proposed to alleviate the insufficient problem of multimodal
+optimization. Finally, two datasets are adopted for experiments, and the popular SOTA baselines are used for comparison. The results
+show that the developed IMKGA-SM achieves much better performance than SOTA baselines on multimodal link prediction datasets of
+different sizes.
+Index Terms—Knowledge graph, link prediction, multimodal, interpretability, sequence modeling, reinforcement learning.
+!
+1
+INTRODUCTION
+T
+HE knowledge graph is the technology and tool for
+carrying and representing background knowledge. It
+structures knowledge in the real world into entities and
+relations in the form of graphs and organizes them into
+networks. In a knowledge graph, knowledge data is rep-
+resented in the form of triples (h, r, t). Among them, h is
+the head entity, r is the relation connecting two entities, and
+t is the tail entity. Knowledge graphs are used in various
+artificial intelligence tasks in different domains [1], such
+as named entity disambiguation [2] in natural language
+processing [3], visual relation detection [4] or collaborative
+filtering [5]. However, it is well known that even state-of-
+the-art knowledge graphs are often incomplete (i.e., lack
+real facts or contain false facts). Therefore, machine learning
+algorithms aimed at addressing this problem attempt to
+infer missing triplets from observed connectivity patterns,
+a task is known as link prediction [6]. For example, given a
+head entity and a relation (h, r), predict a tail entity t.
+In order to solve the problem of link prediction, exist-
+ing problems can be divided into four categories: deduc-
+tive logic and rules, reasoning based on graph structure,
+knowledge graph embedded representation and deep neu-
+ral network model. Rule-based reasoning methods, such as
+AMIE [7], AnyBURL [8], transform natural language queries
+into combinations of logical operators, express such queries
+through combinations, and then implement in a specific
+programming language to get the query. These methods are
+•
+Y.Wen, B.Luo and Y. Zhao are with the School of Automation, Central
+South University, Changsha, China.
+E-mail: yilinwen510@gmail.com, biao.luo@hotmail.com, zyq@csu.edu.cn
+•
+Corresponding Author: Biao Luo.
+accurate and interpretable, but require experts to formulate
+a large number of inference rules, and have poor generaliza-
+tion ability for unknown rules. Reasoning based on graph
+structure has two features: one is the path feature, and
+the representative algorithm is PRA [9]. The path features
+between nodes are extracted by graph traversal algorithm
+or random walk method, and the node connections are
+predicted by path features. Its characteristic is to provide
+path interpretability while reasoning, and the problem is
+that it is difficult to solve the problem because the reasoning
+nodes are not connected. The second is a graph-structure-
+based approach that utilizes a message-passing mechanism
+to extract the structural information of target entities and
+provide subgraph interpretability, and the representative
+algorithm is DeepPath [10]. However, because the knowl-
+edge graph is usually very large, it is extremely compli-
+cated to traverse all the subgraph structures in the graph.
+The knowledge graph embedding representation method
+is to embed the high-dimensional and discrete data of the
+knowledge graph into a low-dimensional continuous vector
+space by designing a certain scoring function, and then
+representing the entities and relations as numerical vectors
+to calculate. Its representative model is the TransE type, for
+example, TransE [11], TransH [12], TransD [13], TransR [14].
+The recent research is bilinear models, e.g., RESCAL [15],
+DisMult [16], TuckER [17], and ComplEx [18]. Its method
+is characterized by a shallow neural network, and the se-
+mantic representation of the knowledge graph is realized
+through a specific structure of the embedded space. The
+deep neural network model, e.g., CoKE [19], ConvE [20],
+is designed by designing entities and relations into query
+pairs, matching query pairs with entities relations, and
+arXiv:2301.02445v1 [cs.AI] 6 Jan 2023
+
+IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
+2
+Table 1
+Summary of Existing Methods for Knowledge Graph Link Prediction
+reasoning algorithm
+logical rules
+graph structure
+knowledge graph embedding
+deep neural network
+reinforcement learning
+interpretability
+✓✓✓✓
+✓✓
+✓
+-
+✓✓✓✓
+performance
+✓
+✓✓
+✓✓✓✓
+✓✓✓✓
+✓✓
+robustness
+✓✓✓✓
+✓✓✓
+✓
+✓
+✓✓
+scale
+✓
+✓
+✓✓✓
+✓✓
+✓✓
+expert experience
+dependence
+partial dependence
+no dependence
+no dependence
+no dependence
+obtaining inference similarity scores through deep neural
+networks to make inference judgments. Both the knowledge
+graph embedding model and the deep network model are
+regarded as neural network models, and the same point
+is that they both design a scoring function, and use the
+gradient backpropagation method for training in a data-
+driven manner. Its advantage is that its generalization per-
+formance is relatively better, and it effectively alleviates
+the problem of graph structure dimensionality disaster. Its
+disadvantage is that it only sees the similarity between
+input and output values, lacks interpretability, and performs
+single-step reasoning. In summary, as shown in Table 1, it is
+found that the methods based on logical deduction rules and
+graph structure are all symbol-based methods, which have
+better interpretability but poor generalization performance.
+Otherwise, based on knowledge graph embedding and deep
+neural network model, its generalization performance is
+better, but it lacks interpretability. Therefore, studying how
+to integrate symbolist and connectionist models is the key
+to obtaining an interpretable knowledge graph reasoning
+model.
+With the development of deep learning, the model struc-
+ture of knowledge reasoning methods is becoming more
+and more complex. Because it is difficult for users to have
+an intuitive understanding of the parameters, structure and
+characteristics in such models, and they also have less un-
+derstanding of the decision-making process and reasoning
+basis, it is difficult for users to trust the prediction results
+of the model. Therefore, in order to establish trust between
+users and reasoning models and balance the contradiction
+between model accuracy and interpretability, multi-hop rea-
+soning methods are used to solve explainable knowledge
+reasoning [21]. The rationale of the multi-hop reasoning
+method is to imitate the multi-hop thinking of human
+beings. A common approach is to apply reinforcement
+learning frameworks to multi-hop reasoning in knowledge
+graphs. Reinforcement learning is a model that has received
+a lot of attention in the past ten years and has been widely
+used in control [22], games [23], and robots [24]. It models a
+learning process as a Markov process and trains the model
+by maximizing long-term cumulative rewards through the
+interaction between the agent and the environment. Mod-
+elling the knowledge map as a reinforcement learning pro-
+cess not only gets the result of reasoning, but also obtains
+the path of reasoning, and explains the reasoning of the
+knowledge graph through the reasoning path. The specific
+fusion method is to regard the knowledge graph as an
+environment, model the agent as a deep neural network,
+combine the advantages and disadvantages of symbolism
+and connectionism, and make the model have both the gen-
+eralization performance and path interpretability of neural
+networks. Methods based on reinforcement learning such as
+DeepPath [10], MINERVA [25], DIVINE [26], and AttnPath
+[27], however, generally have the shortcomings of slow
+convergence and low accuracy, and most of them are inferior
+to some traditional methods. The reason for this may due
+to the sparse rewards of reinforcement learning. Moreover,
+the sparse rewards, sparse data, and insufficient exploration
+of knowledge graphs make reinforcement learning more
+difficult and challenge in multimodal knowledge graph
+reasoning tasks [28]. Therefore, it is meaningful and promis-
+ing to improve the accuracy of reinforcement learning in
+knowledge graph reasoning.
+Recently, the cross-border application of Transformer
+[29] has attracted wide attention, and it has made break-
+throughs in image classification [30], semantic segmenta-
+tion [31], object detection [32] and other fields. Currently,
+Transformer has been employed as a pre-training model
+in offline reinforcement learning, e.g., Decision Transformer
+[33], Trajectory Transformer [34], and Gato [35], etc. These
+methods regard the data of reinforcement learning as a
+string of unstructured sequence data and train with super-
+vised or self-supervised learning methods. It avoids the un-
+stable gradient signal in traditional reinforcement learning
+and performs better in offline reinforcement learning. Deep
+reinforcement learning is a sequential process, therefore,
+the process of multi-hop reasoning is handled by state-
+of-the-art reinforcement learning sequence models, which
+may achieve better results than traditional reinforcement
+learning.
+For knowledge graph reasoning tasks that are complex
+and have the concept of multimodal data, the core idea of
+most existing knowledge graph reasoning algorithms is to
+reason by integrating existing triple structure knowledge,
+so knowledge of the entity is often ignored. However, in-
+formation about entities themselves is usually beneficial for
+link prediction tasks, such as image and textual information.
+As shown in Fig. 1, for example, when performing the
+triple < shoes, style, ? > link prediction task, the answer
+is predicted based on the triple < dress, style, sweet >
+of the similar head entity image, and finally, answer ? is
+sweet. It is worth noting that the text information in the
+image also contains a lot of knowledge, especially when the
+knowledge graph is applied to the e-commerce field, the
+text in the product image is often the brand information of
+the product. Therefore, to address multimodal explainable
+knowledge graph reasoning tasks with high efficiency and
+high performance, a new sequential model IMKGA-SM
+for reinforcement learning is developed, where a reward
+mechanism is designed based on perceptual interaction and
+fine-grained multimodal information extraction.
+
+IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
+3
+Figure 1. When performing triple < shoes, style, ? > link prediction
+tasks, the answer is predicted based on triples < dress, style, sweet >
+which is similar to the head entity image, and finally it is concluded that
+? is sweet.
+2
+RELATED WORKS
+2.1
+Single-modal Knowledge Graph Reasoning
+Single-modal knowledge graph reasoning mainly revolves
+around relational reasoning. The AMIE [7] and AMIE+ [36]
+algorithms are derived from the early inductive logic pro-
+gramming system [37], emphasizing automatic rule learning
+methods. It has strong interpretability, however, all the
+above methods require expert design rules. Graph structure-
+based reasoning methods (e.g., path ranking algorithm [9])
+are also used to tackle such problems, which is interpretable
+but computationally intensive and time-consuming. The
+embedding-based methods include TransE [11], ConvE [20],
+RotatE [38], and TuckER [17]. Each of these models is simple
+and the training speed is fast, but they are not interpretable.
+Reasoning methods based on neural networks include neu-
+ral tensor networks [39], R-GCN [40], implicit ReasoNet
+[41], etc. They are able to learn to reason through implicit
+processing, which results in poor interpretability and unsta-
+ble performance. In addition, there are typical reinforcement
+learning methods, e.g., DeepPath [10], MINERVA [25], RLH
+[42], GussuianPath [43], etc, which have better interpretabil-
+ity and inference performance than representation learning-
+based methods, but the disadvantage is that the effect is
+poor.
+2.2
+Multimodal Knowledge Graph Link Prediction
+Compared with the single-modal knowledge graph link
+prediction task, the main contribution of the multi-modal
+knowledge graph link prediction task is to integrate multi-
+modal data knowledge into the plain text knowledge graph.
+In multimodal knowledge graph link prediction tasks, it
+is very necessary to combine the textual semantics of en-
+tities with multimodalities, such as semantics, vision, and
+hearing. IKRL [44] is the first knowledge representation
+model that includes image information. For each entity, it
+learns two different representations based on triple struc-
+ture information and image information, respectively. DKRL
+[45] is a knowledge representation for fused descriptions.
+Similar to the IKRL model, the DKRL model also learns a
+representation based on structural information and a rep-
+resentation based on text descriptions for each entity. Based
+on the single-modal knowledge graph link prediction model
+TransE [11], an autoencoder is employed in TransAE [46] to
+jointly encode visual information and text information to
+obtain the vector representation of entities. RSME [47] is a
+multimodal knowledge graph reasoning model based on the
+traditional knowledge graph embedding model ComplEx
+[18]. However, most of these multimodal approaches are
+uninterpretable and with low accuracy.
+2.3
+Reinforcement Learning with Transformers
+In [33], Decision Transformer is proposed by modelling
+reinforcement learning tasks as a sequence framework trans-
+former, based on which SQUIRE [48] is employed to han-
+dle single-modal knowledge graph link prediction. How-
+ever, these works are deficient in generalization and the
+reward information is underutilized. Based on Decision
+Transformer [33], Trajectory Transformer [34] uses the beam
+search for model-based planning, while generating new
+trajectories is too complicated. Therefore, a simple ran-
+dom masking mechanism is proposed in this paper, which
+achieves the effect of data enhancement by randomly mask-
+ing historical actions that have been generated in the past.
+Recently, Deepmind proposed a general agent, i.e., Gato
+[35], which made a further breakthrough in multimodal
+tasks. It is promising and potential to extend this model to
+multi-modal multi-hop reasoning.
+3
+METHODOLOGY
+In this section, the overall framework of IMKGA-SM is
+introduced, which treats the multi-hop reasoning problem
+as a sequence-to-sequence task derived from regression
+modelling trajectories and applies it to the task of multi-
+modal link prediction. The hybrid transformer architecture
+of IMKGA-SM mainly includes five stacked modules. (1)
+The underlying multimodal feature extraction module, as
+shown in Fig. 2, aims to obtain basic structural information,
+image information, and text information in images from
+databases and images, and combine the three as a state
+feature. (2) The reinforcement learning sequence module,
+as shown in the bottom part of Fig. 3. The knowledge graph
+link prediction task is modelled as an offline reinforcement
+learning problem, which is then abstracted into a sequen-
+tial framework. (3) The upper multimodal encoder (fusion
+encoder) module, as shown in Fig. 4, fuses the underlying
+features, reward features based on perceptual interaction,
+and action features through a self-attention mechanism. (4)
+The Mask mechanism module, as shown in the upper part
+of Fig. 3, includes three mechanisms to ensure the input and
+output of the encoder and prevent overfitting. (5) The loss
+function module, adopts an autoregressive self-adjusting
+mechanism to maximize the multi-modal performance, as
+shown in Fig. 5.
+In the following subsections, each module of the
+IMKGA-SM will be analyzed and discussed in details. The
+multimodel feature extraction module and the reinforce-
+ment learning sequence architecture are developed in Sub-
+sections 3.1 and 3.2, respectively. The fusion encoder module
+is proposed in Subsection 3.3. The mask and loss function
+modules are designed in Subsection 3.4 and 3.5, respectively.
+
+Saurin素兰
+Associated
+dress up
+dress
+scene
+Associated
+brand
+style
+scene
+sweet
+style
+pink
+foxrabbit
+shoes
+Applicable
+groups
+femaleIEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
+4
+Figure 2. The multimodal feature extraction module.
+Figure 3. Unified interpretable multimodal knowledge graph sequence
+framework.
+3.1
+Multimodal Feature Extraction Module
+In multimodal knowledge graph tasks with only single
+image data, most of the existing methods only learn simple
+image information. However, many visual scenes contain
+text with key information, so understanding text in images
+is crucial for downstream reasoning tasks, such as product
+brand, price, and consumer population. To jointly learn
+multimodal knowledge and inter-entity relations, knowl-
+edge in a single modality is extracted and combined into
+a multimodal transformer. In this paper, two modalities are
+considered: visual and textual, where text is extracted from
+image information. The multimodal part includes image
+input, text input in the image and query input (i.e. head
+entity, relation). Vgg16 pre-trained on ImageNet is used to
+process the head entity image information of the image
+input. Vgg16 consists of several vgg-blocks and three fully
+connected layers, and the vector output by the last fully
+connected layer is used as the image feature vector.
+For image text input, OCR technology is used for image
+text extraction [49]. Generally, OCR technology consists of
+two steps. (1) Text detection: locate the position of the text in
+the image. (2) Text recognition: identify the positioned text
+area, and convert the text area in the image into character
+information. In this paper, the CTPN method [50] is adopted
+for text detection, and the CRNN method [51] is adopted
+for text recognition. If the image information corresponding
+to the head entity is missing, ∅ is used instead. For the
+Figure 4. The fusion encoder.
+Figure 5. Autoregressive dynamic loss regulation.
+query input part, the knowledge graphs corresponding to
+head entities and relations are encoded to form a vector.
+Eq. 1 models an original multimodal feature φ, which is
+specifically manifested as the fusion of structural informa-
+tion (h, r), image information and text information.
+φ : G × G → G
+(1)
+Here hfig, hocr, h, r ∈ G. Then, let ∗ indicates a grouping
+operation, h represents the structure embedding of the
+head entity, hfig represents the image embedding of the
+head entity, and hocr represents the text embedding of
+the head entity after being extracted by the OCR method.
+Thus, as formalized in Eq. 2, a characteristic entity ˜q of
+φ(h, hfig, hocr) and r is written as :
+˜q = φ (h, hfig, hocr) ∗ r
+(2)
+Multimodal fusion is widely used in the fields of computer
+vision [52] and natural language processing [53]. Since the
+currently most popular transformer framework is adopted
+as the core module of IMKGA-SM, according to the charac-
+teristics of the transformer, the number of parameters in the
+learning process largely determines the operation speed, so
+it is very necessary to process the input features of the trans-
+former. Therefore, the module of the multimodal feature is a
+pre-train of the core transformer framework, which aims to
+filter out irrelevant or redundant features from the original
+data of features. Sepecifically, three self-attention blocks
+are used to receive the outputs of the original multimodal
+feature vector φ, and three autoencoders are used to transfer
+them into a 14-dimensional vector in the end.
+Specifically, first, the original multimodal feature φi
+passes through a fully connected feed-forward network to
+obtain different modal features µi, which consists of two
+linear transformations and a ReLu activation function, via
+Eq. 3.
+µi = conv {ReLU [conv (φi)]} , i = 1, . . . , L
+(3)
+
+(,h,r)
+Attention
+Auto-
+datasets
+Block
+encoder
+.vgg16
+Attention
+Auto-
+Fusion
+Block
+encoder
+Encoder
+CTPN+CRNN
+Attention
+Auto-
+Block
+encoderretun to go input
+multimodalfusioninput
+rule path input
+Ro.R1.Re.R-[,q,r,fig1,fig2.-figgocr1,ocr2,ocr3[,r1,t1,r2,t2,r3,t
+ao
+a1
+at
+aN-1
+an
+Autoregréssive dynamic loss.regulation
+IMKGA-SM
+Fusion Encoder
+3
+2N
+Linear Layer + Norm
+Ro
+Us
+ao
+S1
+ai
+at
+SN-1
+aN-1
+RN
+SN
+aNFeedforward
+Self -Attn
+Linear Layer + Norm
+Reward Feature
+Query Feature
+Image Feature
+OCRFeature
+Action FeatureBac ward
+Feed Forward
+limear
+ocr
+FC
+Encoder
+Sauri
+combine
+Cross-
+Encoder
+FC
+entropy loss
+linear
+image
+FC
+Encoder
+Bac wardIEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
+5
+Then, as Eq. 4 shows, different modal features µi are passed
+through a Softmax layer in order to compute the attention
+of each modality ai.
+ai = Softmax (µi) , i = 1, . . . , L
+(4)
+The sum of these attention weights ai multiplied by the
+original multimodal feature embedding µi is called self-
+attention Qs
+φ, formalized in Eq. 5.
+Qs
+φ =
+�L
+i=1 aiµi
+(5)
+Therefore, Qs
+h, Qs
+hfig, Qs
+hocr of h, hfig, hocr are obtained
+respectively. Qs
+φ is used as a query for the corresponding
+feature to calculate the attention weights guided by Qs
+φ
+and put into the softmax layer. Finally, the weights are
+multiplied by the original modal features φk to get the
+filtered vector. Then, the output of the attention block is
+expressed as gφ via Eq. 6.
+pk = W
+�ReLu
+�WsQs
+φ
+� ◦ ReLu (Wxφk)
+� ,
+sk = Softmax (pk) , k = 1, . . . , N,
+gφ =
+�N
+k=1 skφk, φ ∈ G
+(6)
+After feature gφ is obtained, it is input into the autoencoder
+for dimensionality reduction. The final feature hφ is shown
+in Eq. 7, in which hfig is 8-dimensional, and hocr is 3-
+dimensional.
+hφ = σ (W · φ + b)
+(7)
+This part is used as the definition for the state s of reinforce-
+ment learning, as shown in Fig. 3, which will be described
+in detail below.
+3.2
+Reinforcement Learning Sequence Framework
+In this subsection, an offline reinforcement learning frame-
+work is developed for the knowledge graph link prediction
+task. Then, specific Markov triples are designed, and a
+reward expectation mechanism based on perceptual interac-
+tion is proposed. Finally, the whole reinforcement learning
+process is abstracted into a sequential framework, which is
+the core module of IMKGA-SM.
+3.2.1
+Offline Reinforcement Learning Design
+The knowledge graph link prediction problem is mod-
+elled as a Markov decision process. Markov decision pro-
+cess tuple consists of a state s ∈ S, an action a ∈ A,
+a transition dynamic P (s′ |s, a), and a reward function
+r′. sn, an, and r′
+n are used to denote the state, action,
+and reward at time step n, respectively. A trajectory con-
+sists of a sequence of states, actions, and rewards: τ =
+(s0, a0, r′
+0, s1, a1, r′
+1, . . . , sN, aN, r′
+N). The purpose of rein-
+forcement learning is to learn a policy that maximizes
+the expected return E
+��N
+n=1 r′
+n
+�
+in the Markov decision
+process. In this paper, the process of generating a path
+for each triple link in the knowledge graph is regarded
+as a reinforcement learning segment. Since the data in this
+paper are fixed datasets, new data is difficult to be obtained
+through environmental interaction, so it is regarded as an
+offline reinforcement learning problem.
+3.2.2
+Track Representation
+The inference accuracy of reinforcement learning-based in-
+ference methods is usually much lower than that of tradi-
+tional TransE-based methods. This is because the amount
+of data for offline reinforcement learning is very limited,
+and the rewards for the knowledge graph link prediction
+problem are sparse, resulting in serious decision bias in
+reinforcement learning. Therefore, they are not suitable for
+direct transfer to the task of multimodal knowledge graph
+link prediction. To address this problem, a new reinforce-
+ment learning sequence framework with perceptual interac-
+tion expected reward mechanism is proposed in this sub-
+section. Different from traditional reinforcement learning,
+the reward setting here is the expected reward in the future,
+that is, the maximum reward value expected to be obtained
+in the current state. There are two main differences: (1)
+An expected reward mechanism is proposed to eliminate
+the sparsity of rewards by incorporating the perceptual
+similarity of knowledge graph entities. (2) The multi-modal
+perception interface is introduced into the decision trans-
+former framework for the first time, making full use of
+multi-modal features.
+Through the previous pre-training process, the knowl-
+edge graph link prediction process is transformed into a
+Markov decision process. Its purpose is to find a path to
+the target entity, which means that the pathfinding process
+makes multi-hop reasoning interpretable. Therefore, the
+knowledge graph link prediction task is modelled as an
+offline reinforcement learning task, which is then transferred
+to a sequential framework for solving. Similar to offline
+reinforcement learning, the Markov triplet < ˆR, S, A, Pr >
+of IMKGA-SM is defined as follows.
+State sk: For the knowledge graph triple < h, r, t >
+in the dataset, the state of IMKGA-SM is denoted as
+s = (query, hfig, hocr) ∈ S. Among them, S is the state
+space, and query = (< bos >, h, r) is the query of triples,
+representing the beginning, head entity, and relation, respec-
+tively. hfig and hocr represent the image embedding of the
+head entity and the text embedding in the image, which are
+obtained by Eq. 7.
+Action ak: The action space for a given sn is the set of
+all entities, relations and < eos >. The purpose is to infer
+the path from the head entity h, relation r to the tail entity
+t by generating the action output, and ak is the k-th action
+represented by the k-th token of the path generated by the
+rule. Here, AnyBURL [8] is used as the rule-based method
+to find the path between h and t, r (h, t) → r1 (h, t1) ∧
+r2 (t1, t2)∧. . .∧rn (tn, t), decomposing a single relation into
+a combination of multiple relations and entities. If the model
+makes an error during the prediction process, the inferred
+entity or relationship does not conform to the corresponding
+attribute. Then the rule path is modified to remember only
+the last target entity. Taking three-hops as an example, A =
+(0, ∅, t, 0, 0, 0, 0) is modified as a set of actions.
+Transition Pr: The transition function Pr is set to
+map the current state sn to the next state sn+1. For-
+mally, Pr : S × A → S is defined as Pr (sn, An) =
+Pr (query, hfig, hocr, a0, . . . , an−1). When entering the next
+state, the actions of the previous step are added to the
+previous state as history to realize the state change. The
+
+IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
+6
+Mask mechanism I does the specific step. Therefore, unlike
+the traditional Decision Transformer [33], the state transition
+mechanism Pr (sn, An) is designed to let the model focus
+on the state, actions and return-to-go of the previous steps,
+thereby improving the policy.
+Return-to-go ˆR: Since the purpose of knowledge graph
+link prediction is to reason about the tail entity answer, it
+is impossible to know whether the reasoning is success-
+ful unless the last step of the reasoning path is reached.
+Therefore, there is a phenomenon of sparse rewards in the
+reinforcement learning sequence method when solving the
+knowledge graph link prediction task. In order to solve
+these problems, a reward expectation mechanism based on
+perceptual similarity is designed to learn interactively by
+taking the expected reward of the current state as input.
+The definition of ˆR is the maximum reward expected in
+the current state, so after making an action, the value of the
+next ˆR will decrease (or increase) due to the reward from
+the previous action, ˆRn = �T
+n′=n r′
+n. So ˆR changes with
+state and action. After obtaining the initial triple query =
+(< bos >, h, r) and obtaining the corresponding reasoning
+path according to the rules, the return-to-go corresponding
+to each step ˆRn is obtained. τ(h, r, t) is defined as a set of
+triples in the dataset, and ψ(< bos >, r1, t1, r2, . . . , rn, tn) is
+the path obtained by the rule that satisfies τ. All dataset
+entities and relations are stored in collections E and R.
+Specifically, the generation steps of ˆR are as follows:
+(1) At the initial value (n = 0), when no action is taken,
+the maximum expected reward of the task ˆR0 is expected
+to be able to reach the target entity, which is a fixed value
+defined in Eq. 8.
+ˆR0 = r′
+good
+(8)
+(2) When n = 1, the first action is < bos >, which
+means the beginning. As Eq. 9 shows, the first return-
+to-go ˆR1 is defined according to whether the correct tail
+entity is finally successfully inferred. Here, r′
+good represents
+a positive constant and r′
+bad represents a negative constant.
+ˆR1 =
+�
+r′
+good,
+ψ (tn) = t,
+r′
+bad,
+ψ (tn) ̸= t
+(9)
+(3) For each action with n > 1, return-to-go in n-th step
+is defined as Eq. 10, in which a base penalty r′
+step (negative)
+is generated since as few hops as possible are desired to
+be used. When the current action an is the entity tn, an
+additional reward r′
+addn (positive) will be generated.
+r′
+n = r′
+step + r′
+addn, n > 1
+(10)
+(4) Define r′
+addn as:
+r′
+addn =
+�
+�
+�
+�
+�
+ˆR1 × sim (ftn, ft)
+an ∈ E,
+0,
+an ∈ R
+1
+2 ˆR1,
+emtn = ∅|emt = ∅
+(11)
+Here sim(·) denotes the cosine similarity computed with
+sim (u, v) =
+uT v
+∥u∥·∥v∥, ftn and ft are the image embedding
+of the current and target entities, respectively. From Eq. 11,
+it is noted that the more similar the generated entity action
+is to the target entity, the greater the reward for that action
+should be. If the current action is the relation r′
+n, or the target
+entity or the current recommendation entity has no image
+information, the additional reward r′
+addn is 0. If the entity
+action an has no image information or the target entity
+t has no image information, the similarity value takes an
+intermediate value of 0.5.
+(5) An additional penalty r′
+bad will be imposed if the rec-
+ommended action does not conform to the attribute (entity
+or relation) that should be recommended. When performing
+action an in the current state sn, Eq. 12 defines the next
+Return-to-go input ˆRn as the previous step’s Return-to-go
+ˆRn−1 minus the reward r′
+n−1 caused by action an−1.
+ˆRn =
+� ˆRn−1 − r′
+n−1 − r′
+bad,
+an−1 /∈ E, an−1 /∈ R
+ˆRn−1 − r′
+n−1,
+others
+(12)
+(6) Repeated iteration until the end of the round, if it is
+less than three hops, in order to ensure the same embedding
+length, the last ˆR completion vector will be used to length
+7. The trajectory τ is expressed in Eq. 13.
+τ =
+�
+ˆR1, s1, a1, ˆR2, s2, a2, . . . , ˆRN,sN, aN
+�
+(13)
+3.3
+Fusion Encoder Architecture
+Sequences of token embeddings from the three modes,
+return-to-go, state, and action, are concatenated and fed to
+the transformer. Different from the positional embedding
+of the traditional transformer [29], a time step (return-to-
+go, state) shares the same positional embedding and the
+position of them are processed as a complete sequence.
+Therefore, the process of positional embedding is expressed
+as Eq. 14. Here XU
+pc is the projected embedding vector,
+C is the concat operation, Upos, represents the position
+embedding corresponding to the embedding layer, and
+U =
+�
+C
+�
+ˆR, s
+�
+, a
+�
+.
+XC( ˆ
+R,s)
+τ
+= XC( ˆ
+R,s)
+pc
++ C
+�
+ˆR, s
+�
+pos
+Xa
+τ = Xa
+pc + apos
+(14)
+To avoid increased computational complexity due to long
+concatenated sequences, Eq. 15 models ¯XC( ˆ
+R,s)
+τ
+by adding
+an embedding linear layer LN for each modality such that
+the original input is projected to the embedding dimension,
+followed by layer normalization Sigmoid.
+¯XC( ˆ
+R,s)
+τ
+= Sigmoid
+�
+LN
+�
+XC( ˆ
+R,s)
+pc
+��
+¯Xa
+τ = Sigmoid
+�LN
+�Xa
+pc
+��
+(15)
+These tokens are processed by an encoder model that
+predicts future action tokens via autoregressive modelling.
+Since the multi-hop inference is a fixed-length sequence, the
+transformer’s encoder structure is selected, which consists
+of L stacked blocks. As shown in Fig. 4, each block mainly
+includes two types of sublayers: multi-head self-attention
+MHA and fully-connected feed-forward network FFN.
+The transformer model contains many parameters including
+WV , WQ, WK matrices, and the values in each stack and
+head are designed. As formalized in Eq. 16, multi-head
+attention mechanism Attn is introduced to defined headM ′
+r
+i
+.
+Here, r′ and a represent return-to-go and action features
+
+IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
+7
+in reinforcement learning sequence framework, q represents
+knowledge graph query (< bos >, h, r >) feature, and I
+and O represent the image and OCR features in multimodal
+feature extraction module.
+headMr =Attn
+�xrW r
+Q ,
+�
+xrW r
+K, xqW q
+K, xIW I
+K, xoW o
+K
+�
+,
+�
+xrW r
+V , xqW q
+V , xIW I
+V , xoW o
+V
+��
+(16)
+The calculation method of headMq
+i
+, headMI
+i
+and headMo
+i
+is
+similar to that of headMr
+i
+, where headMa
+i
+is redefined via
+Eq. 17.
+headMa = Attn
+�xaW a
+Q, xaW a
+K, xaW a
+V
+�
+(17)
+Hence, Eqs. 18 and 19 model the hidden state of the encoder
+layer l.
+¯XU
+l = MHA
+�
+LN
+�
+¯XU
+τ
+��
++ XU
+l−1
+(18)
+XU
+l = FFN
+�
+LN
+�
+¯XU
+l
+��
++ ¯XU
+l
+(19)
+Next, with the mask shown in Fig. 3, the encoder only
+focus on previous labels a < k of the current return-to-
+go, multimodal fusion input and output paths. The specific
+details of the mask will be described in the next subsection.
+3.4
+Mask Mechanism Design
+In the link prediction task of the recommendation sys-
+tem, due to feature redundancy, lack of sufficient train-
+ing data and complex model design, the recommendation
+system is extremely prone to the one-epoch phenomenon,
+that is, the over-fitting phenomenon. Therefore, three mask
+mechanisms are designed to overcome the overfitting phe-
+nomenon, and they are also used to solve the input and out-
+put requirements of the reinforcement learning framework
+established in Subsection 3.2.
+Mask mechanism I: As shown in the shaded area in
+Fig. 3, Mask mechanism I is used to ensure the input and
+output of the transformer and realize the Markov decision
+process. Through step-by-step prediction, the path is pre-
+dicted sequentially to obtain the final target entity. When
+predicting the next action, the previous action history will
+be added to the state to achieve state transition, that is,
+the real result of the previous step will be used as input
+to predict the output of the next path token. In this way, the
+context information is effectively used by the model, thereby
+ensuring the accuracy of the model, so that the final result
+will not have a large error due to a one-step error.
+Mask mechanism II: As shown in the blue dots in Fig. 3,
+Mask mechanism II is used to solve the problem of model
+overfitting. After multimodal information dimensionality
+reduction, data of training features is sparse, which easily
+leads to fast immature convergence of the model, resulting
+in overfitting. Therefore, it is necessary to perform data
+dropout on this part of the embedded input. The double
+data dropout mechanism is introduced for data enhance-
+ment, which is conducive to retaining the original high-
+quality samples as much as possible. Specifically, for a given
+sequence, the data loss scheme is enabled with a certain
+probability pk, and when applying the data loss scheme,
+tokens in the sequence are randomly masked with a certain
+probability pm.
+Mask mechanism III: As shown in the red dot in Fig. 3,
+the Mask mechanism III is used to make the model generate
+more new trajectories by itself, which makes the trajectories
+generated by the transformer in the past randomly masked
+out for the next action prediction task such that the model
+gradually learns from the self-generated trajectories. Mask
+mechanism III is simple and easy to implement, and it does
+not add any additional computational cost and parameters.
+In the masking mechanism, as formalized in Eq. 20,
+multiplying each token xi ∈ x by the mask gets its au-
+toregressive log maximum likelihood. Here η represents the
+mask ratio.
+log p (x) = log
+n
+�
+i=1
+p
+�
+xi |[I [mj ≤ η] · xj] i=1
+j=0
+�
+, η ∈ [0, 1]
+(20)
+Mask mechanism I adopts a complete mask, so η = 1.
+Mask mechanism II and III are random masks, which are
+randomly masked according to a certain probability value
+mj.
+3.5
+Loss Function Design
+The multimodal information of entities (features of images
+and text features in images) is expected to enhance learning
+through fusion. However, experiments have found that after
+incorporating multiple modalities, the model will suffer
+from a lack of optimization, which is caused by the dom-
+inance of one mode in some scenarios. For example, image
+information dominates when inferring relations is relevant
+to tasks such as “colour, item type”. When reasoning about
+relations is a relevant task like “brand”, textual information
+in images dominates. Therefore, inspired by [54], a dynamic
+gradient adjustment mechanism is introduced to train three
+models separately, taking two modes and their concat as
+three inputs. By monitoring the contribution of each mode
+to the learning objective, each mode optimization is adap-
+tively controlled, thereby alleviating the imbalance of mode
+optimization.
+Three transformer’s encoders, represented by Enc (·),
+are accepted three modal features. When decoding ψk :=
+(r1, t1, . . . , rnt, < eos >), Softmax is used in Eq. 21 to
+calculate the distribution pχ
+i , where χ ∈ (fig, ocr), b is the
+bias of the prediction model [55], and the addition of b/2 is
+used as a bias compensation of single-modal prediction.
+pχ
+i =
+|ψ|
+�
+k=1
+Soft max (MLP
+�
+Enc
+�
+ψχ
+n
+�
+θχ, xχ
+n + b
+2
+�
+· W χ
+n )))k
+(21)
+In the same way, the distribution of concat feature pconcat
+i
+is
+shown in Eq. 22.
+pconcat
+i
+=
+|ψ|
+�
+k=1
+Softmax
+�MLP
+�Enc
+��ψconcat
+n
+�θconcat ,
+xconcat) ·W concat
+n
+���
+k
+(22)
+As Eq. 23-25 shows, a cross-entropy loss is used, where ε
+is a label smoothing hyperparameter ranging from 0 to 1 to
+
+IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
+8
+avoid overfitting. A single-modal image feature, a single-
+modal OCR feature, and a multi-modal feature which is
+defined as the concat of both are fed respectively to compute
+three different losses. At the same time, since the previously
+designed mask mechanism shields some features, it is also
+necessary to exclude the loss caused by the mask token.
+Also, to prevent the model from giving higher scores to
+shorter paths, the sum of log-likelihoods divided by the
+length of the path is used.
+Lfig = −
+1
+���pfig
+mask
+��� · N
+N
+�
+i=1
+�
+ε log pfig
+t
++
+� 1 − ε
+N − 1
+�
+log pfig
+i
+�
+(23)
+Locr = −
+1
+|pocr
+mask| · N
+N
+�
+i=1
+�
+ε log pocr
+t
++
+� 1 − ε
+N − 1
+�
+log pocr
+i
+�
+(24)
+Lconcat = −
+β
+|pconcat
+mask | · N
+N
+�
+i=1
+�
+ε log pconcat
+t
++
+� 1 − ε
+N − 1
+�
+log pconcat
+i
+�
+(25)
+Here β represents the weight value to encourage explo-
+ration. If during the training process, Mask mechanism III is
+activated, that is, the part of the input path is masked out,
+it means that a new trajectory is generated, so the weight of
+pconcat
+t
+should be increased.
+To optimize the multimodal contribution imbalance
+problem, the modal contribution difference ratio parameter
+(ρfig
+t
+, ρocr
+t
+) is introduced in Eq. 26 to adaptively adjust the
+gradient of each modality, where ρfig
+t
+is ρocr
+t
+. As shown in
+Fig. 5, the coefficient coeff u
+n is integrated into the network
+corresponding to the modal association via Eq. 27 based on
+[54]. At the same time, Gaussian noise N is introduced to
+enhance the generalization ability of the model.
+ρocr
+n
+= Locr
+Lfig
+(26)
+coeff u
+n =
+�
+1 − tanh (α · relu (ρu
+n)) ,
+ρu
+n > 1
+1,
+others
+(27)
+∇Wn+1
+u =∇Wn
+u × coeff u
+n + N
+�
+0,
+�
+std (∇Wn
+u)
++e−8�
+(28)
+Here u ∈ {fig, ocr}, and α is a hyperparameter that controls
+the degree of modulation.
+4
+EXPERIMENTS
+In this section, experiments are implemented based on
+two newly established datasets, and some state-of-the-art
+(SOTA) baselines are used for comparison. The experiment
+is mainly divided into four parts: link prediction main
+experiment, ablation experiment, training set mask exper-
+iment, and parameter interpretability analysis experiment.
+4.1
+Datasets
+In this subsection, two newly established datasets, OpenBG-
+IMG+ and OpenBG-Complete-IMG+, are introduced.
+4.1.1
+OpenBG-IMG+
+A new dataset OpenBG-IMG+ is created, derived from a
+part of the OpenBG-IMG dataset [56], which is a multimodal
+dataset in the field of e-commerce. This dataset is released
+in the CCKS2022 Task Three competition. Since the data
+set released by the competition has no correct answer, the
+OpenBG-IMG valid set is used as the test set in this paper,
+and the training set is divided into several parts as the
+valid set. The used dataset contains 28,891 entities and
+136 relations, where only some of the head entities have
+image information, while none of the tail entities has image
+information. Each image corresponds to only one entity, and
+there is no duplication. Table 2 shows specific statistics.
+4.1.2
+OpenBG-Complete-IMG+
+A new repository OpenBG-Complete-IMG+ is created based
+on the already created database OpenBG-IMG+. Its training
+set and valid set are obtained from OpenBG-IMG+ deleting
+the triplet data with no image information in the head entity,
+and the test set remains unchanged. Like OpenBG-IMG+,
+the tail entities of all data do not contain image information,
+but all head entities of the training set of OpenBG-Complete-
+IMG+ contain image information. This new dataset contains
+136 relations and 22297 entities. Table 2 shows specific
+statistics.
+4.2
+Baselines
+To study the performance of IMKGA-SM, three categories
+of methods are used for comparison: (1) Translation-based
+models, TransE [11], TransH [12], and TransD [13]. (2)
+Nonlinear-based models, DistMult [16], ComplEx [18], and
+TuckER [17]. (3) Multimodal knowledge graph linking
+model, TransAE [46].
+4.3
+Evaluation Protocol
+To further analyze the influence of image information, part
+of the training data is masked, and IMKGA-SM is used for
+experiments. According to inferences, the current dataset
+OpenBG-IMG+ may contain enough structural information
+for prediction, which interferes with the analysis of visual
+information. In order to highlight the role of visual infor-
+mation, a part of the training data is masked to create a
+dataset. Similar to recent works [10] [25], as formalized in
+Eq. 29 and Eq. 30, the mean reciprocal rank MRR and the
+average proportion of triples with rank less than n Hits@n
+are used to evaluate inference performance.
+MRR = 1
+|Q|
+|Q|
+�
+i=1
+1
+ranki
+=
+1
+|Q|
+�
+1
+rank1
++
+1
+rank2
++ . . .
++
+1
+rank|S|
+�
+(29)
+Here Q is the set of test queries, |Q| represents the number
+of queries, ranki is the link prediction rank of the i-th triple
+[57]. The larger the MRR indicator, the better the prediction
+effect.
+HIT@n =
+1
+|Q|
+|Q|
+�
+i=1
+I (ranki ⩽ n)
+(30)
+
+IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
+9
+Table 2
+Statistics of The Experimental Datasets
+Dataset
+#Ent
+#Rel
+#Train
+#Valid
+#Test
+#num of image
+OpenBG-IMG+
+28891
+136
+197269
+10383
+10930
+14718
+OpenBG-Complete-IMG+
+22297
+136
+138479
+7289
+10930
+14718
+Table 3
+Results of Knowledge Graph Link Prediction on OpenBG-IMG+ and OpenBG-Complete-IMG+ Datasets
+OpenBG-IMG+
+OpenBG-Complete-IMG+
+Model
+Interpretability
+Multi-model
+MRR
+Hit@1
+Hit@3
+Hit@10
+MRR
+Hit@1
+Hit@3
+Hit@10
+TransE [11]
+No
+No
+0.50858
+0.33769
+0.64876
+0.83037
+0.39092
+0.24419
+0.50155
+0.67520
+TransH [12]
+No
+No
+0.40132
+0.15946
+0.61454
+0.81280
+0.31544
+0.10356
+0.49533
+0.68197
+TransD [13]
+No
+No
+0.39173
+0.15416
+0.59158
+0.80850
+0.30779
+0.09954
+0.47209
+0.69075
+DistMult [16]
+No
+No
+0.14469
+0.07529
+0.17191
+0.35123
+0.13496
+0.07474
+0.16715
+0.29560
+ComplEx [18]
+No
+No
+0.19756
+0.12314
+0.23568
+0.38435
+0.14017
+0.08673
+0.15361
+0.31015
+TransAE [46]
+No
+Yes
+0.47107
+0.33897
+0.56376
+0.75086
+0.51174
+0.38261
+0.60054
+0.77913
+TuckER [17]
+No
+No
+0.42024
+0.31985
+0.49613
+0.60795
+0.39482
+0.29624
+0.48215
+0.57557
+IMKGA-SM(No Img)
+No
+No
+0.64224
+0.53284
+0.73595
+0.83284
+0.59361
+0.48948
+0.68069
+0.78270
+IMKGA-SM(MKG)
+No
+Yes
+0.64652
+0.53705
+0.73998
+0.83678
+0.60110
+0.49332
+0.68893
+0.79780
+IMKGA-SM(RL)
+Yes
+No
+0.64341
+0.53312
+0.73816
+0.83678
+0.59582
+0.49222
+0.68097
+0.78948
+IMKGA-SM(MKG+RL)
+Yes
+Yes
+0.64860
+0.53998
+0.74089
+0.83714
+0.60177
+0.49680
+0.68692
+0.79030
+Improv.
+-
+-
+17.76%
+20.11%
+17.72%
+8.63%
+9.01%
+11.5%
+8.64%
+1.12%
+Here I (·) is the indicator function, if the condition is true,
+the function value is 1, otherwise, it is 0. The three indicators
+HIT@1, HIT@3, and HIT@10 describe the probability
+that the top K(K = 1, 3, 10) entities with the highest score
+in the link prediction contains the correct entity [11].
+4.4
+Implementation Details
+Next, the experiments are mainly based on the knowledge
+graph link prediction task. To augment the training data,
+each original training set triplet is reversed to generate an
+inverse triplet. The knowledge graph triplets in the test
+dataset are sorted by all entities in descending order of
+probability value, leaving the top ten predicted entities.
+Models are trained using the Adam [58] optimizer and
+analyzed for hyperparameters, eigenvectors.
+4.5
+Link Predict Results
+Link prediction results are shown in Table 3 (all scores
+are expressed as percentages), where the most competitive
+baseline TransAE [46] results are underlined and the best
+results are highlighted in bold. The following points are
+observed.
+Table 3 is studied for link prediction tasks. It is seen that
+the accuracy performance of IMKGA-SM is better than all
+other models, and IMKGA-SM uses multi-hop reasoning,
+which is interpretable. It is shown that the model is proven
+to be both interpretable and highly accurate, achieving
+state-of-the-art performance. The introduction of a multi-
+modal knowledge graph in the OpenBG-IMG+ dataset is
+not obvious enough, so only the triplet data with image
+information in the head entity is retained in the new dataset.
+The results show that the improvement effect of IMKGA-SM
+in OpenBG-Complete-IMG+ is generally better than that of
+OpenBG-IMG+. This is speculated to be due to being over-
+whelmed with the help of other multimodal information
+when the dataset already has rich structural information.
+To further analyze the influence of image information, a
+new dataset OpenBG-Complete-IMG+ is created. All head
+entities in the dataset have image information, and the
+experiment is performed again.
+4.6
+Ablation Learning
+In this section, ablation learning is divided into three parts:
+the influence of multimodality, reward expectation, and
+mask mechanism on the model, and perform specific data
+analysis on the complete IMKGA-SM model.
+4.6.1
+IMKGA-SM (No Img) vs IMKGA-SM (MKG)
+To further explore the role of image information, IMKGA-
+SM (No image) and IMKGA-SM (MKG) are compared on
+two datasets, and the link prediction experiment results are
+shown in Table 3. Image embedding and ocr embedding are
+added to IMKGA-SM (MKG), and a dynamic loss function
+adjustment mechanism is adopted. The experiment found
+that the improvement effect on OpenBG-Complete-IMG+ is
+more stable than OpenBG-IMG+.
+4.6.2
+IMKGA-SM(No Img) vs IMKGA-SM(RL)
+In order to verify the improvement of the model by the
+introduction of reward expectation, IMKGA-SM (No image)
+and IMKGA-SM (RL) are compared on two data sets, and
+the link prediction experiment results are shown in the Ta-
+ble 3. A reward expectation mechanism based on perceptual
+interaction and a reward-related mask is added to IMKGA-
+SM (RL). The experiment results show that after adding the
+reward expectation mechanism ˆR, the model effect has been
+improved, but the improvement is not as much as that of
+MKG.
+4.6.3
+IMKGA-SM (MKG+RL)
+IMKGA-SM (MKG+RL) adds a masking mechanism on the
+basis of IMKGA-SM(MKG) and IMKGA-SM(RL) and ver-
+ified its effect on OpenBG-IMG+ and OpenBG-Complete-
+IMG+ datasets, which are shown in Table 3. Compared
+
+IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
+10
+Table 4
+Statistics of Datasets in Training Data Masking Experience
+Dataset
+#Ent
+#Rel
+#Train
+#Valid
+#Test
+OpenBG-IMG+80%
+27839
+136
+158527
+8344
+10930
+OpenBG-IMG+70%
+22297
+136
+138479
+7289
+10930
+OpenBG-IMG+47%
+21817
+136
+92648
+4877
+10930
+OpenBG-IMG+35%
+21469
+136
+68599
+3611
+10930
+OpenBG-IMG+28%
+21215
+136
+55397
+2916
+10930
+Figure 6. Link prediction results (MRR) on OpenBG-Complete-IMG+x%.
+with the multimodal knowledge graph linking baseline
+TransAE [46], IMKGA-SM has a 17.67% improvement on
+the OpenBG-IMG+ dataset and a 9.01% improvement on
+the OpenBG-Complete-IMG+ dataset. Compared with the
+nonlinear-based baseline DistMult [16], IMKGA-SM has a
+50.39% improvement on the OpenBG-IMG+ dataset and
+a 46.681% improvement on the OpenBG-Complete-IMG+
+dataset.
+4.7
+Training Data Masking
+To further explore the impact of image and structural in-
+formation on the results, masking is performed on part of
+the training data, and IMKGA-SM is used for experiments.
+It is speculated that the current dataset OpenBG-IMG+
+may contain enough structural information for prediction,
+interfering with the analysis of visual information. In order
+to highlight the role of visual information, a part of the
+training data is masked out to create datasets by controlling
+the frequency of a head entity with image information,
+which is OpenBG-IMG+28%, OpenBG-IMG+35%, OpenBG-
+IMG+47%,
+OpenBG-IMG+70%,
+OpenBG-IMG+80%
+and
+OpenBG-IMG+100%, respectively, and the dataset informa-
+tion is shown in Table 4. Then, the link prediction exper-
+iment is carried out again, and the results are shown in
+Fig. 6,
+7. It is seen that IMKGA-SM has shown obvious
+advantages in data sets of different scales. The traditional
+method has a significant increase after adding structural
+information, while IMKGA-SM is still relatively stable in
+the improvement of the scale. IMKGA-SM not only has
+comparable generalization ability to neural network models,
+but also has stronger interpretability than other baseline
+methods.
+Figure 7. Link prediction results (HIT@1) on OpenBG-Complete-
+IMG+x%.
+Figure 8. Link prediction results on OpenBG-Complete-IMG+ in different
+N.
+4.8
+Parameter Interpretability
+In this subsection, the parameter interpretability is divided
+into three parts, mainly analyzing the impact of different
+batch sizes, label smooth and modulation impact.
+4.8.1
+The Influence of Different Batch Sizes N
+Fig. 8 investigates the effect of different batch sizes N.
+The batch size is set to N ∈ [16, 32, 64, 128, 256, 512]. It is
+observed that as N increases, the performance of IMKGA-
+SM rises first and then declines steadily in most cases, pre-
+sumably because undertraining and overfitting negatively
+affect the model. The results show that an appropriate
+training parameter size improves the effectiveness of the
+inference model. From the experimental results, the optimal
+parameter is N = 16.
+4.8.2
+The Influence of Different Labels Smooth ε
+In this paper, the method of label smoothing is used in the
+loss function, which is a regularization method to prevent
+overfitting. By setting α = 0.5, the effect of the label smooth-
+ing parameter ε is shown in Fig. 9. The batch size is set
+to ε ∈ [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9]. It is observed
+that with the decrease of ε, the performance of IMKGA-SM
+rises first and then declines steadily in most cases, which
+is probably because MRR is the main evaluation indicator,
+
+0.7
+DistMult
+-CompEx
+-TuckER
+IMK GA-SM
+0.6
+0.5
+0.4
+MRR
+0.3
+0.2
+0.1
+0
+2.896
+35%
+47 %
+7 0%
+80%
+100%
+OPENBG-COMPLETE-IMG+ X%0.7
+DistMult
+-compEx
+TuckER
+IMK GA-SM
+0.6
+0.5
+0.4
+HIR@1
+0.3
+0.2
+0.1
+28%
+%SE
+47%
+70%
+80%
+100%
+OPENBG-COMPLETE-IMG+ X%Hit@1 MRR
+0.7
+0.58922
+80009'0
+0.60177
+0.58892
+0.6
+0.58391
+0.56145
+0.4785
+0.49231
+0.4968
+0.5
+0.48216
+0.47667
+0.4519
+0.4
+0.3
+0.2
+0.1
+0
+N=16
+N=32
+N=64
+N=128
+N=256
+N=512IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
+11
+Figure 9. Link prediction results on OpenBG-Complete-IMG+ in different
+ε.
+Figure 10. Link prediction results on OpenBG-Complete-IMG+ in differ-
+ent α.
+and the shrinking of its proportion has a negative effect on
+the final result influences. The results show that the optimal
+parameter is ε = 0.7.
+4.8.3
+The Influence of Different Modulation Impact α
+With ε = 0.9, the effect of the modulation impact α on
+IMKGA-SM is demonstrated in Fig. 10. From the results,
+α = 0.6 is observed to be the optimal value on the data set.
+5
+CONCLUSIONS AND DISCUSSION
+In this paper, how to effectively utilize multi-modal aux-
+iliary features for multi-hop knowledge graph inference
+is investigated, aiming to improve the accuracy of model
+inference and achieve interpretability simultaneously. An
+efficient IMKGA-SM model is proposed, which outperforms
+existing methods on the multimodal knowledge graph in-
+ference task. In IMKGA-SM, structural features and multi-
+modal data are first extracted in-depth, and then a return-to-
+go mechanism based on perceptual similarity is constructed
+and applied to the large sequence framework. In addition,
+three mask mechanisms are designed to alleviate the prob-
+lem of data sparsity. Next, a multimodal autoregressive loss
+function adjustment mechanism is introduced to take full
+advantage of multimodality. Finally, experimental results
+show that IMKGA-SM achieves higher effectiveness and
+interpretable ability versus other trending rivals in knowl-
+edge graph link prediction tasks. To conclude, IMKGA-SM
+requires effective methods to minimize the negative impact
+of sparse data. These tasks are left for future work.
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+
+Hit@1 MRR
+0.7
+0.59219
+90.5911
+0.59993
+0.6
+0.485
+0.4924
+0.49469
+0.4853
+0.4968
+0.49771
+0.48957
+0.48527
+0.49341
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+α=0.1
+α=0.2
+α=0.3
+α=0.4
+α=0.5
+α=0.6
+α=0.7
+α=0.8
+α=0.9Hit@1 MRR
+0.7
+0.6
+0.4968
+0.49561
+0.49634
+0.49186
+0.5
+0.48884
+0.482
+0.48783
+0.4761
+0.47667
+0.4
+0.3
+0.2
+0.1
+0
+2=0.9
+=0.8
+=0.7
+ε=0.6
+ε=0.5
+=0.4
+=0.3
+E=0.2
+ε=0.1IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
+12
+[20] T. Dettmers, P. Minervini, P. Stenetorp, and S. Riedel, “Convo-
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diff --git a/KNE0T4oBgHgl3EQfigEg/content/tmp_files/load_file.txt b/KNE0T4oBgHgl3EQfigEg/content/tmp_files/load_file.txt
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@@ -0,0 +1,1266 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf,len=1265
+page_content='IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 1 IMKGA-SM: Interpretable Multimodal Knowledge Graph Answer Prediction via Sequence Modeling Yilin Wen, Biao Luo, Senior Member, IEEE, and Yuqian Zhao Abstract—Multimodal knowledge graph link prediction aims to improve the accuracy and efficiency of link prediction tasks for multimodal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' However, for complex multimodal information and sparse training data, it is usually difficult to achieve interpretability and high accuracy simultaneously for most methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' To address this difficulty, a new model is developed in this paper, namely Interpretable Multimodal Knowledge Graph Answer Prediction via Sequence Modeling (IMKGA-SM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' First, a multi-modal fine-grained fusion method is proposed, and Vgg16 and Optical Character Recognition (OCR) techniques are adopted to effectively extract text information from images and images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Then, the knowledge graph link prediction task is modelled as an offline reinforcement learning Markov decision model, which is then abstracted into a unified sequence framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' An interactive perception-based reward expectation mechanism and a special causal masking mechanism are designed, which “converts” the query into an inference path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Then, an autoregressive dynamic gradient adjustment mechanism is proposed to alleviate the insufficient problem of multimodal optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Finally, two datasets are adopted for experiments, and the popular SOTA baselines are used for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The results show that the developed IMKGA-SM achieves much better performance than SOTA baselines on multimodal link prediction datasets of different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Index Terms—Knowledge graph, link prediction, multimodal, interpretability, sequence modeling, reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 1 INTRODUCTION T HE knowledge graph is the technology and tool for carrying and representing background knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' It structures knowledge in the real world into entities and relations in the form of graphs and organizes them into networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' In a knowledge graph, knowledge data is rep- resented in the form of triples (h, r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Among them, h is the head entity, r is the relation connecting two entities, and t is the tail entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Knowledge graphs are used in various artificial intelligence tasks in different domains [1], such as named entity disambiguation [2] in natural language processing [3], visual relation detection [4] or collaborative filtering [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' However, it is well known that even state-of- the-art knowledge graphs are often incomplete (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=', lack real facts or contain false facts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Therefore, machine learning algorithms aimed at addressing this problem attempt to infer missing triplets from observed connectivity patterns, a task is known as link prediction [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' For example, given a head entity and a relation (h, r), predict a tail entity t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' In order to solve the problem of link prediction, exist- ing problems can be divided into four categories: deduc- tive logic and rules, reasoning based on graph structure, knowledge graph embedded representation and deep neu- ral network model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Rule-based reasoning methods, such as AMIE [7], AnyBURL [8], transform natural language queries into combinations of logical operators, express such queries through combinations, and then implement in a specific programming language to get the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' These methods are Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='Wen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='Luo and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Zhao are with the School of Automation, Central South University, Changsha, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' E-mail: yilinwen510@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='com, biao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='luo@hotmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='com, zyq@csu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='cn Corresponding Author: Biao Luo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' accurate and interpretable, but require experts to formulate a large number of inference rules, and have poor generaliza- tion ability for unknown rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Reasoning based on graph structure has two features: one is the path feature, and the representative algorithm is PRA [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The path features between nodes are extracted by graph traversal algorithm or random walk method, and the node connections are predicted by path features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Its characteristic is to provide path interpretability while reasoning, and the problem is that it is difficult to solve the problem because the reasoning nodes are not connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The second is a graph-structure- based approach that utilizes a message-passing mechanism to extract the structural information of target entities and provide subgraph interpretability, and the representative algorithm is DeepPath [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' However, because the knowl- edge graph is usually very large, it is extremely compli- cated to traverse all the subgraph structures in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The knowledge graph embedding representation method is to embed the high-dimensional and discrete data of the knowledge graph into a low-dimensional continuous vector space by designing a certain scoring function, and then representing the entities and relations as numerical vectors to calculate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Its representative model is the TransE type, for example, TransE [11], TransH [12], TransD [13], TransR [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The recent research is bilinear models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=', RESCAL [15], DisMult [16], TuckER [17], and ComplEx [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Its method is characterized by a shallow neural network, and the se- mantic representation of the knowledge graph is realized through a specific structure of the embedded space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The deep neural network model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=', CoKE [19], ConvE [20], is designed by designing entities and relations into query pairs, matching query pairs with entities relations, and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='02445v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='AI] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='6 Jan 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='Table 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='Summary of Existing Methods for Knowledge Graph Link Prediction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='reasoning algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='logical rules ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='graph structure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='knowledge graph embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='deep neural network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='reinforcement learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
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+page_content='expert experience ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='dependence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='partial dependence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='no dependence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='no dependence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='no dependence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='obtaining inference similarity scores through deep neural ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='networks to make inference judgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Both the knowledge graph embedding model and the deep network model are regarded as neural network models, and the same point is that they both design a scoring function, and use the gradient backpropagation method for training in a data- driven manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Its advantage is that its generalization per- formance is relatively better, and it effectively alleviates the problem of graph structure dimensionality disaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Its disadvantage is that it only sees the similarity between input and output values, lacks interpretability, and performs single-step reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' In summary, as shown in Table 1, it is found that the methods based on logical deduction rules and graph structure are all symbol-based methods, which have better interpretability but poor generalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Otherwise, based on knowledge graph embedding and deep neural network model, its generalization performance is better, but it lacks interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Therefore, studying how to integrate symbolist and connectionist models is the key to obtaining an interpretable knowledge graph reasoning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' With the development of deep learning, the model struc- ture of knowledge reasoning methods is becoming more and more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Because it is difficult for users to have an intuitive understanding of the parameters, structure and characteristics in such models, and they also have less un- derstanding of the decision-making process and reasoning basis, it is difficult for users to trust the prediction results of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Therefore, in order to establish trust between users and reasoning models and balance the contradiction between model accuracy and interpretability, multi-hop rea- soning methods are used to solve explainable knowledge reasoning [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The rationale of the multi-hop reasoning method is to imitate the multi-hop thinking of human beings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' A common approach is to apply reinforcement learning frameworks to multi-hop reasoning in knowledge graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Reinforcement learning is a model that has received a lot of attention in the past ten years and has been widely used in control [22], games [23], and robots [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' It models a learning process as a Markov process and trains the model by maximizing long-term cumulative rewards through the interaction between the agent and the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Mod- elling the knowledge map as a reinforcement learning pro- cess not only gets the result of reasoning, but also obtains the path of reasoning, and explains the reasoning of the knowledge graph through the reasoning path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The specific fusion method is to regard the knowledge graph as an environment, model the agent as a deep neural network, combine the advantages and disadvantages of symbolism and connectionism, and make the model have both the gen- eralization performance and path interpretability of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Methods based on reinforcement learning such as DeepPath [10], MINERVA [25], DIVINE [26], and AttnPath [27], however, generally have the shortcomings of slow convergence and low accuracy, and most of them are inferior to some traditional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The reason for this may due to the sparse rewards of reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Moreover, the sparse rewards, sparse data, and insufficient exploration of knowledge graphs make reinforcement learning more difficult and challenge in multimodal knowledge graph reasoning tasks [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Therefore, it is meaningful and promis- ing to improve the accuracy of reinforcement learning in knowledge graph reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Recently, the cross-border application of Transformer [29] has attracted wide attention, and it has made break- throughs in image classification [30], semantic segmenta- tion [31], object detection [32] and other fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Currently, Transformer has been employed as a pre-training model in offline reinforcement learning, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=', Decision Transformer [33], Trajectory Transformer [34], and Gato [35], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' These methods regard the data of reinforcement learning as a string of unstructured sequence data and train with super- vised or self-supervised learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' It avoids the un- stable gradient signal in traditional reinforcement learning and performs better in offline reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Deep reinforcement learning is a sequential process, therefore, the process of multi-hop reasoning is handled by state- of-the-art reinforcement learning sequence models, which may achieve better results than traditional reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' For knowledge graph reasoning tasks that are complex and have the concept of multimodal data, the core idea of most existing knowledge graph reasoning algorithms is to reason by integrating existing triple structure knowledge, so knowledge of the entity is often ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' However, in- formation about entities themselves is usually beneficial for link prediction tasks, such as image and textual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 1, for example, when performing the triple < shoes, style, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' > link prediction task, the answer is predicted based on the triple < dress, style, sweet > of the similar head entity image, and finally, answer ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' is sweet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' It is worth noting that the text information in the image also contains a lot of knowledge, especially when the knowledge graph is applied to the e-commerce field, the text in the product image is often the brand information of the product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Therefore, to address multimodal explainable knowledge graph reasoning tasks with high efficiency and high performance, a new sequential model IMKGA-SM for reinforcement learning is developed, where a reward mechanism is designed based on perceptual interaction and fine-grained multimodal information extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' When performing triple < shoes, style, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' > link prediction tasks, the answer is predicted based on triples < dress, style, sweet > which is similar to the head entity image, and finally it is concluded that ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' is sweet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 2 RELATED WORKS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='1 Single-modal Knowledge Graph Reasoning Single-modal knowledge graph reasoning mainly revolves around relational reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The AMIE [7] and AMIE+ [36] algorithms are derived from the early inductive logic pro- gramming system [37], emphasizing automatic rule learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' It has strong interpretability, however, all the above methods require expert design rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Graph structure- based reasoning methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=', path ranking algorithm [9]) are also used to tackle such problems, which is interpretable but computationally intensive and time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The embedding-based methods include TransE [11], ConvE [20], RotatE [38], and TuckER [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Each of these models is simple and the training speed is fast, but they are not interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Reasoning methods based on neural networks include neu- ral tensor networks [39], R-GCN [40], implicit ReasoNet [41], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' They are able to learn to reason through implicit processing, which results in poor interpretability and unsta- ble performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' In addition, there are typical reinforcement learning methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=', DeepPath [10], MINERVA [25], RLH [42], GussuianPath [43], etc, which have better interpretabil- ity and inference performance than representation learning- based methods, but the disadvantage is that the effect is poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='2 Multimodal Knowledge Graph Link Prediction Compared with the single-modal knowledge graph link prediction task, the main contribution of the multi-modal knowledge graph link prediction task is to integrate multi- modal data knowledge into the plain text knowledge graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' In multimodal knowledge graph link prediction tasks, it is very necessary to combine the textual semantics of en- tities with multimodalities, such as semantics, vision, and hearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' IKRL [44] is the first knowledge representation model that includes image information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' For each entity, it learns two different representations based on triple struc- ture information and image information, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' DKRL [45] is a knowledge representation for fused descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Similar to the IKRL model, the DKRL model also learns a representation based on structural information and a rep- resentation based on text descriptions for each entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Based on the single-modal knowledge graph link prediction model TransE [11], an autoencoder is employed in TransAE [46] to jointly encode visual information and text information to obtain the vector representation of entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' RSME [47] is a multimodal knowledge graph reasoning model based on the traditional knowledge graph embedding model ComplEx [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' However, most of these multimodal approaches are uninterpretable and with low accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='3 Reinforcement Learning with Transformers In [33], Decision Transformer is proposed by modelling reinforcement learning tasks as a sequence framework trans- former, based on which SQUIRE [48] is employed to han- dle single-modal knowledge graph link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' How- ever, these works are deficient in generalization and the reward information is underutilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Based on Decision Transformer [33], Trajectory Transformer [34] uses the beam search for model-based planning, while generating new trajectories is too complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Therefore, a simple ran- dom masking mechanism is proposed in this paper, which achieves the effect of data enhancement by randomly mask- ing historical actions that have been generated in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Recently, Deepmind proposed a general agent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=', Gato [35], which made a further breakthrough in multimodal tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' It is promising and potential to extend this model to multi-modal multi-hop reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 3 METHODOLOGY In this section, the overall framework of IMKGA-SM is introduced, which treats the multi-hop reasoning problem as a sequence-to-sequence task derived from regression modelling trajectories and applies it to the task of multi- modal link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The hybrid transformer architecture of IMKGA-SM mainly includes five stacked modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' (1) The underlying multimodal feature extraction module, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 2, aims to obtain basic structural information, image information, and text information in images from databases and images, and combine the three as a state feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' (2) The reinforcement learning sequence module, as shown in the bottom part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The knowledge graph link prediction task is modelled as an offline reinforcement learning problem, which is then abstracted into a sequen- tial framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' (3) The upper multimodal encoder (fusion encoder) module, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 4, fuses the underlying features, reward features based on perceptual interaction, and action features through a self-attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' (4) The Mask mechanism module, as shown in the upper part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 3, includes three mechanisms to ensure the input and output of the encoder and prevent overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' (5) The loss function module, adopts an autoregressive self-adjusting mechanism to maximize the multi-modal performance, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' In the following subsections, each module of the IMKGA-SM will be analyzed and discussed in details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The multimodel feature extraction module and the reinforce- ment learning sequence architecture are developed in Sub- sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The fusion encoder module is proposed in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The mask and loss function modules are designed in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Saurin素兰 Associated dress up dress scene Associated brand style scene sweet style pink foxrabbit shoes Applicable groups femaleIEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 4 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The multimodal feature extraction module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Unified interpretable multimodal knowledge graph sequence framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='1 Multimodal Feature Extraction Module In multimodal knowledge graph tasks with only single image data, most of the existing methods only learn simple image information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' However, many visual scenes contain text with key information, so understanding text in images is crucial for downstream reasoning tasks, such as product brand, price, and consumer population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' To jointly learn multimodal knowledge and inter-entity relations, knowl- edge in a single modality is extracted and combined into a multimodal transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' In this paper, two modalities are considered: visual and textual, where text is extracted from image information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The multimodal part includes image input, text input in the image and query input (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' head entity, relation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Vgg16 pre-trained on ImageNet is used to process the head entity image information of the image input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Vgg16 consists of several vgg-blocks and three fully connected layers, and the vector output by the last fully connected layer is used as the image feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' For image text input, OCR technology is used for image text extraction [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Generally, OCR technology consists of two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' (1) Text detection: locate the position of the text in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' (2) Text recognition: identify the positioned text area, and convert the text area in the image into character information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' In this paper, the CTPN method [50] is adopted for text detection, and the CRNN method [51] is adopted for text recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' If the image information corresponding to the head entity is missing, ∅ is used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' For the Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The fusion encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Autoregressive dynamic loss regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' query input part, the knowledge graphs corresponding to head entities and relations are encoded to form a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 1 models an original multimodal feature φ, which is specifically manifested as the fusion of structural informa- tion (h, r), image information and text information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' φ : G × G → G (1) Here hfig, hocr, h, r ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Then, let ∗ indicates a grouping operation, h represents the structure embedding of the head entity, hfig represents the image embedding of the head entity, and hocr represents the text embedding of the head entity after being extracted by the OCR method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Thus, as formalized in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 2, a characteristic entity ˜q of φ(h, hfig, hocr) and r is written as : ˜q = φ (h, hfig, hocr) ∗ r (2) Multimodal fusion is widely used in the fields of computer vision [52] and natural language processing [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Since the currently most popular transformer framework is adopted as the core module of IMKGA-SM, according to the charac- teristics of the transformer, the number of parameters in the learning process largely determines the operation speed, so it is very necessary to process the input features of the trans- former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Therefore, the module of the multimodal feature is a pre-train of the core transformer framework, which aims to filter out irrelevant or redundant features from the original data of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Sepecifically, three self-attention blocks are used to receive the outputs of the original multimodal feature vector φ, and three autoencoders are used to transfer them into a 14-dimensional vector in the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Specifically, first, the original multimodal feature φi passes through a fully connected feed-forward network to obtain different modal features µi, which consists of two linear transformations and a ReLu activation function, via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' µi = conv {ReLU [conv (φi)]} , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' , L (3) (,h,r) Attention Auto- datasets Block encoder .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='vgg16 Attention Auto- Fusion Block encoder Encoder CTPN+CRNN Attention Auto- Block encoderretun to go input multimodalfusioninput rule path input Ro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='R-[,q,r,fig1,fig2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='-figgocr1,ocr2,ocr3[,r1,t1,r2,t2,r3,t ao a1 at aN-1 an Autoregréssive dynamic loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='regulation IMKGA-SM Fusion Encoder 3 2N Linear Layer + Norm Ro Us ao S1 ai at SN-1 aN-1 RN SN aNFeedforward Self -Attn Linear Layer + Norm Reward Feature Query Feature Image Feature OCRFeature Action FeatureBac ward Feed Forward limear ocr FC Encoder Sauri combine Cross- Encoder FC entropy loss linear image FC Encoder Bac wardIEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 5 Then, as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 4 shows, different modal features µi are passed through a Softmax layer in order to compute the attention of each modality ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' ai = Softmax (µi) , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' , L (4) The sum of these attention weights ai multiplied by the original multimodal feature embedding µi is called self- attention Qs φ, formalized in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Qs φ = �L i=1 aiµi (5) Therefore, Qs h, Qs hfig, Qs hocr of h, hfig, hocr are obtained respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Qs φ is used as a query for the corresponding feature to calculate the attention weights guided by Qs φ and put into the softmax layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Finally, the weights are multiplied by the original modal features φk to get the filtered vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Then, the output of the attention block is expressed as gφ via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' pk = W �ReLu �WsQs φ � ◦ ReLu (Wxφk) � , sk = Softmax (pk) , k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' , N, gφ = �N k=1 skφk, φ ∈ G (6) After feature gφ is obtained, it is input into the autoencoder for dimensionality reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The final feature hφ is shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 7, in which hfig is 8-dimensional, and hocr is 3- dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' hφ = σ (W · φ + b) (7) This part is used as the definition for the state s of reinforce- ment learning, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 3, which will be described in detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='2 Reinforcement Learning Sequence Framework In this subsection, an offline reinforcement learning frame- work is developed for the knowledge graph link prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Then, specific Markov triples are designed, and a reward expectation mechanism based on perceptual interac- tion is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Finally, the whole reinforcement learning process is abstracted into a sequential framework, which is the core module of IMKGA-SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='1 Offline Reinforcement Learning Design The knowledge graph link prediction problem is mod- elled as a Markov decision process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Markov decision pro- cess tuple consists of a state s ∈ S, an action a ∈ A, a transition dynamic P (s′ |s, a), and a reward function r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' sn, an, and r′ n are used to denote the state, action, and reward at time step n, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' A trajectory con- sists of a sequence of states, actions, and rewards: τ = (s0, a0, r′ 0, s1, a1, r′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' , sN, aN, r′ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The purpose of rein- forcement learning is to learn a policy that maximizes the expected return E ��N n=1 r′ n � in the Markov decision process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' In this paper, the process of generating a path for each triple link in the knowledge graph is regarded as a reinforcement learning segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Since the data in this paper are fixed datasets, new data is difficult to be obtained through environmental interaction, so it is regarded as an offline reinforcement learning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='2 Track Representation The inference accuracy of reinforcement learning-based in- ference methods is usually much lower than that of tradi- tional TransE-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' This is because the amount of data for offline reinforcement learning is very limited, and the rewards for the knowledge graph link prediction problem are sparse, resulting in serious decision bias in reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Therefore, they are not suitable for direct transfer to the task of multimodal knowledge graph link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' To address this problem, a new reinforce- ment learning sequence framework with perceptual interac- tion expected reward mechanism is proposed in this sub- section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Different from traditional reinforcement learning, the reward setting here is the expected reward in the future, that is, the maximum reward value expected to be obtained in the current state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' There are two main differences: (1) An expected reward mechanism is proposed to eliminate the sparsity of rewards by incorporating the perceptual similarity of knowledge graph entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' (2) The multi-modal perception interface is introduced into the decision trans- former framework for the first time, making full use of multi-modal features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Through the previous pre-training process, the knowl- edge graph link prediction process is transformed into a Markov decision process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Its purpose is to find a path to the target entity, which means that the pathfinding process makes multi-hop reasoning interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Therefore, the knowledge graph link prediction task is modelled as an offline reinforcement learning task, which is then transferred to a sequential framework for solving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Similar to offline reinforcement learning, the Markov triplet < ˆR, S, A, Pr > of IMKGA-SM is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' State sk: For the knowledge graph triple < h, r, t > in the dataset, the state of IMKGA-SM is denoted as s = (query, hfig, hocr) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Among them, S is the state space, and query = (< bos >, h, r) is the query of triples, representing the beginning, head entity, and relation, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' hfig and hocr represent the image embedding of the head entity and the text embedding in the image, which are obtained by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Action ak: The action space for a given sn is the set of all entities, relations and < eos >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The purpose is to infer the path from the head entity h, relation r to the tail entity t by generating the action output, and ak is the k-th action represented by the k-th token of the path generated by the rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Here, AnyBURL [8] is used as the rule-based method to find the path between h and t, r (h, t) → r1 (h, t1) ∧ r2 (t1, t2)∧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='∧rn (tn, t), decomposing a single relation into a combination of multiple relations and entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' If the model makes an error during the prediction process, the inferred entity or relationship does not conform to the corresponding attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Then the rule path is modified to remember only the last target entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Taking three-hops as an example, A = (0, ∅, t, 0, 0, 0, 0) is modified as a set of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Transition Pr: The transition function Pr is set to map the current state sn to the next state sn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' For- mally, Pr : S × A → S is defined as Pr (sn, An) = Pr (query, hfig, hocr, a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' , an−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' When entering the next state, the actions of the previous step are added to the previous state as history to realize the state change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 6 Mask mechanism I does the specific step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Therefore, unlike the traditional Decision Transformer [33], the state transition mechanism Pr (sn, An) is designed to let the model focus on the state, actions and return-to-go of the previous steps, thereby improving the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Return-to-go ˆR: Since the purpose of knowledge graph link prediction is to reason about the tail entity answer, it is impossible to know whether the reasoning is success- ful unless the last step of the reasoning path is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Therefore, there is a phenomenon of sparse rewards in the reinforcement learning sequence method when solving the knowledge graph link prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' In order to solve these problems, a reward expectation mechanism based on perceptual similarity is designed to learn interactively by taking the expected reward of the current state as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The definition of ˆR is the maximum reward expected in the current state, so after making an action, the value of the next ˆR will decrease (or increase) due to the reward from the previous action, ˆRn = �T n′=n r′ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' So ˆR changes with state and action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' After obtaining the initial triple query = (< bos >, h, r) and obtaining the corresponding reasoning path according to the rules, the return-to-go corresponding to each step ˆRn is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' τ(h, r, t) is defined as a set of triples in the dataset, and ψ(< bos >, r1, t1, r2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' , rn, tn) is the path obtained by the rule that satisfies τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' All dataset entities and relations are stored in collections E and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Specifically, the generation steps of ˆR are as follows: (1) At the initial value (n = 0), when no action is taken, the maximum expected reward of the task ˆR0 is expected to be able to reach the target entity, which is a fixed value defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' ˆR0 = r′ good (8) (2) When n = 1, the first action is < bos >, which means the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' As Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 9 shows, the first return- to-go ˆR1 is defined according to whether the correct tail entity is finally successfully inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Here, r′ good represents a positive constant and r′ bad represents a negative constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' ˆR1 = � r′ good, ψ (tn) = t, r′ bad, ψ (tn) ̸= t (9) (3) For each action with n > 1, return-to-go in n-th step is defined as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 10, in which a base penalty r′ step (negative) is generated since as few hops as possible are desired to be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' When the current action an is the entity tn, an additional reward r′ addn (positive) will be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' r′ n = r′ step + r′ addn, n > 1 (10) (4) Define r′ addn as: r′ addn = � � � � � ˆR1 × sim (ftn, ft) an ∈ E, 0, an ∈ R 1 2 ˆR1, emtn = ∅|emt = ∅ (11) Here sim(·) denotes the cosine similarity computed with sim (u, v) = uT v ∥u∥·∥v∥, ftn and ft are the image embedding of the current and target entities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 11, it is noted that the more similar the generated entity action is to the target entity, the greater the reward for that action should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' If the current action is the relation r′ n, or the target entity or the current recommendation entity has no image information, the additional reward r′ addn is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' If the entity action an has no image information or the target entity t has no image information, the similarity value takes an intermediate value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' (5) An additional penalty r′ bad will be imposed if the rec- ommended action does not conform to the attribute (entity or relation) that should be recommended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' When performing action an in the current state sn, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 12 defines the next Return-to-go input ˆRn as the previous step’s Return-to-go ˆRn−1 minus the reward r′ n−1 caused by action an−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' ˆRn = � ˆRn−1 − r′ n−1 − r′ bad, an−1 /∈ E, an−1 /∈ R ˆRn−1 − r′ n−1, others (12) (6) Repeated iteration until the end of the round, if it is less than three hops, in order to ensure the same embedding length, the last ˆR completion vector will be used to length 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The trajectory τ is expressed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' τ = � ˆR1, s1, a1, ˆR2, s2, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' , ˆRN,sN, aN � (13) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='3 Fusion Encoder Architecture Sequences of token embeddings from the three modes, return-to-go, state, and action, are concatenated and fed to the transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Different from the positional embedding of the traditional transformer [29], a time step (return-to- go, state) shares the same positional embedding and the position of them are processed as a complete sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Therefore, the process of positional embedding is expressed as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Here XU pc is the projected embedding vector, C is the concat operation, Upos, represents the position embedding corresponding to the embedding layer, and U = � C � ˆR, s � , a � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' XC( ˆ R,s) τ = XC( ˆ R,s) pc + C � ˆR, s � pos Xa τ = Xa pc + apos (14) To avoid increased computational complexity due to long concatenated sequences, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 15 models ¯XC( ˆ R,s) τ by adding an embedding linear layer LN for each modality such that the original input is projected to the embedding dimension, followed by layer normalization Sigmoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' ¯XC( ˆ R,s) τ = Sigmoid � LN � XC( ˆ R,s) pc �� ¯Xa τ = Sigmoid �LN �Xa pc �� (15) These tokens are processed by an encoder model that predicts future action tokens via autoregressive modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Since the multi-hop inference is a fixed-length sequence, the transformer’s encoder structure is selected, which consists of L stacked blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 4, each block mainly includes two types of sublayers: multi-head self-attention MHA and fully-connected feed-forward network FFN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The transformer model contains many parameters including WV , WQ, WK matrices, and the values in each stack and head are designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' As formalized in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 16, multi-head attention mechanism Attn is introduced to defined headM ′ r i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Here, r′ and a represent return-to-go and action features IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 7 in reinforcement learning sequence framework, q represents knowledge graph query (< bos >, h, r >) feature, and I and O represent the image and OCR features in multimodal feature extraction module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' headMr =Attn �xrW r Q , � xrW r K, xqW q K, xIW I K, xoW o K � , � xrW r V , xqW q V , xIW I V , xoW o V �� (16) The calculation method of headMq i , headMI i and headMo i is similar to that of headMr i , where headMa i is redefined via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' headMa = Attn �xaW a Q, xaW a K, xaW a V � (17) Hence, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 18 and 19 model the hidden state of the encoder layer l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' ¯XU l = MHA � LN � ¯XU τ �� + XU l−1 (18) XU l = FFN � LN � ¯XU l �� + ¯XU l (19) Next, with the mask shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 3, the encoder only focus on previous labels a < k of the current return-to- go, multimodal fusion input and output paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The specific details of the mask will be described in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='4 Mask Mechanism Design In the link prediction task of the recommendation sys- tem, due to feature redundancy, lack of sufficient train- ing data and complex model design, the recommendation system is extremely prone to the one-epoch phenomenon, that is, the over-fitting phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Therefore, three mask mechanisms are designed to overcome the overfitting phe- nomenon, and they are also used to solve the input and out- put requirements of the reinforcement learning framework established in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Mask mechanism I: As shown in the shaded area in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 3, Mask mechanism I is used to ensure the input and output of the transformer and realize the Markov decision process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Through step-by-step prediction, the path is pre- dicted sequentially to obtain the final target entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' When predicting the next action, the previous action history will be added to the state to achieve state transition, that is, the real result of the previous step will be used as input to predict the output of the next path token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' In this way, the context information is effectively used by the model, thereby ensuring the accuracy of the model, so that the final result will not have a large error due to a one-step error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Mask mechanism II: As shown in the blue dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 3, Mask mechanism II is used to solve the problem of model overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' After multimodal information dimensionality reduction, data of training features is sparse, which easily leads to fast immature convergence of the model, resulting in overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Therefore, it is necessary to perform data dropout on this part of the embedded input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The double data dropout mechanism is introduced for data enhance- ment, which is conducive to retaining the original high- quality samples as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Specifically, for a given sequence, the data loss scheme is enabled with a certain probability pk, and when applying the data loss scheme, tokens in the sequence are randomly masked with a certain probability pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Mask mechanism III: As shown in the red dot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 3, the Mask mechanism III is used to make the model generate more new trajectories by itself, which makes the trajectories generated by the transformer in the past randomly masked out for the next action prediction task such that the model gradually learns from the self-generated trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Mask mechanism III is simple and easy to implement, and it does not add any additional computational cost and parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' In the masking mechanism, as formalized in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 20, multiplying each token xi ∈ x by the mask gets its au- toregressive log maximum likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Here η represents the mask ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' log p (x) = log n � i=1 p � xi |[I [mj ≤ η] · xj] i=1 j=0 � , η ∈ [0, 1] (20) Mask mechanism I adopts a complete mask, so η = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Mask mechanism II and III are random masks, which are randomly masked according to a certain probability value mj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='5 Loss Function Design The multimodal information of entities (features of images and text features in images) is expected to enhance learning through fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' However, experiments have found that after incorporating multiple modalities, the model will suffer from a lack of optimization, which is caused by the dom- inance of one mode in some scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' For example, image information dominates when inferring relations is relevant to tasks such as “colour, item type”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' When reasoning about relations is a relevant task like “brand”, textual information in images dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Therefore, inspired by [54], a dynamic gradient adjustment mechanism is introduced to train three models separately, taking two modes and their concat as three inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' By monitoring the contribution of each mode to the learning objective, each mode optimization is adap- tively controlled, thereby alleviating the imbalance of mode optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Three transformer’s encoders, represented by Enc (·), are accepted three modal features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' When decoding ψk := (r1, t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' , rnt, < eos >), Softmax is used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 21 to calculate the distribution pχ i , where χ ∈ (fig, ocr), b is the bias of the prediction model [55], and the addition of b/2 is used as a bias compensation of single-modal prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' pχ i = |ψ| � k=1 Soft max (MLP � Enc � ψχ n � θχ, xχ n + b 2 � W χ n )))k (21) In the same way, the distribution of concat feature pconcat i is shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' pconcat i = |ψ| � k=1 Softmax �MLP �Enc ��ψconcat n �θconcat , xconcat) ·W concat n ��� k (22) As Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 23-25 shows, a cross-entropy loss is used, where ε is a label smoothing hyperparameter ranging from 0 to 1 to IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 8 avoid overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' A single-modal image feature, a single- modal OCR feature, and a multi-modal feature which is defined as the concat of both are fed respectively to compute three different losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' At the same time, since the previously designed mask mechanism shields some features, it is also necessary to exclude the loss caused by the mask token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Also, to prevent the model from giving higher scores to shorter paths, the sum of log-likelihoods divided by the length of the path is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Lfig = − 1 ���pfig mask ��� · N N � i=1 � ε log pfig t + � 1 − ε N − 1 � log pfig i � (23) Locr = − 1 |pocr mask| · N N � i=1 � ε log pocr t + � 1 − ε N − 1 � log pocr i � (24) Lconcat = − β |pconcat mask | · N N � i=1 � ε log pconcat t + � 1 − ε N − 1 � log pconcat i � (25) Here β represents the weight value to encourage explo- ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' If during the training process, Mask mechanism III is activated, that is, the part of the input path is masked out, it means that a new trajectory is generated, so the weight of pconcat t should be increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' To optimize the multimodal contribution imbalance problem, the modal contribution difference ratio parameter (ρfig t , ρocr t ) is introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 26 to adaptively adjust the gradient of each modality, where ρfig t is ρocr t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 5, the coefficient coeff u n is integrated into the network corresponding to the modal association via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 27 based on [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' At the same time, Gaussian noise N is introduced to enhance the generalization ability of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' ρocr n = Locr Lfig (26) coeff u n = � 1 − tanh (α · relu (ρu n)) , ρu n > 1 1, others (27) ∇Wn+1 u =∇Wn u × coeff u n + N � 0, � std (∇Wn u) +e−8� (28) Here u ∈ {fig, ocr}, and α is a hyperparameter that controls the degree of modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 4 EXPERIMENTS In this section, experiments are implemented based on two newly established datasets, and some state-of-the-art (SOTA) baselines are used for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The experiment is mainly divided into four parts: link prediction main experiment, ablation experiment, training set mask exper- iment, and parameter interpretability analysis experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='1 Datasets In this subsection, two newly established datasets, OpenBG- IMG+ and OpenBG-Complete-IMG+, are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='1 OpenBG-IMG+ A new dataset OpenBG-IMG+ is created, derived from a part of the OpenBG-IMG dataset [56], which is a multimodal dataset in the field of e-commerce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' This dataset is released in the CCKS2022 Task Three competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Since the data set released by the competition has no correct answer, the OpenBG-IMG valid set is used as the test set in this paper, and the training set is divided into several parts as the valid set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The used dataset contains 28,891 entities and 136 relations, where only some of the head entities have image information, while none of the tail entities has image information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Each image corresponds to only one entity, and there is no duplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Table 2 shows specific statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='2 OpenBG-Complete-IMG+ A new repository OpenBG-Complete-IMG+ is created based on the already created database OpenBG-IMG+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Its training set and valid set are obtained from OpenBG-IMG+ deleting the triplet data with no image information in the head entity, and the test set remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Like OpenBG-IMG+, the tail entities of all data do not contain image information, but all head entities of the training set of OpenBG-Complete- IMG+ contain image information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' This new dataset contains 136 relations and 22297 entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Table 2 shows specific statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='2 Baselines To study the performance of IMKGA-SM, three categories of methods are used for comparison: (1) Translation-based models, TransE [11], TransH [12], and TransD [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' (2) Nonlinear-based models, DistMult [16], ComplEx [18], and TuckER [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' (3) Multimodal knowledge graph linking model, TransAE [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='3 Evaluation Protocol To further analyze the influence of image information, part of the training data is masked, and IMKGA-SM is used for experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' According to inferences, the current dataset OpenBG-IMG+ may contain enough structural information for prediction, which interferes with the analysis of visual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' In order to highlight the role of visual infor- mation, a part of the training data is masked to create a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Similar to recent works [10] [25], as formalized in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 29 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 30, the mean reciprocal rank MRR and the average proportion of triples with rank less than n Hits@n are used to evaluate inference performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' MRR = 1 |Q| |Q| � i=1 1 ranki = 1 |Q| � 1 rank1 + 1 rank2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' + 1 rank|S| � (29) Here Q is the set of test queries, |Q| represents the number of queries, ranki is the link prediction rank of the i-th triple [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The larger the MRR indicator, the better the prediction effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='HIT@n = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='|Q| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='|Q| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='I (ranki ⩽ n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='(30) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
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+page_content='Statistics of The Experimental Datasets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='Dataset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='#Ent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='#Rel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='#Train ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='#Valid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='#Test ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='#num of image ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
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+page_content='136 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
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+page_content='10930 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
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+page_content='Results of Knowledge Graph Link Prediction on OpenBG-IMG+ and OpenBG-Complete-IMG+ Datasets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
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+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='76% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
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+page_content='5% 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='64% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='12% Here I (·) is the indicator function, if the condition is true, the function value is 1, otherwise, it is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The three indicators HIT@1, HIT@3, and HIT@10 describe the probability that the top K(K = 1, 3, 10) entities with the highest score in the link prediction contains the correct entity [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='4 Implementation Details Next, the experiments are mainly based on the knowledge graph link prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' To augment the training data, each original training set triplet is reversed to generate an inverse triplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The knowledge graph triplets in the test dataset are sorted by all entities in descending order of probability value, leaving the top ten predicted entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Models are trained using the Adam [58] optimizer and analyzed for hyperparameters, eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='5 Link Predict Results Link prediction results are shown in Table 3 (all scores are expressed as percentages), where the most competitive baseline TransAE [46] results are underlined and the best results are highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The following points are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Table 3 is studied for link prediction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' It is seen that the accuracy performance of IMKGA-SM is better than all other models, and IMKGA-SM uses multi-hop reasoning, which is interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' It is shown that the model is proven to be both interpretable and highly accurate, achieving state-of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The introduction of a multi- modal knowledge graph in the OpenBG-IMG+ dataset is not obvious enough, so only the triplet data with image information in the head entity is retained in the new dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The results show that the improvement effect of IMKGA-SM in OpenBG-Complete-IMG+ is generally better than that of OpenBG-IMG+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' This is speculated to be due to being over- whelmed with the help of other multimodal information when the dataset already has rich structural information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' To further analyze the influence of image information, a new dataset OpenBG-Complete-IMG+ is created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' All head entities in the dataset have image information, and the experiment is performed again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='6 Ablation Learning In this section, ablation learning is divided into three parts: the influence of multimodality, reward expectation, and mask mechanism on the model, and perform specific data analysis on the complete IMKGA-SM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='1 IMKGA-SM (No Img) vs IMKGA-SM (MKG) To further explore the role of image information, IMKGA- SM (No image) and IMKGA-SM (MKG) are compared on two datasets, and the link prediction experiment results are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Image embedding and ocr embedding are added to IMKGA-SM (MKG), and a dynamic loss function adjustment mechanism is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The experiment found that the improvement effect on OpenBG-Complete-IMG+ is more stable than OpenBG-IMG+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='2 IMKGA-SM(No Img) vs IMKGA-SM(RL) In order to verify the improvement of the model by the introduction of reward expectation, IMKGA-SM (No image) and IMKGA-SM (RL) are compared on two data sets, and the link prediction experiment results are shown in the Ta- ble 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' A reward expectation mechanism based on perceptual interaction and a reward-related mask is added to IMKGA- SM (RL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The experiment results show that after adding the reward expectation mechanism ˆR, the model effect has been improved, but the improvement is not as much as that of MKG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='3 IMKGA-SM (MKG+RL) IMKGA-SM (MKG+RL) adds a masking mechanism on the basis of IMKGA-SM(MKG) and IMKGA-SM(RL) and ver- ified its effect on OpenBG-IMG+ and OpenBG-Complete- IMG+ datasets, which are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Compared IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 10 Table 4 Statistics of Datasets in Training Data Masking Experience Dataset #Ent #Rel #Train #Valid #Test OpenBG-IMG+80% 27839 136 158527 8344 10930 OpenBG-IMG+70% 22297 136 138479 7289 10930 OpenBG-IMG+47% 21817 136 92648 4877 10930 OpenBG-IMG+35% 21469 136 68599 3611 10930 OpenBG-IMG+28% 21215 136 55397 2916 10930 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Link prediction results (MRR) on OpenBG-Complete-IMG+x%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' with the multimodal knowledge graph linking baseline TransAE [46], IMKGA-SM has a 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='67% improvement on the OpenBG-IMG+ dataset and a 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='01% improvement on the OpenBG-Complete-IMG+ dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Compared with the nonlinear-based baseline DistMult [16], IMKGA-SM has a 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='39% improvement on the OpenBG-IMG+ dataset and a 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='681% improvement on the OpenBG-Complete-IMG+ dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='7 Training Data Masking To further explore the impact of image and structural in- formation on the results, masking is performed on part of the training data, and IMKGA-SM is used for experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' It is speculated that the current dataset OpenBG-IMG+ may contain enough structural information for prediction, interfering with the analysis of visual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' In order to highlight the role of visual information, a part of the training data is masked out to create datasets by controlling the frequency of a head entity with image information, which is OpenBG-IMG+28%, OpenBG-IMG+35%, OpenBG- IMG+47%, OpenBG-IMG+70%, OpenBG-IMG+80% and OpenBG-IMG+100%, respectively, and the dataset informa- tion is shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Then, the link prediction exper- iment is carried out again, and the results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 6, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' It is seen that IMKGA-SM has shown obvious advantages in data sets of different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The traditional method has a significant increase after adding structural information, while IMKGA-SM is still relatively stable in the improvement of the scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' IMKGA-SM not only has comparable generalization ability to neural network models, but also has stronger interpretability than other baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Link prediction results (HIT@1) on OpenBG-Complete- IMG+x%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Link prediction results on OpenBG-Complete-IMG+ in different N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='8 Parameter Interpretability In this subsection, the parameter interpretability is divided into three parts, mainly analyzing the impact of different batch sizes, label smooth and modulation impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='1 The Influence of Different Batch Sizes N Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 8 investigates the effect of different batch sizes N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The batch size is set to N ∈ [16, 32, 64, 128, 256, 512].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' It is observed that as N increases, the performance of IMKGA- SM rises first and then declines steadily in most cases, pre- sumably because undertraining and overfitting negatively affect the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The results show that an appropriate training parameter size improves the effectiveness of the inference model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' From the experimental results, the optimal parameter is N = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='2 The Influence of Different Labels Smooth ε In this paper, the method of label smoothing is used in the loss function, which is a regularization method to prevent overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' By setting α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='5, the effect of the label smooth- ing parameter ε is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The batch size is set to ε ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' It is observed that with the decrease of ε, the performance of IMKGA-SM rises first and then declines steadily in most cases, which is probably because MRR is the main evaluation indicator, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='7 DistMult CompEx TuckER IMK GA-SM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
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+page_content='1 0 N=16 N=32 N=64 N=128 N=256 N=512IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 11 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Link prediction results on OpenBG-Complete-IMG+ in different ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Link prediction results on OpenBG-Complete-IMG+ in differ- ent α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' and the shrinking of its proportion has a negative effect on the final result influences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' The results show that the optimal parameter is ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='3 The Influence of Different Modulation Impact α With ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='9, the effect of the modulation impact α on IMKGA-SM is demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' From the results, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content='6 is observed to be the optimal value on the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' 5 CONCLUSIONS AND DISCUSSION In this paper, how to effectively utilize multi-modal aux- iliary features for multi-hop knowledge graph inference is investigated, aiming to improve the accuracy of model inference and achieve interpretability simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' An efficient IMKGA-SM model is proposed, which outperforms existing methods on the multimodal knowledge graph in- ference task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' In IMKGA-SM, structural features and multi- modal data are first extracted in-depth, and then a return-to- go mechanism based on perceptual similarity is constructed and applied to the large sequence framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' In addition, three mask mechanisms are designed to alleviate the prob- lem of data sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Next, a multimodal autoregressive loss function adjustment mechanism is introduced to take full advantage of multimodality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' Finally, experimental results show that IMKGA-SM achieves higher effectiveness and interpretable ability versus other trending rivals in knowl- edge graph link prediction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' To conclude, IMKGA-SM requires effective methods to minimize the negative impact of sparse data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
+page_content=' These tasks are left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE0T4oBgHgl3EQfigEg/content/2301.02445v1.pdf'}
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+Peculiar velocity effects on the Hubble constant from time-delay cosmography
+Charles Dalang,1, ∗ Martin Millon,2, † and Tessa Baker1, ‡
+1Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom
+2Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, Stanford, CA 94305, USA
+Two major challenges of contemporary cosmology are the Hubble tension and the cosmic dipole tension. At
+the crossroad of these, we investigate the impact of peculiar velocities on estimations of the Hubble constant
+from time-delay cosmography. We quantify the bias on the inference of the Hubble constant due to peculiar
+velocities of the lens, the source and of the observer. The former two, which may cancel from one system to
+another, affect the determination of the angular diameter distances in the time-delay formula, and reconstructed
+quantities like the angle to the source, via a lens model. On the other hand, the peculiar velocity of the observer,
+which is a debated quantity in the context of the cosmic dipole tension, systematically affects observed angles
+through aberration, redshifts, angular diameter distance and reconstructed quantities. We compute in detail the
+effect of these peculiar velocities on the inference of the Hubble constant to linear order in the peculiar velocities
+for the seven lenses of the H0LiCOW/TDCOSMO collaboration. The bias generated by the observer’s peculiar
+velocity alone can reach 1.15% for the lenses which are well aligned with it. This results in a 0.25% bias for the
+seven combined lenses. Assuming a typical peculiar velocity of 300 km s−1 for the lens and the source galaxies,
+these add an additional random uncertainty, which can reach 1% for an individual lens, but reduces to 0.24% for
+the full TDCOSMO sample. The picture may change if peculiar velocities turn out to be larger than expected.
+Any time-delay cosmography program which aims for percent precision on the Hubble constant may need to take
+this source of systematic bias into account. This is especially so for future ground-based surveys which cover a
+fraction of the celestial sphere that is well aligned with the observer’s peculiar velocity.
+I.
+INTRODUCTION
+P
+ersistent tensions in cosmological datasets may be indicators of new physics or of unknown systematics. While the former
+is very exciting, excluding confidently the latter is notoriously difficult. On the theoretical side, this is mostly because in
+extracting cosmological parameters, approximations are needed, which require a set of assumptions that may be broken. Two of
+these tensions include disagreement on the kinematic cosmic dipole, which can be translated into a tension on the peculiar velocity
+of the observer [1] and on the Hubble constant [2]. Both of these tensions are between the Cosmic Microwave Background (CMB)
+and other datasets.
+The CMB dipole allows one to extract the velocity of the observer, which effectively Doppler shifts the black body radiation of
+angular average temperature ⟨𝑇⟩ from the CMB to higher and lower temperatures (𝛿𝑇/⟨𝑇⟩)dip ∼ O(10−3) in opposite hemispheres
+aligned with the observer’s velocity. This works well provided the intrinsic CMB dipole, which is expected to be of the order
+O(10−5), is small in comparison. This is expected from a nearly scale-invariant power spectrum of primordial fluctuations of the
+inflaton generated at the end of a period of quasi-de Sitter expansion during inflation. Under the assumption that the intrinsic
+CMB dipole vanishes, known as the entirely kinematic interpretation of the CMB dipole, one obtains ||𝒗dip|| = 369.82 ± 0.11
+km s−1 towards ˆ𝒗dip = (264.021◦ ± 0.011◦, 48.253◦ ± 0.005◦) in galactic coordinates [3–6]. This defines a reference frame known
+as the CMB frame. If the interpretation is correct, the same velocity should induce correlations between the 𝑙 and 𝑙 ± 1 multipoles
+of the CMB, which was checked in [6–8] and gives consistent results, albeit the relatively large error bars still leave room for
+an intrinsic dipole which can make up to 40% of the CMB dipole [9]. It should be noted that spectral distortions of the CMB
+monopole, dipole and quadrupole should let one separate the intrinsic dipole from its kinematic counterpart with a sufficiently
+advanced detectors [10].
+Alternatively, the peculiar velocity of the observer can be extracted from source number counts of relatively high redshift
+sources (𝑧 ≥ 0.1), such as quasars, to avoid contamination from local structures [11, 12]. This was pioneered by G. Ellis and
+J. Baldwin for flux-limited surveys of sources with a flux density following a powerlaw frequency spectrum [13]. Aberration of
+angles and Doppler shift then affect these number counts per unit solid angle in such a way that permits the extraction of the
+observer’s peculiar velocity with respect to these sources. This has led to a number count dipole, which is well aligned with the
+CMB dipole but about 2 − 5 times as large as expected from 𝒗dip and which has reached a ∼ 5𝜎 tension [14–18]. In [12], it was
+suggested that the redshift evolution of the population of sources may, at least partially, explain the discrepancy. This was further
+investigated by the authors of [19], who also find large variations in the theoretical expectation of the number count dipole in the
+presence of parameter evolution when using different quasar luminosity function models. The authors of Ref. [18] reanalyzed the
+∗ c.dalang@qmul.ac.uk
+† millon@stanford.edu
+‡ t.baker@qmul.ac.uk
+arXiv:2301.05574v1 [astro-ph.CO] 13 Jan 2023
+
+2
+data of [14, 15] and concluded that neither masking nor parameter evolution can fully explain the discrepancy, although the latter
+is subject to further assumptions. If the dominantly kinematic interpretation of this number count dipole is correct, it should show
+up in the correlations between the 𝑙 and 𝑙 ± 1 multipoles of the number counts, which require high-angular resolution surveys
+[20, 21]. An observer offset from the center of an ultra-large void was also suggested in [22] as a solution. This would imply
+effective large source peculiar velocities as a result of working with a homogeneous and isotropic background. In any case, this
+problem requires further studies [1].
+The Hubble tension is somehow more popular [23] and has been established for a longer period of time. It is the disagreement
+between direct measurements of the Hubble constant and inference of 𝐻0 from the CMB, if assuming a flat homogeneous and
+isotropic Universe dominated by Cold Dark Matter and a cosmological constant, the so-called ΛCDM model. The Hubble
+constant is inferred from the angle upon which the scale associated to the horizon at the last scattering surface is seen in the CMB,
+which is extracted from the temperature fluctuations. This results in 𝐻0 = 67.4 ± 0.5 km s−1 Mpc−1 at 68% confidence level
+[24]. In contrast, two of the most competitive local measurements of the Hubble constant come from supernovae type Ia, which
+requires calibration via the distance ladder and from time-delay cosmography with strongly lensed quasars. Teams performing
+these experiments reported relatively high 𝐻0 = 74.03 ± 1.42 km s−1 Mpc−1 [25, 26] and 𝐻0 = 73.3+1.7
+−1.8 km s−1 Mpc−1 [27],
+respectively. Combining these two direct measurements leads to a 5.3𝜎 tension on 𝐻0 with the CMB. This has led to a plethora of
+alternative models, with various levels of complexity and success, as demonstrated by the existence of the 𝐻0-olympics [28] (see
+also [29]). Importantly, it should be noted that using only stellar kinematics instead of assumptions about the mass profile of
+the lensing galaxies to break the so-called mass-sheet degeneracy [30], led to 𝐻0 = 74.5+5.6
+−6.1 km s−1 Mpc−1 with the seven same
+strongly-lensed systems [31], which is consistent with Planck [24].
+The peculiar velocity of the observer plays the role of a foundational stone for cosmological experiments which work in
+the CMB frame [32–35]. It is therefore alarming that some experiments disagree on the peculiar velocity of the observer 𝒗𝑜.
+For example, directional dependencies of 2 − 3𝜎 level on cosmological parameters extracted from the CMB were reported in
+[36]. A remnant of anisotropies on 𝐻0 determined from supernovae type Ia data was reported in [37] even when working in
+the CMB frame. The authors of [38] found 4 km s−1 Mpc−1 difference in 𝐻0 in opposite hemispheres aligned with the CMB
+dipole, although such variations are expected. Similar hints of 𝐻0 anisotropies from strongly lensed quasars were noticed in [39],
+although these observations are not corrected for the peculiar velocities of the observer, lenses or sources. In particular, the
+H0LiCOW/TDCOSMO collaboration pointed out a mild 1.8𝜎 significance for an 𝐻0 which decreases with observed lens redshift
+𝑧′
+𝑙 [27, 40]. Two of the lowest lens redshift systems, which give the highest Hubble constant estimates, turn out to also be well
+aligned with the CMB dipole, as remarked in [39].
+In this work, we focus on the determination of the Hubble constant from the time delay of strongly lensed quasars and study
+the impact of peculiar velocities on this measurement. One may expect that peculiar velocities of the order of 𝑣/𝑐 ≃ O(10−3)
+do not affect the 𝐻0 measurement beyond O(10−3). However, to our knowledge, there has not been any rigorous study of the
+accumulation of effects of aberration and Doppler shift on time-delay cosmography, and propagation of biases through the lens
+model, which may inflate the proportionality constant in front of 𝑣/𝑐. Our goal is to fill this gap and study if there can be any
+relation between the Hubble and dipole tensions.
+The paper is structured as follows. In Sec. II, we review the basics of time-delay cosmography for a singular isothermal sphere,
+which allows us to fix notation. In Sec. III, we detail all effects of peculiar velocities on the observables and also how these
+propagate through the lens model and to the Hubble constant determination. In Sec. IV, we apply our findings to the seven lenses of
+TDCOSMO1, compute the bias on the Hubble constant for each lens as a function of the peculiar velocities and discuss our results.
+Finally, in Sec. V, we conclude. We suggest that a busy reader principally interested in the total impact on 𝐻0 measurements
+should review the form of Eqs. (75) and (77), then move directly to section IV. Throughout the article, bold symbols denote 2 or 3
+dimensional vectors, hats indicate unit vectors || ˆ𝒏|| = 1. Sometimes unit vectors in R3 are expressed in spherical coordinates
+ˆ𝒏 = (cos 𝜃 cos 𝜑, cos 𝜃 sin 𝜑, sin 𝜃) �= (𝜃, 𝜑), where 𝜃 ∈ [0, 𝜋] and 𝜑 ∈ [0, 2𝜋[ indicate the polar and azimuthal angle, respectively.
+We note the speed of light 𝑐 and Newton’s constant 𝐺N.
+II.
+SINGULAR ISOTHERMAL SPHERE
+In this section, we review time-delay cosmography for an isothermal sphere and fix our notation. Derivations may be found
+in [42] and the reader experienced in lensing time delay formalism can skip to Sec. III. The cosmic time delay Δ𝑡𝑖 𝑗 ≡ 𝑡𝑖 − 𝑡 𝑗
+variations in lensed images 𝑖 and 𝑗 for a comoving observer, lens and source can be expressed as [42]
+𝑐Δ𝑡𝑖 𝑗 = (1 + 𝑧𝑙) 𝑑𝑙𝑑𝑠
+𝑑𝑙𝑠
+� ˆ𝜙(𝜽𝑖, 𝜷) − ˆ𝜙(𝜽 𝒋, 𝜷)
+�
+,
+(1)
+1 Six lenses come from H0LiCOW [27] and one from STRIDES [41]. These seven systems are now analysed jointly by the TDCOSMO collaboration [40].
+
+3
+image
+source
+lens
+AC2HicjVHLSsNAFD2Nr1pf0S7dBIv
+gqQq6rLoxmUF+8C2lCSdtsG8yEyEUAruxK0/4Fa/SPwD/QvjCmoRX
+RCZs6ce8+ZuXPtyHO5M3XnDY3v7C4lF8urKyurW/om1sNHiaxw+pO6
+IVxy7Y489yA1YUrPNaKYmb5tsea9vWZjDdvWMzdMLgUacS6vjUM3IHr
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+wclqnWavz2MYO9qifx6jiHDXUyTvFI57wrF1pt9qdv+ZquUyTRHfh
+vbwAc0+l+8=✓
+AC13icjVHLSsNAFD2Nr1pfsS7dBIv
+gqiQq6rLoxmUF+5BWSpJOa2heZCZiKcWduPUH3OofiX+gf+GdMQW1iE
+7IzJlz7zkzd64T+x4Xpvma02Zm5+YX8ouFpeWV1TV9vVjnUZq4rOZGf
+pQ0HZsz3wtZTXjCZ804YXbg+KzhDE5kvHNEu5F4bkYxuwysPuh1/Nc
+WxDV0YtJ/K7fBjQMmo7TNjl4y6YaxjSwMlBCNqR/oI2uojgIk
+UAhCsA8bnL4WLJiIibvEiLiEkKfiDGMUSJtSFqMm9gBzX3atTI2p
+L305Ert0ik+/QkpDWyTJqK8hLA8zVDxVDlL9jfvkfKUdxvS6mReAbEC
+V8T+pZtk/lcnaxHo4UjV4FNsWJkdW7mkqpXkTc3vlQlyCEmTuIuxRP
+CrlJO3tlQGq5ql29rq/ibypSs3LtZbop3eUtqsPWzndOgvlu2Dsp7Z/
+ulynHW6jw2sYUd6uchKjhFTXyvsEjnvCsXWi32p12/5mq5TLNBr4N7
+eEDfaiXaw=�
+ACxnicjVHLSsNAFD2Nr1pfVZdugkV
+wVRIVdVl02VF+4BaSjKd1qFpEiYTpRTBH3Crnyb+gf6Fd8YU1CI6Ic
+mZc+85M/dePw5EohznNWfNzS8sLuWXCyura+sbxc2tRhKlkvE6i4JIt
+nwv4YEIeV0JFfBWLk38gPe9IfnOt685TIRUXilxjHvjLxBKPqCeYqo
+y1436BZLTtkxy54FbgZKyFYtKr7gGj1EYEgxAkcIRTiAh4SeNlw4iI
+nrYEKcJCRMnOMeBdKmlMUpwyN2SN8B7doZG9JeyZGzeiUgF5JSht7p
+IkoTxLWp9kmnhpnzf7mPTGe+m5j+vuZ14hYhRti/9JNM/+r07Uo9HFq
+ahBU2wYXR3LXFLTFX1z+0tVihxi4jTuUVwSZkY57bNtNImpXfWM/E
+3k6lZvWdZbop3fUsasPtznLOgcVB2j8uHF0elylk26jx2sIt9mucJKq
+ihjp5D/CIJzxbVSu0UuvuM9XKZptfFvWwdtH5BQdl
+A
+CxnicjVHLSsNAFD2Nr1pfVZdugkVwVRIVdVl02VF+4BaSjKd1qFpEiYTpRTBH3Crnyb+gf6Fd8YU1CI6IcmZc+85M/dePw5EohznNWfNzS8sL
+uWXCyura+sbxc2tRhKlkvE6i4JItnwv4YEIeV0JFfBWLk38gPe9IfnOt685TIRUXilxjHvjLxBKPqCeYqoy1436RZLTtkxy54FbgZKyFYtKr7
+gGj1EYEgxAkcIRTiAh4SeNlw4iInrYEKcJCRMnOMeBdKmlMUpwyN2SN8B7doZG9JeyZGzeiUgF5JSht7pIkoTxLWp9kmnhpnzf7mPTGe+m5j+
+vuZ14hYhRti/9JNM/+r07Uo9HFqahBU2wYXR3LXFLTFX1z+0tVihxi4jTuUVwSZkY57bNtNImpXfWM/E3k6lZvWdZbop3fUsasPtznLOgcVB
+2j8uHF0elylk26jx2sIt9mucJKqihjp5D/CIJzxbVSu0UuvuM9XKZptfFvWwd9v5BXds
+A
+CyXicjVHLSsNAFD2Nr1pfVZdugkVwVRIVdVl0I7ipYB9QS0nSaY3m5WQi1tKVP+BWf0z8A/0L74xTUIvohCRnzr3nzNx73STwU2FZrzljanpmd
+i4/X1hYXFpeKa6u1dM4x6reXEQ86brpCzwI1YTvghYM+HMCd2ANdzrYxlv3DKe+nF0LgYJa4dOP/J7vucIourdzjBIR51iySpbapmTwNagBL2
+qcfEF+gihocMIRgiCMIBHKT0tGDQkJcG0PiOCFfxRlGKJA2oyxGQ6x1/Tt06l2Yj20jNVao9OCejlpDSxRZqY8jheZqp4plyluxv3kPlK
+e82oL+rvUJiBS6J/Us3zvyvTtYi0MOhqsGnmhLFyOo87ZKprsibm1+qEuSQECdxl+KcsKeU4z6bSpOq2mVvHRV/U5mSlXtP52Z4l7ekAds/xzk
+J6jtle7+8e7ZXqhzpUexgU1s0zwPUMEJqiR9xUe8YRn49S4Me6M+89UI6c16/i2jIcPQdKR2Q=dls
+observer
+A
+AC3HicjVHLSsNAFD2Nr1pfURcu3ASL4KqkKu
+qy6MZlBfuAtpY8pm1w8iCZCKV0507c+gNu9
+XvEP9C/8M6YglpEJyRz5tx7Tu6da0fcS4Rpv
+ua0mdm5+YX8YmFpeWV1TV/fqCdhGjus5oQ8j
+Ju2lTDuBawmPMFZM4qZ5ducNezrMxlv3LA48
+cLgUgwj1vGtfuD1PMcSRHX1LbfLjbYdcjcZ
++rSN2mLAhDXu6kWzZKplTINyBorIVjXUX9CG
+ixAOUvhgCAIc1hI6GmhDBMRcR2MiIsJeSrO
+MEaBtClMcqwiL2mb59OrYwN6Cw9E6V26C+
+c3piUBnZJE1JeTFj+zVDxVDlL9jfvkfKUtQ1
+ptzMvn1iBAbF/6SaZ/9XJXgR6OFE9eNRTpBj
+ZnZO5pOpWZOXGl64EOUTESexSPCbsKOXkng
+2lSVTv8m4tFX9TmZKVZyfLTfEuq6QBl3+Ocx
+rU90vlo9LBxWGxcpqNOo9t7GCP5nmMCs5RU
+3V/4gnPGtX2q12p91/pmq5TLOJb0t7+ACcZ
+lmdl✓
+A
+C2HicjVHLSgMxFD0dX7W+ql26GSyCqzJVUZdFNy4r2Ae2pWSmaTuYeTCTEUopuBO3/oBb/SLxD/QvIlTUItohklOzr3nJDfXDoUbS8t6zRhz8
+wuLS9nl3Mrq2vpGfnOrHgdJ5PCaE4gatos5sL1eU26UvBmGHm2YI37OszFW/c8Ch2A/9SjkLe8djAd/uwyR3XyhbQeiF48WsZtJsIhm3T
+zRatk6WHOgnIKikhHNci/oI0eAjhI4IHDhyQswBDT10IZFkLiOhgTFxFydZxjghxpE8rilMGIvaZ5QLtWyvq0V56xVjt0iqA/IqWJXdIElBcRV
+qeZOp5oZ8X+5j3WnupuI1rt1MsjVmJI7F+6aeZ/daoWiT5OdA0u1RqRlXnpC6JfhV1c/NLVZIcQuIU7lE8Iuxo5fSdTa2Jde3qbZmOv+lMxaq
+9k+YmeFe3pAaXf7ZzFtT3S+Wj0sHFYbFymrY6i23sYI/6eYwKzlFjbxHeMQTno0r49a4M+4/U41Mqing2zAePgCm+Jf↵
+Figure 1. We sketch the lensing configuration. The observer, on the left, sees an image at a small angle 𝜽 from the optical axis, which connects
+via a null geodesic the observer to the lens’ center of mass. The unobservable angles to the source 𝜷 and the deflection angle 𝛼 are also displayed.
+The angular diameter distance 𝑑𝑙, 𝑑𝑠 and 𝑑𝑙𝑠 at play are displayed in the intuitive Euclidean case where 𝑑𝑙 + 𝑑𝑙𝑠 = 𝑑𝑠, although that equality
+does in general not hold.
+where 𝑧𝑙 is the lens redshift, 𝑑𝑙, 𝑑𝑠 and 𝑑𝑙𝑠 are angular diameter distances to the lens, to the source and between the lens and the
+source respectively. A sketch of the lensing configuration is displayed in Fig. 1. Contrary to Euclidean intuition, 𝑑𝑙 + 𝑑𝑙𝑠 ≠ 𝑑𝑠, in
+general. See also [43] for a derivation of Eq. (1) in arbitrary spacetimes and with arbitrary peculiar velocity configurations. The
+dimensionless Fermat potential is given by
+ˆ𝜙(𝜽, 𝜷) = (𝜽 − 𝜷)2
+2
+− 𝜓(𝜽) ,
+(2)
+where 𝜽 = (𝜃𝑥, 𝜃𝑦) is a 2 dimensional vector indicating small observed angles to the images, typically of the order of a few arcsec
+on the sky, where the origin is the center of mass of the lens, which defines the optical axis. The unobservable 2 dimensional
+angle 𝜷 = (𝛽𝑥, 𝛽𝑦) indicates the source position. This first part of the time delay comes from the geometric difference in the paths
+followed by photons, emitted simultaneously and deflected by the lens. The lensing potential is indicated by 𝜓(𝜽) and tracks the
+time delay accumulated by Shapiro time dilation, and requires a lens model to compute. The images form at sky locations which
+extremize the Fermat potential. In other words, these are solutions of the lens equation:
+𝜷 = 𝜽 − 𝜶(𝜽) ,
+(3)
+where the 2 dimensional deflection angle is 𝜶(𝜽) = (𝛼𝑥(𝜽), 𝛼𝑦(𝜽)) = ∇𝜓(𝜽). Gravitational lenses at cosmological distances have
+a thickness along the optical axis which can be considered much smaller than the distance between the lens, the source and the
+observer. In this case, one can make a thin lens approximation to find [42]
+𝜶(𝜽) = 1
+𝜋
+∫
+R2 d2𝜽′𝜅(𝜽′)
+𝜽 − 𝜽′
+||𝜽 − 𝜽′||2 ,
+(4)
+where 𝜅(𝜽) is the convergence, defined as
+𝜅(𝜽) ≡ Σ(𝜽)
+Σc
+,
+(5)
+where Σ(𝜽) is the mass surface density (in kg m−2) and the critical surface density is given by
+Σc ≡
+𝑐2
+4𝜋𝐺N
+𝑑𝑠
+𝑑𝑙𝑠𝑑𝑙
+.
+(6)
+Eq. (4) expresses that the deflection angle at an angle 𝜽 is more affected by the massive regions in the lens plane which are close to
+𝜽. For a thin lens, the lensing potential can be expressed as
+𝜓(𝜽) = 1
+𝜋
+∫
+R2 d2𝜽′𝜅(𝜽′) log ||𝜽 − 𝜽′|| + const ,
+(7)
+up to an integration constant, which cancels in the time-delay formula. Note that a photon travelling closer (low ||𝜽 − 𝜽′|| ≪ 1) to
+a region with higher mass density (higher 𝜅(𝜽′)) will experience more Shapiro time delay (more negative 𝜓(𝜽)) than if it travels
+far from this region. In the following and throughout the article, we work with a singular isothermal sphere (SIS), which can be
+described by the following mass density
+𝜌(𝑟) =
+𝜎2
+𝑣
+2𝜋𝐺N𝑟2 ,
+(8)
+
+4
+in (kg m−3) where 𝑟 is the distance from the center of mass of the lens, 𝜎𝑣 is the line-of-sight velocity dispersion, which is assumed
+to be constant. This lens model is spherically symmetric, singular at 𝑟 = 0 and its mass formally extends to infinite radius. Until
+the end of the 90’s, this was the most popular model for strong lensing time-delay cosmography because all lensing quantities can
+be derived analytically. The TDCOSMO collaboration has now adopted more sophisticated models to describe the mass profile of
+the lens galaxy such as the power-law elliptical mass distribution [44] and composite models [45], which explicitly includes a
+baryonic and dark matter component. Nevertheless, we do not expect these more sophisticated lens models to change significantly
+our results, while the simplicity of the SIS grants us analytic control. The mass surface density can be obtained by integrating
+along the optical axis, between the source and the observer. This is most easily done in cylindrical coordinates (𝑟, 𝜑, 𝑧), centered
+on the lens center of mass. In that case, 𝜌(𝑟) = 𝜌(𝑑𝑙𝜃, 0, 𝑙) with 𝜃 = ||𝜽|| and we get
+Σ(𝜽) =
+∫ 𝑙𝑜
+𝑙𝑠
+d𝑙𝜌(𝑑𝑙𝜃, 0, 𝑙) =
+𝜎2
+𝑣
+2𝜋𝐺N𝑑𝑙𝜃 Arccot
+� 𝑑𝑙𝜃
+𝑙
+���
+𝑙𝑜
+𝑙𝑠
+�
+.
+(9)
+Taking the limit of far away source and observer, compared to the impact parameter |𝑙𝑜|, |𝑙𝑠| ≫ 𝑑𝑙𝜃, one finds
+Σ(𝜃) =
+𝜎2
+𝑣
+𝐺N𝑑𝑙𝜃 .
+(10)
+Making use of axial symmetry, (i.e. 𝜅(𝜽) = 𝜅(𝜃)), one finds, using Eq. (4), that 𝜶(𝜽) = 𝛼(𝜃)𝜽/𝜃 with
+𝛼(𝜃) = 2
+𝜃
+∫
+𝜃
+0
+d𝜃′𝜃′𝜅(𝜃′) .
+(11)
+For an SIS, this integral reduces to a constant deflection angle
+𝛼(𝜃) = 4𝜋𝜎2
+𝑣
+𝑐2
+𝑑𝑙𝑠
+𝑑𝑠
+≡ 𝛼0 .
+(12)
+This implies that the source angle 𝜷 for an SIS can be reconstructed from only one image 𝜽𝒊,
+𝜷 = 𝜽𝒊
+�
+1 − 𝛼0
+𝜃𝑖
+�
+.
+(13)
+This can also be read as a quadratic equation for 𝜽𝒊, which gives at most 2 images2. In practice, external shear or deviations from
+spherical symmetry of the lens can lead to the formation of 𝑁images > 2 images. This implies that if one attempts to reconstruct 𝜷
+for these systems, one may get slightly different results for each image, which affect the determination of the Hubble constant.
+Therefore, for practical purposes, one rather estimates a source angle for each image 𝜷 = 𝜷(𝜽𝒊). Similarly, the lensing potential for
+an axially symmetric thin lens can be expressed as
+𝜓(𝜽) = 2
+∫
+𝜃
+0
+d𝜃′𝜃′𝜅(𝜃′) log(𝜃/𝜃′) + const. .
+(14)
+which reduces to
+𝜓(𝜃) = 𝛼0𝜃 ,
+(15)
+for an SIS. One can recognize the primitive of 𝛼 (Eq. (12)), where the integration constant has been set to zero. In practice, the
+angular diameter distances are not measured directly but can be inferred from the lens and source redshifts 𝑧𝑙 and 𝑧𝑠, by assuming
+a cosmological model.3 Throughout the article, we assume a flat ΛCDM model with Ω𝑚0 = 0.3 and 𝐻0 = 70 km s−1 Mpc−1. Of
+course, the determination of 𝐻0 from observables does not require an assumption on 𝐻0 but the relative bias, as we will find in
+Sec. III, does depend on 𝐻0. The angular diameter distances can be expressed as
+𝑑𝑙 = 𝑑𝑙[𝑧𝑙] =
+𝑐
+𝐻0(1 + 𝑧𝑙) 𝜒[𝑧𝑙] ,
+(16)
+𝑑𝑠 = 𝑑𝑠[𝑧𝑠] =
+𝑐
+𝐻0(1 + 𝑧𝑠) 𝜒[𝑧𝑠] ,
+(17)
+𝑑𝑙𝑠 = 𝑑𝑙[𝑧𝑙, 𝑧𝑠] =
+𝑐
+𝐻0(1 + 𝑧𝑠) 𝜒[𝑧𝑙, 𝑧𝑠] ,
+(18)
+2 There is also a third image at 𝜃𝑖 = 0, which is infinitely demagnified.
+3 Note that if one would measure the angular diameter distances directly, one could check that the time-delay formula holds for arbitrary peculiar velocity
+configurations [43].
+
+5
+where 𝐻0 is the present-day Hubble constant, 𝜒[𝑧1, 𝑧2] is the dimensionless integral
+𝜒[𝑧1, 𝑧2] ≡
+∫
+𝑧2
+𝑧1
+d𝑧
+𝐸(𝑧) ,
+(19)
+with 𝐻[𝑧] = 𝐻0𝐸(𝑧) ≡ 𝐻0
+√︁
+Ω𝑚0(1 + 𝑧)3 + (1 − Ω𝑚0) and 𝜒[𝑧] = 𝜒[0, 𝑧], which should not be confused with comoving distances.
+One can solve Eq. (1) for 𝐻0 to get
+𝐻0 = 𝜒[𝑧𝑙]𝜒[𝑧𝑠]
+𝜒[𝑧𝑙, 𝑧𝑠]
+ˆ𝜙(𝜽𝒊, 𝜷(𝜽𝒊)) − ˆ𝜙(𝜽 𝒋, 𝜷(𝜽 𝒋))
+Δ𝑡𝑖 𝑗
+.
+(20)
+The present-day Hubble constant is expressed in terms of the lens and source redshifts 𝑧𝑙, 𝑧𝑠, the time delay Δ𝑡𝑖 𝑗, the images 𝜽𝑖,
+𝑖, 𝑗 ∈ [1, . . . , 𝑁images] and the velocity dispersion of the lens 𝜎𝑣. Nearly all of these observables are directly affected by peculiar
+velocities to some extent; some are also indirectly affected through the lens model, and we detail how in the next section.
+III.
+PECULIAR VELOCITY BIAS
+The previous section outlined how one may relate the present-day Hubble rate to time-delayed images of a lensed source,
+assuming a comoving observer, lens and source. In this section, we relax this assumption and compute the bias that the
+non-relativistic peculiar velocities of the observer 𝒗𝑜, the lens 𝒗𝑙 and the source 𝒗𝑠 generate on 𝐻0 to linear order in 𝑣/𝑐 ≪ 1,
+where 𝑣 indicates any of the three peculiar velocities. In particular, we detail the computation of the biases, which are quite
+straightforward for time delays, redshift and angular diameter distances as a function of redshift. On the other hand, the effect of
+aberration of angles turns out to be quite subtle, especially to infer reconstructed quantities like the source angle or the lensing
+potential. Time-pressured readers may directly skip to Eq. (75) and (77), which constitute the main results of this section. We
+denote the quantities that are observed with a prime, while the quantities that comoving (virtual) observers4 would measure are
+left without a prime.
+A.
+Time dilation
+The motion of the observer induces a special relativistic time dilation, which prevents them from measuring cosmic time
+directly. However, this effect is second order in the velocity of the observer and we neglect it:
+Δ𝑡′
+𝑖 𝑗 =
+Δ𝑡𝑖 𝑗
+√︁
+1 − 𝒗2𝑜/𝑐2 = Δ𝑡𝑖 𝑗 [1 + O(𝒗2
+𝑜/𝑐2)] .
+(21)
+The velocity of the source does not affect the observed time delay because one observes the time delay between flux variations of
+the quasar that have been emitted simultaneously.
+B.
+Redshifts
+The motion of the observer, lens and source affect the lens and source observed redshifts with respect to background
+(cosmological) redshifts through Doppler shift. The observed redshifts 𝑧′
+𝑙, 𝑧′
+𝑠 relate to cosmological (or background) redshift 𝑧𝑙, 𝑧𝑠
+in the following way
+(1 + 𝑧𝑙) = (1 + 𝑧′
+𝑙)
+�
+1 + 𝑍𝐿
+𝑣𝑜
+𝑐
+�
+,
+(22)
+(1 + 𝑧𝑠) = (1 + 𝑧′
+𝑠)
+�
+1 + 𝑍𝑆
+𝑣𝑜
+𝑐
+�
+(23)
+with
+𝑍𝐿 = ˆ𝒏′ · (𝒗𝑜 − 𝒗𝑙)
+𝑣𝑜
+,
+(24)
+𝑍𝑆 = ˆ𝒏′ · (𝒗𝑜 − 𝒗𝑠)
+𝑣𝑜
+.
+(25)
+4 It turns out that it is extremely unlikely to be a comoving observer. In particular, in a Universe with structures such as galaxies and filaments, the probability for
+a massive observer to be comoving is zero. Observers on Earth are certainly not.
+
+6
+This apparent expansion in 𝑣𝑜/𝑐 in Eqs. (29)-(31) is practical for book-keeping, but one should keep in mind that it really is a
+simultaneous expansion in 𝑣𝑜/𝑐, 𝑣𝑙/𝑐 and 𝑣𝑠/𝑐. This affects the time delay (Eq. (1)) via the lens redshift 𝑧𝑙 and via the background
+angular diameter distances, which can be computed from the redshift information. Throughout this work, we denote biases on a
+quantity by a corresponding capital letter, which carries the same units (e.g. 𝑍𝑆 is the bias generated by peculiar velocities on 𝑧𝑠).
+C.
+Angular diameter distances
+One can compute the background angular diameter distances from the observed redshift by assuming a cosmological model,
+provided one corrects for the peculiar motion of the emitter and receiver. By background angular diameter distance, we mean the
+distance that would be inferred by a comoving observer that would measure the subtended angle on the sky of a standard ruler. For
+example, the background angular diameter distance to the source can be expressed as a function of observed redshift 𝑧′
+𝑠
+𝑑𝑠 =
+𝑐
+1 + 𝑧𝑠(𝑧′𝑠)
+∫
+𝑧𝑠 (𝑧′
+𝑠)
+0
+d𝑧
+𝐻(𝑧) =
+1
+1 + 𝑧𝑠(𝑧′𝑠)
+∫
+𝑧′
+𝑠+(1+𝑧′
+𝑠) ˆ𝒏′·(𝒗𝑜−𝒗𝑠)
+0
+d𝑧
+𝐻(𝑧)
+(26)
+=
+𝑐
+1 + 𝑧′𝑠
+�
+1 − ˆ𝒏′ · (𝒗𝑜 − 𝒗𝑠)
+𝑐
+� �∫
+𝑧′
+𝑠
+0
+d𝑧
+𝐻(𝑧) +
+∫
+𝑧′
+𝑠+(1+𝑧′
+𝑠) ˆ𝒏′·(𝒗𝑜−𝒗𝑠)
+𝑧′𝑠
+d𝑧
+𝐻(𝑧)
+�
+(27)
+≃ 𝑑[𝑧′
+𝑠] + ˆ𝒏′ · (𝒗𝑜 − 𝒗𝑠)
+𝑐
+�
+𝑐
+𝐻(𝑧′𝑠) − 𝑑[𝑧′
+𝑠]
+�
+,
+(28)
+where 𝑑[𝑧′
+𝑠] indicates the naive background angular diameter distance as a function of observed redshift, as given in Eq. (33).
+Therefore, one can compute the background angular diameter distances 𝑑𝑙, 𝑑𝑠 and 𝑑𝑙𝑠 as follows
+𝑑𝑙 = 𝑑[𝑧′
+𝑙] + 𝐷𝐿
+𝑣𝑜
+𝑐 ,
+(29)
+𝑑𝑠 = 𝑑[𝑧′
+𝑙] + 𝐷𝑆
+𝑣𝑜
+𝑐 ,
+(30)
+𝑑𝑙𝑠 = 𝑑[𝑧′
+𝑙, 𝑧′
+𝑠] + 𝐷𝐿𝑆
+𝑣𝑜
+𝑐 ,
+(31)
+with
+𝑑[𝑧′
+𝑙] =
+𝑐
+𝐻0(1 + 𝑧′
+𝑙) 𝜒[𝑧′
+𝑙] ,
+(32)
+𝑑[𝑧′
+𝑠] =
+𝑐
+𝐻0(1 + 𝑧′𝑠) 𝜒[𝑧′
+𝑠] ,
+(33)
+𝑑[𝑧′
+𝑙, 𝑧′
+𝑠] =
+𝑐
+𝐻0(1 + 𝑧′𝑠) 𝜒[𝑧′
+𝑠, 𝑧′
+𝑠] ,
+(34)
+where the function 𝜒 was defined explicitly in Eq. (19) and where the lens and source peculiar velocities are included in the
+corrections
+𝐷𝐿 = ˆ𝒏′ · (𝒗𝑜 − 𝒗𝑙)
+𝑣𝑜
+�
+𝑐
+𝐻[𝑧′
+𝑙] − 𝑑[𝑧′
+𝑙]
+�
+,
+(35)
+𝐷𝑆 = ˆ𝒏′ · (𝒗𝑜 − 𝒗𝑠)
+𝑣𝑜
+�
+𝑐
+𝐻[𝑧′𝑠] − 𝑑[𝑧′
+𝑠]
+�
+,
+(36)
+𝐷𝐿𝑆 = ˆ𝒏′ · (𝒗𝑜 − 𝒗𝑠)
+𝑣𝑜
+�
+𝑐
+𝐻[𝑧′𝑠] − 𝑑[𝑧′
+𝑙, 𝑧′
+𝑠]
+�
+−
+𝑐
+𝐻[𝑧′
+𝑙]
+1 + 𝑧′
+𝑙
+1 + 𝑧′𝑠
+ˆ𝒏′ · (𝒗𝑜 − 𝒗𝑙)
+𝑣𝑜
+.
+(37)
+The distances 𝑑[𝑧′
+𝑙], 𝑑[𝑧′
+𝑠] and 𝑑[𝑧′
+𝑙, 𝑧′
+𝑠] are the naive background angular diameter distances, which can be computed from
+Eqs. (32)-(34). Note that it is only the background angular diameter distances as a function of observed redshifts which are biased
+in this way. As encountered with Eq. (22)-(??), we remind the reader that this apparent expansion in 𝑣𝑜/𝑐 really is a simultaneous
+expansion in 𝑣𝑜/𝑐, 𝑣𝑙/𝑐 and 𝑣𝑠/𝑐. The projection of the lens and source peculiar velocities along the line of sight are unknown
+and difficult to measure. We shall vary 𝑣 ∥
+𝑙 ≡ ˆ𝒏′ · 𝒗𝑙 and 𝑣 ∥
+𝑠 ≡ ˆ𝒏′ · 𝒗𝑠 to quantify their impact.
+D.
+Aberration of angles
+In this technical subsection, we give explicit expressions to compute the bias generated by peculiar velocities on the measured
+angles to the images, the Einstein angle and on the inferred source angle. The main results are the biases on these three angles,
+which can be found in Eqs. (46), (50) and (68).
+
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+ANUVmLc=
+ˆn0
+Figure 2. We plot here the coordinate systems involved. Both observers sit at the origin O. One observer is at rest in this coordinate system and
+would observe comoving quantities, which have no primes. The observer moving with peculiar velocity 𝒗𝑜 which is aligned with ˆ𝒛 works in the
+observation coordinate system, spanned by the two vectors { ˜𝜽′𝒙, ˜𝜽′𝒚} which are denoted with primes. The vector ˜𝜽′𝒚 is the projection of the
+Earth’s North pole direction in the plane orthogonal to ˆ𝒏′, while ˜𝜽′𝒙 points East. The moving observer sees the lensed system center of mass
+in the direction ˆ𝒏′ = (𝜃′𝑐𝑚, 𝜑′𝑐𝑚). The more convenient basis is the hatted one, which is spanned by { ˆ𝜽′𝒙, ˆ𝜽′𝒚}. This convenient coordinate
+system is such that ˆ𝜽′𝒙 belongs to the plane orthogonal to ˆ𝒛. As such, it is unaffected by the boost. The angle 𝛿′ relates the two basis such that
+cos 𝛿′ = ˆ𝜽′𝒙 · ˜𝜽′𝒙.
+Observed angles on the sky are affected by the peculiar velocity of the observer. It appears simpler to compute the effect of
+aberration in a frame in which the ˆ𝒛 axis coincides with the direction of the peculiar velocity of the observer 𝒗𝑜. In this special
+case, only the polar angle 𝜃 is affected by the boost, while the azimuthal angle 𝜑 is left unaffected to first order
+𝜃′ = 𝜃 − sin(𝜃) 𝑣𝑜
+𝑐 ,
+(38)
+𝜑′ = 𝜑 .
+(39)
+Note that to first order in 𝑣𝑜/𝑐, one can easily invert the system
+𝜃 = 𝜃′ + sin(𝜃′) 𝑣𝑜
+𝑐 ,
+(40)
+𝜑 = 𝜑′ .
+(41)
+While this is convenient from a calculational point of view, it requires to translate the observations into that coordinate system,
+which we call the calculation coordinate system. To this end, we also introduce an observation coordinate system, which carries
+tildes, which are 2-dimensional angles on the sky in the neighborhood of the lens’ center of mass, which corresponds to the origin
+that points towards ˆ𝒏′. The ˜𝜽′
+𝒚 vector is the projection of the North pole (J2000) in the plane orthogonal to ˆ𝒏′, while ˜𝜽′
+𝒙 points
+East. This is the coordinate system in which strong lensing observations are made. Images are couples ˜𝜽′
+𝒊 = ( ˜𝜃′
+𝑖𝑥, ˜𝜃′
+𝑖𝑦) in that
+coordinate system. There is one such coordinate system for observers with peculiar velocity 𝒗𝑜, which carries primes on top of
+tildes and one for comoving observers (that have 𝒗𝑜 = 0), which is free of primes.
+1.
+Distortion of the images
+Each image appears to a boosted observer with polar and azimuthal angles {𝜃′
+𝑖, 𝜑′
+𝑖}. These can be computed, given an observed
+center of mass lens ˆ𝒏′ = (𝜃′
+𝑐𝑚, 𝜑′
+𝑐𝑚), a rotation angle 𝛿′, which can be computed for a given ˆ𝒏′ following App. A and image
+
+8
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++
++
++
++
++
++
++
++
+-1.0
+-0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+-1.5
+-1.0
+-0.5
+0.0
+0.5
+Figure 3. We plot the 4 images ˜𝜽′
+𝒊 of RXJ1131-1231 (in pink) and the corresponding images ˜𝜽𝒊 (in black) that would be seen by a comoving
+observer if 𝒗𝑜 = 40 𝒗dip. Each image is displaced by 𝚯𝑖𝑣𝑜/𝑐, as should be clear from Eq. (46). The origin on this plot corresponds to the
+directions of ˆ𝒏′ and ˆ𝒏 in the appropriate cases. Note that the ˜𝜽𝒙 and the ˜𝜽′𝒙 axis point in different directions which are captured by 𝛿 and 𝛿′, as
+in Eqs.(42)-(45).
+coordinates ˜𝜽′
+𝒊
+𝜃′
+𝑖 = 𝜃′
+𝑐𝑚 − ˆ𝜃′
+𝑖𝑦 = 𝜃′
+𝑐𝑚 −
+�
+˜𝜃′
+𝑖𝑥 sin 𝛿′ + ˜𝜃′
+𝑖𝑦 cos 𝛿′�
+,
+(42)
+𝜑′
+𝑖 = 𝜑′
+𝑐𝑚 − ˆ𝜃′
+𝑖𝑥 = 𝜑′
+𝑐𝑚 −
+�
+˜𝜃′
+𝑖𝑥 cos 𝛿′ − ˜𝜃′
+𝑖𝑦 sin 𝛿′�
+.
+(43)
+Applying Eq. (40)-(41) to infer ˆ𝒏 and {𝜃𝑖, 𝜑𝑖}, one can solve the following system for ˜𝜽𝒊
+𝜃𝑖 = 𝜃𝑐𝑚 − � ˜𝜃𝑖𝑥 sin 𝛿 + ˜𝜃𝑖𝑦 cos 𝛿� ,
+(44)
+𝜑𝑖 = 𝜑𝑐𝑚 − � ˜𝜃𝑖𝑥 cos 𝛿 − ˜𝜃𝑖𝑦 sin 𝛿� ,
+(45)
+where in particular 𝛿 ≠ 𝛿′, in general (see App. A). One then defines the bias 𝚯𝑖 = (Θ𝑖𝑥, Θ𝑖𝑦) on image 𝑖 implicitly as
+˜𝜽𝑖 = ˜𝜽′
+𝒊 + 𝚯𝒊
+𝑣𝑜
+𝑐 .
+(46)
+This equation can be used to compute 𝚯𝒊 from the observed images ˜𝜽′
+𝒊 together with the solutions ˜𝜽𝒊 of Eqs. (44)-(45), 𝑣𝑜 and
+rotation angles 𝛿, 𝛿′ given in App. A. Note that this bias is independent of the peculiar velocity of the lens and source. The
+images are affected in slightly different ways, due to their different sky positions relative to ˆ𝒗𝑜. This can not be captured by
+an image-independent translation for one lens, as can be seen from Fig. 3, where we plot the displaced images for the system
+RXJ1131-1231 for the exaggerated case 𝒗𝑜 = 40𝒗dip. This is why we rather speak of image distortion, rather than translation.
+2.
+The velocity dispersion from the Einstein angle
+The central velocity dispersion of the lens galaxy traces its total mass and can be either measured directly from spectroscopic
+observation or deduced from the Einstein radius with some assumptions about the mass profile of the lens. In the former case,
+this quantity can in principle be measured independently of peculiar velocities, since these would only affect the position of the
+spectral lines while leaving their width unchanged. The velocity dispersion inferred from the spectral lines’ width would therefore
+
+9
+be unaffected. However, velocity dispersions obtained with this technique are limited to a precision of ∼ 10 %, which is not
+sufficient to precisely constrain the mass profile of the lens galaxies. In fact, most of the constraints on the mass profile in recent
+time-delay cosmography analysis come from the lensing observables, including the Einstein radius. Since the Einstein radius is
+affected by the aberration on the measured angle described in the previous section, this error propagates to the mass profile. In this
+subsection, we use the central velocity dispersion of the lens, 𝜎𝑣 as a proxy to quantify the error on the mass profile due to the
+aberration on the measured Einstein angle. The Einstein angle can be related to 𝜎𝑣 from the following relation [42]
+𝜃𝐸 = 4𝜋𝜎2
+𝑣
+𝑐2
+𝑑𝑙𝑠
+𝑑𝑠
+(47)
+for an SIS. This angle corresponds to the angle under which an observer perfectly aligned with the lens and a point-like source
+would see an Einstein ring. Note that it matches 𝛼0, defined in Eq. (12). For simplicity, we assume that one measures the Einstein
+angle in a plane which is spanned by ˆ𝒗𝑜 and ˆ𝒏′ (that is, in direction ˆ𝜽′
+𝒚 (see Fig. 2)). In this case, the aberration of the Einstein ring
+is maximal. One finds
+𝜃𝐸 = 𝜃′
+𝐸 + 𝑣𝑜
+𝑐 (sin(𝜃′
+𝑐𝑚 + 𝜃′
+𝐸) − sin(𝜃′
+𝑐𝑚))
+(48)
+= 𝜃′
+𝐸 + 𝑣𝑜
+𝑐 cos(𝜃′
+𝑐𝑚)𝜃′
+𝐸 .
+(49)
+In this case, biased measurements of 𝑧′
+𝑙, 𝑧′
+𝑠 and 𝜃′
+𝐸 of 𝑧𝑙, 𝑧𝑠 and 𝜃𝐸 induce a bias on the inference 𝜎′
+𝑣 of 𝜎𝑣. It can be estimated to
+first order in the peculiar velocities by
+𝜎𝑣 = 𝜎′
+𝑣 + 𝑆𝑣
+𝑣𝑜
+𝑐 ,
+(50)
+with
+𝜎′
+𝑣 =
+√︄
+𝑑[𝑧′𝑠]
+𝑑[𝑧′
+𝑙, 𝑧′𝑠]
+𝜃′
+𝐸
+4𝜋 𝑐 ,
+(51)
+𝑆𝑣 =
+𝜎′
+𝑣
+2𝑑[𝑧′
+𝑙, 𝑧′𝑠]𝑑[𝑧′𝑠]
+�
+𝑑[𝑧′
+𝑙, 𝑧′
+𝑠]𝐷𝑆 − 𝑑[𝑧′
+𝑠]𝐷𝐿𝑆 + 𝑑[𝑧′
+𝑙, 𝑧′
+𝑠]𝑑[𝑧′
+𝑠] cos 𝜃′
+𝑐𝑚
+�
+.
+(52)
+Here the distances 𝑑[𝑧′
+𝑠], 𝑑[𝑧′
+𝑙, 𝑧′
+𝑠] and their related biases 𝐷𝑆 and 𝐷𝐿𝑆 can be computed using Eqs. (32)-(37), which depend on
+the source, lens and observer’s peculiar velocities. Note that we use capital letters to denote biases, not the angular diameter
+distances themselves. This is rather an overestimation of the bias on 𝜎𝑣, if estimated from the observed Einstein angle. This is
+because angles, including the Einstein angle, are unaffected5 in the direction ˆ𝜽′
+𝒙. It turns out that the bias 𝑆𝑣 on 𝜎𝑣 increases the
+bias on 𝐻0 generated by peculiar velocities. In the quantitative analysis presented in Sec. IV, we shall also study what happens if
+one measures 𝜎𝑣 independently (setting 𝑆𝑣 = 0), by direct peculiar velocity dispersion measurements in redshift space. This
+would also correspond to the situation in which the Einstein angle is measured in the direction ˆ𝜽′
+𝒙. In practice, one can measure
+the azimuthally averaged Einstein radius. A perfect circle Einstein ring seen by a comoving observer would be unaffected in the
+direction ˆ𝜽′
+𝒙 and maximally affected in the direction ˆ𝜽′
+𝒚. Whether the enclosed area of the deformed circle is larger or smaller
+depends on the sign of cos(𝜃′
+𝑐𝑚). We expect the practical case to lie somewhat in between these two situations.
+3.
+The reconstructed source angle
+Reconstructing the source angle ˜𝜷 is subtle. This is because it is a quantity which is inferred, as opposed to observed, from
+biased observations like ˜𝜽′, 𝑧′
+𝑙 and 𝑧′
+𝑠 and that it appears directly in the time-delay formula (Eq. (1)). Here, we write a tilde, to
+remind the reader that it is a two-dimensional angle in the observation coordinate system. The reconstruction of ˜𝜷 consists of two
+steps. The first one consists in estimating the angle ˜𝜷′′ directly from the observed quantities ˜𝜽′, 𝑧′
+𝑙, 𝑧′
+𝑠. The angle ˜𝜷′ to the source
+which would be observed in absence of the lens can also be computed from these observables and knowledge of the peculiar
+velocities. In the second step, one can reconstruct the angle to the source ˜𝜷 that a comoving observer would observe in absence of
+the lens. We carry on with the first step. The lens equation for a singular isothermal sphere and comoving observer, source and
+lens reads
+˜𝜷 = ˜𝜽
+�
+1 − 𝛼0
+|| ˜𝜽||
+�
+.
+(53)
+5 This is the reason why the intermediate coordinate system spanned by { ˆ𝜽′𝒙, ˆ𝜽′𝒚 } was introduced.
+
+10
+This equation allows, through the observation of images ˜𝜽𝑖 and an estimate of 𝛼0 to reconstruct ˜𝜷. However, all of these quantities
+are affected by the boost and so is the reconstruction of ˜𝜷. By measuring 𝜃′
+𝐸, 𝑧′
+𝑙 and 𝑧′
+𝑠, one estimates 𝛼′
+0, which is related to a
+comoving deflection angle 𝛼0 by
+𝛼0 = 𝛼′
+0 + 𝐴0
+𝑣𝑜
+𝑐 ,
+(54)
+with
+𝛼′
+0 = 4𝜋(𝜎′
+𝑣)2
+𝑐2
+𝑑[𝑧′
+𝑙, 𝑧′
+𝑠]
+𝑑[𝑧′𝑠]
+,
+(55)
+𝐴0 =
+4𝜋𝜎′
+𝑣
+𝑐2𝑑2[𝑧′𝑠]
+�2𝑑[𝑧′
+𝑙, 𝑧′
+𝑠]𝑑[𝑧′
+𝑠]𝑆𝑣 − 𝑑[𝑧′
+𝑙, 𝑧′
+𝑠]𝐷𝑆𝜎′
+𝑣 + 𝐷𝐿𝑆𝑑[𝑧′
+𝑠]𝜎′
+𝑣
+� .
+(56)
+Here the distances 𝑑[𝑧′
+𝑠], 𝑑[𝑧′
+𝑙, 𝑧′
+𝑠], their biases 𝐷𝑆, 𝐷𝐿𝑆 and 𝑆𝑣 can be calculated directly from the observables, using Eqs. (32)-
+(36) and (52). The deflection angle is therefore biased by the distance biases and the bias on the veloctiy dispersion. The inference
+of ˜𝜷
+′ as should be made by an observer with peculiar velocity 𝑣𝑜 is biased because of the bias in all images ˜𝜽𝒊
+′, redshift of the lens
+and source and because of the bias in 𝛼0. There is only one true source angle ˜𝜷. However, since we use an isothermal sphere,
+which has only 2 images; for systems which have 3 or 4 images, the ˜𝜷 inferred via Eq. (53) may give different results depending on
+which image is used. This turns out to impact significantly the determination of the Hubble constant. Therefore, we compute ˜𝜷′
+𝒊
+and its corresponding bias 𝑩′
+𝒊 for each image. We get
+˜𝜷′
+𝒊 = ˜𝜷′′
+𝒊 + 𝑩′
+𝒊
+𝑣𝑜
+𝑐 ,
+(57)
+with
+˜𝜷′′
+𝒊 = ˜𝜽′
+𝒊
+�
+1 −
+𝛼′
+0
+|| ˜𝜽𝒊||
+�
+,
+(58)
+and
+𝐵′
+𝑖𝑥 = Θ𝑖𝑥 +
+1
+|| ˜𝜽𝒊||3
+�
+𝛼′
+0 ˜𝜃′
+𝑖𝑦(Θ𝑖𝑦 ˜𝜃′
+𝑖𝑥 − Θ𝑖𝑥 ˜𝜃′
+𝑖𝑦) − 𝐴0 ˜𝜃′
+𝑖𝑥|| ˜𝜽𝒊||2�
+,
+(59)
+𝐵′
+𝑖𝑦 = Θ𝑖𝑦 +
+1
+|| ˜𝜽𝒊||3
+�
+𝛼′
+0 ˜𝜃′
+𝑖𝑥(Θ𝑖𝑥 ˜𝜃′
+𝑖𝑦 − Θ𝑖𝑦 ˜𝜃′
+𝑖𝑥) − 𝐴0 ˜𝜃′
+𝑖𝑦|| ˜𝜽𝒊||2�
+,
+(60)
+where 𝚯𝒊 and 𝐴0 were defined in Eq. (46) and (56). Those can be computed directly from the observables. A moving observer
+makes a biased inference ˜𝜷
+′′ of ˜𝜷
+′, which differs from image to image. We wish to express this source angle on the sky
+𝜷′ = (𝜃′
+𝛽, 𝜑′
+𝛽) for a comoving observer, which would rather observe 𝜷 = (𝜃𝛽, 𝜑𝛽), given by
+𝜃𝛽 = 𝜃′
+𝛽 + 𝑣𝑜
+𝑐 sin 𝜃′
+𝛽 ,
+(61)
+𝜑𝛽 = 𝜑′
+𝛽 ,
+(62)
+where the right hand side can be computed directly by the measured quantities
+𝜃′
+𝛽 = 𝜃′
+𝑐𝑚 − ( ˜𝛽′
+𝑥 sin 𝛿′ + ˜𝛽′
+𝑦 cos 𝛿′) ,
+(63)
+𝜑′
+𝛽 = 𝜑′
+𝑐𝑚 − ( ˜𝛽′
+𝑥 cos 𝛿′ − ˜𝛽′
+𝑦 sin 𝛿′) ,
+(64)
+together with the rotation angle 𝛿′, which can be computed for a given direction following App. A. Once the left hand side of
+Eq. (61) is determined, one can infer ˜𝜷 that would be infered by a comoving observer by solving the following equations for ˜𝜷
+𝜃𝛽 = 𝜃𝑐𝑚 − ( ˜𝛽𝑥 sin 𝛿 + ˜𝛽𝑦 cos 𝛿) ,
+(65)
+𝜑𝛽 = 𝜑𝑐𝑚 − ( ˜𝛽𝑥 cos 𝛿 − ˜𝛽𝑦 sin 𝛿) ,
+(66)
+where 𝛿 ≠ 𝛿′ can also be computed following App. A. The solutions can be expressed as
+˜𝜷𝒊 = ˜𝜷′
+𝒊 + 𝑩𝒊
+𝑣𝑜
+𝑐 ,
+(67)
+
+11
+where the image index 𝑖 was reintroduced and which defines implicitly the bias 𝑩𝒊. Note that in general, 𝑩𝒊 ≠ 𝑩′
+𝒊. In this way,
+˜𝜷𝒊 = ˜𝜷′′
+𝒊 + 𝑩′′
+𝒊
+𝑣𝑜
+𝑐 ,
+(68)
+𝑩′′
+𝒊 ≡ 𝑩′
+𝒊 + 𝑩𝒊 ,
+(69)
+where 𝑩′
+𝒊 was defined in Eqs. (59)-(60) and 𝑩𝑖 was defined implicitly in Eq. (67). Those can be computed directly from the
+observables. In this sense, one pays twice the price in neglecting peculiar velocities in the determination of ˜𝜷. That is because it is
+a quantity which is inferred from biased quantities like ˜𝜽′
+𝒊, 𝑧′
+𝑙 and 𝑧′
+𝑠. One first needs to reconstruct the angle to the source ˜𝜷′ that
+the moving observer would see in absence of the lens. Only then, one can compute the angle to the source ˜𝜷 that would be seen by
+a comoving observer. Eqs. (68) and (69) are the final results of this section, which we use for the remainder of this work. The
+source angle is biased by the source, lens and observer’s peculiar velocities.
+E.
+The lensing potential
+The lensing potential for an isothermal sphere reads (see Eq. (15))
+𝜓(𝜽𝑖) = 𝛼0||𝜽𝑖|| .
+(70)
+Expanding this expression to linear order in 𝑣𝑜/𝑐, one finds
+𝜓(𝜽𝑖) = 𝜓′
+𝑖 + 𝑃𝑖
+𝑣𝑜
+𝑐 ,
+(71)
+where
+𝜓′
+𝑖 = 𝛼′
+0|| ˜𝜽′
+𝒊 || ,
+(72)
+𝑃𝑖 =
+1
+|| ˜𝜽′
+𝒊 ||
+(𝛼′
+0𝚯𝑖 · ˜𝜽′
+𝒊 + 𝐴0|| ˜𝜽′
+𝒊 ||2) ,
+(73)
+where 𝚯𝒊 and 𝐴0 were defined in Eq. (46) and (56). It is affected directly by the peculiar velocity bias on the images and indirectly
+by the bias on 𝛼0, which comes from the bias on the distances and on the velocity dispersion. As such, it is sensitive to the peculiar
+velocities of the source, lens and observer.
+F.
+The time delay and Hubble constant
+At this point, all necessary contributions to the bias on the time delay have been computed and we expand the right-hand side of
+Eq. (1) to first order in 𝑣𝑜/𝑐, while the left-hand side is invariant, up to O(𝑣2
+𝑜/𝑐2) (see Eq. (21)). We get
+𝑐Δ𝑡𝑖 𝑗 ≃ 𝑐Δ𝑡′
+𝑖 𝑗 = (1 + 𝑧′
+𝑙)
+𝑑[𝑧′
+𝑙]𝑑[𝑧′
+𝑠]
+𝑑[𝑧′
+𝑙, 𝑧′𝑠]
+��
+(𝜽′
+𝑖 − 𝜷′′)2
+2
+− (𝜽′
+𝑖 − 𝜷′′)2
+2
+�
+− [𝜓′
+𝑖 − 𝜓′
+𝑗]
+�
++ 𝑐Δ𝑇𝑖 𝑗
+𝑣𝑜
+𝑐 ,
+(74)
+where the (distance) time-delay bias is given by
+𝑐Δ𝑇𝑖 𝑗 = (1 + 𝑧′
+𝑙)
+𝑑[𝑧′
+𝑙]𝑑[𝑧′
+𝑠]
+𝑑[𝑧′
+𝑙, 𝑧′𝑠]
+�
+( ˜𝜽
+′
+𝑖 − ˜𝜷′′
+𝒊 )(𝚯𝒊 − 𝑩′′
+𝒊 ) − ( ˜𝜽
+′
+𝑗 − ˜𝜷′′
+𝒋 )(𝚯𝒋 − 𝑩′′
+𝒋 ) − (𝑃𝑖 − 𝑃 𝑗)
+�
++
+1 + 𝑧′
+𝑙
+𝑑2[𝑧′
+𝑙, 𝑧′𝑠]
+�
+𝑍𝐿𝑑[𝑧′
+𝑙]𝑑[𝑧′
+𝑙, 𝑧′
+𝑠]𝑑[𝑧′
+𝑠] + 𝑑[𝑧′
+𝑙]𝑑[𝑧′
+𝑙, 𝑧′
+𝑠]𝐷𝑆 − 𝑑[𝑧′
+𝑙]𝐷𝐿𝑆𝑑[𝑧′
+𝑠] + 𝐷𝐿𝑑[𝑧′
+𝑙, 𝑧′
+𝑠]𝑑[𝑧′
+𝑠]
+�
+×
+�
+ˆ𝜙( ˜𝜽′
+𝒊, ˜𝜷′′
+𝒊 ) − ˆ𝜙( ˜𝜽′
+𝒋, ˜𝜷′′
+𝒋 )
+�
+,
+(75)
+which can be computed directly from observables, following the steps provided in subsections III A-III E. In particular, it can be
+computed directly from the observed redshifts 𝑧′
+𝑙, 𝑧′
+𝑠, their associated distances (Eqs. (32)-(34)), images ˜𝜽′
+𝒊, the reconstructed
+source angle ˜𝜷′′ via Eq. (58), the angle biases 𝚯𝒊, 𝑩′′
+𝒊 defined in Eq. (46) and Eq. (69), the lensing potential biases 𝑃𝑖 defined in
+Eq. (73), and the distance biases 𝐷𝐿, 𝐷𝑆 and 𝐷𝐿𝑆 defined in Eq. (35), (36) and (37). One can recognize the contributions coming
+from the bias on angles in the first line, together with the lensing potential. Those are directly affected by the peculiar velocity of
+
+12
+the observer, and indirectly affected by the peculiar velocities of the source and lens through the lens model. The second line is
+due to the direct bias on redshift and angular diameter distances as a function of observed redshift from the peculiar velocity of the
+source, lens and observer. Solving Eq. (74) for 𝐻0, which appears in the angular diameter distance ratio, one gets
+𝐻0 =
+𝜒[𝑧′
+𝑙]𝜒[𝑧′
+𝑠]
+𝜒[𝑧′
+𝑙, 𝑧′𝑠]
+�
+ˆ𝜙(𝜽′
+𝒊, 𝜷′′
+𝒊 ) − ˆ𝜙(𝜽′
+𝒋, 𝜷′′
+𝒋 )
+�
+Δ𝑡′
+𝑖 𝑗
+��������������������������������������������������������������������������������������������
+=𝐻′
+0
+�
+1 + 𝑐Δ𝑇𝑖 𝑗
+𝑐Δ𝑡′
+𝑖 𝑗
+𝑣𝑜
+𝑐
+�
+≡ 𝐻′
+0
+�
+1 + Δ𝐻0
+𝐻′
+0
+�
+,
+(76)
+with
+Δ𝐻0
+𝐻′
+0
+= Δ𝑇𝑖 𝑗
+Δ𝑡′
+𝑖 𝑗
+𝑣𝑜
+𝑐 .
+(77)
+Equation (77) together with Eq. (75) are the main results of this work. For a given pair of images with measured {Δ𝑡′
+𝑖 𝑗, 𝑧′
+𝑙, 𝑧′
+𝑠, ˜𝜽′
+𝒊, ˜𝜽′
+𝒋}
+and given peculiar velocities, one can compute the corresponding bias Δ𝐻0/𝐻′
+0 as a function of 𝐻0. This is because Δ𝑇𝑖 𝑗 given in
+Eq. (75) is inversely proportional to 𝐻0. Alternatively, one can compute Δ𝐻0 independently of 𝐻0 to first order in 𝑣𝑜 since the ratio
+𝐻′
+0/𝐻0 which would appear on the right hand side of Eq. (77) only brings second order corrections. Throughout the manuscript,
+we take 𝐻0 = 70 km s−1 Mpc−1. In the next section, we apply these findings to the seven lenses of TDCOSMO [27, 40, 41]. It
+should be noted also that with this definition, a positive Δ𝐻0 implies that 𝐻′
+0 is an underestimation of 𝐻0. Therefore, a relatively
+high 𝐻′
+0 could be explained by a negative Δ𝐻0/𝐻′
+0.
+IV.
+RESULTS
+In this section, we quantify what is the relative bias on 𝐻0 from the peculiar velocities of the observer, lens and source for the
+seven lenses of H0LiCOW. We first consider our results with expected peculiar velocities, before considering what happens for
+larger peculiar velocities. We get estimations of the Hubble constant 𝐻′
+0 which vary between 47 km s−1 Mpc−1 and 112 km s−1
+Mpc−1. Since the model is relatively crude, we do not expect to make a competitive inference of the Hubble constant. The SIS
+model is spherically symmetric and fixes the logarithmic slope of the mass profile to 𝛾𝑙 = 2. This of course does not contain
+enough azimuthal and radial degrees of freedom to represent accurately massive elliptical galaxies. However, we expect this
+model to be sufficient to capture the leading contributions to a bias on 𝐻0 from peculiar velocities. In Fig. 4, we plot the sky
+distribution of the 7 lenses. Two are well aligned with the velocity ˆ𝒗dip, namely RXJ1131−1231 and PG1115+080. These two
+systems coincidentally also happen to have the lowest lens redshifts and the highest inference of the Hubble constant.
+𝑁
+Lens system
+𝑧′
+𝑙
+𝑧′𝑠
+ˆ𝒏′ (𝑙′, 𝑏′) [◦]
+cos(𝜃′𝑐𝑚)
+𝑁images Δ𝐻0/𝐻′
+0 [%] Reference
+1
+B1608+656
+0.6304 1.394
+(98.339, 40.891)
+0.000706
+4
+0.0006
+[47, 48]
+2
+RXJ1131-1231
+0.295 0.654
+(-85.573,45.888)
+0.991526
+4
+1.1353
+[45, 49]
+3
+HE0435-1223
+0.4546 1.693 (-150.934, -35.060) -0.115625
+4
+-0.2153
+[49, 50]
+4 SDSS1206+4332 0.745 1.789 (148.991, 71.244)
+0.615891
+2
+0.2324
+[51]
+5
+WFI2033-4723
+0.6575 1.662
+(-7.585, -36.556)
+-0.429394
+4
+-0.5008
+[52]
+6
+PG1115+080
+0.311 1.722 (-110.113, 60.644)
+0.966824
+4
+1.1461
+[49]
+7
+DES0408-5354
+0.597 2.375 (-96.447, -45.304) -0.0620664
+3
+-0.0590
+[41]
+Table I. This table contains the system number, their lens systems with observed lens and source redshift, optical axis directions in galactic
+coordinates, projection of the line of sight along the peculiar velocity of the observer 𝒗dip and the number 𝑁images of effective images that can be
+used for time-delay cosmography per system. There is also a column indicating the relative bias Δ𝐻0/𝐻′
+0 generated by the observer’s peculiar
+velocity 𝒗dip alone. The latter is averaged over the non redundant pairs of images.
+First, we compute the bias generated by the peculiar velocity of the observer, assuming that it is known from the entirely
+kinematic interpretation of the CMB dipole. That corresponds to 𝑣𝑜 = 369.82 km s−1 towards (264.021◦, 48.253◦) in galactic
+coordinates. Then, we vary the source and lens peculiar velocities projected on the line of sight in the set {0, ±300, ±600, ±900}
+km s−1, which spans the expected peculiar velocity amplitudes from simulations [53] and from observations [54]. We do this for
+two different cases. In the first case, we assume that 𝜎𝑣 can be measured independently of the peculiar velocities, from the lens
+
+13
+AC1nicjVHLSsNAFD2Nr1pfqS7dBIsgCXxUbsdeOygn1AW0qSTtvQNAnJRC2l7sStP+BWP0n8A/0L74wR1CI6IcmZc
++85M/deK3CdiOv6S0qZm19YXEovZ1ZW19Y31OxmLfLj0GZV23f9sGZEXMdj1W5w13WCEJmjiyX1a3hqYjXL1kYOb53wcBa
+4/Mvuf0HNvkRHXUbGtg+deTslHQi/uF4I27ag5Pa/Lpc0CIwE5JKviq89oQsfNmKMwOCBE3ZhIqKnCQM6AuLamBAXEnJkn
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+/h3Ovp3Dˆvdip
+AC3HicjVHLSsNAFD2N7/qunDhJlgEQxJK1ZdFd10WcFqwUdJ4liHpklIJoKU7tyJW3/ArX6P+Af6F94Zp+AD0QlJz
+px7z5l753pxwFNh2y85Y2h4ZHRsfCI/OTU9M1uYmz9MoyzxWcOPgihpem7KAh6yhuAiYM04YW7XC9iR19mT8aMrlqQ8Cg/Ed
+cxOu2475BfcdwVRrcJirWXvmBXHs96ayVru9/qrZetcr9VKNqWrZb5EzgaFKFXPSo84wTniOAjQxcMIQThAC5Seo7hwEZM3
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+IPM/+pkLwIX2FI9cOopVozsztcumboVWbn5qStBDjFxEp9TPCHsK+Xgnk2lSVXv8m5dFX9VmZKVe1/nZniTVdKAne/j/AkOS
+5azaZX3N4rVXT3qcSxhGas0zwqKGOhqr/AY94Ms6MG+PWuPtINXJas4Avy7h/Bwa8lkM=
+H0 : 71.0+2.9
+�3.3
+AC3HicjVHLSsNAFD2Nr1pfVRcu3ASLIghfWCLq6KbL
+ivYB9S2JOlUQ9MkJBOhlO7ciVt/wK1+j/gH+hfeGVNQi+iEZM6ce8/JvXN37FDruvCWVufmFxKbmcWldW9Ib27VQy8KLFazPMcLmqYRMsd2WY3b3GFNP2DG0HRYwxyciXjhgWh7bkXfOSz9tC4cu2+bRmcqG56p9LVT9RiSct1xod5rTDpjo/kls7om
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+H0 : 78.2+3.4
+�3.4
+AC3HicjVHLSsNAFD3Gd31VXbhwEyCIZEqxVXRTdK
+lgt1LYkcayDaRKSiSClO3fi1h9wq98j/oH+hXfGKahFdEKSM+fec+beuV4c8FTY9uQMTwyOjY+MZmbmp6ZncvPL5ykUZb4rOpHQZTUPDdlAQ9ZVXARsFqcMLfjBezUuzqQ8dNrlqQ8Co/FTcwaHbcd8gvu4KoVn6p0rL3zJjlZrd9aK12t1N4rWdq+VL
+9iWrZY5CBwNCtDrMq/4AzniOAjQwcMIQThAC5SeupwYCMmroEucQkhruIMPeRIm1EWowyX2Cv6tmlX12xIe+mZKrVPpwT0JqQ0sUqaiPISwvI0U8Uz5SzZ37y7ylPWdkN/T3t1iBW4JPYvXT/zvzrZi8AFdlUPnHqKFSO787VLpm5FVm5+6UqQ0ycxOcUT
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+H0 : 71.7+4.8
+�4.5
+AC3HicjVHLSsNAFD3GV31HXbhwEyCIbEiq+V6Malg
+n1AbUsSxqaF5OJUEp37sStP+BWv0f8A/0L74wR1CI6IcmZc+85c+9cNwn8VFjWy5A2PDI6Nl6YmJyanpmd0+cXKmcY+VvTiIec1Uhb4ESsLXwSslnDmhG7Aqm7nSMar14ynfhydiW7CGqHTjvxL3MEUS196bhl7Rs7trnd7K2XzN1+q7exZe71W3rRM
+i21jEFg56CIfJ3E+jPOcYEYHjKEYIgCAdwkNJThw0LCXEN9IjhHwVZ+hjkrQZTHKcIjt0LdNu3rORrSXnqlSe3RKQC8npYFV0sSUxwnL0wVz5SzZH/z7ilPWVuX/m7uFRIrcEXsX7rPzP/qZC8Cl9hVPfjU6IY2Z2Xu2TqVmTlxpeuBDkxEl8QXFO2
+FPKz3s2lCZVvcu7dVT8VWVKVu69PDfDm6ySBmz/HOcgqGya9rZOt0qHhzmoy5gGStYo3nu4ADHOEFZ1f+ARzxpTe1Gu9XuPlK1oVyziG9Lu38HJfuWUA=
+H0 : 71.6+3.8
+�4.9
+AC3HicjVHLSsNAFD2Nr1pfVRcu3ASLIghUbHFVdFNl
+xXsA2pbknRaQ9MkJBOhlO7ciVt/wK1+j/gH+hfeGVNQi+iEJGfOvefMvXOtwHUiruvKWVmdm5+Ib2YWVpeWV3Lrm9UIz8ObVaxfdcP65YZMdfxWIU73GX1IGTmwHJZzeqfi3jthoWR43uXfBiw5sDseU7XsU1OVDu7VWrp2rB0IzWaL+g6eP26CvGeN2N
+qdrulzqNDASkEOyn72BVfowIeNGAMweOCEXZiI6GnAgI6AuCZGxIWEHBlnGCND2piyGWYxPbp26NdI2E92gvPSKptOsWlNySlil3S+JQXEhanqTIeS2fB/uY9kp6itiH9rcRrQCzHNbF/6SaZ/9WJXji6KMgeHOopkIzozk5cYnkronL1S1ecHALiBO5QP
+CRsS+XknlWpiWTv4m5NGX+TmYIVezvJjfEuqQBGz/HOQ2qh5pxoh1dHOeKZ8mo09jGDvZonkUIZFVn/I57wrLSUW+VOuf9MVKJZhPflvLwAQk5lkQ=
+H0 : 81.1+8.0
+�7.1
+AC3HicjVHLSsNAFD2N7/qKunDhJlgEQyJ2vpYiW5cKlgrVC1JOq2DeZFMBCnduRO3/oBb/R7xD/QvDOmoBbRCUnOn
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+4e4hBXcYuiqTNKItRhkPsFX3btKvnbEh76ZkqtUen+PQmpDSwRJqI8hLC8jRDxTPlLNnfvDvKU9Z2Q3839wqIFbgk9i9dL
+/O/OtmLQAtbqgdOPcWKkd15uUumbkVWbnzpSpBDTJzETYonhD2l7N2zoTSp6l3eraPibypTsnLv5bkZ3mWVNGD75zj7wcma
+VfM9aON0u5ePupRLGARyzTPTeziAIeoqvof8YRn7UK71e60+89UrZBr5vBtaQ8fJnOWUA=
+H0 : 68.9+5.4
+�5.1
+AC3HicjVHLSsNAFD2Nr1pfVRcu3ASLIghfWDFVdFNl
+xXsA2pbknRaQ9MkJBOhlO7ciVt/wK1+j/gH+hfeGVNQi+iEJGfOvefMvXN37FDruvCWVufmFxKbmcWldW9Ib27VQi8KLFa1PMcLGqYRMsd2WZXb3GENP2DG0HRY3Ryci3j9hgWh7bmXfOSz1tDou3bPtgxOVCe9U+7op2qxoOXa48OcVpx0xkd5TZ90
+hld0+VSZ0E2BhnEq+KlX3CFLjxYiDAEgwtO2IGBkJ4mstDhE9fCmLiAkC3jDBOkSBtRFqMg9gBfu0a8asS3vhGUq1Rac49AakVLFPGo/yAsLiNFXGI+ks2N+8x9JT1Daivxl7DYnluCb2L9087860QtHDyeyB5t68iUjurNil0jeiqhc/dIVJwefOIG7F
+A8IW1I5vWdVakLZu7hbQ8bfZKZgxd6KcyO8iypwNmf45wFtZyWPdbyF4VM6SwedRK72MBzbOIEsqoCrf8QTnpW2cqvcKfefqUoi1mzj21IePgAG9ZD
+H0 : 74.2+2.7
+�3.0
+Figure 4. Blue dots indicate the sky position in galactic coordinates of the 7 lenses of H0LiCOW together with their corresponding estimation of
+𝐻0 (in km s−1 Mpc−1), extracted from [27, 41]. Their sky positions are given in Table I. We superimpose the CMB temperature map from
+WMAP [46], where the monopole has been removed, leaving the dipole apparent, together with contamination from the galactic plane. The red
+dot indicates the direction of the velocity obtained from the CMB dipole 𝒗dip. The two lenses RXJ1131-1231 and PG1115+080 have the two
+lines of sight which are best aligned with the CMB dipole, with cos(𝜃′𝑐𝑚) > 0.96. Coincidentally, they also have the lower lens redshift and give
+the highest values of 𝐻0: 78.2+3.4
+−3.4 km s−1 Mpc−1 and 81.1+8.0
+−7.1 km s−1 Mpc−1, respectively. This was pointed out in [39]. The CMB dipole in
+celestial coordinates is ˆ𝒗dip ≃ (−7◦, 167◦), which is well aligned with the Earth’s equator. In this sense, North or South hemisphere sky surveys
+are nearly as orthogonal as they can be from the CMB dipole.
+galaxy emission lines’ width. In this case, only the peculiar velocity of the lens on top of the peculiar velocity of the observer
+changes the bias on 𝐻0 in a way that can be seen in Fig. 5, where we plot Δ𝐻0/𝐻′
+0 as a function of the lens redshift. In this plot
+and in the following, Δ𝐻0/𝐻′
+0 is actually the average over the non-redundant image pairs available. In practice, for a given system,
+some time delays are more precisely measured than others and therefore a weighted average may be more sensible to compute the
+relative bias on 𝐻0. The source peculiar velocity only affects the bias subdominantly. The bias for one lens is bounded by 2.5%
+and the bias generated by the observer’s peculiar velocity alone is bounded by 1%.
+In the second case, we assume that 𝜎′
+𝑣 is extracted from the measurement of the Einstein angle, as outlined in Sec. III. In that
+case, both the lens and the source peculiar velocities give significant changes to the bias on 𝐻0, as can be seen in Fig. 6, where we
+plot Δ𝐻0/𝐻′
+0 as a function of the lens redshift 𝑧′
+𝑙. In this case, the bias Δ𝐻0/𝐻′
+0 for a single lens is bounded by 5% for these seven
+lenses. The effect of the peculiar velocity of the observer alone, as extracted from the CMB dipole, is bounded by 1.2%. This
+shows how the effect of 𝑣𝑜/𝑐 = O(10−3) can give an order of magnitude larger bias, as the bias piles up from different observables.
+In table II, we give the maximal relative bias on each quantity that enters Eq. (75) from the velocity of the observer set to 𝑣dip for
+each of the seven systems of TDCOSMO. Combining the seven lenses, we find that the bias generated on 𝐻0 by the observer’s
+peculiar velocity is of order 0.25%. Assuming that the lens and source peculiar velocities are normally distributed around zero
+with standard deviation 300 km s−1, one finds that this results in an additional random uncertainty which can reach 1.00% for a
+single lens. It combines to a 0.24% random uncertainty for the seven lenses of TDCOSMO. This uncertainty is expected to drop
+to zero for a higher number of systems.
+Since the calculation is valid for non-relativistic velocities, one may push to larger peculiar velocities, as long as 𝑣/𝑐 ≪ 1.
+One may be curious to see what peculiar velocities would be necessary to affect the Hubble constant by 10%, which would
+constitute an important correction in the context of the Hubble tension. We plot the bias from the velocity of the observer for
+peculiar velocities which vary in 𝑣𝑜 ∈ {0, ±1000, ±2000, ±3000} km s−1 in Fig. 7. Negative peculiar velocities correspond to
+changing the direction of the peculiar velocity by a rotation of 𝜋. For 𝑣𝑜 = ±3000, the bias for the best-aligned lenses (system
+RXJ1131−1231 and PG1115+080), at observed lens redshift 𝑧′
+𝑙 ∼ 0.3, reaches ±10%. It is intriguing that the number count dipole
+measurements point to higher 𝒗𝑜, which argues in favor of positive Δ𝐻0. This suggests that if the peculiar velocity of the observer
+is higher than expected, even by a factor of 10, then the estimation of the Hubble constant by the H0LICOW collaboration is
+rather an underestimation of 𝐻0, which would enhance the tension. For 𝑣𝑜 = 0, which corresponds to a comoving observer, the
+bias vanishes, as expected. The bias changes in different directions and with different amplitudes for different systems. This
+depends on the sign and value of cos(𝜃′
+𝑐𝑚) together with the lens and source redshifts, which are given in Tab. I. Finally, we play
+the same game with peculiar velocities of the lens and source. We vary them in {0, ±1500, ±3000} km s−1. The assumption on 𝜎𝑣
+determines how less important 𝑣 ∥
+𝑠 matters compared to 𝑣 ∥
+𝑙 for the bias on 𝐻0. Since in practice, 𝜎𝑣 is extracted from the Einstein
+angle, we plot what happens in that case in Fig. 8. These large peculiar velocities, which are expected to be rare, can bias 𝐻0 by
+
+14
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+-2
+-1
+0
+1
+2
+3
+AC2XicjVHLSsNAFD2Nr1pf8bFzEyxFVzVRUZdFXRZwT6gLSVJpzU0L5KJUIsLd+LWH3CrPyT+gf6Fd8
+YU1CI6YSZnzr3nzNy5Vug6Mdf14wyNT0zO5edzy0sLi2vqKtrtThIptV7cANoZlxsx1fFblDndZI4yY6Vkuq1
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+Figure 5. In this plot, we show the normalized bias on 𝐻0, for 𝑣𝑜 extracted from the entirely kinematic interpretation of the CMB and vary 𝑣 ∥
+𝑙 ,
+𝑣 ∥
+𝑠 ∈ {0, ±300, ±600, ±900} km s−1. For these plots, we assumed that 𝜎𝑣 can be measured independently from the peculiar velocities. This
+implies that we set the lens parameter bias 𝑆𝑣 = 0, instead of using the expression for 𝑆𝑣 given in Eq. (52). While varying 𝑣 ∥
+𝑠 does change the
+bias on 𝐻0, the change is much smaller than that of 𝑣 ∥
+𝑙 and the points with different 𝑣 ∥
+𝑠 appear to coincide on this plot. In this case, the amplitude
+of the bias is bounded by 2.5%.
+N
+Lens system
+Angular diameter distances
+Deflection
+angle
+Lensing
+potential
+SIS velocity
+dispersion
+Source
+position angle
+Images
+position angle
+𝐷𝐿
+𝑑[𝑧′
+𝑙 ]
+𝑣dip
+𝑐
+[%]
+𝐷𝑆
+𝑑[𝑧′𝑠 ]
+𝑣dip
+𝑐
+[%]
+𝐷𝐿𝑆
+𝑑[𝑧′
+𝑙,𝑧′𝑠 ]
+𝑣dip
+𝑐
+[%]
+𝐴0
+𝛼0
+𝑣dip
+𝑐
+[%] | 𝑃𝑖
+𝜓′
+𝑖 | 𝑣dip
+𝑐
+[%]
+𝑆𝑣
+𝜎′𝑣
+𝑣dip
+𝑐
+[%]
+||𝑩′′
+𝒊 ||
+|| ˜𝜷′′
+𝒊 ||
+𝑣dip
+𝑐
+[%]
+||𝚯𝒊 ||
+|| ˜𝜽′
+𝒊 ||
+𝑣dip
+𝑐
+[%]
+1
+B1608+656
+0.0001
+0.00001
+-0.0001
+0.0001
+0.0003
+0.0001
+1.1011
+0.2846
+2
+RXJ1131-1231
+0.3739
+0.1325
+-0.1051
+0.1221
+0.2107
+0.1799
+0.6470
+0.0986
+3
+HE0435-1223
+-0.0259
+0.0006
+0.0161
+-0.0143
+0.0282
+-0.0149
+2.3707
+0.1293
+4 SDSS1206+4332
+0.0663
+0.-0.0064
+-0.0925
+0.0759
+0.1516
+0.0810
+0.3607
+0.0880
+5
+WFI2033-4723
+-0.0569
+0.0015
+0.0627
+-0.0530
+0.1054
+-0.0571
+1.2443
+0.1443
+6
+PG1115+080
+0.3431
+-0.0066
+-0.1294
+0.1191
+0.2353
+0.1210
+5.5110
+0.1182
+7
+DES0408-5354
+-0.0095
+0.0021
+0.0094
+-0.0076
+0.0154
+-0.0075
+0.9690
+0.2024
+Table II. We give the relative biases on each quantity assuming that 𝑣𝑙 = 0 = 𝑣𝑠 and that the observer has a peculiar velocity of amplitude
+||𝒗dip|| = 369.82 km s−1, as extracted from the entirely kinematic interpretation of the CMB dipole. Most of the biases are below the percent
+level. When there are several values for a single system, for example for 𝑃𝑖/𝜓′
+𝑖, ||𝑩′′
+𝒊 ||/|| ˜𝜷′′
+𝒊 ||, ||𝑻𝒊||/|| ˜𝜽′
+𝒊 ||, we give only the result for the image
+which maximizes the bias.
+more than 10%.
+V.
+CONCLUSION
+In this work, we quantified the effects of peculiar velocities of the lens, source and observer on the determination of the Hubble
+constant from time-delay cosmography, carefully taking into account all boost effects on the observables and their repercussion
+on the lens model. We showed in detail how to compute the bias, given peculiar velocities and assuming that the lens is well
+described by a singular isothermal sphere. Even if this model alone does not allow for more than two images per lensed quasar,
+and gives crude estimates of 𝐻0, we expect this model to be sufficient to capture the leading effects of peculiar velocities on
+
+15
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+-4
+-2
+0
+2
+4
+6
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+flvLwATxglZ0=DES0408-5354
+Figure 6. In this plot, we show the normalized bias on 𝐻0, for 𝑣𝑜 extracted from the entirely kinematic interpretation of the CMB and vary 𝑣 ∥
+𝑙 ,
+𝑣 ∥
+𝑠 ∈ {0, ±450, ±900} km s−1. Larger dots indicate larger source peculiar velocities 𝑣 ∥
+𝑠. For this plot, we assumed that 𝜎𝑣 is extracted from the
+observed Einstein angle, as outlined in Sec. III. This implies that 𝑆𝑣 is calculated using Eq. (52), contrary to Fig. 5, where it was set to zero.
+While varying 𝑣 ∥
+𝑠 does change the bias on 𝐻0, the change is smaller than that of 𝑣 ∥
+𝑙 . Note that this is different to the situation presented in
+Fig. 5, where the velocity of the source affects less the bias on 𝐻0. The largest bias appears for lens and source peculiar velocities which are
+anti-aligned. In this case, the amplitude of the bias on the Hubble constant can reach 5%. The peculiar velocity of the observer alone gives an
+amplitude bias which is bounded by 1.2%.
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+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+-10
+-5
+0
+5
+10
+Figure 7. In this plot, we show the relative bias Δ𝐻0/𝐻′
+0 on 𝐻0 for an observer with peculiar velocity 𝑣𝑜 ∈ {0, ±1000, ±2000, ±3000} km s−1 as
+a function of the observed lens redshift 𝑧′
+𝑙. Peculiar velocities of the order of ±3000 km s−1 are required to bias the Hubble constant to the
+order of 10% for the systems which are best aligned with 𝒗dip, which are RXJ1131-1231 and PG1115+080. This corresponds to more than 8
+times more than 𝒗dip. One might be intrigued by the sign of the bias for these two lenses. It turns out that a negative velocity −3000 km s−1 is
+required, to bias 𝐻′
+0 to ∼ 10% higher than 𝐻0. In this sense, the higher 𝑣𝑜 expected from number count dipoles works against lowering the
+Hubble constant extracted from the two low redshift lenses RXJ1131-1231 and PG1115+080. Here, peculiar velocities of the lens and source are
+set to zero and 𝜎𝑣 is assumed to be extracted from the observed Einstein angle, as outlined in Sec. III. The system B1608+656 at lens redshift
+𝑧′
+𝑙 ≃ 0.63 is nearly not affected by the boost because it is quasi-orthogonal to the CMB dipole direction. The situation would change if the
+direction of 𝒗𝑜 was also varied.
+
+16
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+-20
+-10
+0
+10
+20
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+Figure 8. We plot the bias on 𝐻0 from the peculiar velocity of the observer, lens and source. Here 𝑣𝑜 is fixed by the entirely kinematic
+interpretation of the CMB dipole, i.e. 𝑣𝑜 = 369.82 km s−1. The lens and source peculiar velocities are allowed to vary in {0, ±1500, ±3000} km
+s−1. Larger dots indicate larger source peculiar velocities 𝑣 ∥
+𝑠. Here 𝜎𝑣 was computed from the observed Einstein angle, as outlined in Sec. III.
+In this case, 𝑆𝑣 ≠ 0. This corresponds to what has mostly been done in practice in [27, 41]. In this case, the velocity of the lens influences
+significantly the bias on 𝐻0 and the source peculiar velocity, less so. The largest bias in magnitude appears for the source and lens peculiar
+velocities which are anti-aligned.
+current time-delay cosmography experiments. For the observer’s peculiar velocities fixed to ||𝒗dip|| = 369.82 km s−1, as extracted
+from the entirely kinematic interpretation of the CMB dipole, the bias on 𝐻0 is, at most, of the order of the percent level for a
+single lens. The sign and amplitude of the bias depends on the direction of the observed lens center of mass, which is captured by
+cos(𝜃′
+𝑐𝑚). The bias on 𝐻0 from the observer’s peculiar velocity for the combined seven lenses, which span different corners of the
+sky is of 0.25%. These cancellations for the observer’s peculiar velocity require an isotropic distribution of lensed quasars, which
+may be jeopardized by the specific footprint of time-domain surveys. This is however mitigated by the fact that the CMB dipole
+points to celestial declination −7◦, which is close to the Earth’s equator. In this sense, North or South hemisphere surveys are
+nearly as orthogonal as they can be from the observer’s peculiar velocity.
+If one includes the effect of the lens and source peculiar velocities projected on the line of sight, up to |𝑣 ∥
+𝑙 |, |𝑣 ∥
+𝑠 | ≤ 900 km
+s−1, then the effect reaches at most 5%. The sign of these contributions depends entirely on the sign and amplitude of these
+peculiar velocities, which vary from one system to another and may be expected to cancel out between a source and another, for a
+sufficiently high number of systems. Assuming that the lens and source peculiar velocities are normally distributed around zero
+with standard variation of 300 km s−1, we found that these generate a random uncertainty on 𝐻0, which can reach 1.00% for a
+single lens and which combines to 0.24% for the seven systems. We also found that the way that the lens model parameter, i.e. the
+velocity dispersion 𝜎𝑣 is determined, affects how subdominant the source peculiar velocities are in the Hubble constant bias. If
+one can determine the velocity dispersion independently of the peculiar velocities of the observer and lens, from spectroscopic
+measurements, then the bias from peculiar velocities on the Hubble constant is reduced. This can bring the bias from 5% to 2.5%
+in the most extreme cases with 𝑣 ∥
+𝑙 = −900 km s−1 antialigned with 𝑣 ∥
+𝑠 = 900 km s−1.
+Finally, we studied what peculiar velocities are required to bias the Hubble constant determination to the order of 10%. We
+found that peculiar velocities projected on the line of sight of the order of 3000 km s−1 would do the job. This can be cumulated
+between the observer, the source and the lens. This requires unexpectedly large peculiar velocities. Coincidentally, the two systems
+which are best aligned with 𝒗dip are also the ones which give the higher 𝐻0 estimates in H0LiCOW [27]. It is interesting, in light
+of the number count experiments which favor a larger ||𝒗𝑜||, that a larger observer peculiar velocity works against resolving the
+Hubble tension since it would imply that the H0LiCOW collaboration rather underestimates 𝐻0 for these two lenses, which already
+give the highest 𝐻0 estimates. In other words, lowering the Hubble constant estimates for these two lenses requires an observer
+velocity which goes in the opposite direction of the CMB, with an amplitude roughly 8 times larger than ||𝒗dip||. Future biased
+estimations of the Hubble constant could also be expected if one observes systems consistently in the same hemisphere aligned
+with the observer’s peculiar velocity. Even more so if the observer’s peculiar velocity is larger in magnitude than one expects from
+the entirely kinematic interpretation of the CMB dipole.
+The small number of sources (O(10)) implies that cancellations over many different sources which are distributed isotropically
+may be spoiled by shot noise. If it is clear that these large peculiar velocities are rare in ΛCDM, to rule them out would require
+
+17
+distance estimates, which combined with redshifts, can be used to constrain the lens and source peculiar velocities. To affect the
+Hubble constant consistently over many sources would require large bulk flows of sources which are not expected in homogeneous
+and isotropic cosmologies. In ΛCDM, one expects the bulk flow velocity of sources on a sphere centered on the observer to
+decay with increasing radius. It should be noted that several anomalies have been pointed out in such convergence to the Hubble
+flow [1, 55–57] on scales which can reach up to 800 Mpc. For example, these large peculiar velocities could be expected for an
+observer who is offset from the center of an ultra-large void, which was studied in [22] and proposed as a solution to the cosmic
+dipole tension. In this scenario, these large peculiar velocities could be interpreted as artifacts from working with the wrong
+background equations of motion.
+Finally, we conclude that peculiar velocities of the observer, source and lens play a significant role in time-delay cosmography,
+if one is after percent precision on the Hubble constant. It seems difficult to accomodate a larger observer’s peculiar velocity,
+as suspected from radio source and quasar number counts, as a simultaneous explanation for the bias towards higher 𝐻0 from
+time-delay cosmography. Future independent constraints on the peculiar velocities of the lenses, sources and observer could help
+to constrain the Hubble constant to percent precision using time-delay cosmography.
+VI.
+ACKNOWLEDGMENTS
+We would like to thank Pierre Fleury for interesting discussions and Aymeric Galan and Simon Birrer for valuable feedback
+on a preliminary version of this work. C.D. and T.B. are supported by ERC Starting Grant SHADE (grant no. StG 949572).
+M.M. acknowledges the support of the Swiss National Science Foundation (SNSF) under grant P500PT_203114. T.B. is further
+supported by a Royal Society University Research Fellowship (grant no. URF\ R1\180009).
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+Clusters of Galaxies. ApJ, 712(1):L81–L85, March 2010.
+[56] A. Kashlinsky, F. Atrio-Barandela, D. Kocevski, and H. Ebeling. A Measurement of Large-Scale Peculiar Velocities of Clusters of Galaxies:
+Results and Cosmological Implications. ApJ, 686(2):L49, October 2008.
+[57] James E. Gunn. Hubble’s Deviations from Pure Hubble Flow: A Review. In Sidney van den Bergh and Christopher J. Pritchet, editors, The
+Extragalactic Distance Scale, volume 4 of Astronomical Society of the Pacific Conference Series, page 344, January 1988.
+Appendix A: Rotation angle
+The rotation angle 𝛿′ serves to translate the coordinates in the observation frame for a moving observer to the calculation
+frame. The rotation angle 𝛿 serves to transform these back to the observation frame of a comoving observer. The rotation angle 𝛿′
+depicted in Fig. 2 can be obtained from the lens’ center of mass vectors ˆ𝒏′ and the vector ˆ𝑵′ = (122.932◦, 27.128◦) in galactic
+coordinates, which points in the direction of the Earth’s North pole in J2000. The vector ˜𝜽′
+𝒚 = ( ˜𝑦′
+1, ˜𝑦′
+2, ˜𝑦′
+3) is the projection of the
+North pole direction ˆ𝑵′ in the plane orthogonal to ˆ𝒏′, while ˜𝜽′
+𝒙 points East. That is
+˜𝜽′
+𝒚 = ˆ𝑵′ − ( ˆ𝒏′ · ˆ𝑵′) ˆ𝒏′ .
+(A1)
+The vector ˆ𝜽′
+𝒙 is defined as a vector which is orthogonal both to ˆ𝒗𝒐 and to ˆ𝒏′. There are two such vectors which can be obtained by
+solving the following system for ˆ𝜽′
+𝒙 = ( ˆ𝑥′
+1, ˆ𝑥′
+2, ˆ𝑥′
+3)
+ˆ𝒏′ · ˆ𝜽′
+𝒙 = 0 ,
+(A2)
+ˆ𝒗𝑜 · ˆ𝜽′
+𝒙 = 0 .
+(A3)
+The vector ˆ𝜽′
+𝒚 = ( ˆ𝑦′
+1, ˆ𝑦′
+2, ˆ𝑦′
+3) is orthogonal to ˆ𝜽′
+𝒙 and ˆ𝒏′ and points towards the positive ˆ𝒛 axis, meaning that it is a solution of the
+following system
+ˆ𝜽′
+𝒙 · ˆ𝜽′
+𝒚 = 0 ,
+(A4)
+ˆ𝒏′ · ˆ𝜽′
+𝒚 = 0 ,
+(A5)
+ˆ𝜽′
+𝒚 · ˆ𝒛 > 0 .
+(A6)
+One can compute 𝛿′ in the following way
+cos 𝛿′ = ˜𝜽′
+𝒚 · ˆ𝜽′
+𝒚 .
+(A7)
+Since the comoving North pole ˆ𝑵 = (𝜃𝑁 , 𝜑𝑁 ) and the direction ˆ𝒏 = (𝜃𝑐𝑚, 𝜑𝑐𝑚) can be reconstructed using Eqs. (40)-(41), one
+can repeat these steps to find 𝛿. This defines implicitly the bias 𝐷 on the rotation angle
+𝛿 = 𝛿′ + 𝐷 𝑣𝑜
+𝑐 .
+(A8)
+Recall that 𝛿′ is the angle between the observation coordinate system spanned by { ˜𝜽′
+𝒙, ˜𝜽′
+𝒚} and a convenient coordinate system
+{ ˆ𝜽′
+𝒙, ˆ𝜽′
+𝒚} as depicted in Fig. 2. This rotation angle is used to determine how the images on the sky appear biased to an observer
+who has a peculiar velocity 𝒗𝑜.
+
diff --git a/LdE5T4oBgHgl3EQfYg9W/content/tmp_files/load_file.txt b/LdE5T4oBgHgl3EQfYg9W/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..d505f0955f6c87fc08bdd6e4d5a433ed7460dd71
--- /dev/null
+++ b/LdE5T4oBgHgl3EQfYg9W/content/tmp_files/load_file.txt
@@ -0,0 +1,2056 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf,len=2055
+page_content='Peculiar velocity effects on the Hubble constant from time-delay cosmography Charles Dalang,1, ∗ Martin Millon,2, † and Tessa Baker1, ‡ 1Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom 2Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, Stanford, CA 94305, USA Two major challenges of contemporary cosmology are the Hubble tension and the cosmic dipole tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' At the crossroad of these, we investigate the impact of peculiar velocities on estimations of the Hubble constant from time-delay cosmography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' We quantify the bias on the inference of the Hubble constant due to peculiar velocities of the lens, the source and of the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' The former two, which may cancel from one system to another, affect the determination of the angular diameter distances in the time-delay formula, and reconstructed quantities like the angle to the source, via a lens model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' On the other hand, the peculiar velocity of the observer, which is a debated quantity in the context of the cosmic dipole tension, systematically affects observed angles through aberration, redshifts, angular diameter distance and reconstructed quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' We compute in detail the effect of these peculiar velocities on the inference of the Hubble constant to linear order in the peculiar velocities for the seven lenses of the H0LiCOW/TDCOSMO collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' The bias generated by the observer’s peculiar velocity alone can reach 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='15% for the lenses which are well aligned with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' This results in a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='25% bias for the seven combined lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' Assuming a typical peculiar velocity of 300 km s−1 for the lens and the source galaxies, these add an additional random uncertainty, which can reach 1% for an individual lens, but reduces to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='24% for the full TDCOSMO sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' The picture may change if peculiar velocities turn out to be larger than expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' Any time-delay cosmography program which aims for percent precision on the Hubble constant may need to take this source of systematic bias into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' This is especially so for future ground-based surveys which cover a fraction of the celestial sphere that is well aligned with the observer’s peculiar velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' INTRODUCTION P ersistent tensions in cosmological datasets may be indicators of new physics or of unknown systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' While the former is very exciting, excluding confidently the latter is notoriously difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' On the theoretical side, this is mostly because in extracting cosmological parameters, approximations are needed, which require a set of assumptions that may be broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' Two of these tensions include disagreement on the kinematic cosmic dipole, which can be translated into a tension on the peculiar velocity of the observer [1] and on the Hubble constant [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' Both of these tensions are between the Cosmic Microwave Background (CMB) and other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' The CMB dipole allows one to extract the velocity of the observer, which effectively Doppler shifts the black body radiation of angular average temperature ⟨𝑇⟩ from the CMB to higher and lower temperatures (𝛿𝑇/⟨𝑇⟩)dip ∼ O(10−3) in opposite hemispheres aligned with the observer’s velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' This works well provided the intrinsic CMB dipole, which is expected to be of the order O(10−5), is small in comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' This is expected from a nearly scale-invariant power spectrum of primordial fluctuations of the inflaton generated at the end of a period of quasi-de Sitter expansion during inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' Under the assumption that the intrinsic CMB dipole vanishes, known as the entirely kinematic interpretation of the CMB dipole, one obtains ||𝒗dip|| = 369.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='11 km s−1 towards ˆ𝒗dip = (264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='021◦ ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='011◦, 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='253◦ ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='005◦) in galactic coordinates [3–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' This defines a reference frame known as the CMB frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' If the interpretation is correct, the same velocity should induce correlations between the 𝑙 and 𝑙 ± 1 multipoles of the CMB, which was checked in [6–8] and gives consistent results, albeit the relatively large error bars still leave room for an intrinsic dipole which can make up to 40% of the CMB dipole [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' It should be noted that spectral distortions of the CMB monopole, dipole and quadrupole should let one separate the intrinsic dipole from its kinematic counterpart with a sufficiently advanced detectors [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' Alternatively, the peculiar velocity of the observer can be extracted from source number counts of relatively high redshift sources (𝑧 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='1), such as quasars, to avoid contamination from local structures [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' This was pioneered by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' Ellis and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' Baldwin for flux-limited surveys of sources with a flux density following a powerlaw frequency spectrum [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' Aberration of angles and Doppler shift then affect these number counts per unit solid angle in such a way that permits the extraction of the observer’s peculiar velocity with respect to these sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' This has led to a number count dipole, which is well aligned with the CMB dipole but about 2 − 5 times as large as expected from 𝒗dip and which has reached a ∼ 5𝜎 tension [14–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' In [12], it was suggested that the redshift evolution of the population of sources may, at least partially, explain the discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' This was further investigated by the authors of [19], who also find large variations in the theoretical expectation of the number count dipole in the presence of parameter evolution when using different quasar luminosity function models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' The authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' [18] reanalyzed the ∗ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='dalang@qmul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='uk † millon@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='edu ‡ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='baker@qmul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='uk arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='05574v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='CO] 13 Jan 2023 2 data of [14, 15] and concluded that neither masking nor parameter evolution can fully explain the discrepancy, although the latter is subject to further assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' If the dominantly kinematic interpretation of this number count dipole is correct, it should show up in the correlations between the 𝑙 and 𝑙 ± 1 multipoles of the number counts, which require high-angular resolution surveys [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' An observer offset from the center of an ultra-large void was also suggested in [22] as a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' This would imply effective large source peculiar velocities as a result of working with a homogeneous and isotropic background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' In any case, this problem requires further studies [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' The Hubble tension is somehow more popular [23] and has been established for a longer period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' It is the disagreement between direct measurements of the Hubble constant and inference of 𝐻0 from the CMB, if assuming a flat homogeneous and isotropic Universe dominated by Cold Dark Matter and a cosmological constant, the so-called ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' The Hubble constant is inferred from the angle upon which the scale associated to the horizon at the last scattering surface is seen in the CMB, which is extracted from the temperature fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' This results in 𝐻0 = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='5 km s−1 Mpc−1 at 68% confidence level [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' In contrast, two of the most competitive local measurements of the Hubble constant come from supernovae type Ia, which requires calibration via the distance ladder and from time-delay cosmography with strongly lensed quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' Teams performing these experiments reported relatively high 𝐻0 = 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='03 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='42 km s−1 Mpc−1 [25, 26] and 𝐻0 = 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='3+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='7 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='8 km s−1 Mpc−1 [27], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' Combining these two direct measurements leads to a 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='3𝜎 tension on 𝐻0 with the CMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' This has led to a plethora of alternative models, with various levels of complexity and success, as demonstrated by the existence of the 𝐻0-olympics [28] (see also [29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' Importantly, it should be noted that using only stellar kinematics instead of assumptions about the mass profile of the lensing galaxies to break the so-called mass-sheet degeneracy [30], led to 𝐻0 = 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='5+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='6 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='1 km s−1 Mpc−1 with the seven same strongly-lensed systems [31], which is consistent with Planck [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' The peculiar velocity of the observer plays the role of a foundational stone for cosmological experiments which work in the CMB frame [32–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' It is therefore alarming that some experiments disagree on the peculiar velocity of the observer 𝒗𝑜.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' For example, directional dependencies of 2 − 3𝜎 level on cosmological parameters extracted from the CMB were reported in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' A remnant of anisotropies on 𝐻0 determined from supernovae type Ia data was reported in [37] even when working in the CMB frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' The authors of [38] found 4 km s−1 Mpc−1 difference in 𝐻0 in opposite hemispheres aligned with the CMB dipole, although such variations are expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' Similar hints of 𝐻0 anisotropies from strongly lensed quasars were noticed in [39], although these observations are not corrected for the peculiar velocities of the observer, lenses or sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' In particular, the H0LiCOW/TDCOSMO collaboration pointed out a mild 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content='8𝜎 significance for an 𝐻0 which decreases with observed lens redshift 𝑧′ 𝑙 [27, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' Two of the lowest lens redshift systems, which give the highest Hubble constant estimates, turn out to also be well aligned with the CMB dipole, as remarked in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' In this work, we focus on the determination of the Hubble constant from the time delay of strongly lensed quasars and study the impact of peculiar velocities on this measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' One may expect that peculiar velocities of the order of 𝑣/𝑐 ≃ O(10−3) do not affect the 𝐻0 measurement beyond O(10−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' However, to our knowledge, there has not been any rigorous study of the accumulation of effects of aberration and Doppler shift on time-delay cosmography, and propagation of biases through the lens model, which may inflate the proportionality constant in front of 𝑣/𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' Our goal is to fill this gap and study if there can be any relation between the Hubble and dipole tensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' II, we review the basics of time-delay cosmography for a singular isothermal sphere, which allows us to fix notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' III, we detail all effects of peculiar velocities on the observables and also how these propagate through the lens model and to the Hubble constant determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' IV, we apply our findings to the seven lenses of TDCOSMO1, compute the bias on the Hubble constant for each lens as a function of the peculiar velocities and discuss our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' V, we conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' We suggest that a busy reader principally interested in the total impact on 𝐻0 measurements should review the form of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' (75) and (77), then move directly to section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' Throughout the article, bold symbols denote 2 or 3 dimensional vectors, hats indicate unit vectors || ˆ𝒏|| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' Sometimes unit vectors in R3 are expressed in spherical coordinates ˆ𝒏 = (cos 𝜃 cos 𝜑, cos 𝜃 sin 𝜑, sin 𝜃) �= (𝜃, 𝜑), where 𝜃 ∈ [0, 𝜋] and 𝜑 ∈ [0, 2𝜋[ indicate the polar and azimuthal angle, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' We note the speed of light 𝑐 and Newton’s constant 𝐺N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' SINGULAR ISOTHERMAL SPHERE In this section, we review time-delay cosmography for an isothermal sphere and fix our notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' Derivations may be found in [42] and the reader experienced in lensing time delay formalism can skip to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' The cosmic time delay Δ𝑡𝑖 𝑗 ≡ 𝑡𝑖 − 𝑡 𝑗 variations in lensed images 𝑖 and 𝑗 for a comoving observer, lens and source can be expressed as [42] 𝑐Δ𝑡𝑖 𝑗 = (1 + 𝑧𝑙) 𝑑𝑙𝑑𝑠 𝑑𝑙𝑠 � ˆ𝜙(𝜽𝑖, 𝜷) − ˆ𝜙(𝜽 𝒋, 𝜷) � , (1) 1 Six lenses come from H0LiCOW [27] and one from STRIDES [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' These seven systems are now analysed jointly by the TDCOSMO collaboration [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
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+page_content='observer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE5T4oBgHgl3EQfYg9W/content/2301.05574v1.pdf'}
+page_content=' 0,β > 0, and V(si(t)) usually takes the form
+V(s) =
+�
+�
+�
+�
+�
+0,
+s ≤ sst
+fv(s),
+sst < s < sgo
+vmax,
+s ≥ sgo
+(5)
+and a typical fv(s) takes the form
+fv(s) = vmax
+2
+�
+1−cos
+�
+π s−sst
+sgo −sst
+��
+.
+(6)
+As a result, v∗ = V(s∗),α1 = α ˙V(s∗),α2 = α +β,α3 = β.
+B. Piecewise-constant control
+We consider a system with one piecewise-constant con-
+trolled vehicle i = 1 with hold length ∆, and n − 1 human
+OVM vehicles. At a given time t ∈ [tk,tk+1] where [tk,tk+1] is
+the corresponding holding period, the CFM for the controlled
+vehicle is modeled by
+˙v1(t) = f(u(z(tk),z(t)))
+(7)
+where f is a function described in details below, z(ˆt) =
+[s1(ˆt),v1(ˆt),...,sn(ˆt),vn(ˆt)] is the state vector at time ˆt, and
+u(z(tk),z(t)) represents the control function, which we allow
+a part to be held constant from the input z(tk), and the rest to
+be continuous from the input z(t).
+As an example, a class of piecewise-constant velocity guid-
+ance control in OVM proposes a constant desired velocity
+u(z(tk)) to the controlled vehicle during the holding period;
+the vehicle uses a OVM-like dynamics to reach the desired
+controlled velocity, resulting in the dynamics
+˙v1(t) = α(u(z(tk))−v1(t))+β ˙s1(t)
+(8)
+Meanwhile, a class of piecewise-constant acceleration control
+directly forces the controlled vehicle to take a constant accel-
+eration during the holding period, resulting in the dynamics
+˙v1(t) = u(z(tk))
+(9)
+We follow previous works [7, 8] to focus on the piecewise-
+constant acceleration control in this work.
+Lumping the error state into a vector form with x(t) =
+[˜s1(t), ˜v1(t),..., ˜sn(t), ˜vn(t)]⊺, the error dynamics for the con-
+trolled vehicle is given by
+�
+˙˜s1(t)
+= ˜vn(t)− ˜v1(t)
+˙˜v1(t)
+= ˜u(x(tk))
+(10)
+where t ∈ [tk,tk+1] and tk+1 − tk ≤ ∆ is the corresponding
+holding period.
+The error dynamics of the linearized piecewise-constant
+control system is thus given by
+˙x(t) = Ax(t)+A1x(tk),k = 0,1,...
+(11)
+with
+A =
+�
+��������
+C1
+0
+...
+...
+0
+C2
+D2
+D1
+0
+...
+...
+0
+0
+D2
+D1
+0
+...
+0
+...
+...
+...
+...
+...
+...
+0
+...
+0
+D2
+D1
+0
+0
+...
+...
+0
+D2
+D1
+�
+��������
+,B =
+�
+������
+B1
+B2
+B2
+...
+B2
+�
+������
+(12)
+with
+D1 =
+�
+0
+−1
+α1
+−α2
+�
+,D2 =
+�
+0
+1
+0
+α3
+�
+,
+C1 =
+�
+0
+−1
+0
+0
+�
+,C2 =
+�
+0
+1
+0
+0
+�
+,B1 =
+�
+0
+1
+�
+,B2 =
+�
+0
+0
+�
+(13)
+where A1 = −BK represents the full state feedback piecewise-
+constant control coefficients. We note that the formulation
+is exactly the same as in Zheng et al. [10] except for
+the piecewise-constant control component; we also note that
+the formulation perfectly aligns with the sample-data sys-
+tem framework [13] with zero-order hold. We further note
+that, different classes of piecewise-constant controls result in
+slightly different A and A1 matrices. For example, the velocity
+guidance control, as described in Eq. (8), has
+C1 =
+�
+0
+−1
+0
+−α2
+�
+,C2 =
+�
+0
+1
+0
+α3
+�
+,B1 =
+�
+0
+α
+�
+,
+(14)
+with the rest of the D1,D2,B2 matrices be the same. The
+representations C1 and C2 follow the human vehicle represen-
+tations D1 and D2, except the α1 term representing the desired
+velocity is moved from the uncontrolled system matrix D1 to
+the control matrix B1. The Lyapunov analyses in the following
+Section IV naturally apply to the broader classes of piecewise-
+constant controls, as they are agnostic to the specific form of
+A and A1 matrices for the system.
+
+IV. LYAPUNOV ANALYSIS
+A. A Lyapunov bound
+We first derive a Lyapunov bound on the hold limit ∆,
+which is defined as the maximum hold length such that the
+traffic system remains stable; while a Lyapunov bound on the
+general nonlinear system with discontinous control has been
+derived in the previous literature [21], we adapt the derivation
+to the linearized system with piecewise-constant control. In
+later sections, we apply the bound to the ring-road optimal
+velocity model to extract meaningful insights into the traffic
+system and controller design.
+Proposition 1. Let there exist n × n matrices P > 0,Q > 0
+such that V(x) = x⊺Px > 0 with ˙V(x) = −x⊺Qx < 0 and
+−Q = (A + A1)P + P(A + A1)⊺ is a valid Lyapunov function
+for the linear continuous system with continuous full-state
+feedback control, ˙x(t) = (A+A1)x(t) where A1 = −BK. Then
+the sample-data system with piecewise constant control (11)
+is asymptotically stable for hold length
+∆ ≤ c′
+σmin(Q)
+σmax(P)(σmax(A)+σmax(A1))2
+(15)
+up to a scaling constant c′ > 0, where σmin(·) and σmax(·)
+are the minimum and maximum singular value of the
+corresponding matrix.
+Proof. Consider a time period [tk,tk+1] with tk+1 −tk ≤ ∆.
+We use the Lyapunov function for the continuous system
+V(x) = x⊺Px, and show that it is a valid Lyapunov function
+for the sample-data system by showing V(x(t)) −V(x(tk)) is
+sufficiently negative, i.e. V(x(t)) decreases as t increases. We
+have for all t ∈ [tk,tk+1]:
+V(x(t))−V(x(tk))
+= ⟨∇V(x(t∗)), ˙x(t∗)⟩(t −tk)
+for some t∗ ∈ (tk,t)
+= ⟨∇V(x(tk)), ˙x(tk)⟩(t −tk)
++ ⟨∇V(x(tk)), ˙x(t∗)− ˙x(tk)⟩(t −tk)
++ ⟨∇V(x(t∗))−∇V(x(tk)), ˙x(tk)⟩(t −tk)
+(16)
+where the first equality holds by the mean value theorem.
+For the three terms in the last equality, the first term gives a
+decrease in Lyapunov value, as at time tk the system behaves
+the same as the continuous system with continuous control
+using the instantaneous state information x(tk). Specifically,
+⟨∇V(x(tk)), ˙x(tk)⟩
+= x(tk)((A+A1)P+P(A+A1)⊺)x(tk)
+= −x(tk)Qx(tk)
+(17)
+Therefore, a lower bound on the decrease of the Lyapunov
+function from the first term gives
+⟨∇V(x(tk)), ˙x(tk)⟩ ≤ −σmin(Q)∥x(tk)∥2
+2 ≤ 0
+The second and third terms represent the perturbation incurred
+by the piecewise-constant control, where
+∇V(x(tk)) = 2x(tk)⊺P
+˙x(t∗)− ˙x(tk) = A(x(t∗)−x(tk))
+∇V(x(t∗))−∇V(x(tk)) = 2(x(t∗)−x(tk))⊺P
+˙x(tk) = (A+A1)x(tk)
+(18)
+where the second equality is due to the same piecewise-
+constant control in the entire period. The following worst-case
+bounds hold:
+∥∇V(x(tk))∥ ≤ 2σmax(P)∥x(tk)∥2
+∥˙x(tk)∥2 ≤ σmax(A+A1)∥x(tk)∥2
+∥∇V(x(t∗))−∇V(x(tk))∥ ≤ 2σmax(P)∥x(t∗)−x(tk)∥2
+∥x(t∗)−x(tk)∥2 =
+����
+� t∗
+tk
+˙x(s)ds
+����
+2
+≤ (t∗ −tk) max
+s∈[tk,t∗]∥˙x(s)∥2
+≤ ∆ max
+s∈[tk,t∗]∥Ax(s)+A1x(tk)∥2
+≤ ∆(σmax(A)+σmax(A1))
+max
+s∈[tk,tk+1]∥x(s)∥2
+(19)
+Taken together, we have
+V(x(t))−V(x(tk))
+≤ (t −tk)
+�
+−σmin(Q)∥x(tk)∥2
+2
++2σmax(P)∥x(tk)∥2σmax(A)∥x(t∗)−x(tk)∥2
++2σmax(P)∥x(t∗)−x(tk)∥2σmax(A+A1)∥x(tk)∥2
+�
+= (t −tk)
+�
+−σmin(Q)∥x(tk)∥2
+2
++2σmax(P)
+�
+σmax(A)+σmax(A+A1)
+�
+×
+∥x(tk)∥2∥x(t∗)−x(tk)∥2
+�
+≤ (t −tk)
+�
+−σmin(Q)+c∆·σmax(P)(σmax(A)+σmax(A1))2�
+∥x(tk)∥2
+max
+s∈[tk,tk+1]∥x(s)∥2
+(20)
+where c > 0 is an appropriate constant. In the last inequal-
+ity, we apply Weyl’s inequality to separate σmax(A + A1) ≤
+σmax(A) + σmax(A1) and substitute the bound on ∥x(t∗) −
+x(tk)∥2 in Eq. (19) to obtain the square term (σmax(A) +
+σmax(A1))2. In order for V(x(t))−V(x(tk)) to have a sufficient
+decrease, e.g. for some d > 1 (d = 2 in Clarke [21]),
+V(x(t))−V(x(tk)) ≤ −(t −tk)σmin(Q)
+d
+∥x(tk)∥2
+max
+s∈[tk,tk+1]∥x(s)∥2,
+(21)
+the following gives a sufficient condition
+c∆·σmax(P)(σmax(A)+σmax(A1))2 ≤ d −1
+d
+σmin(Q)
+⇔ ∆ ≤ c′
+σmin(Q)
+σmax(P)(σmax(A)+σmax(A1))2
+(22)
+for some c′ > 0. ■
+While the above bound can be loose due to the worst case
+singular-value bounds, it still provides a way to qualitatively
+analyze the system. As an interpretation, let us suppose P = I
+
+results in Q > 0. Then loosely speaking, an unstable uncon-
+trolled system A with larger σmax(A) makes the bound smaller.
+The contribution of the control is more complicated with a
+trade-off involved: on one hand, the larger control makes the
+continuous controlled system A + A1 more stable, increasing
+the σmin(Q) term in the numerator; on the other hand, it also
+increases σmax(A1) and hence increases the denominator.
+B. A Lyapunov-Krasovskii functional
+As the human-compatible system with piecewise constant
+control perfectly aligns with the sample-data system frame-
+work, we seek to find a tighter bound on the hold limit using
+theory developed for sample-data systems. A few works [23,
+24, 25] view the sample-data system as a special case of
+the time-delay system with delay τ(t) = t −tk, which has a
+constant rate of change ˙τ(t) = 1 for all t. Lyapunov-Krasovskii
+functionals are commonly used to analyze the performance of
+time-delay systems, and naturally extend to the sample-data
+system (11), which can be equivalently written in the form
+˙x(t) = (A+A1)x(t)−A1
+� t
+tk
+˙x(s)ds
+(23)
+as x(tk) = x(t) −
+� t
+tk ˙x(s)ds. In Fridman [23], the following
+Lyapunov-Krasovskii functional for sample-data system is
+proposed
+V(t,x(t), ˙x(t)) = x⊺(t)Px(t)+(∆−τ(t))
+� t
+t−τ(t) ˙x⊺(s)U ˙x(s)ds
+(24)
+where
+τ(t) = t − tk,
+P > 0,
+U > 0.
+The
+first
+term
+x⊺(t)Px(t)
+in
+the
+above
+functional
+is
+the
+regular
+Lyapunov function for the unperturbed nominal system
+˙x(t) = (A + A1)x(t),
+whereas
+the
+second
+integral
+term
+handles
+the
+integral
+perturbation
+−
+� t
+tk ˙x(s)ds.
+Jensen’s
+inequality, descriptor method [24], and state-augmentation
+with η1(t) = col{x(t), ˙x(t),
+1
+τ(t)
+� t
+t−τ(t) ˙x(s)ds} are applied to
+arrive at the following proposition on a given hold length ∆
+with Linear Matrix Inequalities (LMIs):
+Proposition 2. Let there exist n × n matrices P > 0,U > 0;
+P2 and P3 such that the LMIs (25) are feasible. Then (11)
+is asymptotically stable for all variable sampling instants
+tk+1 −tk ≤ ∆.
+�
+Φ11
+P−P⊺
+2 +(A+A1)⊺P3
+∗
+−P3 −P⊺
+3 +∆U
+�
+< 0,
+�
+�
+Φ11
+P−P⊺
+2 +(A+A1)⊺P3
+−∆P⊺
+2 A1
+∗
+−P3 −P⊺
+3
+−∆P⊺
+3 A1
+∗
+∗
+−∆U
+�
+� < 0.
+(25)
+where Φ11 = P⊺
+2 (A + A1) + (A + A1)⊺P2 and ∗ denotes the
+symmetric elements of the symmetric matrix.
+Proof. See [24].
+Comparing with the previous Lyapunov bound which upper
+bounds the perturbation
+� t
+tk ˙x(s)ds by the minimax singular
+value ratio of the controlled system A + A1, represented by
+σmin(Q)
+σmax(P), divided a function of the maximum singular values
+of the uncontrolled system A and the control A1, represented
+by σmax(A) and σmax(A1), the Lyapunov-Krasovskii bound
+solves for matrices P, U, P2, P3 to account for the interactions
+among A, A1, and A + A1, and hence can possibly render a
+tighter bound. Additionally, while the above proposition takes
+a fixed controller K as given to verify if such a controller
+can stabilize the system with a hold length ∆, we can in fact
+solve for a possibly better controller K using the following
+corollary that takes the sample-data system property into
+consideration.
+Corollary 1. Let there exist n × n matrices ¯P > 0, ¯U > 0,
+Q and an nu × n-matrix L and a tuning parameter ε such
+that the LMIs (26) are feasible. Then (11) is asymptotically
+stable for all variable sampling instants tk+1 −tk ≤ ∆ with the
+stabilizing gain given by K = LQ−1.
+� ¯Φ11
+¯P−Q+εQ⊺A⊺ +L⊺B⊺
+∗
+−ε(Q+Q⊺)+∆ ¯U
+�
+< 0,
+�
+�
+¯Φ11
+¯P−Q+ε(Q⊺A⊺ +L⊺B⊺)
+−∆BL
+∗
+−ε(Q+Q⊺)
+−∆εBL
+∗
+∗
+−∆ ¯U
+�
+� < 0.
+(26)
+where ¯Φ11 = Q⊺A⊺ +AQ+BL+L⊺B⊺.
+Proof. From above and following [25], we can perform
+full state-feedback controller design by substituting P3 = εP2
+where ε
+is a tuning parameter, Q = P−1
+2 ,
+¯P = Q⊺PQ,
+¯U = Q⊺UQ
+and
+L = KQ.
+Multiplying
+LMIs
+(26)
+by
+diag{Q⊺,...,Q⊺} and diag{Q,...,Q} from the left and right,
+we recover LMIs (25). ■
+V. EXPERIMENTS
+In the following Experiments section, we compare the
+Lyapunov analysis and the Lyapunov-Krasovskii analysis with
+the hold limit from empirical simulation. We aim to answer
+the following questions:
+1) How well does the theory match simulation? Moreover,
+to what extent do simplified theoretical analyses explain
+integrated human-compatible traffic flow stability?
+2) What relationships emerge from the problem parameters
+and how do they affect stability?
+3) Can we derive better piecewise-constant controllers us-
+ing the Lyapunov or Lyapunov-Krasovskii analysis?
+A. Experimental Setup and Results on the default parameters
+We adopt the implementation from Zheng et al. [10] in
+Python and extend it to the piecewise-constant control setting.
+In the default scenario, we use the same parameter for OVM,
+with n = 20,L = 400,α = 0.6,β = 0.9,sst = 5,sgo = 35,vmax =
+30. Vehicles are initialized by a uniform perturbation around
+the equilibrium, with the ith vehicle’s position and velocity
+(xi
+0,vi
+0) = (is∗ + δs,v∗ + δv) where δs ∼ Unif[−7.5,7.5],δv ∼
+Unif[−4.5,4.5], and v∗ =V(s∗) from Eq. (5) is the equilibrium
+velocity corresponding to the equilibrium spacing s∗ = L/n.
+By default, we apply the same H2 optimal full state-feedback
+
+controller for the continuous system to the sample-data system
+by holding it piecewise-constant. The controller
+u(t) = −Kx(t),
+(27)
+where K ∈ R1×2n, can be obtained by the following convex
+program with K = ZX−1:
+min
+X,Y,Z
+Trace(QX)+Trace(RY)
+subject to
+(AX −BZ)+(AX −BZ)⊺ +HH⊺ ≼ 0,
+�
+Y
+Z
+Z⊺
+X
+�
+≽ 0,X ≻ 0.
+(28)
+where
+Q
+1
+2 = diag(γs,γv,...,γs,γv), R
+1
+2 = γu, H = I
+(29)
+with the default γs = 0.03,γv = 0.15,γu = 1, corresponding to
+the performance state
+z(t) =
+�
+Q
+1
+2
+0
+�
+x(t)+
+� 0
+R
+1
+2
+�
+u(t)
+(30)
+We simulate the system by integrating the ordinary differ-
+ential equation (11) using the forward Euler method, with
+a discretization of Tstep = 0.01s. We say a system (either
+uncontrolled, or with continuous / piecewise constant con-
+trol) is stable in simulation if 50 simulated trajectories from
+different initial perturbations all converge to the equilibrium
+within TotalTime = 300s, and no vehicle collides within the
+trajectory (given by negative spacings). To mitigate collisions,
+we follow Zheng et al. to equip all vehicles with a standard
+automatic emergency braking system
+˙v(t) = amin, if v2
+i (t)−v2
+i−1(t)
+2(si(t)−sd) ≥ |amin|
+(31)
+where amin = −5m/s2 is the maximum deceleration rate of
+each vehicle, and sd = 0.5m is the safe distance.
+(a)
+(b)
+Fig. 2: The traffic system with all human vehicles and no
+controlled vehicle is unstable under the default parameters
+in Section V-A. The equilibrium spacing and velocity are
+20m and 15m/s. (a) The time-space diagram. Darker colors
+represent lower velocities. (b) The time-velocity diagram. The
+initial perturbation on the velocities get amplified, leading to
+the formation of stop-and-go waves in the system.
+We study the behavior of the system by putting a piecewise-
+constant hold on the controller for ∆ ≫ Tstep seconds. Without
+any controlled vehicle, the default OVM system is unstable
+(see Fig. 2), forming stop-and-go waves gradually. Zheng et
+al. [10] show that introducing one autonomous vehicle with
+the continuous H2 optimal controller is able to stabilize the
+continuous system. In Fig. 3, we show the behavior of the
+sample-data traffic system by holding the same H2 optimal
+control for ∆ = 1.59s (left) and ∆ = 2.29s (right). With a
+smaller hold length of 1.59s, the controller is able to stabilize
+the system. Such a controller translates to a human-compatible
+driving design where a new instruction is issued to the human
+driver every 1.59s; introducing a guided human vehicle is able
+to stabilize the traffic. However, with a slightly larger hold
+length of 2.29s, we observe unstable system behavior, where
+holding the control piecewise-constant introduces an exces-
+sive amount of noise that breaks the system’s stability. It is
+interesting to observe the sawtooth pattern in the time-velocity
+diagram in Fig. 3d, where errors are accumulated within
+each piecewise-constant holding period, but get corrected at
+the beginning of the next holding period when we update
+the control. While there is system slowdown, the velocity
+perturbation is constrained within a range between [7.5,20]
+m/s, instead of getting amplified and diverging as in Fig. 2.
+We note that, although the previous work [7] reports a
+longer hold limit in simulation, it is sensible that the hold
+limits are smaller in our settings, mainly because we consider
+the worst case scenario where we declare instability of a
+system if any of the 50 trajectories is unstable, whereas
+the previous work considers an average case that declares
+stability of a system if the average vehicle velocity of all
+trajectories is above a reasonable value. The previous work
+also considers a more advanced car following model, the
+intelligent driver model (IDM), for human drivers, and uses
+nonlinear controls represented by neural networks and trained
+by reinforcement learning. We choose to bound the worst-
+case scenario to provide certificates to the piecewise-constant
+controller even under adversarial settings, and focus on linear
+controls for the ease of theoretical analysis. In practice, we
+may encounter more stable human driving behaviors and
+provide more advanced nonlinear instructions to the guided
+vehicle to enable longer hold lengths; even when shorter hold
+lengths are required than human drivers can handle, we can
+still trade practicality for efficiency by issuing longer hold
+lengths at the cost of mitigating traffic less effectively.
+B. How well does the theory match simulation?
+In this section, we examine to what extent the theoretical
+hold limits from Eq. (15) and (25) are able to match the
+hold limits in simulation. In Fig. 4, we vary seven OVM
+system parameters (L,n,vmax,sst,sgo,vmax,α,β), as well as
+three control parameters (kmult,γs,γv). For each scenario, we
+vary one parameter while fixing the others to default values;
+we solve for an continuous H2 optimal controller using the
+corresponding system and control parameters. A summary of
+all parameters is listed in Table I.
+For each scenario, We perform a binary search within
+[0s,10s] with a granularity of Tstep = 0.01s in simulation to
+find the empirical hold limit for the system to be stable.
+The hold limit for each parameter set informs the designs
+of transportation system and controller to be more human-
+
+400
+15
+Position m]
+300
+s
+10
+3
+200
+Velocity
+5
+100
+0
+0
+20
+40
+60
+80
+t [s]25
+OVM
+s
+20
+-Average velocity
+m
+Velocity
+15
+10
+5
+0
+20
+40
+60
+80
+[s] (a)
+(b)
+(c)
+(d)
+Fig. 3: The traffic system consists of n − 1 human vehicles (gray) and 1 piecewise-constant controlled vehicle (red) with
+different hold lengths ∆s. The controlled vehicle applies the same H2 optimal control gain matrix for the continuous system to
+the sample-data system, under the default parameters in Section V-A. (a) and (b): The time-space and time-velocity diagrams
+when ∆ = 1.59s. The traffic is stabilized to the equilibrium velocity 15m/s after a short amount of time. (c) and (d): The
+time-space and time-velocity diagrams when ∆ = 2.29s. The system becomes unstable when the hold length is too long.
+Symbol
+Default
+Description
+System parameters
+L
+400m
+Circumference of the ring-road, where the equilibrium spacing s∗ = L/n
+n
+20
+Number of vehicles in the ring-road system, where the equilibrium spacing s∗ = L/n
+sst
+5m
+Small spacing threshold such that the optimal velocity = 0 below the threshold, see Eq. (5)
+sgo
+35m
+Large spacing threshold such that the optimal velocity = vmax above the threshold, see Eq. (5)
+vmax
+30m/s
+Maximum optimal velocity, see Eq. (5) and (6)
+α
+0.6
+Driver’s sensitivity to the difference between the current velocity and the desired spacing-dependent optimal velocity, see Eq. (4)
+β
+0.9
+Driver’s sensitivity to the difference between the velocities of the ego vehicle and the preceding vehicle, see Eq. (4)
+Control parameters
+kmult
+1
+Scale the H2 optimal controller Kcont by a constant: Knew = kmult ·Kcont
+γs
+0.03
+weight on the position derivation from equilibrium in the H2 optimal control objective, see Eq. (28) and Eq. (29)
+γv
+0.15
+weight on the velocity derivation from equilibrium in the H2 optimal control objective, see Eq. (28) and Eq. (29)
+γu
+1
+weight on the control magnitude in the H2 optimal control objective, see Eq. (28) and Eq. (30)
+TABLE I: System And Control Parameters In The Optimal Velocity Model.
+compatible. However, empirical trajectory simulations are in-
+feasible or computationally expensive due to the large and
+continuous space of initial conditions (starting positions and
+velocities of all vehicles), even for the ring road. This problem
+will undoubtedly be exacerbated in real-world settings. Hence,
+accurate theoretical guarantees on hold limit are essential to
+system design. To this end, we provide theoretical estimates
+of the hold limit using the following three methods on the
+linearized traffic system:
+1) The Lypaunov analysis: see Eq. (15). Due to redundancy
+in headway representation with ˜s1 + ˜s2... + ˜sn = 0, we
+first obtain the reduced representation by omitting ˜s1
+from the state vector and replacing it with −˜s2 −...− ˜sn
+to construct the reduced system matrices A†,B†,K†.
+Then, we set Q = I(n−1)×(n−1) which has σmin(Q) = 1,
+and solve for P from the Lyapunov equation (A† −
+B†K†)P+P(A† −B†K†)⊺ = −Q to obtain σmax(P) in the
+denominator of Eq. (15). The detailed matrix represen-
+tations of A†,B†,K† can be found in Appendix VI-A.
+2) The Lyapunov-Krasovskii analysis: see LMIs (25). We
+perform a binary search within [0s,10s] with a granu-
+larity of Tstep = 0.01s to find the theoretical hold limit
+such that the LMIs are feasible.
+3) The OVM stability: stability theory of the linearized,
+uncontrolled system. Previous work [9] uses string sta-
+bility to analyze the linearized, uncontrolled continuous
+OVM model, and derive the stability criteria α +2β ≥
+2 ˙V(s∗) = 2 ˙V(L/n). Equivalently, for s∗ = L/n ∈ [sst,sgo],
+the OVM system is stable if
+α +2β −vmax
+π
+sgo −sst
+sin
+�
+π L/n−sst
+sgo −sst
+�
+≥ 0
+(32)
+We plot the value of the left hand side in Fig. 4, which
+takes on negative values because we choose parameter
+values so that the uncontrolled system is unstable.
+Examining the extent to which piecewise-constant control
+is effective under different traffic conditions is a complex
+problem. Following the motivation of “All models are wrong,
+but some are useful,” it is attractive to consider whether
+reduced-order linearized models, such as the uncontrolled
+OVM system stability or the direct Lyapunov analysis, can
+lend themselves as proxies to analyzing the true traffic prob-
+lem. Through validating the work in simulation, we ultimately
+find that it is important to capture both the role of the
+controller (insufficiency of OVM stability) and the effect of
+the Lyapunov-Krasovskii integral (insufficiency of Lyapunov
+analysis). Perhaps surprisingly, both OVM stability and the
+Lyapunov do generally capture the trends quite well. More-
+over, the Lypaunov-Krasovskii analysis captures not only the
+trend but also the absolute hold limit, indicating that the effect
+of linearizing the system dynamics is not a strong limitation
+of the approach.
+Specifically, in Fig. 4, we plot the theoretical against the
+empirical hold limits. We display the scale of the y-axis
+
+400
+15
+Position m
+300
+S
+10
+m
+200
+Velocity
+5
+100
+0
+0
+0
+20
+40
+60
+80
+[s] 25
+OVM
+s
+.20
+-Controlled vehicle
+Average velocity
+Velocity
+15
+10
+5
+0
+20
+40
+60
+80
+[s] 400
+15
+Position m
+300
+s
+10
+200
+Velocity
+5
+100
+0
+20
+40
+60
+80
+[s] 25
+OVM
+s
+20
+Controlled vehidle
+Average/veldcity
+Velocity
+15
+10
+5
+0
+20
+40
+60
+80
+[s] Fig. 4: The hold limit that stabilizes the system from simulation (solid blue), Lyapunov-Krasovskii analysis in Eq. (25) (dashed
+orange), Lyapunov analysis in Eq. (15) (dash-dotted gray), and uncontrolled OVM stability criterion in Eq. (32) (dotted green).
+Default parameter values are shown as the black vertical lines in each plot. Left axis is for simulation and Lyapunov-Krasovskii
+analysis, while the y-axis scale for the Lyapunov analysis and the OVM stability are displayed in Table II.
+for Lyapunov analysis and OVM stability in Table II; the
+Lyapunov-Krasovskii analysis shares the same scale as the
+simulation, which is depicted as the numbers on the left of
+the y-axis. The Lyapunov-Krasovskii analysis is remarkably
+accurate in general, matching both the trend of the simulation
+and the absolute scale of all parameters, whereas the other two
+theoretical methods only provide relative trend estimates. The
+Lyapunov-Krasovskii analysis overestimates the simulation
+hold limit for large vmax and small β, however, where the
+uncontrolled system is more unstable that leads to collisions
+in the system. In such cases, the Lyapunov analysis gives
+a more accurate bound by more aggressively penalizing the
+worst-case uncontrolled system behavior given by σmax(A).
+The Lyapunov-Krasovskii analysis also overestimates the sim-
+ulation hold limit for small kmult, where the effect of the
+controller on the system is too small. In such a case, again,
+we obtain a more accurate trend estimate from the Lyapunov
+analysis, which more conservatively estimates the stability of
+the controlled continuous system ∝ σmin(Q)/σmax(P) when the
+controller has very small magnitude.
+The Lyapunov analysis matches the trend of the simulation
+hold limit decently well, despite the difference in absolute
+scale, and slight misalignment for the system parameters
+(n,vmax,α,β), and the control parameters (kmult,γs,γv). In the
+cases of (n,vmax,α,β), the worst-case singular-value bounds
+Symbol
+Lyapunov analysis
+OVM stability
+System parameters
+L
+(1.07×10-3,1.81×10-2)
+(-7.74×10-1,-5.99×10-2)
+n
+(7.70×10-4,3.19×10-3)
+(-7.62×10-1,2.94×10-2)
+sst
+(6.82×10-4,1.68×10-3)
+(-1.29,-1.67×10-1)
+sgo
+(6.88×10-4,.56×10-3)
+(-1.28,-2.79×10-1)
+vmax
+(6.79×10-5,1.72×10-3)
+(-5.17,1.76×10-2)
+α
+(9.65×10-4,1.89×10-3)
+(-1.30,-8.66×10-2)
+β
+(7.44×10-5,1.56×10-3)
+(-2.45,-3.16×10-2)
+Control parameters
+kmult
+(0,3.02×10-3)
+-
+γs
+(1.08×10-4,1.47×10-3)
+-
+γv
+(5.60×10-4,1.17×10-3)
+-
+TABLE II: Scales of hold limits (the minimum and maximum
+of y-axis in Fig. 4) for Lyapunov analysis and OVM stability.
+become too aggressive; a more fine-grained theoretical analy-
+sis given by Lyapunov-Krasovskii renders a better estimate by
+considering the interaction of A (the uncontrolled system), BK
+(the control) and A−BK (the controlled system). In the cases
+of (kmult,γs,γv), the Lyapunov analysis captures the correct
+trend in general, but fails to capture the correct absolute slopes
+for three parameters, and the correct peak for (kmult,γv). This is
+understandable because the analysis only holds up to a scaling
+constant that decides the slope, and the location of the peak
+can change by scaling σmax(A) and σmax(A1) differently. We
+keep equal scaling in the analysis for clarity of interpretation,
+
+Simulation
+Lyapunov-Krasovskiianalysis
+Lyapunov analysis
+OVMstability
+L
+n
+Sst
+Sgo
+1.9 -
+2.0 -
+1.8 -
+1.8 -
+1.7 -
+1.8 -
+1.8 -
+1.6 -
+1.6
+1.7 -
+1.5
+300
+400
+500
+15
+20
+25
+maximum hold length (
+5
+10
+15
+25
+30
+35
+Vmax
+α
+β
+1.9
+1
+1.8
+1
+1.7
+40
+60
+0.5
+1.0
+0.5
+1.0
+Ys
+Yv
+2.5 -
+10 -
+1.5 -
+2.0 -
+5 -
+:
+1.5 -
+1.0 -
+1.0 -
+0-
+-
+0
+2
+0
+1
+2
+0
+1
+2
+parameterand leave finding more accurate scalings to future work.
+To our (slight) surprise, the uncontrolled OVM stability
+matches the trend of the simulation hold limits particularly
+well for a few parameters (L,sst,sgo). However, mismatches
+occur when the controller BK has a significant effect on the
+system. Slight trend mismatch occurs for vmax and β, and
+opposite trends are observed for the α parameter. As seen
+in Fig. 5, in these cases, the controller σmax(−BK) displays a
+non-linear trend different from σmax(A), making the resulting
+controlled hold limit nonlinear and even displaying an opposite
+trend for α; such a behavior is in contrast with the three
+aligned cases, where the trends of σmax(A) and σmax(−BK)
+match (see the L plot in Fig. 5). In the misaligned cases, the
+missing information of the controller is necessary for a more
+accurate trend estimate.
+Fig. 5: A visualization of different components in the
+Lyapunov analysis (Eq. (15)) for four system parameters
+L,n,vmax,α. We plot the denominator components σmax(P)
+that represents the continuous controlled system (dashed
+green), σmax(A) (dotted red) that represents the continuous
+uncontrolled system, σmax(A1) = σmax(−BK) that represents
+the control (dash-dotted orange), and the final theory bound
+on the hold limit ∆ that stabilizes the system (solid blue). Note
+that numerator component σmin(Q) = 1 by construction. The
+absolute scales of the different components are omitted.
+Overall, the closeness in both trend and absolute-scale
+makes Lyapunov-Krasovskii theory a reliable theoretical sur-
+rogate for quantitative estimation of the simulation hold limit;
+meanwhile, the clean expression from the Lyapunov analysis
+makes it a reliable tool for qualitative interpretation of the
+system’s behavior, especially when the controller’s behavior,
+given by σmax(−BK), and the uncontrolled system’s stability,
+given by σmax(A), do not completely align. When the two
+align, the OVM stability criteria for the uncontrolled system
+can be used to analyze the sample-data system.
+C. How do traffic conditions affect the hold limit?
+In this section, we interpret relationships between traffic
+system parameters, which represent different traffic conditions,
+and their hold limits, as detailed in Fig. 4. Specifically, we fix
+the control parameters (kmult,γs,γv), and vary the OVM system
+parameters (L,n,sst,sgo,vmax,α,β). Overall, we observe three
+main types of traffic situations that promote longer hold
+limits by means of low driver sensitivity: (1) traffic conditions
+(density, speed limit, and spacing thresholds) that promote
+a smoother spacing response, i.e., the flatter region of the
+optimal velocity function (through various combinations of
+L,n,sgo,sstop,vmax), (2) low sensitivity of drivers to relative
+position (low α), and (3) high sensitivity of drivers to relative
+speed, which tends towards equilibrium (high β).
+Why low driver sensitivity? Because a controlled vehicle
+can exert more fine-grained control with a shorter hold length,
+we can think of a longer hold length as exerting a larger
+”blunter” change to the environment. The primary mechanism
+by which the hold length interacts with the traffic condition is
+the extent to which these larger changes to the environment
+affect the drivers’ spacing (headway). If the drivers’ are more
+sensitive to changes in spacing, then ”blunter” control is
+more likely to cause large deviations in the environment, thus
+preferring shorter hold lengths to maintain stability.
+Why not OVM system stability? Indeed, greater uncon-
+trolled system stability often, but not always, corresponds
+to longer hold limits. Thus, the discrepancy between OVM
+system stability and the empirical analysis is illustrative for
+understanding the importance of low driver sensitivity. OVM
+system stability (Eq. (32)) has a strong negative correlation
+with drivers’ sensitivity, with the exception of the α parameter:
+larger α corresponds to a more stable uncontrolled OVM
+system as it corresponds to strong compliance of drivers to the
+optimal velocity; however, the OVM optimal velocity might
+conflict with the controlled vehicle, especially when errors
+are incurred with the piecewise-constant holds. For example,
+with a longer hold length and larger α, the controlled vehicle
+may open up wider gaps, resulting in a stronger response
+from the following driver, in turn causing system instabilities.
+Hence, larger α increases the uncontrolled system’s stability
+but reduces the hold limit of the controlled system.
+Smoother
+spacing
+response:
+We
+observe
+that
+(L,n,sst,sgo,vmax) determines various aspects of the optimal
+velocity function, as shown in Eq. (5) and illustrated in
+Fig. 6. The parameters L and n are related to the density of
+the traffic. Their ratio s∗ = L/n determines the equilibrium
+spacing, which further determines the desired optimal velocity
+v∗ = V(s∗), with the value clipped within [0,vmax]. When the
+spacing is either too small (close to sst) or too large (close
+to sgo), the uncontrolled system is more stable, since the
+desired optimal velocity is easier to follow by the drivers,
+who can drive either very slowly (v∗ is near 0) or follow the
+maximum speed (v∗ is near vmax). However, when the spacing
+is close to the sgo−sst
+2
+, as depicted by the red star in Fig. 6, the
+original system becomes more unstable, since slight changes
+in spacing would lead to large changes in the desired optimal
+velocity. In fact, the default sst = 5,sgo = 35 directly place
+the default spacing L/n = 20m at the most unstable inflection
+point (the red star).
+Similarly, the two boundary values sst and sgo determine
+the length of the region; varying them would vary both the
+location of the equilibrium spacing on the curve and the
+
+theory
+Omax(P)
+Omax(-BK)
+Omax(A)
+L
+n
+300
+400
+500
+15
+20
+25
+Vmax
+α
+30
+40
+50
+60
+70
+0.25
+0.50
+0.75
+1.00
+1.25Fig. 6: The Optimal Velocity function V(s) in Eq. 5 and 6 with
+default parameters in Section V-A. The red star represents the
+equilibrium spacing and velocity with the default parameters,
+where the function attains maximum slope. Changing system
+parameters moves the red star to different positions on the
+curve, affecting the stability of the uncontrolled system and
+the hold limit to stabilize the system.
+slope of the curve. When we increase sst or decrease sgo,
+we make the curve steeper, and hence more unstable, but
+we also move the location of the equilibrium spacing away
+from the inflection point. Such a trade-off reflects the slight
+asymmetry in the sst and sgo curves. In general, we observe
+that the stability of the uncontrolled system, which is closely
+tied to the the rate of change of the optimal velocity function
+at the equilibrium spacing, translates well to the hold limit of
+the piecewise-constant system for (L,n,sst,sgo), following an
+upward parabola shape resulting from the cosine wave of the
+desired optimal velocity curve.
+We also observe that the variation of the hold limit in
+(L,n,sst,sgo) is mild, ranging from 1.5s to 2s in simulation,
+as these four variables are all encapsulated within the cosine
+function in the desired optimal velocity equation. On the
+other hand, the maximum desired velocity parameter vmax,
+as the multiplier to the cosine wave, affects the hold limit
+substantially more, from 2s down to 0s when vmax goes from
+25m/s to 60m/s. Increasing the maximum desired velocity
+effectively stretches the desired velocity curve taller, resulting
+in sharper changes of the desired optimal velocity when the
+spacing changes. Hence, for larger vmax, the uncontrolled
+system becomes more unstable, resulting in a shorter hold
+limit. While the stability of the uncontrolled system only
+explains a linear decrease of hold limit, we observe a super-
+linear decrease in simulation due to the following two addi-
+tional reasons: (1) the larger magnitude of the controller, as
+illustrated in Fig. 5, incurs more errors to the system from the
+piecewise-constant hold, and (2) the unstable system leads to
+collisions of the vehicles, making the system even harder to
+stabilize with the noisy controller.
+Low sensitivity to relative position, high sensitivity to
+relative speed: The remaining two parameters, α and β,
+reflect the sensitivity of human drivers between the current
+velocity and the desired optimal velocity (α), and the velocity
+of the vehicle in front (β). Interestingly, we see different
+trends of the simulation hold limit for the two parameters,
+despite larger α and β both make the original uncontrolled
+system more stable (Eq. (32)). For β, the expected velocity
+dissipation term makes the human vehicle more observant of
+the surroundings, hence increasing stability of the system. The
+sharp super-linear decrease in the hold limit for small values
+of β is a result of the analogous reasons as for vmax, which
+a combination of uncontrolled system’s stability, additional
+errors induced due to large magnitude of the controller, and
+vehicle collisions when the system is excessively unstable.
+On the other hand, for the α parameter, the simulation hold
+limit exhibits an opposite trend of the uncontrolled system
+stability, despite the mild variation in hold limit (1.6s to 2s).
+Examining the controller magnitude in Fig. 5, we observe that
+the magnitude is larger for the more stable uncontrolled system
+with larger α; as a result, the piecewise-constant control adds
+more noise to the system when α is large, despite the original,
+uncontrolled system is in fact more stable.
+D. Controller design for human-compatible driving
+∆in(s)
+1
+2
+3
+4
+5
+6
+7
+8
+∆sim(s)
+2.71
+3.33
+4.55
+4.54
+4.3
+4.19
+4.09
+3.87
+TABLE III: The simulation hold limit ∆sim with the Lyapunov-
+Krasovskii control gain matrix in LMIs (26) when we set ∆in ∈
+{1,2,3,...,9} in the LMIs.
+Thus far, we have focused on analyzing a given controller;
+in particular, the continuous H2 optimal controller, which
+achieves a simulation hold limit of 1.66s by default. In this
+section, we consider several approaches to intentionally design
+controllers for human-compatible control to achieve system-
+level traffic flow stability.
+Lyapunov-Krasovskii controller search: Recall that the
+Lyapunov-Krasovskii analysis in Section IV-B provides a
+method to obtain piecewise-constant controllers. We thus
+examine the quality of the resulting controllers via simulation.
+We fix the OVM system parameters (L,n,sst,sgo,vmax,α,β) to
+the same default value in Section V-A, and solve LMIs (26)
+for a control gain matrix K = LQ−1, with a grid search of
+input hold length parameter ∆in ∈ {1,2,3,4,5,6,7,8}. We fix
+the tuning parameter ε = 1 in LMI (26) where we substitute
+P3 = εP2 from (25), as we empirically find such a ε gives the
+best controller with the longest simulation hold limit. Given
+the resulting control gain matrix KLK, we perform simulation
+via a binary search with a granularity of Tstep = 0.01s to
+examine the empirical hold limit ∆sim.
+Table III depicts the actual simulation hold limits ∆sims
+for different input parameters ∆ins. We observe that, initially
+as ∆in increases, the Lyapunov-Krasovskii analysis is able
+to find better controllers with longer hold limits; however,
+as ∆in further increases over 4s, the simulation hold limit
+∆sim decreases, causing discrepancies between the theory and
+the actual simulation. The fundamental reason comes from
+the collision constraint in simulation: while we declare a
+trajectory with collisions (negative spacings) as unstable in
+simulation, the Lyapunov-Krasovskii analysis ignores such a
+constraint, and allows the spacing and velocity variables to
+take negative values. In fact, if we omit the constraint in
+simulation, we would achieve substantially larger hold limits
+
+Desired Velocity V(s) [m/s]
+max
+５
+0
+0
+5
+10
+15
+20
+25
+30
+35
+40
+Spacing s [m]
+S
+go∆sim > 10s for the Lyapunov-Krasovskii controllers with large
+∆ins. However, ignoring such a constraint makes the traffic
+scenario unrealistic; an important direction of future work is
+to incorporate control barrier functions [26] to the Lyapunov-
+Krasovskii analysis to explicitly consider the collisions.
+H2 re-scaling: Next, we propose and examine a heuristic
+controller design policy where we fix the OVM system param-
+eters (L,n,sst,sgo,vmax,α,β) and vary the control parameters
+(kmult,γs,γv) in order to find scaled controllers more suitable
+for the sample-data system. In Fig. 4, we observe controllers of
+smaller magnitudes than the default continuous H2 controllers,
+given by smaller kmult < 1, γs < 0.03, γv < 0.15, result in longer
+hold limit. Noticably, the longest hold limit is achieved at
+4.78s when kmult = 0.2, offering a 2.78x improvement from
+the default at 1.66s when kmult = 1. Such a behavior can be
+explained by the σmax(A1) term in the denominator of the
+Lyapunov analysis, where controllers of larger magnitudes
+incur larger errors from the piecewise-constant hold. However,
+when the controller is too small, the controller is not powerful
+enough to stabilize the system, resulting in a drop in hold limit
+from 4.78s when kmult = 0.2 to 2.84s when kmult = 0.005, and
+finally to 0s when kmult = 0.001; such a phenomenon can be
+explained by the ratio σmin(Q)/σmax(P) in the the Lyapunov
+analysis, which represents the stability of the controlled system
+A − kmultBK. Hence, there is a trade-off between the power
+of the controller and the noise incurred from the piecewise-
+constant hold. In general, when we design controllers for
+the sample-data system, a reasonable penalization on the
+magnitude of the controller could improve the hold limit. In
+the meantime, we also observe similar discrepancies between
+simulation and Lyapunov-Krasovskii theoretical hold limit in
+the kmult plot of Fig. 4, due to the omission of collision
+constraints in the LMIs (25).
+Meanwhile, as the best simulation hold limit of the best
+scaled continuous H2 optimal controllers, 4.78s, is around
+the same level of the Lyapunov-Krasovskii controllers with
+a 4.55s hold limit, we make an interesting observation that
+the piecewise-constant controller obtained by scaling down a
+reasonable continuous control may obtain decent performance
+for human-compatible driving. As abundant reinforcement
+learning controllers have been developed for the continuous
+traffic systems [4, 15], a few promising strategies for human-
+compatible driving are to take the down-scaled version of
+the same controllers, or to finetune these controllers with a
+magnitude penalization to avoid expensive re-training of the
+piecewise-constant controllers.
+VI. CONCLUSIONS AND FUTURE WORK
+This work presents an integrated Lyapunov analysis frame-
+work of human-compatible, piecewise-constant control to sta-
+bilize the traffic flow. We derive both a Lyapunov analysis
+for qualitative interpretation of the relationships between traf-
+fic system parameters and the hold limit, and a Lyapunov-
+Krasovskii analysis for quantitative estimation of the hold limit
+to stabilize traffic. The Lyapunov-Krasovskii analysis can also
+be used for piecewise-constant controller design. Our theory
+provides certificates to human-compatible driving, a class of
+policies that aim to guide human drivers to stabilize the traffic,
+bypassing the difficulty of autonomous vehicle deployment
+and having the potential to achieve desirable traffic outcomes
+with low cost and quick timeline. Our work highlights the
+power of the Lyapunov analysis framework as an impor-
+tant integrated theoretical tool for obtaining efficient, safe,
+and sustainable transportation systems with human-compatible
+control.
+We propose a few important directions for future research.
+First, we would like to tighten the derivation of the Lyapunov
+analysis (Eq. (15)) to obtain absolute scales of different
+components in the bound. The correct scaling will enable us
+to pinpoint the exact slope and location of the optimum of
+the curves in Fig. 4, while the current bound is only able to
+describe the relative trend. Next, we would like to incorporate
+control barrier functions to the Lyapunov-Krasovskii analysis
+(LMIs (25) and (25)) to tighten the bound under unsafe
+events such as collision. Finally, we would like to consider
+expanding our theory to a broader class of human-compatible
+driving policies that consist of other easy-to-follow driving
+instructions, as well as to more complex traffic scenarios.
+APPENDIX
+A. Reduced state-space representation
+Due to redundancy in headway representation with
+˜s1(t)+ ˜s2(t)...+ ˜sn(t) =
+n
+∑
+i=1
+si(t)−ns∗ = L−L = 0,
+(33)
+we obtain the reduced state-space representation for the system
+by first omitting ˜s1(t) from the state vector and obtain
+x†(t) = [˜v1(t), ˜s2(t), ˜v2(t)..., ˜sn(t), ˜vn(t)].
+(34)
+Then, we omit the first rows of A and B which correspond to
+the system equation for ˜s1(t). We also omit the first column
+of A, which is all zero as none of the other state equations for
+˜si(t), ∀i ̸= 1 and ˜vj(t) ∀j depends on ˜s1(t). So, we have
+A† =
+�
+��������
+C†
+1
+0
+...
+...
+0
+C†
+2
+D†
+2
+D1
+0
+...
+...
+0
+0
+D2
+D1
+0
+...
+0
+...
+...
+...
+...
+...
+...
+0
+...
+0
+D2
+D1
+0
+0
+...
+...
+0
+D2
+D1
+�
+��������
+,B† =
+�
+������
+B†
+1
+B2
+B2
+...
+B2
+�
+������
+(35)
+with D1,D2,C1,C2,B2 the same as in Eq. (13), and
+D†
+2 =
+�
+1
+α3
+�
+,C†
+1 =
+�
+0
+�
+,C†
+2 =
+�
+0
+0
+�
+,B†
+1 =
+�
+1
+�
+,
+(36)
+Similarly, we have
+u(t) = −Kx(t)
+= −K1 ˜s1(t)−...−Kn ˜sn(t)
+= −K1(−˜s2(t)−...− ˜sn(t))−...−Kn ˜sn(t)
+= −(K2 −K1)˜s2(t)−...−(Kn −K1)˜sn(t)
+= −K†x†(t)
+(37)
+Hence, the new control gain matrix
+K† = [K2 −K1,...,Kn −K1] ∈ R1×(2n−1).
+(38)
+
+ACKNOWLEDGMENT
+This work was supported by the National Science Founda-
+tion (NSF) under grant number 2149548, the MIT Amazon
+Science Hub, the MIT Energy Initiative (MITEI) Mobility
+Systems Center, MIT’s Research Support Committee, as well
+as a gift from Mathworks.
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+
diff --git a/NtE2T4oBgHgl3EQfqwjT/content/tmp_files/load_file.txt b/NtE2T4oBgHgl3EQfqwjT/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..56c9c81d7e007f8e149eb078f6c404b72a8ac772
--- /dev/null
+++ b/NtE2T4oBgHgl3EQfqwjT/content/tmp_files/load_file.txt
@@ -0,0 +1,721 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf,len=720
+page_content='Integrated Analysis of Human-compatible Control for Traffic Flow Stability Sirui Li1, Roy Dong2, Cathy Wu3 Abstract—Autonomous vehicles (AVs) enable more efficient and sustainable transportation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Ample studies have shown that controlling a small fraction of AVs can smooth traffic flow and mitigate traffic congestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' However, deploying AVs to real- world systems is challenging due to safety and cost concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' An alternative approach deployable in the imminent future is human- compatible control, where human drivers are guided by real-time instructions to stabilize the traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' To respect drivers’ cognitive load, a class of piecewise-constant policies is considered, where periodic instructions are given every ∆ seconds to human drivers, who hold the instructed action constant until the next instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' While previous works separately consider stability analysis for continuous AV control or the extent to which human drivers can follow guidance, this article is the first to consider an integrated theoretical analysis, directly relating the guidance provided to the human drivers to the traffic flow stability outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Casting the problem into the Lyapunov stability framework, sufficient conditions are derived for piecewise-constant controls with hold length ∆ to stabilize the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Numerical simulations reveal that the theoretical analysis closely matches simulated results, and, importantly, classical stability concepts are insufficient for explaining hold lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Additionally, the theoretical and empiri- cal analyses can be leveraged to derive improved controllers with greater maximum hold length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' INTRODUCTION Transportation is one of the largest contributors to the greenhouse gas (GHG) emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In 2020, it accounted for 27% of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' GHG emissions, of which over 80% was due to land transportation including light-duty vehicles as well as medium- and heavy-duty trucks [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' To reduce emissions in the future, studies have shown that mitigating traffic congestions such as stop-and-go waves allow up to almost 20% reduction of CO2 emissions [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The introduction of autonomous vehicles (AVs) provides a solution to traffic congestion mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Previous work has shown that, with only 4-7% adoption of reinforcement- learning controlled AVs, the system level average velocity can increase up to 57% [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Real world experiments have also been conducted, where a single AV is able to dampen stop-and-go waves in a circular track with 20 vehicles [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' However, it is challenging to deploy AVs at scale in the real world due to the lack of safety and robustness guarantees, the difficulty of interpretation as the actions are rapidly changing, and the enormous cost and long timeline for actual deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Sirui Li is with the Institute for Data, Systems, and Society, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' siruil@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='edu Roy Dong is with the Department of Electrical and Computer Engineer- ing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' the Coordinated Science Laboratory, University of Illinois at Urbana- Champaign, Urbana, IL, 61801, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' roydong@illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='edu Cathy Wu is with the Laboratory for Information & Decision Systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' the Institute for Data, Systems, and Society;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' and the Department of Civil and En- vironmental Engineering, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' cathywu@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='edu Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 1: An illustration of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' We study the ring-road traffic system with 1 controlled vehicle (the AV / Guided Vehi- cle) and n−1 human vehicles following the Optimal Velocity Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Previous works control the AV using reinforcement learning to smooth traffic, with the controller continuously updated (discretized up to the simulation granularity of the underlying ordinary differential equation ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='01s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' we study the human-compatible driving scenario with a piecewise- constant controller for the guided vehicle held constant for a substantially longer period (∆ = 4s and longer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Our work use Lyapunov analysis to provide theoretical guarantees on the maximum hold length to stabilize the traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Human-compatible driving provides a middle ground so- lution to deploy traffic stabilization controls at scale in the imminent future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' It relies on human drivers to mitigate traffic jams by following periodic instructions, which can be provided by minimally intrusive real-time apps (similar to Google Maps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Intuitively speaking, human-compatible driving studies the question of to what extent we need AVs to achieve certain desirable outcomes for traffic, or can we leverage instructed human drivers instead?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' If the latter is the case, it has huge implication for the cost and timeline to realize the desired traffic outcomes by providing effective alternatives that can be deployed in a short time frame, while AV deployment is currently out of reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Due to the reaction times ∆ of human drivers to comprehend the instruction and the traffic situation (≈ 5-8s on average, based on human-factor research [6]), a class of piecewise- constant driving policies has been proposed by Sridhar and Wu [7, 8] to realize human-compatible driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' It consists of holding each instructed control constant for ∆ seconds before updating to a new instructed control, subject to safety constraints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' it assumes that the human driver will override the instruction if needed to drive safely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Empirical results have shown that the traffic can be stabilized even with a long hold length (up to 24 seconds on average) with the intelligent driver model (IDM), where the term hold length is used to refer to the duration of each constant holding period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Theoretical analysis has also been provided, but it considers a simplified system arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='04043v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='SY] 10 Jan 2023 Ours: Human-Compatible Guided Vehicle Control (Acceleration) A = 4s Ours: Certified by Lyapunov Theory C Previous: Continuous AV Control (Acceleration) discretized up to ODE simulation granularity △ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='01s C AV / Guided Vehicle CD Human vehiclewith a single AV and does not consider interactions with other vehicles in the traffic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In this paper, we provide a principled control-theoretic analysis of the traffic system with the piecewise-constant policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' While previous works separately consider 1) stability analysis for continuous AV control [9, 10], and 2) the extent to which human drivers can follow a variety of different kinds of instructions [11], this work is the first to consider an integrated analysis that directly relates the guidance provided to the human drivers to the system-level stability outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The theory takes into account the interaction between a single piecewise-constant controlled vehicle and the rest of human vehicles governed by the optimal velocity model (OVM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Both Lyapunov functions and Lyapunov-Krasovskii functionals are used to provide sufficient conditions for the stability of the traffic system under the piecewise-constant control with a hold length ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Through numerical analysis, we further demonstrate that the theoretical conditions closely match empirical simula- tions under a variety of OVM parameters, and hence the theory serves as a reliable certificate of the hold limit for the system, which we define as the maximum hold length that guarantees system’s stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' We note that, different from the average case simulation results reported from Sridhar and Wu [7, 8], we consider the worst-case scenario to ensure stability on any arbitrary initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' As a result, shorter hold limits are observed in our setting with stronger theoretical certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In summary, our contributions are We propose an integrated theoretical framework with Lyapunov functions and Lyapunov-Krasovskii functionals to provide sufficient conditions for a single piecewise- constant controlled vehicle to stabilize the traffic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' We perform extensive numerical analysis to show that the theory closely match empirical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Noticably, the Lyapunov-Krasovskii functionals closely match the empirical hold limits in both trend and absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' As an extension of [12], we derive detailed insights into the relationship between OVM parameters and stability criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In particular, we observe the primary mechanism by which the hold length interacts with the traffic con- dition is the extent to which the changes to the traffic environment affect the drivers’ spacing (headway).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' As an extension of [12], we further use our theory to design piecewise-constant controllers with longer hold limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' As an extension of [12], we additionally discuss appli- cations of the analyses to a broader class of human- compatible driving such as piecewise-constant velocity guidance in addition to acceleration control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Human-compatible driving Sridhar and Wu [7, 8] proposes the class of piecewise- constant driving policies that allow human drivers to mitigate traffic congestion by following periodic instructions provided to them every ∆ seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Such a class of policies is conceptu- ally similar to the zero-order hold sample-data systems [13], where a continuous system is controlled by a digital holding device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The device takes a digital input every ∆ seconds to produce a digital control being held constant for the entire holding period of length ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In contrast to such systems, which typically are designed for hold lengths of milliseconds or less, we consider longer hold lengths of tens of seconds to respect human reaction times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The piecewise-constant driving policies belong to a broader class of human-compatible driving policies, where simple and easy-to-follow interventions are used to achieve desirable traffic outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Real-world field studies demonstrate the ef- fectiveness of human-compatible driving in fuel-saving, safety, congestion mitigation, and emission reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In particular, the Greek eco-driving pilot program [14] provides eco-driving training such as anticipating traffic flow and maintaining a steady speed at low RPM to bus drivers, and demonstrates an average decrease of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='2% in fuel consumption among the drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Similarly, fieldwork in south California [3] shows speed management techniques that reduce excessively high speed to safe speeds can help mitigate congestion, resulting in a 12% CO2 reduction when combined with traffic flow smoothing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Traffic stabilization with autonomous vehicles Recently, there has been a surge of interest to control autonomous vehicles to stabilize mixed traffic systems of autonomous and human vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' A few works use rein- forcement learning to design controls in various scenarios such as stabilizing stop-and-go waves in the low AV-adoption regime [4], coordinating AVs to exhibit traffic light behav- iors [15], and designing eco-driving Lagrangian controls to reduce fuel consumption [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Theoretical studies for the ring road traffic setting have been conducted on the linearized continuous system, with two primary approaches of analysis: 1) string stability [17, 18, 9, 19], and 2) state-space Lya- punov stability [10, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Our work follows the second line of approaches with Lyapunov analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Continuous optimal controllers have also been derived, with numerical simulations to demonstrate their abilities to stabilize traffic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' However, these controllers are continually updated due to continual changes in traffic conditions, and hence it is challenging to deploy them in the real-world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Lyapunov stability analysis Lyapunov functions have been used to analyze general control systems with discontinuous feedback [21], of which our human-compatible piecewise-constant policy is a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' To incorporate delays in human driver reaction times, Lyapunov-Krasovskii functionals have been used [22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' how- ever, the work considers a different scenario where the human drivers issue continuous controls according to delayed input states, and hence, the controlled system is still continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Our work, instead, considers piecewise-constant controls that are updated every ∆ seconds, and hence belongs to the sample- data system paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Previous works [23, 24, 25] adopt Lyapunov-Krasovskii functionals to general sample-data sys- tems, and show that tailored Lyapunov-Krasovskii functionals perform better than general time-delay Lyapunov-Krasovskii functionals on toy sample-data control examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Our work is the first to apply a sample-data Lyapunov-Krasovskii func- tional to analyze system-level stability of human-compatible control, and show by simulation that the theoretical guarantees indeed closely match simulated results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' PRELIMINARIES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Ring-road optimal velocity model (OVM) Following Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' [10], we consider a single-lane ring road with circumference L and n vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Let the position of i-th vehicle be pi(t), the velocity be vi(t) = ˙pi(t), the spacing be si(t) = pi−1(t)− pi(t), and the acceleration be ai(t) = ˙vi(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The standard car following model (CFM) for human vehi- cles takes the nonlinear form ˙vi(t) = F(si(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' ˙si(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='vi(t)) (1) where the uniform flow equilibrium achieved at spacing s∗ and velocity v∗ such that F(s∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='v∗) = 0 (2) Let the error state be defined as ˜si(t) = si(t)−s∗ and ˜vi(t) = vi(t)−v∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' the linearization of the CFM around the equilibrium is � ˙˜si(t) = ˜vi−1(t)− ˜vi(t) ˙˜vi(t) = α1 ˜si(t)−α2 ˜vi(t)+α3 ˜vi−1(t) (3) where α1 = ∂F ∂s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='α2 = ∂F ∂ ˙s − ∂F ∂v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='α3 = ∂F ∂ ˙s evaluated at (s∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='v∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The optimal velocity model (OVM) follows the form F(si(t), ˙si(t),vi(t)) = α(V(si(t))−vi(t))+β ˙si(t) (4) where α > 0,β > 0, and V(si(t)) usually takes the form V(s) = � � � � � 0, s ≤ sst fv(s), sst < s < sgo vmax, s ≥ sgo (5) and a typical fv(s) takes the form fv(s) = vmax 2 � 1−cos � π s−sst sgo −sst �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (6) As a result, v∗ = V(s∗),α1 = α ˙V(s∗),α2 = α +β,α3 = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Piecewise-constant control We consider a system with one piecewise-constant con- trolled vehicle i = 1 with hold length ∆, and n − 1 human OVM vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' At a given time t ∈ [tk,tk+1] where [tk,tk+1] is the corresponding holding period, the CFM for the controlled vehicle is modeled by ˙v1(t) = f(u(z(tk),z(t))) (7) where f is a function described in details below, z(ˆt) = [s1(ˆt),v1(ˆt),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=',sn(ˆt),vn(ˆt)] is the state vector at time ˆt, and u(z(tk),z(t)) represents the control function, which we allow a part to be held constant from the input z(tk), and the rest to be continuous from the input z(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' As an example, a class of piecewise-constant velocity guid- ance control in OVM proposes a constant desired velocity u(z(tk)) to the controlled vehicle during the holding period;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' the vehicle uses a OVM-like dynamics to reach the desired controlled velocity, resulting in the dynamics ˙v1(t) = α(u(z(tk))−v1(t))+β ˙s1(t) (8) Meanwhile, a class of piecewise-constant acceleration control directly forces the controlled vehicle to take a constant accel- eration during the holding period, resulting in the dynamics ˙v1(t) = u(z(tk)) (9) We follow previous works [7, 8] to focus on the piecewise- constant acceleration control in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Lumping the error state into a vector form with x(t) = [˜s1(t), ˜v1(t),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=', ˜sn(t), ˜vn(t)]⊺, the error dynamics for the con- trolled vehicle is given by � ˙˜s1(t) = ˜vn(t)− ˜v1(t) ˙˜v1(t) = ˜u(x(tk)) (10) where t ∈ [tk,tk+1] and tk+1 − tk ≤ ∆ is the corresponding holding period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The error dynamics of the linearized piecewise-constant control system is thus given by ˙x(t) = Ax(t)+A1x(tk),k = 0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (11) with A = � �������� C1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 0 C2 D2 D1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 0 0 D2 D1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 0 D2 D1 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 0 D2 D1 � �������� ,B = � ������ B1 B2 B2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' B2 � ������ (12) with D1 = � 0 −1 α1 −α2 � ,D2 = � 0 1 0 α3 � , C1 = � 0 −1 0 0 � ,C2 = � 0 1 0 0 � ,B1 = � 0 1 � ,B2 = � 0 0 � (13) where A1 = −BK represents the full state feedback piecewise- constant control coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' We note that the formulation is exactly the same as in Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' [10] except for the piecewise-constant control component;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' we also note that the formulation perfectly aligns with the sample-data sys- tem framework [13] with zero-order hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' We further note that, different classes of piecewise-constant controls result in slightly different A and A1 matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' For example, the velocity guidance control, as described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (8), has C1 = � 0 −1 0 −α2 � ,C2 = � 0 1 0 α3 � ,B1 = � 0 α � , (14) with the rest of the D1,D2,B2 matrices be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The representations C1 and C2 follow the human vehicle represen- tations D1 and D2, except the α1 term representing the desired velocity is moved from the uncontrolled system matrix D1 to the control matrix B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The Lyapunov analyses in the following Section IV naturally apply to the broader classes of piecewise- constant controls, as they are agnostic to the specific form of A and A1 matrices for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' LYAPUNOV ANALYSIS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' A Lyapunov bound We first derive a Lyapunov bound on the hold limit ∆, which is defined as the maximum hold length such that the traffic system remains stable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' while a Lyapunov bound on the general nonlinear system with discontinous control has been derived in the previous literature [21], we adapt the derivation to the linearized system with piecewise-constant control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In later sections, we apply the bound to the ring-road optimal velocity model to extract meaningful insights into the traffic system and controller design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Let there exist n × n matrices P > 0,Q > 0 such that V(x) = x⊺Px > 0 with ˙V(x) = −x⊺Qx < 0 and −Q = (A + A1)P + P(A + A1)⊺ is a valid Lyapunov function for the linear continuous system with continuous full-state feedback control, ˙x(t) = (A+A1)x(t) where A1 = −BK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Then the sample-data system with piecewise constant control (11) is asymptotically stable for hold length ∆ ≤ c′ σmin(Q) σmax(P)(σmax(A)+σmax(A1))2 (15) up to a scaling constant c′ > 0, where σmin(·) and σmax(·) are the minimum and maximum singular value of the corresponding matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Consider a time period [tk,tk+1] with tk+1 −tk ≤ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' We use the Lyapunov function for the continuous system V(x) = x⊺Px, and show that it is a valid Lyapunov function for the sample-data system by showing V(x(t)) −V(x(tk)) is sufficiently negative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' V(x(t)) decreases as t increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' We have for all t ∈ [tk,tk+1]: V(x(t))−V(x(tk)) = ⟨∇V(x(t∗)), ˙x(t∗)⟩(t −tk) for some t∗ ∈ (tk,t) = ⟨∇V(x(tk)), ˙x(tk)⟩(t −tk) + ⟨∇V(x(tk)), ˙x(t∗)− ˙x(tk)⟩(t −tk) + ⟨∇V(x(t∗))−∇V(x(tk)), ˙x(tk)⟩(t −tk) (16) where the first equality holds by the mean value theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' For the three terms in the last equality, the first term gives a decrease in Lyapunov value, as at time tk the system behaves the same as the continuous system with continuous control using the instantaneous state information x(tk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Specifically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' ⟨∇V(x(tk)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' ˙x(tk)⟩ = x(tk)((A+A1)P+P(A+A1)⊺)x(tk) = −x(tk)Qx(tk) (17) Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' a lower bound on the decrease of the Lyapunov function from the first term gives ⟨∇V(x(tk)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' ˙x(tk)⟩ ≤ −σmin(Q)∥x(tk)∥2 2 ≤ 0 The second and third terms represent the perturbation incurred by the piecewise-constant control,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' where ∇V(x(tk)) = 2x(tk)⊺P ˙x(t∗)− ˙x(tk) = A(x(t∗)−x(tk)) ∇V(x(t∗))−∇V(x(tk)) = 2(x(t∗)−x(tk))⊺P ˙x(tk) = (A+A1)x(tk) (18) where the second equality is due to the same piecewise- constant control in the entire period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The following worst-case bounds hold: ∥∇V(x(tk))∥ ≤ 2σmax(P)∥x(tk)∥2 ∥˙x(tk)∥2 ≤ σmax(A+A1)∥x(tk)∥2 ∥∇V(x(t∗))−∇V(x(tk))∥ ≤ 2σmax(P)∥x(t∗)−x(tk)∥2 ∥x(t∗)−x(tk)∥2 = ���� � t∗ tk ˙x(s)ds ���� 2 ≤ (t∗ −tk) max s∈[tk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='t∗]∥˙x(s)∥2 ≤ ∆ max s∈[tk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='t∗]∥Ax(s)+A1x(tk)∥2 ≤ ∆(σmax(A)+σmax(A1)) max s∈[tk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='tk+1]∥x(s)∥2 (19) Taken together,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' we have V(x(t))−V(x(tk)) ≤ (t −tk) � −σmin(Q)∥x(tk)∥2 2 +2σmax(P)∥x(tk)∥2σmax(A)∥x(t∗)−x(tk)∥2 +2σmax(P)∥x(t∗)−x(tk)∥2σmax(A+A1)∥x(tk)∥2 � = (t −tk) � −σmin(Q)∥x(tk)∥2 2 +2σmax(P) � σmax(A)+σmax(A+A1) � × ∥x(tk)∥2∥x(t∗)−x(tk)∥2 � ≤ (t −tk) � −σmin(Q)+c∆·σmax(P)(σmax(A)+σmax(A1))2� ∥x(tk)∥2 max s∈[tk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='tk+1]∥x(s)∥2 (20) where c > 0 is an appropriate constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In the last inequal- ity, we apply Weyl’s inequality to separate σmax(A + A1) ≤ σmax(A) + σmax(A1) and substitute the bound on ∥x(t∗) − x(tk)∥2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (19) to obtain the square term (σmax(A) + σmax(A1))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In order for V(x(t))−V(x(tk)) to have a sufficient decrease, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' for some d > 1 (d = 2 in Clarke [21]), V(x(t))−V(x(tk)) ≤ −(t −tk)σmin(Q) d ∥x(tk)∥2 max s∈[tk,tk+1]∥x(s)∥2, (21) the following gives a sufficient condition c∆·σmax(P)(σmax(A)+σmax(A1))2 ≤ d −1 d σmin(Q) ⇔ ∆ ≤ c′ σmin(Q) σmax(P)(σmax(A)+σmax(A1))2 (22) for some c′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' ■ While the above bound can be loose due to the worst case singular-value bounds, it still provides a way to qualitatively analyze the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' As an interpretation, let us suppose P = I results in Q > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Then loosely speaking, an unstable uncon- trolled system A with larger σmax(A) makes the bound smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The contribution of the control is more complicated with a trade-off involved: on one hand, the larger control makes the continuous controlled system A + A1 more stable, increasing the σmin(Q) term in the numerator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' on the other hand, it also increases σmax(A1) and hence increases the denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' A Lyapunov-Krasovskii functional As the human-compatible system with piecewise constant control perfectly aligns with the sample-data system frame- work, we seek to find a tighter bound on the hold limit using theory developed for sample-data systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' A few works [23, 24, 25] view the sample-data system as a special case of the time-delay system with delay τ(t) = t −tk, which has a constant rate of change ˙τ(t) = 1 for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Lyapunov-Krasovskii functionals are commonly used to analyze the performance of time-delay systems, and naturally extend to the sample-data system (11), which can be equivalently written in the form ˙x(t) = (A+A1)x(t)−A1 � t tk ˙x(s)ds (23) as x(tk) = x(t) − � t tk ˙x(s)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In Fridman [23], the following Lyapunov-Krasovskii functional for sample-data system is proposed V(t,x(t), ˙x(t)) = x⊺(t)Px(t)+(∆−τ(t)) � t t−τ(t) ˙x⊺(s)U ˙x(s)ds (24) where τ(t) = t − tk, P > 0, U > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The first term x⊺(t)Px(t) in the above functional is the regular Lyapunov function for the unperturbed nominal system ˙x(t) = (A + A1)x(t), whereas the second integral term handles the integral perturbation − � t tk ˙x(s)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Jensen’s inequality, descriptor method [24], and state-augmentation with η1(t) = col{x(t), ˙x(t), 1 τ(t) � t t−τ(t) ˙x(s)ds} are applied to arrive at the following proposition on a given hold length ∆ with Linear Matrix Inequalities (LMIs): Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Let there exist n × n matrices P > 0,U > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' P2 and P3 such that the LMIs (25) are feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Then (11) is asymptotically stable for all variable sampling instants tk+1 −tk ≤ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' � Φ11 P−P⊺ 2 +(A+A1)⊺P3 ∗ −P3 −P⊺ 3 +∆U � < 0, � � Φ11 P−P⊺ 2 +(A+A1)⊺P3 −∆P⊺ 2 A1 ∗ −P3 −P⊺ 3 −∆P⊺ 3 A1 ∗ ∗ −∆U � � < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (25) where Φ11 = P⊺ 2 (A + A1) + (A + A1)⊺P2 and ∗ denotes the symmetric elements of the symmetric matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' See [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Comparing with the previous Lyapunov bound which upper bounds the perturbation � t tk ˙x(s)ds by the minimax singular value ratio of the controlled system A + A1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' represented by σmin(Q) σmax(P),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' divided a function of the maximum singular values of the uncontrolled system A and the control A1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' represented by σmax(A) and σmax(A1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' the Lyapunov-Krasovskii bound solves for matrices P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' P2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' P3 to account for the interactions among A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' A1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' and A + A1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' and hence can possibly render a tighter bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Additionally, while the above proposition takes a fixed controller K as given to verify if such a controller can stabilize the system with a hold length ∆, we can in fact solve for a possibly better controller K using the following corollary that takes the sample-data system property into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Let there exist n × n matrices ¯P > 0, ¯U > 0, Q and an nu × n-matrix L and a tuning parameter ε such that the LMIs (26) are feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Then (11) is asymptotically stable for all variable sampling instants tk+1 −tk ≤ ∆ with the stabilizing gain given by K = LQ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' � ¯Φ11 ¯P−Q+εQ⊺A⊺ +L⊺B⊺ ∗ −ε(Q+Q⊺)+∆ ¯U � < 0, � � ¯Φ11 ¯P−Q+ε(Q⊺A⊺ +L⊺B⊺) −∆BL ∗ −ε(Q+Q⊺) −∆εBL ∗ ∗ −∆ ¯U � � < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (26) where ¯Φ11 = Q⊺A⊺ +AQ+BL+L⊺B⊺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' From above and following [25], we can perform full state-feedback controller design by substituting P3 = εP2 where ε is a tuning parameter, Q = P−1 2 , ¯P = Q⊺PQ, ¯U = Q⊺UQ and L = KQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Multiplying LMIs (26) by diag{Q⊺,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=',Q⊺} and diag{Q,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=',Q} from the left and right, we recover LMIs (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' ■ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' EXPERIMENTS In the following Experiments section, we compare the Lyapunov analysis and the Lyapunov-Krasovskii analysis with the hold limit from empirical simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' We aim to answer the following questions: 1) How well does the theory match simulation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Moreover, to what extent do simplified theoretical analyses explain integrated human-compatible traffic flow stability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 2) What relationships emerge from the problem parameters and how do they affect stability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 3) Can we derive better piecewise-constant controllers us- ing the Lyapunov or Lyapunov-Krasovskii analysis?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Experimental Setup and Results on the default parameters We adopt the implementation from Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' [10] in Python and extend it to the piecewise-constant control setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In the default scenario, we use the same parameter for OVM, with n = 20,L = 400,α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='6,β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='9,sst = 5,sgo = 35,vmax = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Vehicles are initialized by a uniform perturbation around the equilibrium, with the ith vehicle’s position and velocity (xi 0,vi 0) = (is∗ + δs,v∗ + δv) where δs ∼ Unif[−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='5,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='5],δv ∼ Unif[−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='5,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='5], and v∗ =V(s∗) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (5) is the equilibrium velocity corresponding to the equilibrium spacing s∗ = L/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' By default, we apply the same H2 optimal full state-feedback controller for the continuous system to the sample-data system by holding it piecewise-constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The controller u(t) = −Kx(t), (27) where K ∈ R1×2n, can be obtained by the following convex program with K = ZX−1: min X,Y,Z Trace(QX)+Trace(RY) subject to (AX −BZ)+(AX −BZ)⊺ +HH⊺ ≼ 0, � Y Z Z⊺ X � ≽ 0,X ≻ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (28) where Q 1 2 = diag(γs,γv,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=',γs,γv), R 1 2 = γu, H = I (29) with the default γs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='03,γv = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='15,γu = 1, corresponding to the performance state z(t) = � Q 1 2 0 � x(t)+ � 0 R 1 2 � u(t) (30) We simulate the system by integrating the ordinary differ- ential equation (11) using the forward Euler method, with a discretization of Tstep = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='01s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' We say a system (either uncontrolled, or with continuous / piecewise constant con- trol) is stable in simulation if 50 simulated trajectories from different initial perturbations all converge to the equilibrium within TotalTime = 300s, and no vehicle collides within the trajectory (given by negative spacings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' To mitigate collisions, we follow Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' to equip all vehicles with a standard automatic emergency braking system ˙v(t) = amin, if v2 i (t)−v2 i−1(t) 2(si(t)−sd) ≥ |amin| (31) where amin = −5m/s2 is the maximum deceleration rate of each vehicle, and sd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='5m is the safe distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 2: The traffic system with all human vehicles and no controlled vehicle is unstable under the default parameters in Section V-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The equilibrium spacing and velocity are 20m and 15m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (a) The time-space diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Darker colors represent lower velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (b) The time-velocity diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The initial perturbation on the velocities get amplified, leading to the formation of stop-and-go waves in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' We study the behavior of the system by putting a piecewise- constant hold on the controller for ∆ ≫ Tstep seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Without any controlled vehicle, the default OVM system is unstable (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 2), forming stop-and-go waves gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' [10] show that introducing one autonomous vehicle with the continuous H2 optimal controller is able to stabilize the continuous system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 3, we show the behavior of the sample-data traffic system by holding the same H2 optimal control for ∆ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='59s (left) and ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='29s (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' With a smaller hold length of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='59s, the controller is able to stabilize the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Such a controller translates to a human-compatible driving design where a new instruction is issued to the human driver every 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='59s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' introducing a guided human vehicle is able to stabilize the traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' However, with a slightly larger hold length of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='29s, we observe unstable system behavior, where holding the control piecewise-constant introduces an exces- sive amount of noise that breaks the system’s stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' It is interesting to observe the sawtooth pattern in the time-velocity diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 3d, where errors are accumulated within each piecewise-constant holding period, but get corrected at the beginning of the next holding period when we update the control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' While there is system slowdown, the velocity perturbation is constrained within a range between [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='5,20] m/s, instead of getting amplified and diverging as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' We note that, although the previous work [7] reports a longer hold limit in simulation, it is sensible that the hold limits are smaller in our settings, mainly because we consider the worst case scenario where we declare instability of a system if any of the 50 trajectories is unstable, whereas the previous work considers an average case that declares stability of a system if the average vehicle velocity of all trajectories is above a reasonable value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The previous work also considers a more advanced car following model, the intelligent driver model (IDM), for human drivers, and uses nonlinear controls represented by neural networks and trained by reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' We choose to bound the worst- case scenario to provide certificates to the piecewise-constant controller even under adversarial settings, and focus on linear controls for the ease of theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In practice, we may encounter more stable human driving behaviors and provide more advanced nonlinear instructions to the guided vehicle to enable longer hold lengths;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' even when shorter hold lengths are required than human drivers can handle, we can still trade practicality for efficiency by issuing longer hold lengths at the cost of mitigating traffic less effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' How well does the theory match simulation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In this section, we examine to what extent the theoretical hold limits from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (15) and (25) are able to match the hold limits in simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 4, we vary seven OVM system parameters (L,n,vmax,sst,sgo,vmax,α,β), as well as three control parameters (kmult,γs,γv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' For each scenario, we vary one parameter while fixing the others to default values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' we solve for an continuous H2 optimal controller using the corresponding system and control parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' A summary of all parameters is listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' For each scenario, We perform a binary search within [0s,10s] with a granularity of Tstep = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='01s in simulation to find the empirical hold limit for the system to be stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The hold limit for each parameter set informs the designs of transportation system and controller to be more human- 400 15 Position m] 300 s 10 3 200 Velocity 5 100 0 0 20 40 60 80 t [s]25 OVM s 20 Average velocity m Velocity 15 10 5 0 20 40 60 80 [s] (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 3: The traffic system consists of n − 1 human vehicles (gray) and 1 piecewise-constant controlled vehicle (red) with different hold lengths ∆s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The controlled vehicle applies the same H2 optimal control gain matrix for the continuous system to the sample-data system, under the default parameters in Section V-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (a) and (b): The time-space and time-velocity diagrams when ∆ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='59s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The traffic is stabilized to the equilibrium velocity 15m/s after a short amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (c) and (d): The time-space and time-velocity diagrams when ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='29s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The system becomes unstable when the hold length is too long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Symbol Default Description System parameters L 400m Circumference of the ring-road, where the equilibrium spacing s∗ = L/n n 20 Number of vehicles in the ring-road system, where the equilibrium spacing s∗ = L/n sst 5m Small spacing threshold such that the optimal velocity = 0 below the threshold, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (5) sgo 35m Large spacing threshold such that the optimal velocity = vmax above the threshold, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (5) vmax 30m/s Maximum optimal velocity, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (5) and (6) α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='6 Driver’s sensitivity to the difference between the current velocity and the desired spacing-dependent optimal velocity, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (4) β 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='9 Driver’s sensitivity to the difference between the velocities of the ego vehicle and the preceding vehicle, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (4) Control parameters kmult 1 Scale the H2 optimal controller Kcont by a constant: Knew = kmult ·Kcont γs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='03 weight on the position derivation from equilibrium in the H2 optimal control objective, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (28) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (29) γv 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='15 weight on the velocity derivation from equilibrium in the H2 optimal control objective, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (28) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (29) γu 1 weight on the control magnitude in the H2 optimal control objective, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (28) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (30) TABLE I: System And Control Parameters In The Optimal Velocity Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' However, empirical trajectory simulations are in- feasible or computationally expensive due to the large and continuous space of initial conditions (starting positions and velocities of all vehicles), even for the ring road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' This problem will undoubtedly be exacerbated in real-world settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Hence, accurate theoretical guarantees on hold limit are essential to system design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' To this end, we provide theoretical estimates of the hold limit using the following three methods on the linearized traffic system: 1) The Lypaunov analysis: see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Due to redundancy in headway representation with ˜s1 + ˜s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' + ˜sn = 0, we first obtain the reduced representation by omitting ˜s1 from the state vector and replacing it with −˜s2 −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='− ˜sn to construct the reduced system matrices A†,B†,K†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Then, we set Q = I(n−1)×(n−1) which has σmin(Q) = 1, and solve for P from the Lyapunov equation (A† − B†K†)P+P(A† −B†K†)⊺ = −Q to obtain σmax(P) in the denominator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The detailed matrix represen- tations of A†,B†,K† can be found in Appendix VI-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 2) The Lyapunov-Krasovskii analysis: see LMIs (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' We perform a binary search within [0s,10s] with a granu- larity of Tstep = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='01s to find the theoretical hold limit such that the LMIs are feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 3) The OVM stability: stability theory of the linearized, uncontrolled system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Previous work [9] uses string sta- bility to analyze the linearized, uncontrolled continuous OVM model, and derive the stability criteria α +2β ≥ 2 ˙V(s∗) = 2 ˙V(L/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Equivalently, for s∗ = L/n ∈ [sst,sgo], the OVM system is stable if α +2β −vmax π sgo −sst sin � π L/n−sst sgo −sst � ≥ 0 (32) We plot the value of the left hand side in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 4, which takes on negative values because we choose parameter values so that the uncontrolled system is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Examining the extent to which piecewise-constant control is effective under different traffic conditions is a complex problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Following the motivation of “All models are wrong, but some are useful,” it is attractive to consider whether reduced-order linearized models, such as the uncontrolled OVM system stability or the direct Lyapunov analysis, can lend themselves as proxies to analyzing the true traffic prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Through validating the work in simulation, we ultimately find that it is important to capture both the role of the controller (insufficiency of OVM stability) and the effect of the Lyapunov-Krasovskii integral (insufficiency of Lyapunov analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Perhaps surprisingly, both OVM stability and the Lyapunov do generally capture the trends quite well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' More- over, the Lypaunov-Krasovskii analysis captures not only the trend but also the absolute hold limit, indicating that the effect of linearizing the system dynamics is not a strong limitation of the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Specifically, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 4, we plot the theoretical against the empirical hold limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' We display the scale of the y-axis 400 15 Position m 300 S 10 m 200 Velocity 5 100 0 0 0 20 40 60 80 [s] 25 OVM s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='20 Controlled vehicle Average velocity Velocity 15 10 5 0 20 40 60 80 [s] 400 15 Position m 300 s 10 200 Velocity 5 100 0 20 40 60 80 [s] 25 OVM s 20 Controlled vehidle Average/veldcity Velocity 15 10 5 0 20 40 60 80 [s] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 4: The hold limit that stabilizes the system from simulation (solid blue), Lyapunov-Krasovskii analysis in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (25) (dashed orange), Lyapunov analysis in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (15) (dash-dotted gray), and uncontrolled OVM stability criterion in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (32) (dotted green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Default parameter values are shown as the black vertical lines in each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Left axis is for simulation and Lyapunov-Krasovskii analysis, while the y-axis scale for the Lyapunov analysis and the OVM stability are displayed in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' for Lyapunov analysis and OVM stability in Table II;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' the Lyapunov-Krasovskii analysis shares the same scale as the simulation, which is depicted as the numbers on the left of the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The Lyapunov-Krasovskii analysis is remarkably accurate in general, matching both the trend of the simulation and the absolute scale of all parameters, whereas the other two theoretical methods only provide relative trend estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The Lyapunov-Krasovskii analysis overestimates the simulation hold limit for large vmax and small β, however, where the uncontrolled system is more unstable that leads to collisions in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In such cases, the Lyapunov analysis gives a more accurate bound by more aggressively penalizing the worst-case uncontrolled system behavior given by σmax(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The Lyapunov-Krasovskii analysis also overestimates the sim- ulation hold limit for small kmult, where the effect of the controller on the system is too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In such a case, again, we obtain a more accurate trend estimate from the Lyapunov analysis, which more conservatively estimates the stability of the controlled continuous system ∝ σmin(Q)/σmax(P) when the controller has very small magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The Lyapunov analysis matches the trend of the simulation hold limit decently well, despite the difference in absolute scale, and slight misalignment for the system parameters (n,vmax,α,β), and the control parameters (kmult,γs,γv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In the cases of (n,vmax,α,β), the worst-case singular-value bounds Symbol Lyapunov analysis OVM stability System parameters L (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='07×10-3,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='81×10-2) (-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='74×10-1,-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='99×10-2) n (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='70×10-4,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='19×10-3) (-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='62×10-1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='94×10-2) sst (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='82×10-4,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='68×10-3) (-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='29,-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='67×10-1) sgo (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='88×10-4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='56×10-3) (-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='28,-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='79×10-1) vmax (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='79×10-5,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='72×10-3) (-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='17,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='76×10-2) α (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='65×10-4,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='89×10-3) (-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='30,-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='66×10-2) β (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='44×10-5,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='56×10-3) (-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='45,-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='16×10-2) Control parameters kmult (0,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='02×10-3) γs (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='08×10-4,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='47×10-3) γv (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='60×10-4,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='17×10-3) TABLE II: Scales of hold limits (the minimum and maximum of y-axis in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 4) for Lyapunov analysis and OVM stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' become too aggressive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' a more fine-grained theoretical analy- sis given by Lyapunov-Krasovskii renders a better estimate by considering the interaction of A (the uncontrolled system), BK (the control) and A−BK (the controlled system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In the cases of (kmult,γs,γv), the Lyapunov analysis captures the correct trend in general, but fails to capture the correct absolute slopes for three parameters, and the correct peak for (kmult,γv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' This is understandable because the analysis only holds up to a scaling constant that decides the slope, and the location of the peak can change by scaling σmax(A) and σmax(A1) differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' We keep equal scaling in the analysis for clarity of interpretation, Simulation Lyapunov-Krasovskiianalysis Lyapunov analysis OVMstability L n Sst Sgo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='9 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='0 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='8 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='8 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='7 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='8 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='8 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='6 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='7 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='5 300 400 500 15 20 25 maximum hold length ( 5 10 15 25 30 35 Vmax α β 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='7 40 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='0 Ys Yv 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='5 - 10 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='5 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='0 - 5 - : 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='5 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='0 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='0 - 0- 0 2 0 1 2 0 1 2 parameterand leave finding more accurate scalings to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' To our (slight) surprise, the uncontrolled OVM stability matches the trend of the simulation hold limits particularly well for a few parameters (L,sst,sgo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' However, mismatches occur when the controller BK has a significant effect on the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Slight trend mismatch occurs for vmax and β, and opposite trends are observed for the α parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 5, in these cases, the controller σmax(−BK) displays a non-linear trend different from σmax(A), making the resulting controlled hold limit nonlinear and even displaying an opposite trend for α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' such a behavior is in contrast with the three aligned cases, where the trends of σmax(A) and σmax(−BK) match (see the L plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In the misaligned cases, the missing information of the controller is necessary for a more accurate trend estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 5: A visualization of different components in the Lyapunov analysis (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (15)) for four system parameters L,n,vmax,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' We plot the denominator components σmax(P) that represents the continuous controlled system (dashed green), σmax(A) (dotted red) that represents the continuous uncontrolled system, σmax(A1) = σmax(−BK) that represents the control (dash-dotted orange), and the final theory bound on the hold limit ∆ that stabilizes the system (solid blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Note that numerator component σmin(Q) = 1 by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The absolute scales of the different components are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Overall, the closeness in both trend and absolute-scale makes Lyapunov-Krasovskii theory a reliable theoretical sur- rogate for quantitative estimation of the simulation hold limit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' meanwhile, the clean expression from the Lyapunov analysis makes it a reliable tool for qualitative interpretation of the system’s behavior, especially when the controller’s behavior, given by σmax(−BK), and the uncontrolled system’s stability, given by σmax(A), do not completely align.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' When the two align, the OVM stability criteria for the uncontrolled system can be used to analyze the sample-data system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' How do traffic conditions affect the hold limit?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In this section, we interpret relationships between traffic system parameters, which represent different traffic conditions, and their hold limits, as detailed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Specifically, we fix the control parameters (kmult,γs,γv), and vary the OVM system parameters (L,n,sst,sgo,vmax,α,β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Overall, we observe three main types of traffic situations that promote longer hold limits by means of low driver sensitivity: (1) traffic conditions (density, speed limit, and spacing thresholds) that promote a smoother spacing response, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=', the flatter region of the optimal velocity function (through various combinations of L,n,sgo,sstop,vmax), (2) low sensitivity of drivers to relative position (low α), and (3) high sensitivity of drivers to relative speed, which tends towards equilibrium (high β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Why low driver sensitivity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Because a controlled vehicle can exert more fine-grained control with a shorter hold length, we can think of a longer hold length as exerting a larger ”blunter” change to the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The primary mechanism by which the hold length interacts with the traffic condition is the extent to which these larger changes to the environment affect the drivers’ spacing (headway).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' If the drivers’ are more sensitive to changes in spacing, then ”blunter” control is more likely to cause large deviations in the environment, thus preferring shorter hold lengths to maintain stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Why not OVM system stability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Indeed, greater uncon- trolled system stability often, but not always, corresponds to longer hold limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Thus, the discrepancy between OVM system stability and the empirical analysis is illustrative for understanding the importance of low driver sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' OVM system stability (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (32)) has a strong negative correlation with drivers’ sensitivity, with the exception of the α parameter: larger α corresponds to a more stable uncontrolled OVM system as it corresponds to strong compliance of drivers to the optimal velocity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' however, the OVM optimal velocity might conflict with the controlled vehicle, especially when errors are incurred with the piecewise-constant holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' For example, with a longer hold length and larger α, the controlled vehicle may open up wider gaps, resulting in a stronger response from the following driver, in turn causing system instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Hence, larger α increases the uncontrolled system’s stability but reduces the hold limit of the controlled system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Smoother spacing response: We observe that (L,n,sst,sgo,vmax) determines various aspects of the optimal velocity function, as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (5) and illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The parameters L and n are related to the density of the traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Their ratio s∗ = L/n determines the equilibrium spacing, which further determines the desired optimal velocity v∗ = V(s∗), with the value clipped within [0,vmax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' When the spacing is either too small (close to sst) or too large (close to sgo), the uncontrolled system is more stable, since the desired optimal velocity is easier to follow by the drivers, who can drive either very slowly (v∗ is near 0) or follow the maximum speed (v∗ is near vmax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' However, when the spacing is close to the sgo−sst 2 , as depicted by the red star in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 6, the original system becomes more unstable, since slight changes in spacing would lead to large changes in the desired optimal velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In fact, the default sst = 5,sgo = 35 directly place the default spacing L/n = 20m at the most unstable inflection point (the red star).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Similarly, the two boundary values sst and sgo determine the length of the region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' varying them would vary both the location of the equilibrium spacing on the curve and the theory Omax(P) Omax(-BK) Omax(A) L n 300 400 500 15 20 25 Vmax α 30 40 50 60 70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='25Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 6: The Optimal Velocity function V(s) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 5 and 6 with default parameters in Section V-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The red star represents the equilibrium spacing and velocity with the default parameters, where the function attains maximum slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Changing system parameters moves the red star to different positions on the curve, affecting the stability of the uncontrolled system and the hold limit to stabilize the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' slope of the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' When we increase sst or decrease sgo, we make the curve steeper, and hence more unstable, but we also move the location of the equilibrium spacing away from the inflection point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Such a trade-off reflects the slight asymmetry in the sst and sgo curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In general, we observe that the stability of the uncontrolled system, which is closely tied to the the rate of change of the optimal velocity function at the equilibrium spacing, translates well to the hold limit of the piecewise-constant system for (L,n,sst,sgo), following an upward parabola shape resulting from the cosine wave of the desired optimal velocity curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' We also observe that the variation of the hold limit in (L,n,sst,sgo) is mild, ranging from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='5s to 2s in simulation, as these four variables are all encapsulated within the cosine function in the desired optimal velocity equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' On the other hand, the maximum desired velocity parameter vmax, as the multiplier to the cosine wave, affects the hold limit substantially more, from 2s down to 0s when vmax goes from 25m/s to 60m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Increasing the maximum desired velocity effectively stretches the desired velocity curve taller, resulting in sharper changes of the desired optimal velocity when the spacing changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Hence, for larger vmax, the uncontrolled system becomes more unstable, resulting in a shorter hold limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' While the stability of the uncontrolled system only explains a linear decrease of hold limit, we observe a super- linear decrease in simulation due to the following two addi- tional reasons: (1) the larger magnitude of the controller, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 5, incurs more errors to the system from the piecewise-constant hold, and (2) the unstable system leads to collisions of the vehicles, making the system even harder to stabilize with the noisy controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Low sensitivity to relative position, high sensitivity to relative speed: The remaining two parameters, α and β, reflect the sensitivity of human drivers between the current velocity and the desired optimal velocity (α), and the velocity of the vehicle in front (β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Interestingly, we see different trends of the simulation hold limit for the two parameters, despite larger α and β both make the original uncontrolled system more stable (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (32)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' For β, the expected velocity dissipation term makes the human vehicle more observant of the surroundings, hence increasing stability of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The sharp super-linear decrease in the hold limit for small values of β is a result of the analogous reasons as for vmax, which a combination of uncontrolled system’s stability, additional errors induced due to large magnitude of the controller, and vehicle collisions when the system is excessively unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' On the other hand, for the α parameter, the simulation hold limit exhibits an opposite trend of the uncontrolled system stability, despite the mild variation in hold limit (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='6s to 2s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Examining the controller magnitude in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 5, we observe that the magnitude is larger for the more stable uncontrolled system with larger α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' as a result, the piecewise-constant control adds more noise to the system when α is large, despite the original, uncontrolled system is in fact more stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Controller design for human-compatible driving ∆in(s) 1 2 3 4 5 6 7 8 ∆sim(s) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='71 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='33 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='55 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='54 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='09 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='87 TABLE III: The simulation hold limit ∆sim with the Lyapunov- Krasovskii control gain matrix in LMIs (26) when we set ∆in ∈ {1,2,3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=',9} in the LMIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Thus far, we have focused on analyzing a given controller;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' in particular, the continuous H2 optimal controller, which achieves a simulation hold limit of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='66s by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In this section, we consider several approaches to intentionally design controllers for human-compatible control to achieve system- level traffic flow stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Lyapunov-Krasovskii controller search: Recall that the Lyapunov-Krasovskii analysis in Section IV-B provides a method to obtain piecewise-constant controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' We thus examine the quality of the resulting controllers via simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' We fix the OVM system parameters (L,n,sst,sgo,vmax,α,β) to the same default value in Section V-A, and solve LMIs (26) for a control gain matrix K = LQ−1, with a grid search of input hold length parameter ∆in ∈ {1,2,3,4,5,6,7,8}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' We fix the tuning parameter ε = 1 in LMI (26) where we substitute P3 = εP2 from (25), as we empirically find such a ε gives the best controller with the longest simulation hold limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Given the resulting control gain matrix KLK, we perform simulation via a binary search with a granularity of Tstep = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='01s to examine the empirical hold limit ∆sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Table III depicts the actual simulation hold limits ∆sims for different input parameters ∆ins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' We observe that, initially as ∆in increases, the Lyapunov-Krasovskii analysis is able to find better controllers with longer hold limits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' however, as ∆in further increases over 4s, the simulation hold limit ∆sim decreases, causing discrepancies between the theory and the actual simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The fundamental reason comes from the collision constraint in simulation: while we declare a trajectory with collisions (negative spacings) as unstable in simulation, the Lyapunov-Krasovskii analysis ignores such a constraint, and allows the spacing and velocity variables to take negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In fact, if we omit the constraint in simulation, we would achieve substantially larger hold limits Desired Velocity V(s) [m/s] max ５ 0 0 5 10 15 20 25 30 35 40 Spacing s [m] S go∆sim > 10s for the Lyapunov-Krasovskii controllers with large ∆ins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' However, ignoring such a constraint makes the traffic scenario unrealistic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' an important direction of future work is to incorporate control barrier functions [26] to the Lyapunov- Krasovskii analysis to explicitly consider the collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' H2 re-scaling: Next, we propose and examine a heuristic controller design policy where we fix the OVM system param- eters (L,n,sst,sgo,vmax,α,β) and vary the control parameters (kmult,γs,γv) in order to find scaled controllers more suitable for the sample-data system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 4, we observe controllers of smaller magnitudes than the default continuous H2 controllers, given by smaller kmult < 1, γs < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='03, γv < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='15, result in longer hold limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Noticably, the longest hold limit is achieved at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='78s when kmult = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='2, offering a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='78x improvement from the default at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='66s when kmult = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Such a behavior can be explained by the σmax(A1) term in the denominator of the Lyapunov analysis, where controllers of larger magnitudes incur larger errors from the piecewise-constant hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' However, when the controller is too small, the controller is not powerful enough to stabilize the system, resulting in a drop in hold limit from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='78s when kmult = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='2 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='84s when kmult = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='005, and finally to 0s when kmult = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' such a phenomenon can be explained by the ratio σmin(Q)/σmax(P) in the the Lyapunov analysis, which represents the stability of the controlled system A − kmultBK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Hence, there is a trade-off between the power of the controller and the noise incurred from the piecewise- constant hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In general, when we design controllers for the sample-data system, a reasonable penalization on the magnitude of the controller could improve the hold limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' In the meantime, we also observe similar discrepancies between simulation and Lyapunov-Krasovskii theoretical hold limit in the kmult plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 4, due to the omission of collision constraints in the LMIs (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Meanwhile, as the best simulation hold limit of the best scaled continuous H2 optimal controllers, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='78s, is around the same level of the Lyapunov-Krasovskii controllers with a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='55s hold limit, we make an interesting observation that the piecewise-constant controller obtained by scaling down a reasonable continuous control may obtain decent performance for human-compatible driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' As abundant reinforcement learning controllers have been developed for the continuous traffic systems [4, 15], a few promising strategies for human- compatible driving are to take the down-scaled version of the same controllers, or to finetune these controllers with a magnitude penalization to avoid expensive re-training of the piecewise-constant controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' CONCLUSIONS AND FUTURE WORK This work presents an integrated Lyapunov analysis frame- work of human-compatible, piecewise-constant control to sta- bilize the traffic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' We derive both a Lyapunov analysis for qualitative interpretation of the relationships between traf- fic system parameters and the hold limit, and a Lyapunov- Krasovskii analysis for quantitative estimation of the hold limit to stabilize traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The Lyapunov-Krasovskii analysis can also be used for piecewise-constant controller design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Our theory provides certificates to human-compatible driving, a class of policies that aim to guide human drivers to stabilize the traffic, bypassing the difficulty of autonomous vehicle deployment and having the potential to achieve desirable traffic outcomes with low cost and quick timeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Our work highlights the power of the Lyapunov analysis framework as an impor- tant integrated theoretical tool for obtaining efficient, safe, and sustainable transportation systems with human-compatible control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' We propose a few important directions for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' First, we would like to tighten the derivation of the Lyapunov analysis (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (15)) to obtain absolute scales of different components in the bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' The correct scaling will enable us to pinpoint the exact slope and location of the optimum of the curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 4, while the current bound is only able to describe the relative trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Next, we would like to incorporate control barrier functions to the Lyapunov-Krasovskii analysis (LMIs (25) and (25)) to tighten the bound under unsafe events such as collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Finally, we would like to consider expanding our theory to a broader class of human-compatible driving policies that consist of other easy-to-follow driving instructions, as well as to more complex traffic scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' Reduced state-space representation Due to redundancy in headway representation with ˜s1(t)+ ˜s2(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='+ ˜sn(t) = n ∑ i=1 si(t)−ns∗ = L−L = 0, (33) we obtain the reduced state-space representation for the system by first omitting ˜s1(t) from the state vector and obtain x†(t) = [˜v1(t), ˜s2(t), ˜v2(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=', ˜sn(t), ˜vn(t)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (34) Then, we omit the first rows of A and B which correspond to the system equation for ˜s1(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' We also omit the first column of A, which is all zero as none of the other state equations for ˜si(t), ∀i ̸= 1 and ˜vj(t) ∀j depends on ˜s1(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' So, we have A† = � �������� C† 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 0 C† 2 D† 2 D1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 0 0 D2 D1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 0 D2 D1 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' 0 D2 D1 � �������� ,B† = � ������ B† 1 B2 B2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' B2 � ������ (35) with D1,D2,C1,C2,B2 the same as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (13), and D† 2 = � 1 α3 � ,C† 1 = � 0 � ,C† 2 = � 0 0 � ,B† 1 = � 1 � , (36) Similarly, we have u(t) = −Kx(t) = −K1 ˜s1(t)−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='−Kn ˜sn(t) = −K1(−˜s2(t)−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='− ˜sn(t))−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='−Kn ˜sn(t) = −(K2 −K1)˜s2(t)−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='−(Kn −K1)˜sn(t) = −K†x†(t) (37) Hence, the new control gain matrix K† = [K2 −K1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=',Kn −K1] ∈ R1×(2n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' (38) ACKNOWLEDGMENT This work was supported by the National Science Founda- tion (NSF) under grant number 2149548, the MIT Amazon Science Hub, the MIT Energy Initiative (MITEI) Mobility Systems Center, MIT’s Research Support Committee, as well as a gift from Mathworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
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+page_content=' Control barrier function based quadratic programs for safety critical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
+page_content=' IEEE Transactions on Automatic Control, 62(8):3861–3876, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE2T4oBgHgl3EQfqwjT/content/2301.04043v1.pdf'}
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+EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH
+COMPASS
+CERN-EP-2022–292
+December 27, 2022
+Transverse-spin-dependent azimuthal asymmetries of pion and kaon pairs
+produced in muon-proton and muon-deuteron semi-inclusive
+deep inelastic scattering
+The COMPASS Collaboration
+Abstract
+A set of measurements of azimuthal asymmetries in the production of pairs of identified hadrons
+in deep-inelastic scattering of muons on transversely polarised 6LiD (deuteron) and NH3 (proton)
+targets is presented. All available data collected in the years 2002–2004 and 2007/2010 with the
+COMPASS spectrometer using a muon beam of 160 GeV/c at the CERN SPS were analysed. The
+asymmetries provide access to the transversity distribution functions via a fragmentation function
+that in principle may be independently obtained from e+e− annihilation data. Results are presented,
+discussed and compared to existing measurements as well as to model predictions. Asymmetries of
+π+π− pairs measured with the proton target as a function of the Bjorken scaling variable are sizeable
+in the range x > 0.032, indicating non-vanishing transversity distribution and di-hadron interference
+fragmentation functions. As already pointed out by several authors, the small asymmetries of π+π−
+measured on the 6LiD target can be interpreted as indication for a cancellation of u and d-quark
+transversity distributions.
+(to be submitted to Phys. Lett. B)
+arXiv:2301.02013v1 [hep-ex] 5 Jan 2023
+
+Transverse-spin-dependent azimuthal asymmetries of pion and kaon pairs produced ...
+1
+1
+Introduction
+The description of the nucleon spin structure remains one of the main challenges in hadron physics.
+For a polarised nucleon, a leading-twist description comprises eight transverse-momentum-dependent
+(TMD) parton distribution functions (PDFs), describing the distributions of longitudinal and transverse
+momenta of partons and their correlations with nucleon and quark polarisations [1]. After integration
+over quark intrinsic transverse momentum kT, three PDFs fully describe the nucleon, i.e. the momentum
+distribution function f q
+1 (x), the helicity distribution function gq
+1(x) and the transversity distribution func-
+tion hq
+1(x), where x denotes the Bjorken scaling variable. For simplicity, the latter will be referred to as
+”transversity” throughout this paper. While the momentum and the helicity distribution functions have
+been measured with good accuracy, the knowledge on transversity is inferior but steadily increasing. Un-
+like f q
+1 and gq
+1, transversity is not accessible at leading twist in inclusive deep-inelastic scattering (DIS)
+because it is related to soft processes correlating quarks with opposite chirality, making it a chiral-odd
+function. Transversity can be accessed through observables that conserve chirality, i.e. when it is cou-
+pled to a chiral-odd partner. In this regard, measuring semi-inclusive deep-inelastic scattering (SIDIS) is
+advantageous as transversity is coupled to the chiral-odd fragmentation functions (FFs) that describe the
+hadronisation of a transversely polarised quark q into unpolarised final-state hadrons.
+The major source of information on transversity has been measurements of transverse-spin-dependent
+asymmetries (TSAs) in single-hadron production in SIDIS (ℓN↑ → ℓ′hX), where transversity is coupled
+to the Collins FF [2]. Transverse-spin asymmetries define the size of the transverse-target-spin-dependent
+azimuthal modulations of the SIDIS cross section. The first measurement of the Collins asymmetries
+was performed by the HERMES Collaboration [3] using a transversely polarised hydrogen target. Size-
+able asymmetries were observed, suggesting non-zero transversity and Collins FFs. The COMPASS
+Collaboration has the highest statistics data in this field, e.g. 28M pion pairs taken with the NH3 (pro-
+ton) target and 4M pion pairs taken with the 6LiD (deuteron) target. The COMPASS collaboration has
+delivered a full set of measurements, i.e. both TSAs for unidentified charged hadrons [4] as well as
+pions and kaons [5] using the transversely polarised deuteron target, and TSAs for unidentified charged
+hadrons [6, 7] as well as pions and kaons [8] using the transversely polarised proton target. The TSAs
+measured with the polarised proton target showed a non-zero signal for Collins asymmetries with high
+statistics and a wide kinematic coverage. The TSAs measured with the polarised deuteron target are
+compatible with zero indicating a possible cancellation between up and down quark contributions to
+transversity. Despite the lower accuracy of these data, they were shown to play a key role in the ex-
+traction of flavour-dependent transversity distribution functions and remain the only SIDIS measurement
+ever performed using a transversely polarised deuteron target. In order to complete the COMPASS pro-
+gramme [9], a new high statistics measurement of TSAs using a transversely polarised deuteron target
+was performed in the last data taking campaign, in 2022.
+A promising alternative approach to access transversity is the measurement of TSAs in semi-inclusive
+production of pairs of hadrons of opposite charge (ℓ N↑ → ℓ′h+h−X). Following this approach, in this
+work π+π− and K+K− as well as π+K− and K+π− pairs will be studied. In this case, transversity is
+coupled to the chiral-odd interference fragmentation function (IFF) H∢
+1 [10–12], which describes the
+hadronisation of a transversely polarised quark into a pair of unpolarised hadrons. At leading twist, and
+after integration over total transverse momentum, the differential cross section on a transversely polarised
+target comprises two terms and can be written as [13]
+d7σ
+dcosθ dMhh dφR dzdxdydφS
+=
+α2
+2πQ2y
+�
+(1−y+ y2
+2 )∑
+q
+e2
+q f q
+1 (x) D1,q(z,M2
+hh,cosθ)
++S⊥(1−y)∑
+q
+e2
+q
+|p1 −p2|
+2Mhh
+sinθ sinφRS hq
+1(x) H∢
+1,q(z,M2
+hh,cosθ)
+�
+.
+(1)
+
+2
+The COMPASS Collaboration
+Here α is the fine-structure constant, D1,q(z,M2
+hh,cosθ) is the spin-independent dihadron fragmentation
+function (DiFF), y is the fraction of the lepton energy in the laboratory frame transferred to the exchanged
+virtual-photon and Q2 the negative square of the four-momentum transfer. Here, z is the fraction of the
+virtual-photon energy carried by the hadron pair, Mhh its invariant mass and θ the polar angle of the
+positive hadron with respect to the two-hadron boost axis in the two-hadron rest frame. The symbol
+S⊥ denotes the component of the target spin vector S perpendicular to the virtual-photon direction, with
+φS the azimuthal angle of the initial nucleon spin, φS′ the azimuthal angle of the spin vector of the
+fragmenting quark and φRS = φR −φS′ = φR +φS −π. The azimuthal angle φR is given as
+φR = (q×l)·R
+|(q×l)·R| arccos (q×l)·(q×R)
+|q×l||q×R| ,
+(2)
+where l is the incoming lepton momentum, q the virtual-photon momentum and R the relative hadron
+momentum defined as R = (z2p1 −z1p2)/(z1 +z2). The TSAs are experimentally accessible through the
+measured number of hadron pairs written as
+Nhh(x,y,z,M2
+hh,cosθ,φRS) ∝ σUU(1+ f(x,y)PTDnn(y)AφRS
+UT sinθ sinφRS),
+(3)
+where f(x,y) is the target polarisation dilution factor, PT is the transverse polarisation of the target nu-
+cleons and Dnn the transverse-spin transfer coefficient. A more detailed discussion about the theoretical
+framework can be found in Ref. [14]. As shown in Ref. [14], the asymmetry
+AsinφRS
+UT
+= |p1 −p2|
+2Mhh
+∑q e2
+q hq
+1(x) H∢
+1,q(z,M2
+hh,cosθ)
+∑q e2
+q f q
+1 (x) D1,q(z,M2
+hh,cosθ)
+(4)
+is proportional to the product of the transversity distribution function hq
+1(x) and the polarised two-hadron
+interference fragmentation function H∢
+1,q(z,M2
+hh,cosθ), summed over the quark flavours q with charge
+eq.
+Transverse-spin-dependent asymmetries of hadron pairs were first measured by the HERMES Collabo-
+ration [15] for pion pairs using a transversely polarised hydrogen target. A sizeable signal was seen as
+a function of x, indicating a sizeable u-quark transversity and non-vanishing interference fragmentation
+functions. The COMPASS collaboration has published measurements of transverse spin asymmetries
+for pairs of unidentified hadrons produced on polarised deuterons [14] and polarised protons [16]. The
+COMPASS results obtained with the proton target showed significantly sizeable asymmetries and a clear
+slope in their x-dependence thanks to the high accuracy of the proton data set, while those extracted
+from deuteron-target data were found to be compatible with zero. An intriguing similarity between
+Collins-like single-hadron asymmetries for the positive and negative hadrons extracted from the SIDIS
+hadron-pair data and the standard Collins asymmetries is observed as a function of x, suggesting that
+both single hadron and hadron-pair transverse-spin dependent fragmentation functions are generated by
+the same elementary mechanism, as presented and discussed in Ref. [17].
+In this paper, we present a new measurement of TSAs for identified hadron-pairs using the full data set
+collected by the COMPASS Collaboration on transversely polarised deuteron (2002-2004) and proton
+(2007 and 2010) targets. The paper is organised as follows. Only a brief description of the experimental
+setup and data analysis are given in Sec. 3, as the same setup and methods of data cleaning, selection and
+extraction of TSAs as in previous COMPASS analyses [14,16] are used. The measured asymmetries are
+presented in Sec. 3 and discussed in Sec. 4.
+2
+Experimental data and analysis
+The analysis presented in this paper is based on data collected in the years 2002-2004 and 2007/2010
+using the COMPASS spectrometer [18] by scattering the naturally polarised µ+ beam of 160 GeV/c
+
+Transverse-spin-dependent azimuthal asymmetries of pion and kaon pairs produced ...
+3
+delivered by the CERN SPS off transversely polarised 6LiD and NH3 targets, respectively. For 6LiD, the
+average dilution factor calculated for semi-inclusive reactions is ⟨f⟩ ∼ 0.38 and the average polarisation
+is ⟨PT⟩ ∼ 0.47, while for NH3 the corresponding values are ⟨f⟩ ∼ 0.15 and ⟨PT⟩ ∼ 0.83, respectively.
+The target consisted of two or three cylindrical cells assembled in a row, which can be independently
+polarised. In 2002–2004, two cells were used, each 60 cm long and 3 cm in diameter, separated by a
+10 cm gap. In 2007 and 2010, the target consisted of three cells of 4 cm diameter, with gaps of 5 cm
+in between. The middle cell was 60 cm long and the two outer ones 30 cm long each. From 2006 on,
+a new solenoidal magnet was used to polarise the target with a polar angle acceptance of 180 mrad as
+seen from the upstream end of the target, while in the earlier measurements with the 6LiD target the
+polar angle acceptance was 70 mrad. For the measurement of transverse spin effects, the target material
+was polarised along the vertical direction. In order to reduce systematic effects, neighbouring cells were
+polarised in opposite directions allowing for simultaneous data taking with both target spin directions to
+reduce flux-dependent systematic uncertainties. Furthermore, the polarisation was destroyed and built
+up in reversed direction every four to five days, in order to cancel residual acceptance effects associated
+with the longitudinal position of the target cells (i.e. position along the beam line). For the data collected
+using a proton target, in the analysis, the central cell is divided into two parts, providing four data samples
+with two different orientations of polarisation. Note that for the measurements in 2007 and in 2010 a
+similar spectrometer configuration was used.
+In the analysis, events with incoming and outgoing muons and at least two reconstructed charged hadrons
+originating from the interaction vertex inside the target cells are selected. Equal flux through the whole
+target is obtained by requiring that the extrapolated beam tracks pass through all three cells. In order
+to select events in the DIS regime, requirements are applied on the squared four-momentum transfer,
+Q2 > 1 (GeV/c)2, and on the invariant mass of the final hadronic state, W > 5 GeV/c2. Furthermore,
+the fractional energy transfer to the virtual photon is required to be y > 0.1 and y < 0.9 to remove events
+with poorly reconstructed virtual-photon energy and events with large radiative corrections, respectively.
+For a selected DIS event, all reconstructed hadrons originating from the interaction vertex are consid-
+ered. Only hadrons produced in the current fragmentation region are selected by requiring z > 0.1 for
+the fractional energy and xF > 0.1 for the Feynmann-x variable. The two-hadron sample consists of all
+combinations of oppositely charged hadrons built from the same DIS event. Exclusive dihadron produc-
+tion is suppressed by requiring the missing energy for each hadron pair to be greater than 3 GeV. As the
+azimuthal angle φR is only defined for non-collinear vectors R and q, a minimum value is required on the
+component of R perpendicular to q, i.e. R⊥ > 0.07 GeV/c. After the application of all requirements, 0.56
+×107 h+h− combinations remain for the deuteron data and 3.5 ×107 h+h− pairs for the proton data.
+The RICH detector information is used to identify charged hadrons as pions or kaons in the momentum
+range between the Cherenkov threshold (about 2.6 GeV/c and 9 GeV/c, respectively) and 50 GeV/c.
+The detector set-up after the upgrade of 2005 and the particle identification (PID) procedure are fully
+described in Ref. [19], while details on the likelihood PID method and the purity of identified samples
+are explained in Ref. [5] and Ref. [8] for deuteron and proton targets, respectively. In the kinematic
+domain of the COMPASS experiment, about 67% of the final-state charged hadrons are identified as
+Table 1: Final statistics for unidentified and identified charged-hadron pairs in deuteron (2002-2004) and
+proton (2007 and 2010) data.
+Year
+Number of pairs (×106)
+h+h−
+π+π−
+π+K−
+K+π−
+K+K−
+2002-2004 (deuteron)
+5.65
+3.97
+0.26
+0.30
+0.10
+2007 (proton)
+10.91
+7.41
+0.38
+0.53
+0.22
+2010 (proton)
+34.56
+20.60
+1.10
+1.53
+0.60
+
+4
+The COMPASS Collaboration
+pairs/0.02 GeV/c2
+π+π−
+×103
+0.0
+2.0
+1.5
+1.0
+0.5
+2.5
+20
+60
+40
+Mhh (GeV/c2)
+pairs/0.02 GeV/c2
+K+K−
+×103
+0.0
+2.0
+1.5
+1.0
+0.5
+2.5
+Mhh (GeV/c2)
+20
+40
+pairs/0.02 GeV/c2
+π+K−
+×103
+0.0
+2.0
+1.5
+1.0
+0.5
+2.5
+Mhh (GeV/c2)
+10
+K+π−
+5
+pairs/0.02 GeV/c2
+π+π−
+×106
+0.0
+2.0
+1.5
+1.0
+0.5
+2.5
+0.2
+0.8
+0.6
+0.4
+Mhh (GeV/c2)
+pairs/0.02 GeV/c2
+K+K−
+×103
+0.0
+2.0
+1.5
+1.0
+0.5
+2.5
+2
+4
+Mhh (GeV/c2)
+pairs/0.02 GeV/c2
+π+K−
+×103
+0.0
+2.0
+1.5
+1.0
+0.5
+2.5
+Mhh (GeV/c2)
+60
+K+π−
+40
+20
+Fig. 1: Distributions of invariant mass Mhh for 2002-2004 deuteron data (top row) and combined
+2007/2010 proton data (bottom row): π+π− pairs (1st column) , K+K− pairs (2nd column), π+K− and
+K+π− pairs (3rd column).
+pions and about 10% as kaons. The remaining particles are either protons, electrons or not clearly
+identified. About 60% are pion pairs (π+π−), about 2% are kaon pairs (K+K−) and about 8% are mixed
+pairs (π+K−, K+π−). The missing fraction refers to cases where at least one of the two hadrons cannot
+be accurately identified. The resulting statistics for unidentified and identified hadron pairs after applying
+all requirements are shown in Table 1.
+The invariant-mass distributions for the four opposite-charge combinations that can be formed using
+identified charged pions and kaons (π+π−, K+K−, π+K−, K+π−) are shown in Fig.1 for deuteron and
+proton targets. In the π+π− spectrum, the mass signatures of some mesons decays, such as K0 around
+500MeV/c2, ρ0 around 770MeV/c2, f0 around 980MeV/c2 and f2 around 1270MeV/c2, respectively,
+are clearly visible in both deuteron and proton data as expected from Ref. [20]. Other decays with more
+than two hadrons in the final state (like the decays of ω, η and η′) generate broader peaks and contribute
+less to the overall pion-pair invariant-mass spectra [20]. The K+K− invariant-mass distribution shows
+a very pronounced signal of the φ(1020) resonance close to its production threshold. The φ meson can
+also contribute to the pion pair spectra via the two-step decay φ(1020) → ρπ → π+π−π0. The invariant-
+mass distribution of K+K− pairs in the proton data shows indications of further broad peaks around
+1300MeV/c2 and 1500MeV/c2, which might be caused by f2(1270) and f ′
+2(1525). The invariant-mass
+distributions of π+K− and K+π− also show in each case one dominant channel caused by the decays
+of K∗(892). Further possible candidates for peaks in the Mhh spectra of the π+K− and K+π− pairs are
+K∗(1430) and K∗
+4(2045).
+3
+Results
+The asymmetries extracted from 6LiD and NH3 targets are presented in Figs. 2 and 3, respectively. They
+were evaluated in bins of x, z and Mhh as given in Table 2. For 6LiD, no significant asymmetry is observed
+in any variable for all pair combinations. For NH3, large negative asymmetries up to −0.07 are obtained
+for π+π− pairs in the region x > 0.03, which implies that both transversity distributions and polarised
+two-hadron interference fragmentation functions do not vanish, as already observed in Refs. [15, 16].
+For x < 0.03, these asymmetries are compatible with zero.
+The asymmetry measured with the 6LiD
+
+Transverse-spin-dependent azimuthal asymmetries of pion and kaon pairs produced ...
+5
+COMPASS proton data
+x
+2
+−
+10
+1
+−
+10
+〉
+θ
+sin
+RS
+φ
+sin
+UT,D
+A
+〈
+0.2
+−
+0.1
+−
+0
+0.1
+z
+0.5
+1
+0.2
+−
+0.1
+−
+0
+0.1
+)
+2
+c
+(GeV/
+hh
+M
+0.5
+1
+1.5
+2
+0.2
+−
+0.1
+−
+0
+0.1
+x
+2
+−
+10
+1
+−
+10
+〉
+θ
+sin
+RS
+φ
+sin
+UT,D
+A
+〈
+0.2
+−
+0
+0.2
+0.4
+z
+0.5
+1
+0.2
+−
+0
+0.2
+0.4
+)
+2
+c
+(GeV/
+hh
+M
+0.5
+1
+1.5
+2
+0.2
+−
+0
+0.2
+0.4
+x
+2
+−
+10
+1
+−
+10
+〉
+θ
+sin
+RS
+φ
+sin
+UT,D
+A
+〈
+0.2
+−
+0
+0.2
+0.4
+z
+0.5
+1
+0.2
+−
+0
+0.2
+0.4
+)
+2c
+(GeV/
+hh
+M
+0.5
+1
+1.5
+2
+0.2
+−
+0
+0.2
+0.4
+x
+2
+−
+10
+1
+−
+10
+〉
+θ
+sin
+RS
+φ
+sin
+UT,D
+A
+〈
+0.2
+−
+0
+0.2
+0.4
+z
+0.5
+1
+0.2
+−
+0
+0.2
+0.4
+)
+2c
+(GeV/
+hh
+M
+0.5
+1
+1.5
+2
+0.2
+−
+0
+0.2
+0.4
+π+π−
+π+K−
+K+π−
+K+K−
+Fig. 2: Hadron-pair transverse-spin-dependent asymmetries as a function of x, z and Mhh, extracted from
+the full data set collected with the 6LiD (deuteron) target. Systematic uncertainties are shown by the gray
+bands.
+target is compatible with zero within uncertainties over the whole x range. For both targets, no clear
+dependence on z can be observed, and for the NH3 target the asymmetry is observed to be negative in
+the whole range. For both targets, the Mhh-dependence shows negative asymmetry values in the region
+of the ρ0 mass.
+For K+K− pairs, the proton data show negative asymmetries in all three variables, while the deuteron
+data show indications for a positive signal. In particular the Mhh-dependence shows opposite signs for the
+asymmetries measured with the NH3 and 6LiD target, with an indication of a mirror-symmetric shape.
+In the case of π+K− and K+π− pairs, the deuteron data show asymmetries compatible with zero, while
+the proton data show slightly negative asymmetries.
+The HERMES Collaboration measured TSAs for π+π− pairs using electron-proton scattering [15].
+Given the wider kinematic coverage by COMPASS, the π+π− COMPASS asymmetry was re-evaluated
+in the region x > 0.032 to allow for a direct comparison. The comparison is shown in Fig. 4. The results
+are in very good agreement within statistical uncertainties.
+4
+Interpretation of the results
+The dihadron fragmentation functions entering the SIDIS cross section in Eq. (1) are non-perturbative
+objects. As such, they can not be calculated from first principles. Two classes of models have been
+proposed to describe them. In spectator-jet type models a mechanism different from that of the Collins FF
+is invoked to produce a non-vanishing H∢
+1 function. Such a mechanism involves the interference between
+
+6
+The COMPASS Collaboration
+COMPASS proton data
+x
+2
+−
+10
+1
+−
+10
+〉
+θ
+sin
+RS
+φ
+sin
+UT,p
+A
+〈
+0.1
+−
+0.05
+−
+0
+0.05
+0.1
+z
+0.5
+1
+0.1
+−
+0.05
+−
+0
+0.05
+0.1
+)
+2
+c
+(GeV/
+hh
+M
+0.5
+1
+1.5
+2
+0.1
+−
+0.05
+−
+0
+0.05
+0.1
+x
+2
+−
+10
+1
+−
+10
+〉
+θ
+sin
+RS
+φ
+sin
+UT,p
+A
+〈
+0.2
+−
+0.1
+−
+0
+0.1
+z
+0.5
+1
+0.2
+−
+0.1
+−
+0
+0.1
+)
+2
+c
+(GeV/
+hh
+M
+0.5
+1
+1.5
+2
+0.2
+−
+0.1
+−
+0
+0.1
+x
+2
+−
+10
+1
+−
+10
+〉
+θ
+sin
+RS
+φ
+sin
+UT,p
+A
+〈
+0.2
+−
+0.1
+−
+0
+0.1
+z
+0.5
+1
+0.2
+−
+0.1
+−
+0
+0.1
+)
+2c
+(GeV/
+hh
+M
+0.5
+1
+1.5
+2
+0.2
+−
+0.1
+−
+0
+0.1
+x
+2
+−
+10
+1
+−
+10
+〉
+θ
+sin
+RS
+φ
+sin
+UT,p
+A
+〈
+0.2
+−
+0.1
+−
+0
+0.1
+z
+0.5
+1
+0.2
+−
+0.1
+−
+0
+0.1
+)
+2c
+(GeV/
+hh
+M
+0.5
+1
+1.5
+2
+0.2
+−
+0.1
+−
+0
+0.1
+π+π−
+π+K−
+K+π−
+K+K−
+Fig. 3: Hadron-pair transverse-spin-dependent asymmetries as a function of x, z and Mhh, extracted from
+the full data set collected with the NH3 (proton) target. Systematic uncertainties are shown by the gray
+bands.
+the amplitudes of two competing channels for the production of the hadron pair, e.g. either the amplitude
+for direct production and the amplitude for resonance production [20,21], or the two amplitudes for the
+production of two different resonances [11]. A different approach is followed by the recursive string+3P0
+model of polarised quark fragmentation [22]. It is implemented in the StringSpinner package [23] for the
+simulation of the Collins effect for pseudoscalar mesons produced in the fragmentation of transversely
+Table 2: Bin limits of the variables x, z and Mhh (in units of GeV/c2) for the four types of pairs.
+x bin limits
+ππ
+0.003
+0.008
+0.013
+0.020
+0.032
+0.050
+0.080
+0.130
+0.210
+1.000
+πK/Kπ
+0.003
+0.013
+0.020
+0.032
+0.050
+0.080
+0.130
+1.000
+KK
+0.003
+0.008
+0.013
+0.020
+0.032
+0.080
+1.000
+z bin limits
+ππ
+0.20
+0.25
+0.30
+0.35
+0.40
+0.50
+0.65
+0.80
+1.00
+πK/Kπ
+0.20
+0.30
+0.35
+0.40
+0.50
+0.65
+1.00
+KK
+0.20
+0.40
+0.50
+0.65
+0.80
+1.00
+Mhh bin limits
+ππ
+0.0
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+1.2
+1.6
+100
+πK/Kπ
+0.0
+0.8
+0.9
+1.0
+1.2
+100
+KK
+0.9
+1.05
+1.15
+1.30
+1.50
+100
+
+Transverse-spin-dependent azimuthal asymmetries of pion and kaon pairs produced ...
+7
+x
+-2
+10
+-1
+10
+〉
+θ
+ sin
+RS
+φ
+sin
+UT,p
+A
+〈
+-0.1
+-0.05
+0
+0.05
+0.1
+z
+0.5
+1
+-0.1
+-0.05
+0
+0.05
+0.1
+x > 0.032
+x < 0.032
+−
+π
++
+π
+HERMES
+)
+2c
+(GeV/
+−
+π
++
+π
+M
+0.5
+1
+1.5
+2
+-0.1
+-0.05
+0
+0.05
+0.1
+Fig. 4: Comparison of π+π− pair asymmetries measured by the HERMES Collaboration [15] (blue
+open squares) with the results of the COMPASS Collaboration re-evaluated in the x > 0.032 region
+(black dots).
+polarised quarks in SIDIS with the PYTHIA 8 event generator [24].
+The classical string+3P0 model for the fragmentation of a transversely polarised quark qA is illustrated
+in Fig. 5. The string is stretched between the scattered quark qA and the target remnants along the quark
+direction and the string fragmentation occurs via tunneling of quark-antiquark pairs in the 3P0 state, i.e.
+with spin S = 1 and relative orbital angular momentum L = 1, such that the total angular momentum J is
+zero. Given the polarisation of qA, taken here along the normal to the figure plane, at the string breakings
+the spin and the transverse momentum of the quark and antiquark, as well as the transverse momentum
+of the produced hadron are fixed. The rank r indicates how far the hadron hr is produced from the
+fragmenting quark qA, with h1 being the hadron which contains qA. For odd (even) r the hadron hr is
+emitted to the left (right) with respect to the plane spanned by the momentum and polarisation vectors of
+the fragmenting quark. As an example, if the flavor of the fragmenting quark is qA = u and h1 = π+, it
+can be h2 = π− and opposite Collins asymmetries for oppositely charged hadrons are generated. Also,
+a dihadron asymmetry with the same sign as for positive hadrons is produced. StringSpinner uses the
+quantum mechanical formulation of this model, in which the spin effects depend on a complex parameter,
+tuned as in Ref. [23]. The initial quark polarisation is given by a parametrisation of the transversity
+PDF for valence u and d quarks. For this work we have used the default parametrisations, which were
+tuned to reproduce the π+ and π− Collins asymmetries measured by COMPASS on an NH3 target.
+The simulations were performed neglecting the intrinsic transverse momentum of the quarks, but it was
+checked that the dihadron asymmetries are not affected [25].
+In Fig. 6 the measured dihadron asymmetries (closed points) are compared to the simulated asymmetries
+(open points) for proton data. As can be seen, the simulation describes the data particularly well for
+π+π− and K+K− pairs, in all kinematic variables. The trend of the asymmetries as a function of x is
+Fig. 5: The string+3P0 mechanism of polarised quark fragmentation [25]. The closed (open) circles
+represent quark (antiquarks) at the string ends. The circular arrows above quarks show the orientation of
+their spins whereas the arrows at each string breaking L2,L3 ... represent the orientation of the relative
+orbital angular momenta of the q ¯q pairs. The straight arrows indicate the quark transverse momenta.
+
+C=qA
+.37
+(b)
+h18
+The COMPASS Collaboration
+2
+−
+10
+1
+−
+10
+x
+0.2
+−
+0.1
+−
+0
+0.1
+〉
+θ
+sin
+UT,p
+RS
+φ
+sin
+ A
+〈
+2
+−
+10
+1
+−
+10
+x
+0.2
+−
+0.1
+−
+0
+0.1
+〉
+θ
+sin
+UT,p
+RS
+φ
+sin
+ A
+〈
+2
+−
+10
+1
+−
+10
+x
+0.2
+−
+0.1
+−
+0
+0.1
+〉
+θ
+sin
+UT,p
+RS
+φ
+sin
+ A
+〈
+2
+−
+10
+1
+−
+10
+x
+0.1
+−
+0.05
+−
+0
+0.05
+0.1
+〉
+θ
+sin
+UT,p
+RS
+φ
+sin
+ A
+〈
+0.5
+1
+z
+0.2
+−
+0.1
+−
+0
+0.1
+0.5
+1 z
+0.2
+−
+0.1
+−
+0
+0.1
+0.5
+1 z
+0.2
+−
+0.1
+−
+0
+0.1
+0.5
+1 z
+0.1
+−
+0.05
+−
+0
+0.05
+0.1
+these results
+StringSpinner
+0.5
+1
+1.5
+2
+)
+ 2
+/c
+GeV
+ (
+hh
+M
+0.2
+−
+0.1
+−
+0
+0.1
+0.5
+1
+1.5
+2 )
+ 2
+/c
+GeV
+ (
+hh
+M
+0.2
+−
+0.1
+−
+0
+0.1
+0.5
+1
+1.5
+2 )
+ 2
+/c
+GeV
+ (
+hh
+M
+0.2
+−
+0.1
+−
+0
+0.1
+0.5
+1
+1.5
+2
+)
+ 2
+/c
+GeV
+ (
+hh
+M
+0.1
+−
+0.05
+−
+0
+0.05
+0.1
+π+π−
+π+K−
+K+π−
+K+K−
+Fig. 6: Comparison between π+π−, π+K−, K+π− and K+K− asymmetries for proton data (closed
+points) and results from simulations using StringSpinner (open points).
+mainly driven by the x-shape of the implemented transversity PDFs. While the z and Mhh dependences
+are predictions of the model. The large signal for π+π− and K+K− pairs can be understood in the
+approximation of u-quark dominance considering the fact that π+ or K+ are most likely produced at
+rank one, whereas π− or K− are produced at rank two. Regarding the π+K− and K+π− pairs, the
+simulated asymmetries are small and compatible with the data within uncertainties. This is expected
+considering the fact that, e.g., the π+ and the K− of a π+K− pair are most likely produced at rank one
+and three separated by a rank two neutral kaon. Thus the π+ and the K− are most likely emitted on the
+same side producing a small dihadron asymmetry.
+In corresponding simulations for deuteron data, dihadron asymmetries compatible with zero were found
+for all types of hadron pairs. This is in agreement with the data and is expected from the fact that the
+transversity PDFs for valence u and d-quarks have almost the same size but opposite sign.
+5
+Conclusions
+In this paper we present the results of a new measurement of transverse-spin-dependent asymmetries
+in hadron pair production in DIS of 160 GeV/c muons off transversely polarised deuteron (6LiD) and
+proton (NH3) targets. The measurement covers all possible combinations of oppositely charged pions
+and kaons observed in the COMPASS kinematic range.
+The deuteron data used in the analysis were collected during 2002 and 2004, while the proton data
+include two separate parts collected in 2007 and 2010. Both data sets were already used earlier to
+
+Transverse-spin-dependent azimuthal asymmetries of pion and kaon pairs produced ...
+9
+extract the Collins and Sivers asymmetries for semi-inclusively measured single hadrons, with separate
+publications for charged hadrons as well for identified pions and kaons. These two data sets are the
+largest ones available on this process, including e.g. 28M (4M) pion pairs in the proton (deuteron) data,
+and they provide important input for global analyses.
+The proton data show significant non-zero asymmetries. For π+π− pairs, values reach −7% in the region
+x > 0.032 and −2.5% in the invarinat-mass region around the ρ0-meson mass. Slightly negative asym-
+metries are observed for K+K− and K+π− pairs. The deuteron data show for all hadron combinations
+asymmetries compatible with zero, within statistical uncertainties.
+Acknowledgements
+This work was made possible thanks to the financial support of our funding agencies. We also acknowl-
+edge the support of the CERN management and staff, as well as the skills and efforts of the technicians
+of the collaborating institutes.
+
+10
+The COMPASS Collaboration
+References
+[1] M. Anselmino, A. Mukherjee and A. Vossen, Prog. Part. Nucl. Phys. 114 (2020) 103806,
+arXiv:2001.05415 [hep-ph].
+[2] J. C. Collins Nucl. Phys. B 396 (1993) 161–182, arXiv:hep-ph/9208213.
+[3] HERMES Collaboration, A. Airapetian et al., Phys. Rev. Lett. 94 (2005) 012002,
+arXiv:hep-ex/0408013.
+[4] COMPASS Collaboration, V. Alexakhin et al., Phys. Rev. Lett. 94 (2005) 202002,
+arXiv:hep-ex/0503002.
+[5] COMPASS Collaboration, M. Alekseev et al., Phys. Lett. B 673 (2009) 127–135,
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+[6] COMPASS Collaboration, M. Alekseev et al., Phys. Lett. B 692 (2010) 240–246,
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+[13] A. Bacchetta and M. Radici, Phys. Rev. D 67 (2003) 094002, arXiv:hep-ph/0212300.
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+arXiv:1202.6150 [hep-ex].
+[15] HERMES Collaboration, A. Airapetian et al., JHEP 06 (2008) 017,
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+arXiv:1401.7873 [hep-ex].
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+arXiv:hep-ex/0703049.
+[19] P. Abbon et al., Nucl. Instrum. and Meth. A631 (2011) no. 1, 26 – 39.
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+
diff --git a/S9A0T4oBgHgl3EQfD__i/content/tmp_files/load_file.txt b/S9A0T4oBgHgl3EQfD__i/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..3d9a525465c145f6928315ed5eed8c65b5b4451f
--- /dev/null
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@@ -0,0 +1,688 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf,len=687
+page_content='EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH COMPASS CERN-EP-2022–292 December 27, 2022 Transverse-spin-dependent azimuthal asymmetries of pion and kaon pairs produced in muon-proton and muon-deuteron semi-inclusive deep inelastic scattering The COMPASS Collaboration Abstract A set of measurements of azimuthal asymmetries in the production of pairs of identified hadrons in deep-inelastic scattering of muons on transversely polarised 6LiD (deuteron) and NH3 (proton) targets is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' All available data collected in the years 2002–2004 and 2007/2010 with the COMPASS spectrometer using a muon beam of 160 GeV/c at the CERN SPS were analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The asymmetries provide access to the transversity distribution functions via a fragmentation function that in principle may be independently obtained from e+e− annihilation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Results are presented, discussed and compared to existing measurements as well as to model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Asymmetries of π+π− pairs measured with the proton target as a function of the Bjorken scaling variable are sizeable in the range x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='032, indicating non-vanishing transversity distribution and di-hadron interference fragmentation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' As already pointed out by several authors, the small asymmetries of π+π− measured on the 6LiD target can be interpreted as indication for a cancellation of u and d-quark transversity distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' (to be submitted to Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' B) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='02013v1 [hep-ex] 5 Jan 2023 Transverse-spin-dependent azimuthal asymmetries of pion and kaon pairs produced .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' 1 1 Introduction The description of the nucleon spin structure remains one of the main challenges in hadron physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' For a polarised nucleon, a leading-twist description comprises eight transverse-momentum-dependent (TMD) parton distribution functions (PDFs), describing the distributions of longitudinal and transverse momenta of partons and their correlations with nucleon and quark polarisations [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' After integration over quark intrinsic transverse momentum kT, three PDFs fully describe the nucleon, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' the momentum distribution function f q 1 (x), the helicity distribution function gq 1(x) and the transversity distribution func- tion hq 1(x), where x denotes the Bjorken scaling variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' For simplicity, the latter will be referred to as ”transversity” throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' While the momentum and the helicity distribution functions have been measured with good accuracy, the knowledge on transversity is inferior but steadily increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Un- like f q 1 and gq 1, transversity is not accessible at leading twist in inclusive deep-inelastic scattering (DIS) because it is related to soft processes correlating quarks with opposite chirality, making it a chiral-odd function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Transversity can be accessed through observables that conserve chirality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' when it is cou- pled to a chiral-odd partner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' In this regard, measuring semi-inclusive deep-inelastic scattering (SIDIS) is advantageous as transversity is coupled to the chiral-odd fragmentation functions (FFs) that describe the hadronisation of a transversely polarised quark q into unpolarised final-state hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The major source of information on transversity has been measurements of transverse-spin-dependent asymmetries (TSAs) in single-hadron production in SIDIS (ℓN↑ → ℓ′hX), where transversity is coupled to the Collins FF [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Transverse-spin asymmetries define the size of the transverse-target-spin-dependent azimuthal modulations of the SIDIS cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The first measurement of the Collins asymmetries was performed by the HERMES Collaboration [3] using a transversely polarised hydrogen target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Size- able asymmetries were observed, suggesting non-zero transversity and Collins FFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The COMPASS Collaboration has the highest statistics data in this field, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' 28M pion pairs taken with the NH3 (pro- ton) target and 4M pion pairs taken with the 6LiD (deuteron) target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The COMPASS collaboration has delivered a full set of measurements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' both TSAs for unidentified charged hadrons [4] as well as pions and kaons [5] using the transversely polarised deuteron target, and TSAs for unidentified charged hadrons [6, 7] as well as pions and kaons [8] using the transversely polarised proton target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The TSAs measured with the polarised proton target showed a non-zero signal for Collins asymmetries with high statistics and a wide kinematic coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The TSAs measured with the polarised deuteron target are compatible with zero indicating a possible cancellation between up and down quark contributions to transversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Despite the lower accuracy of these data, they were shown to play a key role in the ex- traction of flavour-dependent transversity distribution functions and remain the only SIDIS measurement ever performed using a transversely polarised deuteron target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' In order to complete the COMPASS pro- gramme [9], a new high statistics measurement of TSAs using a transversely polarised deuteron target was performed in the last data taking campaign, in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' A promising alternative approach to access transversity is the measurement of TSAs in semi-inclusive production of pairs of hadrons of opposite charge (ℓ N↑ → ℓ′h+h−X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Following this approach, in this work π+π− and K+K− as well as π+K− and K+π− pairs will be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' In this case, transversity is coupled to the chiral-odd interference fragmentation function (IFF) H∢ 1 [10–12], which describes the hadronisation of a transversely polarised quark into a pair of unpolarised hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' At leading twist, and after integration over total transverse momentum, the differential cross section on a transversely polarised target comprises two terms and can be written as [13] d7σ dcosθ dMhh dφR dzdxdydφS = α2 2πQ2y � (1−y+ y2 2 )∑ q e2 q f q 1 (x) D1,q(z,M2 hh,cosθ) +S⊥(1−y)∑ q e2 q |p1 −p2| 2Mhh sinθ sinφRS hq 1(x) H∢ 1,q(z,M2 hh,cosθ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' (1) 2 The COMPASS Collaboration Here α is the fine-structure constant, D1,q(z,M2 hh,cosθ) is the spin-independent dihadron fragmentation function (DiFF), y is the fraction of the lepton energy in the laboratory frame transferred to the exchanged virtual-photon and Q2 the negative square of the four-momentum transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Here, z is the fraction of the virtual-photon energy carried by the hadron pair, Mhh its invariant mass and θ the polar angle of the positive hadron with respect to the two-hadron boost axis in the two-hadron rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The symbol S⊥ denotes the component of the target spin vector S perpendicular to the virtual-photon direction, with φS the azimuthal angle of the initial nucleon spin, φS′ the azimuthal angle of the spin vector of the fragmenting quark and φRS = φR −φS′ = φR +φS −π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The azimuthal angle φR is given as φR = (q×l)·R |(q×l)·R| arccos (q×l)·(q×R) |q×l||q×R| , (2) where l is the incoming lepton momentum, q the virtual-photon momentum and R the relative hadron momentum defined as R = (z2p1 −z1p2)/(z1 +z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The TSAs are experimentally accessible through the measured number of hadron pairs written as Nhh(x,y,z,M2 hh,cosθ,φRS) ∝ σUU(1+ f(x,y)PTDnn(y)AφRS UT sinθ sinφRS), (3) where f(x,y) is the target polarisation dilution factor, PT is the transverse polarisation of the target nu- cleons and Dnn the transverse-spin transfer coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' A more detailed discussion about the theoretical framework can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' As shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' [14], the asymmetry AsinφRS UT = |p1 −p2| 2Mhh ∑q e2 q hq 1(x) H∢ 1,q(z,M2 hh,cosθ) ∑q e2 q f q 1 (x) D1,q(z,M2 hh,cosθ) (4) is proportional to the product of the transversity distribution function hq 1(x) and the polarised two-hadron interference fragmentation function H∢ 1,q(z,M2 hh,cosθ), summed over the quark flavours q with charge eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Transverse-spin-dependent asymmetries of hadron pairs were first measured by the HERMES Collabo- ration [15] for pion pairs using a transversely polarised hydrogen target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' A sizeable signal was seen as a function of x, indicating a sizeable u-quark transversity and non-vanishing interference fragmentation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The COMPASS collaboration has published measurements of transverse spin asymmetries for pairs of unidentified hadrons produced on polarised deuterons [14] and polarised protons [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The COMPASS results obtained with the proton target showed significantly sizeable asymmetries and a clear slope in their x-dependence thanks to the high accuracy of the proton data set, while those extracted from deuteron-target data were found to be compatible with zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' An intriguing similarity between Collins-like single-hadron asymmetries for the positive and negative hadrons extracted from the SIDIS hadron-pair data and the standard Collins asymmetries is observed as a function of x, suggesting that both single hadron and hadron-pair transverse-spin dependent fragmentation functions are generated by the same elementary mechanism, as presented and discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' In this paper, we present a new measurement of TSAs for identified hadron-pairs using the full data set collected by the COMPASS Collaboration on transversely polarised deuteron (2002-2004) and proton (2007 and 2010) targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Only a brief description of the experimental setup and data analysis are given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' 3, as the same setup and methods of data cleaning, selection and extraction of TSAs as in previous COMPASS analyses [14,16] are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The measured asymmetries are presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' 3 and discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' 2 Experimental data and analysis The analysis presented in this paper is based on data collected in the years 2002-2004 and 2007/2010 using the COMPASS spectrometer [18] by scattering the naturally polarised µ+ beam of 160 GeV/c Transverse-spin-dependent azimuthal asymmetries of pion and kaon pairs produced .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' 3 delivered by the CERN SPS off transversely polarised 6LiD and NH3 targets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' For 6LiD, the average dilution factor calculated for semi-inclusive reactions is ⟨f⟩ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='38 and the average polarisation is ⟨PT⟩ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='47, while for NH3 the corresponding values are ⟨f⟩ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='15 and ⟨PT⟩ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='83, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The target consisted of two or three cylindrical cells assembled in a row, which can be independently polarised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' In 2002–2004, two cells were used, each 60 cm long and 3 cm in diameter, separated by a 10 cm gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' In 2007 and 2010, the target consisted of three cells of 4 cm diameter, with gaps of 5 cm in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The middle cell was 60 cm long and the two outer ones 30 cm long each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' From 2006 on, a new solenoidal magnet was used to polarise the target with a polar angle acceptance of 180 mrad as seen from the upstream end of the target, while in the earlier measurements with the 6LiD target the polar angle acceptance was 70 mrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' For the measurement of transverse spin effects, the target material was polarised along the vertical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' In order to reduce systematic effects, neighbouring cells were polarised in opposite directions allowing for simultaneous data taking with both target spin directions to reduce flux-dependent systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Furthermore, the polarisation was destroyed and built up in reversed direction every four to five days, in order to cancel residual acceptance effects associated with the longitudinal position of the target cells (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' position along the beam line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' For the data collected using a proton target, in the analysis, the central cell is divided into two parts, providing four data samples with two different orientations of polarisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Note that for the measurements in 2007 and in 2010 a similar spectrometer configuration was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' In the analysis, events with incoming and outgoing muons and at least two reconstructed charged hadrons originating from the interaction vertex inside the target cells are selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Equal flux through the whole target is obtained by requiring that the extrapolated beam tracks pass through all three cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' In order to select events in the DIS regime, requirements are applied on the squared four-momentum transfer, Q2 > 1 (GeV/c)2, and on the invariant mass of the final hadronic state, W > 5 GeV/c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Furthermore, the fractional energy transfer to the virtual photon is required to be y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 and y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='9 to remove events with poorly reconstructed virtual-photon energy and events with large radiative corrections, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' For a selected DIS event, all reconstructed hadrons originating from the interaction vertex are consid- ered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Only hadrons produced in the current fragmentation region are selected by requiring z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 for the fractional energy and xF > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 for the Feynmann-x variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The two-hadron sample consists of all combinations of oppositely charged hadrons built from the same DIS event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Exclusive dihadron produc- tion is suppressed by requiring the missing energy for each hadron pair to be greater than 3 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' As the azimuthal angle φR is only defined for non-collinear vectors R and q, a minimum value is required on the component of R perpendicular to q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' R⊥ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='07 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' After the application of all requirements, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='56 ×107 h+h− combinations remain for the deuteron data and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='5 ×107 h+h− pairs for the proton data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The RICH detector information is used to identify charged hadrons as pions or kaons in the momentum range between the Cherenkov threshold (about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='6 GeV/c and 9 GeV/c, respectively) and 50 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The detector set-up after the upgrade of 2005 and the particle identification (PID) procedure are fully described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' [19], while details on the likelihood PID method and the purity of identified samples are explained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' [5] and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' [8] for deuteron and proton targets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' In the kinematic domain of the COMPASS experiment, about 67% of the final-state charged hadrons are identified as Table 1: Final statistics for unidentified and identified charged-hadron pairs in deuteron (2002-2004) and proton (2007 and 2010) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Year Number of pairs (×106) h+h− π+π− π+K− K+π− K+K− 2002-2004 (deuteron) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='65 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
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+page_content='5 Mhh (GeV/c2) 60 K+π− 40 20 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' 1: Distributions of invariant mass Mhh for 2002-2004 deuteron data (top row) and combined 2007/2010 proton data (bottom row): π+π− pairs (1st column) , K+K− pairs (2nd column), π+K− and K+π− pairs (3rd column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' pions and about 10% as kaons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The remaining particles are either protons, electrons or not clearly identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' About 60% are pion pairs (π+π−), about 2% are kaon pairs (K+K−) and about 8% are mixed pairs (π+K−, K+π−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The missing fraction refers to cases where at least one of the two hadrons cannot be accurately identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The resulting statistics for unidentified and identified hadron pairs after applying all requirements are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The invariant-mass distributions for the four opposite-charge combinations that can be formed using identified charged pions and kaons (π+π−, K+K−, π+K−, K+π−) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 for deuteron and proton targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' In the π+π− spectrum, the mass signatures of some mesons decays, such as K0 around 500MeV/c2, ρ0 around 770MeV/c2, f0 around 980MeV/c2 and f2 around 1270MeV/c2, respectively, are clearly visible in both deuteron and proton data as expected from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Other decays with more than two hadrons in the final state (like the decays of ω, η and η′) generate broader peaks and contribute less to the overall pion-pair invariant-mass spectra [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The K+K− invariant-mass distribution shows a very pronounced signal of the φ(1020) resonance close to its production threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The φ meson can also contribute to the pion pair spectra via the two-step decay φ(1020) → ρπ → π+π−π0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The invariant- mass distribution of K+K− pairs in the proton data shows indications of further broad peaks around 1300MeV/c2 and 1500MeV/c2, which might be caused by f2(1270) and f ′ 2(1525).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The invariant-mass distributions of π+K− and K+π− also show in each case one dominant channel caused by the decays of K∗(892).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Further possible candidates for peaks in the Mhh spectra of the π+K− and K+π− pairs are K∗(1430) and K∗ 4(2045).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' 3 Results The asymmetries extracted from 6LiD and NH3 targets are presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' 2 and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' They were evaluated in bins of x, z and Mhh as given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' For 6LiD, no significant asymmetry is observed in any variable for all pair combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' For NH3, large negative asymmetries up to −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='07 are obtained for π+π− pairs in the region x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='03, which implies that both transversity distributions and polarised two-hadron interference fragmentation functions do not vanish, as already observed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' For x < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='03, these asymmetries are compatible with zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The asymmetry measured with the 6LiD Transverse-spin-dependent azimuthal asymmetries of pion and kaon pairs produced .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' 5 COMPASS proton data x 2 − 10 1 − 10 〉 θ sin RS φ sin UT,D A 〈 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 ) 2 c (GeV/ hh M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 x 2 − 10 1 − 10 〉 θ sin RS φ sin UT,D A 〈 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='4 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='4 ) 2 c (GeV/ hh M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='4 x 2 − 10 1 − 10 〉 θ sin RS φ sin UT,D A 〈 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='4 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='4 ) 2c (GeV/ hh M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='4 x 2 − 10 1 − 10 〉 θ sin RS φ sin UT,D A 〈 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='4 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='4 ) 2c (GeV/ hh M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='4 π+π− π+K− K+π− K+K− Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' 2: Hadron-pair transverse-spin-dependent asymmetries as a function of x, z and Mhh, extracted from the full data set collected with the 6LiD (deuteron) target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Systematic uncertainties are shown by the gray bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' target is compatible with zero within uncertainties over the whole x range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' For both targets, no clear dependence on z can be observed, and for the NH3 target the asymmetry is observed to be negative in the whole range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' For both targets, the Mhh-dependence shows negative asymmetry values in the region of the ρ0 mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' For K+K− pairs, the proton data show negative asymmetries in all three variables, while the deuteron data show indications for a positive signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' In particular the Mhh-dependence shows opposite signs for the asymmetries measured with the NH3 and 6LiD target, with an indication of a mirror-symmetric shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' In the case of π+K− and K+π− pairs, the deuteron data show asymmetries compatible with zero, while the proton data show slightly negative asymmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The HERMES Collaboration measured TSAs for π+π− pairs using electron-proton scattering [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Given the wider kinematic coverage by COMPASS, the π+π− COMPASS asymmetry was re-evaluated in the region x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='032 to allow for a direct comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The comparison is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The results are in very good agreement within statistical uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' 4 Interpretation of the results The dihadron fragmentation functions entering the SIDIS cross section in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' (1) are non-perturbative objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' As such, they can not be calculated from first principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Two classes of models have been proposed to describe them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' In spectator-jet type models a mechanism different from that of the Collins FF is invoked to produce a non-vanishing H∢ 1 function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Such a mechanism involves the interference between 6 The COMPASS Collaboration COMPASS proton data x 2 − 10 1 − 10 〉 θ sin RS φ sin UT,p A 〈 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='05 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='05 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 ) 2 c (GeV/ hh M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='05 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 x 2 − 10 1 − 10 〉 θ sin RS φ sin UT,p A 〈 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 ) 2 c (GeV/ hh M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 x 2 − 10 1 − 10 〉 θ sin RS φ sin UT,p A 〈 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 ) 2c (GeV/ hh M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 x 2 − 10 1 − 10 〉 θ sin RS φ sin UT,p A 〈 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 ) 2c (GeV/ hh M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 π+π− π+K− K+π− K+K− Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' 3: Hadron-pair transverse-spin-dependent asymmetries as a function of x, z and Mhh, extracted from the full data set collected with the NH3 (proton) target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Systematic uncertainties are shown by the gray bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' the amplitudes of two competing channels for the production of the hadron pair, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' either the amplitude for direct production and the amplitude for resonance production [20,21], or the two amplitudes for the production of two different resonances [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' A different approach is followed by the recursive string+3P0 model of polarised quark fragmentation [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' It is implemented in the StringSpinner package [23] for the simulation of the Collins effect for pseudoscalar mesons produced in the fragmentation of transversely Table 2: Bin limits of the variables x, z and Mhh (in units of GeV/c2) for the four types of pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' x bin limits ππ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='130 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
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+page_content='50 100 Transverse-spin-dependent azimuthal asymmetries of pion and kaon pairs produced .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' 7 x 2 10 1 10 〉 θ sin RS φ sin UT,p A 〈 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
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+page_content='032 − π + π HERMES ) 2c (GeV/ − π + π M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
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+page_content='1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' 4: Comparison of π+π− pair asymmetries measured by the HERMES Collaboration [15] (blue open squares) with the results of the COMPASS Collaboration re-evaluated in the x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='032 region (black dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' polarised quarks in SIDIS with the PYTHIA 8 event generator [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The classical string+3P0 model for the fragmentation of a transversely polarised quark qA is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The string is stretched between the scattered quark qA and the target remnants along the quark direction and the string fragmentation occurs via tunneling of quark-antiquark pairs in the 3P0 state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' with spin S = 1 and relative orbital angular momentum L = 1, such that the total angular momentum J is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Given the polarisation of qA, taken here along the normal to the figure plane, at the string breakings the spin and the transverse momentum of the quark and antiquark, as well as the transverse momentum of the produced hadron are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The rank r indicates how far the hadron hr is produced from the fragmenting quark qA, with h1 being the hadron which contains qA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' For odd (even) r the hadron hr is emitted to the left (right) with respect to the plane spanned by the momentum and polarisation vectors of the fragmenting quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' As an example, if the flavor of the fragmenting quark is qA = u and h1 = π+, it can be h2 = π− and opposite Collins asymmetries for oppositely charged hadrons are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Also, a dihadron asymmetry with the same sign as for positive hadrons is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' StringSpinner uses the quantum mechanical formulation of this model, in which the spin effects depend on a complex parameter, tuned as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The initial quark polarisation is given by a parametrisation of the transversity PDF for valence u and d quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' For this work we have used the default parametrisations, which were tuned to reproduce the π+ and π− Collins asymmetries measured by COMPASS on an NH3 target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The simulations were performed neglecting the intrinsic transverse momentum of the quarks, but it was checked that the dihadron asymmetries are not affected [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' 6 the measured dihadron asymmetries (closed points) are compared to the simulated asymmetries (open points) for proton data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' As can be seen, the simulation describes the data particularly well for π+π− and K+K− pairs, in all kinematic variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The trend of the asymmetries as a function of x is Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' 5: The string+3P0 mechanism of polarised quark fragmentation [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The closed (open) circles represent quark (antiquarks) at the string ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The circular arrows above quarks show the orientation of their spins whereas the arrows at each string breaking L2,L3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' represent the orientation of the relative orbital angular momenta of the q ¯q pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The straight arrows indicate the quark transverse momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' C=qA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='37 (b) h18 The COMPASS Collaboration 2 − 10 1 − 10 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 〉 θ sin UT,p RS φ sin A 〈 2 − 10 1 − 10 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 〉 θ sin UT,p RS φ sin A 〈 2 − 10 1 − 10 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
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+page_content='1 these results StringSpinner 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
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+page_content='5 2 ) 2 /c GeV ( hh M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
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+page_content='5 2 ) 2 /c GeV ( hh M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
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+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
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+page_content='5 2 ) 2 /c GeV ( hh M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='05 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='1 π+π− π+K− K+π− K+K− Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' 6: Comparison between π+π−, π+K−, K+π− and K+K− asymmetries for proton data (closed points) and results from simulations using StringSpinner (open points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' mainly driven by the x-shape of the implemented transversity PDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' While the z and Mhh dependences are predictions of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The large signal for π+π− and K+K− pairs can be understood in the approximation of u-quark dominance considering the fact that π+ or K+ are most likely produced at rank one, whereas π− or K− are produced at rank two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Regarding the π+K− and K+π− pairs, the simulated asymmetries are small and compatible with the data within uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' This is expected considering the fact that, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=', the π+ and the K− of a π+K− pair are most likely produced at rank one and three separated by a rank two neutral kaon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Thus the π+ and the K− are most likely emitted on the same side producing a small dihadron asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' In corresponding simulations for deuteron data, dihadron asymmetries compatible with zero were found for all types of hadron pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' This is in agreement with the data and is expected from the fact that the transversity PDFs for valence u and d-quarks have almost the same size but opposite sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' 5 Conclusions In this paper we present the results of a new measurement of transverse-spin-dependent asymmetries in hadron pair production in DIS of 160 GeV/c muons off transversely polarised deuteron (6LiD) and proton (NH3) targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The measurement covers all possible combinations of oppositely charged pions and kaons observed in the COMPASS kinematic range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The deuteron data used in the analysis were collected during 2002 and 2004, while the proton data include two separate parts collected in 2007 and 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Both data sets were already used earlier to Transverse-spin-dependent azimuthal asymmetries of pion and kaon pairs produced .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' 9 extract the Collins and Sivers asymmetries for semi-inclusively measured single hadrons, with separate publications for charged hadrons as well for identified pions and kaons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' These two data sets are the largest ones available on this process, including e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' 28M (4M) pion pairs in the proton (deuteron) data, and they provide important input for global analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The proton data show significant non-zero asymmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' For π+π− pairs, values reach −7% in the region x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='032 and −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='5% in the invarinat-mass region around the ρ0-meson mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Slightly negative asym- metries are observed for K+K− and K+π− pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' The deuteron data show for all hadron combinations asymmetries compatible with zero, within statistical uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' Acknowledgements This work was made possible thanks to the financial support of our funding agencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' We also acknowl- edge the support of the CERN management and staff, as well as the skills and efforts of the technicians of the collaborating institutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' 10 The COMPASS Collaboration References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' D 97 (2018) no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content=' 7, 074010, arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
+page_content='00962 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9A0T4oBgHgl3EQfD__i/content/2301.02013v1.pdf'}
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+arXiv:2301.00374v1 [cs.CL] 1 Jan 2023
+Optimizing Readability Using Genetic Algorithms
+Jorge Martinez-Gil
+Software Competence Center Hagenberg GmbH
+Softwarepark 32a, 4232 Hagenberg, Austria
+jorge. martinez-gil@ scch. at
+Abstract
+This research presents ORUGA, a method that tries to automatically optimize the readability of
+any text in English. The core idea behind the method is that certain factors affect the readability of
+a text, some of which are quantifiable (number of words, syllables, presence or absence of adverbs,
+and so on). The nature of these factors allows us to implement a genetic learning strategy to replace
+some existing words with their most suitable synonyms to facilitate optimization. In addition,
+this research seeks to preserve both the original text’s content and form through multi-objective
+optimization techniques. In this way, neither the text’s syntactic structure nor the semantic content
+of the original message is significantly distorted. An exhaustive study on a substantial number
+and diversity of texts confirms that our method was able to optimize the degree of readability in
+all cases without significantly altering their form or meaning. The source code of this approach is
+available at https://github.com/jorge-martinez-gil/oruga
+Keywords:
+Text readability, Text Optimization, Genetic Algorithms
+1. Introduction
+Readability is a measure that tells us how easy it is to read a text. It corresponds to the level
+of literacy that is expected from the readers in the target audience. In this way, readability is
+considered one of the most critical factors that facilitate the user experience when consuming in-
+formation. It is crucial because it is key to establishing a trusting relationship between information
+producers and consumers. It must be considered that some factors, such as complexity, legibility,
+or typography, contribute to making a text readable. However, not all factors are quantifiable and
+cannot be optimized by automatic techniques. In this paper, we focus solely and exclusively on
+factors of a quantifiable nature, which always revolve around basic or advanced statistics associated
+with the text to be optimized.
+Therefore, text readability refers to how simple it is to read and comprehend a given text,
+depending on its unique characteristics.
+These characteristics are usually measurable through
+metrics like the number of syllables in a sentence. The diversity of words used to create a readability
+
+score can be used to gauge this measure (Collins-Thompson, 2014). Therefore, ORUGA is intended
+to facilitate the reader’s ability to understand a text by optimizing its readability.
+In the context of this work, a clear distinction between readability and quality should be made.
+Quality emphasizes essential elements like grammar, spelling, and voice. However, the goal of
+producing textual information as plainly as possible and better matching it with its audience is
+what is meant when a text is said to be readable. In this way, text readability partially overlaps
+with the notion of text difficulty, which is also an essential aspect of human language, and has an
+impact on the daily lives of most people who consume written information.
+Our research is based on the premise that readability can be measured by considering met-
+rics related to the text and then using a specific mathematical formula to calculate it. Because
+different readability scores are calculated using different mathematical formulas, it is possible to
+design a strategy that replaces many of the terms in the text with synonyms using a global opti-
+mization scheme. Such optimization is about maximizing or minimizing the result yielded by such
+mathematical formulas at the user’s convenience. In practice, such a strategy can be implemented
+through a genetic algorithm, as shown throughout this work.
+Therefore, our research offers a viewpoint from the field of computer science, concentrating on
+the fundamental text representations and metrics utilized by readability assessment methods. In
+this way, the most significant contributions of this work to state-of-the-art are the following:
+• We present, for the first time, a technique that can automatically optimize the readability of
+any text, i.e., we can minimize or maximize the degree of readability of a text automatically
+without substantive changes.
+• We study which are the best sources of synonyms currently available for text readability opti-
+mization. Specifically, we have studied Wordnet, word2vec, and web scraping and established
+a classification around optimizing up to ten texts of different natures.
+• We present an additional method based on multi-objective optimization, whose mission is to
+ensure that the minimum number of words needed in the original text is replaced in such a
+way that the structure of the text is not impacted.
+• Last but not least, we studied several strategies that allow us to measure (and therefore
+optimize) the semantic distance between the original text and the generated text. In this
+way, the impact on the original text’s meaning can be controlled.
+This research work is structured as follows: Section 2 shows state-of-the-art methods and tools
+for improving text readability.
+Section 3 presents the technical details of our proposal; these
+technical details are based on the design of a genetic cutting strategy that allows us to explore
+a vast search space while consuming a reasonable amount of resources. Section 4 explains how
+to minimize the impact on the form of the original text using a multi-objective optimization
+2
+
+technique. Section 5 shows how to preserve the essence of the original text by ensuring that the
+distance between the original text and the text obtained is kept as small as possible. It is necessary
+to remark that sections 3, 4, and 5 present information about the design of different experiments
+and the raw results obtained, and their subsequent analysis. Finally, we conclude with the main
+lessons that can be learned from this research.
+2. State-of-the-art
+Let us begin with a formal definition of readability; for instance,(Chall & Dale, 1995) defines
+readability as the ”total number of elements in a given text that affect a reader’s success.” This
+reader’s success is a measure of how well a text that is read at an optimal speed can be understood.
+At the same time, (Mc Laughlin, 1969) defines readability as ”the level at which certain people find
+reading material convincing and understandable.” Beyond these definitions, we are particularly
+interested in the quantifiable aspects of readability, i.e., what can be objectively measured. Oth-
+erwise, it would not be easy to proceed to its optimization using a computer. Let us see what the
+literature says about these quantifiable aspects.
+2.1. Text readability in the scientific literature
+Readability metrics usually use simple features to calculate the degree of readability of a text.
+Some commonly used features are the number of sentences or words, the ratio of unique words,
+the total number of syllables, the proportion of unique words in the text, the number of digits, the
+number of words with many syllables, etc. Although, at first, it may seem that these metrics are
+simplistic, they are very commonly used for two important reasons: they are much cheaper and
+faster to use than the alternatives consisting of human surveys, and according to experts, they
+usually give exceptionally reliable results that are in line with reality.
+The goal of improving readability is to increase the chances that readers can understand the
+thoughts and ideas reflected in the text.
+So that misunderstanding is minimized, information
+processing is facilitated without requiring much effort and energy consumption. With this goal in
+mind, many sources can be found that advise how to improve a text’s readability. However, these
+are manually compiled protocols that a human operator must translate into reality by modifying
+the text manually. For example, for a given metric, it is better to shorten sentences; for another,
+it is better to replace complex words with simpler ones, etc.
+This is precisely where our contribution to the state-of-the-art lies. Optimizing texts automat-
+ically using a metric as a target means we do not have to concern about taking any manual action
+leading to altering the text. The genetic algorithm will find a way to proceed automatically.
+2.2. Why is text readability important?
+There are several contexts and population groups for which readability is critical. Especially
+when it is necessary to convey a written message to an audience. For example,
+3
+
+• Teachers need to be sure of the readability of a text before deciding whether it is appropriate
+for their students. This is particularly important in language learning. With the method
+presented here, the text can be optimized for a certain niche of learners.
+• In the world of advertising, readability allows for building a trust relationship between ad-
+vertisers and potential consumers. Advertising goods or services using texts with high read-
+ability is usually not a good idea since the message might not reach an essential part of the
+population. This is even more important in the search engine optimization sector, as many
+search engines use readability metrics as a ranking factor when responding to user searches.
+• Readability is also relevant for professionals who work on websites (Pantula & Kuppusamy,
+2022), news (Qin, 2021), or even educational materials (Ante, 2022).
+In some countries,
+there is even a legal requirement that government agencies provide textual information with
+certain readability levels to reach the entire population.
+2.3. Readability metrics
+There are several metrics to quantify how readable a text is (Meade & Smith, 1991). Most
+metrics have been designed for the English language (Maqsood et al., 2022), although works also
+explore readability in other languages (Madrazo Azpiazu & Pera, 2020). Without being exhaus-
+tive, we can mention, in chronological order, some of the metrics that enjoy or have enjoyed more
+significant popularity when dealing with the English language.
+Text readability depends not only on the characteristics of the text but also on the educational
+background of the individuals interested in understanding the text.
+We will see this reflected
+below when measuring readability using formulas. We will see how readability metrics take a text
+as input and calculate a numerical score that usually corresponds to the level of education required
+to understand the text.
+2.3.1. Dale-Chall readability
+The Dale-Chall readability formula (Dale & Chall, 1948) requires a list of 3,000 words that
+fourth-grade U.S. students could reliably understand, as shown in Equation 1.
+DCRF = 0.1579
+�difficult words
+total words
+× 100
+�
++ 0.0496
+�
+total words
+total sentences
+�
+(1)
+2.3.2. SMOG readability
+The SMOG readability level (Mc Laughlin, 1969) can be assessed through a formula originally
+used for checking health messages, as shown in Equation 2. It corresponds to the years of education
+necessary to understand the text.
+SMOG = 1.0430
+�
+number of polysyllables ×
+30
+number of sentences + 3.1291
+(2)
+4
+
+2.3.3. ARI readability
+The ARI assesses the U.S. grade level required to read a text (Senter & Smith, 1967). In some
+ways, it is similar to other formulas. Its difference is that rather than counting syllables, it counts
+characters: the more characters, the more complex the word. It also counts sentences as shown in
+Equation 3. This sets it apart from some other formulas.
+ARI = 4.71
+�total Characters
+total Words
+�
++ 0.5
+�
+total Words
+total Sentences
+�
+− 21.43
+(3)
+2.3.4. Flesch Kincaid readability
+Flesch Kincaid’s readability score, as shown in Equation 4, is a metric based on grade levels
+that is used commonly in the insurance industry. Grade levels made it much easier for people
+to understand (Kincaid et al., 1975). A Flesch Kincaid Grade Level (FKGL) between 8 and 10
+means that the text should be accessible to the public.
+FKGL remains the most widely-used
+formula today.
+FKGL = 206.835 − 1.015
+�
+total words
+total sentences
+�
+− 84.6
+�total syllables
+total words
+�
+(4)
+Over the past decade, several natural language processing (NLP) techniques have been pro-
+posed to determine the readability of a text. Thus, as opposed to the classical approach of using
+formulas that measure a limited set of text features, these new variants have attempted to mea-
+sure the difficulty of understanding sentences and words and even the complexity of the syntax
+(Martinc et al., 2021). Even some techniques based on predictors of readability, such as cohesion
+and coherence, have received considerable attention. However, so far, these approaches have yet
+to be able to predict the readability of a text better than the classical techniques discussed here
+(Todirascu et al., 2016).
+2.4. Semantic Similarity
+The field of semantic similarity measurement (Martinez-Gil, 2022) is one of the most ac-
+tive in several different research communities (information retrieval, database integration, nat-
+ural language processing, and so on) (Rus et al., 2013).
+This is due to its significant implica-
+tions on many available frameworks, methods, and tools (Navigli & Martelli, 2019) both in in-
+dustry and academia.
+The literature around this topic has skyrocketed in the last few years
+(Chandrasekaran & Mago, 2021). However, most research works focus on determining the similar-
+ity between words (Zhu & Iglesias, 2017), phrases, or documents (Martinez-Gil & Chaves-Gonzalez,
+2019; Martinez-Gil & Chaves-Gonzalez, 2020).
+In this way, rarely the likelihood of effectively throwing out the most similar words to a
+given one has been discussed. To this effect, there are synonym libraries that have been man-
+ually compiled and some word embedding techniques that do the job well.
+The most promi-
+5
+
+nent solutions in this direction are WordNet (Pedersen et al., 2004), word2vec (Mikolov et al.,
+2013), or BERT (Devlin et al., 2019).
+However, it must be taken into account that the re-
+source requirements for techniques based on the computation of word embeddings are very high
+(Martinez-Gil & Chaves-Gonzalez, 2022).
+2.5. Contribution over the state-of-the-art
+Determining text readability based on a formal analysis of the structures and words used has
+been a recurring theme in the literature over the last few years. As a result, many metrics have
+been proposed to measure text readability. A common denominator of all these metrics is that a
+high score usually means the text is difficult to understand. In other words, a higher degree of
+study is needed to understand it. For this reason, many communication professionals often use
+tools to help them discern whether the text fits a given audience. However, none of these tools
+can do the professional’s job automatically, which is why our contribution is essential.
+To the best of our knowledge, this is the first time anyone has proposed automatically improving
+the readability of text without significantly altering the content or the text’s form. In addition,
+we put this innovative method through its paces using a wide range of texts that varied in subject
+matter and level of readability. Furthermore, we provide the source code of the first implementation
+for anyone interested in experimenting with or improving the method.
+3. Part I: Design and Implementation of a Functional Solution
+In the following, we explain our proposal for text readability optimization using a method
+based on genetic algorithms. In this section, we outline the technical preliminaries, discuss the
+implementation, show an illustrative example of how our approach works in practice, and conduct
+an experimental study using real data and use cases that can help us get an idea of the performance
+of this approach in practice.
+3.1. Technical Preliminaries
+Our hypothesis is that we can find a set of synonyms to replace some words in the original
+text so that the value returned by the readability formula can be optimized. For example, if the
+readability formula rewards or penalizes long words, we have to find synonyms of less length. In
+reality, we only have to concern about understanding which formula best represents readability in
+our specific scenario and indicate it as a fitness function. The genetic algorithm will understand
+how to proceed. Thus, we are faced with a classical optimization problem.
+We intend to act only on the vocabulary since it is one of the most critical parts of that
+language. It is widely assumed that vocabulary is the essential part of a language because, without
+vocabulary, it is impossible to compose any message (Wilkins, 1972). We do not act on proper
+nouns, prepositions, or other stop words to avoid distorting the original message.
+6
+
+While this problem can be solved by a brute-force search over the range of the words of a given
+text w0, w1, ..., wn, the GA method scales very well when dealing with large texts. In this case, a
+brute-force search would be prohibitively expensive. We could act on other aspects, such as the
+structure, but then we would risk distorting the original text’s essence again.
+Our idea is to find the combination of words (which will be encoded in the form of an individual)
+that optimizes the desired objective. The choice of the fitness function is effortless and has the
+advantage that the solution automatically identifies what kind of words lead to the optimization.
+In this work, we work with three main sources of synonyms:
+• WordNet (Miller, 1995), which is a thesaurus that has been manually compiled. It is probably
+one of the most widely used dictionaries in information systems and will give us several
+alternatives to substitute each candidate word.
+No surprises are to be expected in this
+library, except perhaps the substitution of words with a synonym that has a sense far from
+the original. In any case, we have a solution to this problem.
+• wordvec (Mikolov et al., 2013), which is an approach that calculates the vectors associated
+with each word according to a textual corpus of relevance.
+Once each vector has been
+computed, a computation process can be performed by which the N vectors most similar
+to a given one are identified. This way, synonyms of relevance are obtained for the word in
+question. Note that this process is computationally expensive.
+• Web scraping (Mitchell, 2018), which consists of obtaining the synonyms from some websites,
+usually specialized, so that we can shuffle several alternatives per candidate word.
+This
+method obtains many high-quality word candidates, but it should be used responsibly because
+it can cause problems on the server side if many hundreds or thousands of requests are made.
+Therefore, the method is fine for experimentation, but it would be unreasonable to exploit
+it.
+3.2. Implementation
+The implementation of this novel approach is based on a classical optimization scheme using
+genetic algorithms. Algorithm 1 briefly explains in pseudo-code how the whole process is performed
+by adapting the classical structure of the genetic algorithm (Forrest, 1996) where different operators
+capable of implementing selection, cross-over and mutation processes are considered in order to
+evolve a given population towards the desired objective.
+Please note that before starting the evolution process, a pre-processing of the text must be
+done in order to identify the candidate words to be replaced by a synonym. We propose that all
+words are candidates except: proper names, stop words, and prepositions. The reason for this is
+that they are often words for which it is really difficult to find a synonym.
+7
+
+Algorithm 1 Optimizing Readability Using Genetic Algorithm
+1: procedure ORUGA
+2:
+population ← generationRandomIndividual (population)
+3:
+calculateReadabilityScore (population)
+4:
+while (stop condition not reached) do
+5:
+parents ← selectionOfIndividuals (population)
+6:
+offspring ← Crossover (parents)
+7:
+offspring ← Mutation (offspring)
+8:
+offspring ← calculateReadabilityScore (offspring)
+9:
+population ← updatePopulation (offspring)
+10:
+endwhile
+11:
+optimizedText ← optimizedIndividual (population)
+12:
+optimizedText ← correctErrorsIfNecessary (optimizedText)
+13:
+return optimizedText
+In relation to the genetic algorithm itself, the parametric details of the solution will be discussed
+below, but it is possible to see how we implement it in the form of a classical evolutionary process.
+This means that a population of individuals is selected randomly, and their readability score is
+calculated. We then proceed with an iterative process of selection, crossover, and mutation; the
+best individuals are passed from generation to generation until one of the stopping criteria is met:
+the highest possible has been reached (unlikely), or the number of iterations has been exhausted
+(very likely).
+Finally, at the end of the evolutionary process, we correct the text in case grammatical errors
+are produced by substituting a synonym that does not fit the tense and the form of the sentence
+in which it is framed. This way, we obtain a corrected readability score, which may vary slightly
+from the one derived automatically. In return, we ensure that the results are usable, or at least
+close to being usable.
+3.3. Illustrative examples
+Example 1 shows us how ORUGA works with written material about science extracted from
+Wikipedia. In fact, our aim is to observe how ORUGA behaves when trying to minimize the FKGL
+readability score using synonyms from the library WordNet. Let us remember that FKGL (or one
+of its variants) is probably the most widely used metric and its optimization brings advantages
+in several fields. Please note that the text to be treated can be of any length, but to facilitate
+the presentation to the reader, we have chosen (and will choose throughout this paper) one that
+contains only several sentences.
+8
+
+Example 1. Science
+Original text Source: Wikipedia
+“The sea moderates the climate and has important roles in the water cycle, car-
+bon cycle, and nitrogen cycle. Humans harnessing and studying the sea have been
+recorded since ancient times, and evidenced well into prehistory, while its modern
+scientific study is called oceanography. The most abundant solid dissolved in sea-
+water is sodium chloride.
+The water also contains salts of magnesium, calcium,
+potassium, and mercury, amongst many other elements, some in minute concentra-
+tions. Salinity varies widely, being lower near the surface and the mouths of large
+rivers and higher in the depths of the ocean; however, the relative proportions of
+dissolved salts vary little across the oceans.” FKGL score: 14.06.
+ORUGA - minimizing the FKGL score - library Wordnet
+The sea moderates the climate and has important part in the H2O cycle, C cycle,
+and N cycle. Humans harnessing and studying the sea have been taped since ancient
+times, and attested well into prehistory, while its modern scientific study is called
+oceanography. The most abundant solid fade out in brine is Na chloride. The H2O
+too contains salts of magnesium, calcium, potassium, and mercury, amongst many
+other elements, some in min concentrations. Salinity changes widely, being got down
+near the surface and the mouths of large rivers and higher in the depth of the ocean;
+however, the relative proportions of fade out salts change little across the oceans.
+FKGL score: 11.85
+As can be seen, ORUGA can optimize the textual input by first automatically identifying which
+words can be replaced by a synonym, and then undertaking a process of searching for synonyms
+that improve the results of the metric to be optimized. In this way, the impact on the initial
+message is minimal, although it is true that some synonyms can slightly distort the meaning of the
+text, and therefore final supervision by the user is required. However, there is no need to concern
+because this problem will be addressed later in the paper.
+Let us look now at Example 2, which is a written text about the history of Austria that has
+been also extracted from Wikipedia, and we would like to to minimize the FKGL readability score
+using synonyms automatically obtained by web scraping.
+9
+
+Example 2. History
+Original text Source: Wikipedia
+“Austria emerged from the remnants of the Eastern and Hungarian March at the
+end of the first millennium. Originally a margraviate of Bavaria, it developed into
+a duchy of the Holy Roman Empire in 1156 and was later made an archduchy
+in 1453. In the 16th century, Vienna began serving as the empire administrative
+capital and Austria thus became the heartland of the Habsburg monarchy. After
+the dissolution of the Holy Roman Empire in 1806, Austria established its own
+empire, which became a great power and the dominant member of the German
+Confederation. The defeat in the Austro-Prussian War of 1866 led to the end of the
+Confederation and paved the way for the establishment of Austria-Hungary a year
+later.” FKGL score: 13.20
+ORUGA - minimizing the FKGL score - web scraping
+Austria looms from the debris of the Eastern and Hungarian March at the end of
+the first millennium. Originally a margravate of Bavaria, it matured within a duchy
+of the Holy Roman Empire in 1156 and was next made an arch duchy in 1453.
+In the 16th century, Vienna lead plate as the command departmental central and
+Austria thus come the heartland of the Habsburg monarchy. After the divorce of
+the Holy Roman Empire in 1806, Austria settled its own empire, that come a great
+power and the dominant branch of the German Confederation. The defeat in the
+Austro-Prussian War of 1866 led to the end of the Confederation and brick the way
+for the founding of Austria-Hungary a year later. FKGL score: 11.35
+Once again, we can see how the genetic algorithm has acted intelligently to decrease the text
+readability score and therefore make the text accessible to more people.
+It is clear that the
+suitability of some words may be subject to debate, but the first objective of this research, i.e., to
+optimize the readability score, has been achieved. As we mentioned before, we will be concerned
+to outline a final product later in this paper.
+Finally, let us look at the opposite case, i.e., let us see if we can make a text more difficult to
+read without losing its essence. In Example 3, we have a text about sports also extracted from
+Wikipedia, and we do not want to make the text accessible to as many people as possible, but on
+the contrary, we want to increase the level of readability. We will now try a different metric, e.g.,
+ARI.
+10
+
+Example 3. Sports
+Original text Source: Wikipedia
+“Real Madrid Club de Futbol, meaning Royal Madrid Football Club, commonly
+referred to as Real Madrid, is a Spanish professional football club based in Madrid.
+Founded in 1902 as Madrid Football Club, the club has traditionally worn a white
+home kit since its inception. The honorific title real is Spanish for Royal and was
+bestowed to the club by King Alfonso XIII in 1920 together with the royal crown in
+the emblem. Real Madrid have played their home matches in the Santiago Bernabeu
+Stadium in downtown Madrid since 1947. Unlike most European sporting entities,
+Real Madrid members (socios) have owned and operated the club throughout its
+history.” ARI score: 12.69.
+ORUGA - maximizing the ARI score - web scraping
+Real Madrid Club de Futbol, connotation Royal Madrid Football Club, frequently
+referred to as Real Madrid, is a Spanish professional football business established
+in Madrid.
+Founded in 1902 as Madrid Football Club, the business has consis-
+tently timeworn an alabaster home kit since its inception. The appellation title real
+is Spanish for Royal and was entrusted to the business by King Alfonso XIII in
+1920 together alongside the aristocratic culmination in the emblem. Real Madrid
+have played their familiar matches in the Santiago Bernab´eu Stadium in downtown
+Madrid afterward 1947. Unlike most European sporting entities, Real Madrid as-
+semblage (socios) have owned and negotiated the business throughout its history.
+ARI score: 15.53
+As can be seen, after processing the fragment related to Real Madrid extracted from Wikipedia,
+the genetic algorithm selects synonyms that are much longer and more complicated to read to
+increase the ARI score. Although a use case consisting of making the text less accessible to people
+is hard to imagine, it may find application in some specific niches of learning, education, etc.
+3.4. Experimental study
+In this section, we explain the details of an empirical study that we have carried out to know
+the feasibility of our new method. To do so, we first established the conditions of the experi-
+ments by setting up an experimental setup. Secondly, we have run and obtained the raw results.
+Furthermore, thirdly, and lastly, we have proceeded with the analysis of the data obtained.
+11
+
+3.4.1. Experimental setup
+Adjusting the parameters of the genetic algorithm differs from the focus of this work. Therefore,
+we have chosen a classical parameter setting, which has been shown to work quite well. Standard
+tuning of the parameters, e.g., through a grid search, is a possible line of future work. In the
+meantime, the parameters we have used for our experiments are the following:
+• Population size {10, 15, 20}: 20
+• Number of parents mating {10, 15, 20}: 10
+• Number of genes: one per candidate word to be substituted by a synonym
+• Fitness function: the user can choose among Equations 1, 2, 3, 4
+• Stop condition: {100, 200, 300}: 300 generations
+To test whether our approach can optimize text readability, we semi-randomly selected ten texts
+extracted from Wikipedia. These texts are classified into several categories engineering, geography,
+history, science, and sports. Some metrics, such as SMOG, require at least 30 sentences to apply
+their formula. For cases where we do not reach 30 sentences, we will duplicate the text until
+we reach that number of sentences. It should also be noted that depending on which readability
+category each use case represents. For example, magenta represents the highest difficulty (FKGL
+between 15 and 18), equivalent to an academic paper.
+Red represents medium-high difficulty
+(FKGL between 12 and 15). The black color represents a medium difficulty (FKGL between 9
+and 12). Furthermore, the blue color represents a low-medium difficulty. It is estimated that up
+to 80% of the native English-speaking population could understand text with an FKGL between
+6 and 9, which is what the blue value represents.
+Furthermore, every experiment was run on a standard computer with 32 GB of primary memory
+and an Intel Core i7-8700 processor running at 3.20 GHz on Microsoft Windows 10 64-bit. Most of
+the functionality has been implemented using the library PyGAD1, an open-source Python library
+for building genetic algorithms.
+3.4.2. Experiments
+The first experiment is the one that is the most useful in practice. It consists of trying to
+minimize the readability of the ten proposed texts so that the processed text can reach the largest
+possible audience. To do that, we want to use readability metrics which estimate the readability
+of a text based on simple aspects such as syllable and word counts. We use the FKGL score since
+this metric, or some of its variants, has been used for decades on traditional texts, and it is still
+one of the most common and widely used traditional readability measures.
+1https://pypi.org/project/pygad/
+12
+
+01
+02
+03
+04
+05
+06
+07
+08
+09
+10
+1
+1.5
+2
+2.5
+#Use Case
+Improvement (GL)
+Figure 1: Results for the minimization of the FKGL score using WordNet
+Fig 1 shows us the results obtained when reducing the FKGL for the texts under consideration
+using the library WordNet.
+The results shown are the summary of ten independent runs per
+use case. Since we are dealing with cold start methods with random values, and there is even a
+component of randomness in the mutations, we will almost always get a different result, so it is
+essential to show the results in this way.
+Positive results have been obtained in all 100 experiments performed (ten runs for ten use cases).
+In addition, minimum improvements, average improvements, and even maximum improvements
+have been achieved in the range between 0.77 and 2.73 points on the FKGL scale.
+Fig 2 shows us the results obtained using the synonym calculation method using word2vec.
+As it is possible to see, the variance of the results is enormous, which means that using this
+library will make the results not very predictable. At the same time, as in the previous case,
+all 100 experiments were able to obtain readability improvements that ranged between 0.42 and
+3.90-grade levels.
+Fig 3 shows us the results we have obtained by searching for synonyms with web scraping
+techniques. The results this time have a smaller variance than in the previous case. In addition,
+we have again obtained favorable results in all 100 experiments performed.
+The optimization
+achieved ranges between 1.02 and 4.20-grade levels.
+Table 1 shows the summary results of the 300 experiments performed (100 for each synonym
+library). For WordNet, a median optimization of 1.63-grade levels is expected, very similar to
+13
+
+01
+02
+03
+04
+05
+06
+07
+08
+09
+10
+1
+2
+3
+4
+#Use Case
+Improvement (GL)
+Figure 2: Results for the minimization of the FKGL score using word2vec
+01
+02
+03
+04
+05
+06
+07
+08
+09
+10
+1
+1.5
+2
+2.5
+3
+3.5
+4
+#Use Case
+Improvement (GL)
+Figure 3: Results for the minimization of the FKGL score using Web Scraping
+14
+
+Library
+Minimum
+Median
+Maximum
+WordNet
+0.77
+1.63
+2.73
+word2vec
+0.42
+0.96
+3.90
+Web scraping
+1.02
+1.61
+4.20
+Table 1: Summary of the results obtained using the different libraries of synonyms
+Use case
+DCRF
+SMOG
+ARI
+FKGL
+01/science
+1.11 ± 0.17
+2.03 ± 0.13
+2.90 ± 0.27
+2.53 ± 0.15
+02/history
+0.29 ± 0.12
+0.71 ± 0.08
+0.84 ± 0.05
+0.90 ± 0.04
+03/geography
+0.65 ± 0.07
+0.91 ± 0.04
+1.17 ± 0.12
+1.33 ± 0.04
+04/sports
+0.78 ± 0.14
+0.42 ± 0.12
+0.83 ± 0.05
+0.93 ± 0.10
+05/engineering
+0.59 ± 0.06
+0.60 ± 0.10
+0.66 ± 0.10
+0.75 ± 0.09
+06/geography
+0.52 ± 0.15
+0.70 ± 0.10
+0.77 ± 0.19
+0.77 ± 0.18
+07/engineering
+1.03 ± 0.15
+0.51 ± 0.03
+1.12 ± 0.21
+1.32 ± 0.01
+08/history
+0.77 ± 0.09
+0.27 ± 0.06
+0.44 ± 0.00
+0.73 ± 0.17
+09/sports
+0.46 ± 0.06
+1.06 ± 0.28
+0.99 ± 0.12
+0.81 ± 0.10
+10/history
+0.74 ± 0.04
+0.65 ± 0.03
+0.77 ± 0.13
+0.93 ± 0.04
+Table 2: Summary of the results obtained for the experiments performed w.r.t. minimization. The numerical
+values represent absolute improvements after using ORUGA
+that achieved by web scraping. The median optimization using word2vec is the worst of the three
+libraries considered. Please note that we are not judging here the quality of the replacements, but
+the values to optimize the input text independently of the quality of the final result.
+Table 2 shows a summary of all the results we have obtained. The texts have been randomly
+obtained from Wikipedia. The range of values shown indicates the minimum value that could be
+reached and the maximum value (for minimization and maximization problems respectively).
+We analyze the four formulas that we consider most representative, but the analysis of other
+formulas would not be a problem. When working with SMOG, we must be careful because the text
+to be analyzed requires at least 30 sentences. We have duplicated the text for these experiments
+so that these 30 sentences can be used.
+Table 3 shows a summary of all the results we have obtained when maximizing the readability
+scores. As can be seen by comparing with the table above, it is much easier to increase the level
+of readability than to decrease it, i.e., it is easier to make a text difficult to read than the other
+way around.
+3.4.3. Discussion
+We have seen how it is possible to build a solution that optimizes the readability of texts of
+different natures. Moreover, such optimization is done respecting the content and the form of
+such texts, trying to minimize the impact of word replacements by synonyms that better fit the
+readability criteria of the different formulas we have studied. We have seen how optimization can
+occur in two ways. First, it can be done in such a way as to reduce the readability score, which
+15
+
+Use case
+DCRF
+SMOG
+ARI
+FKGL
+01/science
+1.98 ± 0.16
+1.30 ± 0.06
+1.80 ± 0.08
+2.09 ± 0.14
+02/history
+1.85 ± 0.25
+1.99 ± 0.09
+3.88 ± 0.28
+3.02 ± 0.08
+03/geography
+2.38 ± 0.12
+1.23 ± 0.08
+4.44 ± 0.13
+4.00 ± 0.44
+04/sports
+2.09 ± 0.40
+2.79 ± 0.07
+4.63 ± 0.40
+4.26 ± 0.04
+05/engineering
+2.10 ± 0.43
+2.16 ± 0.11
+4.55 ± 0.10
+4.23 ± 0.35
+06/geography
+1.53 ± 0.21
+1.81 ± 0.22
+2.96 ± 0.04
+2.90 ± 0.41
+07/engineering
+1.49 ± 0.38
+1.71 ± 0.04
+3.87 ± 0.46
+3.34 ± 0.16
+08/history
+1.43 ± 0.18
+3.09 ± 0.13
+4.11 ± 0.63
+3.70 ± 0.21
+09/sports
+1.60 ± 0.21
+2.29 ± 0.16
+3.54 ± 0.28
+2.56 ± 0.04
+10/history
+2.06 ± 0.08
+2.64 ± 0.04
+3.99 ± 0.25
+3.27 ± 0.52
+Table 3: Summary of the results obtained for the experiments performed w.r.t. maximization. The numerical
+values represent absolute improvements after using ORUGA
+will allow more people to understand the text perfectly. This is undoubtedly the most practical
+option in practice. Furthermore, secondly, it can be done to increase the readability score, allowing
+a smaller number of people to understand the processed text unambiguously. This option has less
+practical utility than can be discerned at first glance.
+Based on the research we have carried out, possible improvements can be made. For example,
+a multi-objective optimization algorithm could simultaneously optimize the readability score by
+affecting as few words as possible. In this way, the impact of our method on the original text would
+be even more negligible. A solution front could allow the human operator to decide the trade-off
+between the score modification and the replaced words.
+Finally, and as a limitation of our method, it can be observed that sometimes the processed
+text contains minor grammatical errors. This is because we have yet to use techniques that allow,
+for example, to choose the appropriate verb tense for the context. Here there are two alternatives.
+On the one hand, we can implement better methods for correcting grammatical errors to obtain a
+corrected readability score; on the other hand, we can give the user the option to edit or choose
+the word that best fits each moment.
+4. Part II: Minimizing the impact on the form of the original text
+While in Section 3, we have seen that it is possible to design a functional solution to optimize
+the text readability, we have also seen that the approach can be intrusive at times. That is, the
+replacement of many words by synonyms can lead to a distortion of the original message. For
+this reason, in this section, we focus on minimizing the impact of ORUGA on the original text
+by replacing as few words as possible. To do so, we will build a solution based on multi-objective
+optimization (MOO) that allows us to optimize readability and simultaneously minimize the num-
+ber of replacements. This section comprises the technical preliminaries, the implementation, some
+illustrative examples, and an empirical study to test several texts of different natures.
+16
+
+4.1. Technical Preliminaries
+We have already seen how in the field of text readability, there has been a great effort to build
+straightforward formulas that can be understood by people and have a good correlation to how
+easily a text can be read from a human perspective. Now we go one step further to obtain a higher
+quality result. MOO is a strategy in which two or more objectives are simultaneously optimized.
+This is the situation we find ourselves in, given that we want: on the one hand, to improve the
+readability of a text, and on the other hand, we want to reduce as much as possible the number of
+words that need to be replaced to improve readability, and thus to minimize the impact that our
+approach has on the original text form.
+Furthermore, MOO is useful when decisions must be made despite potential trade-offs between
+more than one orthogonal objective. Again, this is our situation because our goals of maximizing
+readability while simultaneously replacing the fewest possible words require us to pursue two
+completely different goals. In situations like this, different solutions can simultaneously fulfill all
+objectives. As a result, all optimal solutions ought to be regarded as equivalent merit without
+any external evaluation from a human operator (Martinez-Gil & Chaves-Gonzalez, 2021). More
+formally, we can model a MOO problem as expressed in Equation 5.
+min (s1(⃗x), s2(⃗x), . . . , sn(⃗x))
+subject to ⃗x ∈ X
+(5)
+In MOO, no solution addresses all objective functions simultaneously. As a result, the priority
+should be placed on finding solutions that cannot make any goals better without making at least
+one of the other goals worse. Therefore, a solution ⃗x1 ∈ X is said to dominate another one ⃗x2 ∈ X
+if the conditions expressed in Equation 6 are met.
+si(⃗x1) ≤ si(⃗x2) ∀i ∈ {1, 2, 3, . . ., n}
+sj(⃗x1) < sj(⃗x2) ∃j ∈ {1, 2, 3, . . ., n}
+(6)
+In this way, a solution ⃗x ∈ X is optimal if no solution might dominate it. An element x is
+said to dominate another element y if x is not worse than y concerning all the goals and is strictly
+better than y for at least one. The elements of the search space that are not dominated give rise
+to a Pareto front, which represents the best possible solutions to the orthogonal objectives.
+4.2. Implementation
+There are several implementations for MOO strategies Kukkonen & Lampinen (2005); Zhang & Li
+(2007). It is beyond the scope of this paper to consider them all. However, we will look at one of
+the best ones, NSGA-II (Deb et al., 2002). This strategy is summarized in Algorithm 2. Its mode
+of operation is based on the concepts of fronts and crowding distance.
+17
+
+NSGA-II adheres to the basic structure of a genetic algorithm but employs a different approach
+to mating and selection for survival. In the NSGA-II, the first step is to select individuals in a
+front-wise fashion. In this way, a situation will arise where it will be necessary to divide a front
+because it will not be possible for all individuals to survive.
+The solutions are chosen according to the crowding distance for this particular splitting front.
+Within the parameters of the objective space, the Manhattan Distance corresponds to the crowding
+distance. On the other hand, it is desired that the extreme points be maintained with each new
+generation, and as a result, an infinite crowding distance is assigned to them. Algorithm 2 shows
+us a possible implementation of this approach.
+Algorithm 2 MOO Technique for Optimizing Text Readability
+1: procedure ORUGA2-MOO
+2:
+population ← initializePopulation ()
+3:
+population ← generationRandomIndividual (population)
+4:
+calculateReadabilityScore (population)
+5:
+assignRankBasedOnPareto (population)
+6:
+auxiliarPopulation ← generationChildPopulation (population)
+7:
+while (stop condition not reached) do
+8:
+for (each individual in population and auxiliarPopulation) do
+9:
+solution ← calculateReadabilityScore (population)
+10:
+solution ← assignRankBasedOnPareto (population)
+11:
+solution ← generateNonDominateSolutions (population)
+12:
+solution ← determiningCrowdingDistance (population)
+13:
+for (each solution) do
+14:
+population ← addingSolutionsNextGeneration (population)
+15:
+end for
+16:
+end for
+17:
+population ← selectPointsLowFrontHighCrowdingDistance (population)
+18:
+population ← generationNextPopulation (population)
+19:
+end while
+20:
+optimizedText ← optimizedIndividual (solution)
+21:
+optimizedText ← correctErrorsIfNecessary (optimizedText)
+22:
+return optimizedText
+NSGA-II is an example of a genetic algorithm, and it possesses the three characteristics listed
+below: It operates based on an elitist principle, which states that only the most privileged mem-
+bers of a population are permitted to be passed down to subsequent generations. In addition, it
+employs a mechanism specifically designed to preserve diversity (crowding distance). As a direct
+consequence of this, it can identify non-dominated solutions.
+4.3. Illustrative examples
+Since we are trying to optimize two orthogonal objectives simultaneously, it is impossible to
+offer a single solution. Nevertheless, we can put in the hands of the human operator a front of
+18
+
+solutions ranging from a total optimization by modifying the most significant number of words to
+a minor optimization by touching a minimum number of words. It is up to the human operator to
+decide which solution to choose.
+Example 4 shows a real trace that controls the number of words to be replaced using the
+MOO technique known as NSGA-II. The text on which it operates is about geography and has
+been extracted from Wikipedia. Once again, we must insist that although, in theory, it would
+be possible to edit the words manually to satisfy the criteria of a given metric, this approach is
+transparent and works automatically for any desired metric. Please note that the words in blue
+are candidates to be replaced by a synonym.
+Example 4. Geography Source: Wikipedia
+Goal: Minimize the FKGL score by using Wordnet synonyms and minimize the words to
+be replaced using NSGA-II.
+“Niagara Falls is a group of three waterfalls at the southern end of Niagara Gorge,
+spanning the border between the province of Ontario in Canada and the state of New
+York in the United States. The largest of the three is Horseshoe Falls, which straddles
+the international border of the two countries. It is also known as the Canadian Falls. The
+smaller American Falls and Bridal Veil Falls lie within the United States. Bridal Veil Falls
+is separated from Horseshoe Falls by Goat Island and from American Falls by Luna Island,
+with both islands situated in New York. Formed by the Niagara River, which drains Lake
+Erie into Lake Ontario, the combined falls have the highest flow rate of any waterfall in
+North America that has a vertical drop of more than 50 m (160 ft). During peak daytime
+tourist hours, more than 168,000 m3 (5.9 million cu ft) of water goes over the crest of the
+falls every minute.” FKGL score: 10.72.
+Words to be replaced
+FKGL expected
+5
+10.43 (▽ 2.71%)
+6
+10.28 (▽ 4.10%)
+7
+10.13 (▽ 5.50%)
+8
+9.98 (▽ 6.90%)
+9
+9.91 (▽ 7.56%)
+10
+9.84 (▽ 8.21%)
+11
+9.76 (▽ 8.96%)
+12
+9.68 (▽ 9.70%)
+13
+9.61 (▽ 10.35%)
+As it is possible to observe, minor changes can be made to the original text and still optimize
+readability. It is still an open question whether minor changes in form are not so minor in meaning.
+But that open question will be addressed later in this paper.
+19
+
+4.4. Experimental study
+This section will explain the specifics of an empirical study we conducted to determine whether
+this novel approach is feasible. To accomplish this, we initially prepared an experimental setup so
+that we could determine the parameters of the experiments. Second, we completed the test and
+collected the unprocessed data. We have moved forward with the analysis of the data that we have
+gathered, which brings us to our third and last point.
+4.4.1. Experimental setup
+As was the case in the preceding part, the meticulous fine-tuning of the MOO strategy’s pa-
+rameters will not be the primary focus of this work. As a result, following a brief preliminary study
+based on a scheme of traditional parameter settings, one configuration works quite well. As a direct
+consequence of this, the following is a list of the parameters that we used for our experiments:
+• Population size {10, 15, 20}: 20
+• Number of parents mating {10, 15, 20}: 20
+• Number of genes: one per candidate word to be substituted by a synonym
+• Fitness function: the user can choose among Equations 1, 2, 3, 4, words to be replaced
+• Stop condition: {300, 600, 900}: 900 generations
+Furthermore, every experiment was run on a standard computer with 32 GB of primary memory
+and an Intel Core i7-8700 processor running at 3.20 GHz on Microsoft Windows 10 64-bit. Most
+of the functionality has been implemented using the library jMetalPy2, an open-source Python
+library for designing and implementing MOO strategies (Ben´ıtez-Hidalgo et al., 2019).
+4.4.2. Experiments
+Now we will proceed with the experiments concerning minimizing the impact on the original
+message’s form. While we focused previously on pure optimization, we focus here on minimizing
+the number of replaced words to affect how the message looks as little as possible.
+In Figure 4, we can see a summary of the results obtained after our experiments. What we
+have done is try to minimize the readability score (FKGL) at the same time as the number of
+words to be replaced in the ten use cases we are using throughout this work. As can be seen,
+we always obtain a Pareto front of solutions which indicates that the fewer words replaced, the
+less optimization is carried out. However, it should also be noted that the fewer words replaced
+also means a more negligible impact on the form of the initial message. It should be noted that
+different colors have been used for the Pareto fronts as in the previous experiments. Each color
+represents the degree of difficulty of each case study.
+2https://jmetal.github.io/jMetalPy/
+20
+
+10
+11
+12
+13
+14
+15
+12.3
+12.4
+12.5
+12.6
+12.7
+12.8
+12.9
+Words to be replaced
+Readability Score
+Use Case #1 - science
+FKGL
+4
+5
+6
+7
+8
+12.4
+12.5
+12.6
+12.7
+12.8
+Words to be replaced
+Readability Score
+Use Case #2 - history
+FKGL
+7
+7.5
+8
+8.5
+9
+9.5
+10
+15.8
+15.9
+16
+16.1
+16.2
+16.3
+16.4
+Words to be replaced
+Readability Score
+Use Case #3 - geography
+FKGL
+2
+3
+4
+5
+6
+7
+8
+10.4
+10.6
+10.8
+11
+Words to be replaced
+Readability Score
+Use Case #4 - sports
+FKGL
+7
+8
+9
+10
+11
+12
+11.4
+11.6
+11.8
+12
+Words to be replaced
+Readability Score
+Use Case #5 - engineering
+FKGL
+8
+9
+10
+11
+12
+9.8
+9.9
+10
+10.1
+10.2
+10.3
+10.4
+Words to be replaced
+Readability Score
+Use Case #6 - geography
+FKGL
+10
+11
+12
+13
+14
+15
+13
+13.2
+13.4
+13.6
+13.8
+Words to be replaced
+Readability Score
+Use Case #7 - engineering
+FKGL
+6
+7
+8
+9
+10
+14.2
+14.3
+14.4
+14.5
+14.6
+Words to be replaced
+Readability Score
+Use Case #8 - history
+FKGL
+6
+7
+8
+9
+10
+11
+12
+18.2
+18.4
+18.6
+18.8
+Words to be replaced
+Readability Score
+Use Case #9 - sports
+FKGL
+13
+14
+15
+16
+17
+7.9
+8
+8.1
+8.2
+8.3
+Words to be replaced
+Readability Score
+Use Case #10 - history
+FKGL
+Figure 4: Non-dominated solutions for ten use cases obtained using NSGA-II
+21
+
+4.4.3. Discussion
+We have shown how it is possible to design a solution that enhances the readability of texts,
+and we implemented that solution through MOO techniques. In addition, this sort of optimization
+is carried out while paying attention to the form of the texts in question to minimize the impact
+of replacing words with synonyms that better fit the readability criteria of the various formulas we
+have researched.
+As we explained earlier the paper, this kind of optimization can also occur in two distinct
+ways: it can be carried out to lower the readability score, making it possible for more people to
+comprehend the text altogether. Also, as a second possibility, it is possible to do so in order to
+improve the readability score, which will enable a lesser number of individuals to comprehend the
+processed text.
+Although the solutions will always be presented as a Pareto front for the human operator to
+choose his solution, it is also possible to define profiles: conservative, medium, and aggressive,
+which will opt for the conservative, medium, or more daring versions respectively without any
+question, without prejudice to the fact that it should be possible to manually edit a word that
+does not fit the context.
+5. Part III: Preserving the essence of the original text
+In Section 3, we have seen how it is possible to design and implement a functional solution
+to optimize the readability of any text. Although such a solution seems to work quite well, it
+cannot be considered fully automatic since a human user still has to evaluate whether any word
+used to replace original terms is not out of place in the context of the original message. In Section
+4, we have used a MOO strategy to ensure that we only replace a minimum number of words in
+the original text in order to preserve its form, and we even go a step further and allow a human
+operator to decide the degree of impact of ORUGA on the form of the original text. In Section 5,
+we will address a remaining problem, which is to keep under control the semantic distance between
+the original text and the generated text so that a short distance guarantees that the original text
+has not been distorted, while a long-distance means that we have been able to optimize readability
+by a large amount, but at a high cost in terms of altering the meaning of the original text. The
+importance of this third and final part is that success in our strategy is what can guarantee that
+the solution is entirely unsupervised.
+5.1. Technical Preliminaries
+The additional element we will add in this part is to measure the semantic distance between
+the original text and the text to be delivered. In this way, we can control any significant change
+in the meaning of the message. This semantic distance will be the third objective in our MOO
+strategy.
+22
+
+In the literature, there are methods that allow us to determine the semantic distance between
+two pieces of text in a meaningful way, even when those pieces do not share any words. To do
+that, words are embedded as vectors using this method. It has been demonstrated to perform
+better than many of the methods considered to be state-of-the-art in the k-nearest neighbor’s
+classification.
+With the help of Word Mover’s Distance (WMD) (Kusner et al., 2015), and given pre-trained
+word embeddings, it is possible to automatically assess the semantic distance between two texts
+by computing the minimum distance that the embedded words of one text need to travel to reach
+the ones of another text. So, for example, we can use the WordNet library to calculate synonyms
+that allow us to optimize text readability. At the same time, we can use word2vec to supervise
+that the semantic between the generated text and the original text is minimized. We could use as
+a metaphor that it is an adversarial process.
+The beauty of this approach is that the different synonym libraries now do not compete with
+each other but collaborate to try to measure (and therefore facilitate control) the semantic distance
+between the original text and the final text. Values close to zero will indicate that the meaning of
+the texts under consideration is practically equivalent, while distances approaching infinity indicate
+that the texts are incredibly different.
+Moreover, we do not have to concern about whether the genetic algorithm replaces a word with
+one or more words (e.g., ’considering’ by ’taking into account’) since the WMD is prepared for
+this contiguity, as it assumes by design that the texts will not have the same length. Since when
+using WMD, each word is matched against all other words, but weighted by a flow matrix T that
+ensures the semantic distance will be symmetric, even when an unequal number of words must be
+matched.
+5.2. Implementation
+In this work, we have decided to use WMD to facilitate the measurement of the semantic
+distance between the initial text and the text that will be delivered at the end of the process. This
+ensures that the synonyms used to replace candidate words are coherent. This choice is because
+WMD can measure the amount of semantic distance that separates two pieces of text by comparing
+the words that are important to each other. This is true even if the two pieces of text do not share
+any words.
+In addition to that, the method makes use of a representation known as the bag-of-words
+representation. The idea behind this method is that it should be possible to figure out how far
+apart two different texts are by figuring out the best way to move the distribution of the source
+text and the text being targeted.
+We can formally define our strategy, so that let d and d′ be the embedding representation of
+two texts, and T ∈ Rn×n where Tij ≥ 0 means how much of word i in d travels to word j in d′.
+Furthermore, the distance between i and j might be c(i, j) = ∥xi − xj∥. By c(i, j), we denote the
+23
+
+cost of moving from one word to another. In order to transform d into d′, it is necessary to be sure
+that the flow from i is equivalent to di so that �
+j Tij = di. In this way, the minimum cumulative
+cost of moving d to d′, given all these constraints, is provided by the solution shown in Equation
+7.
+arg min
+n
+�
+i,j=1
+Tijc(i, j)
+subject to
+n
+�
+j=1
+Tij = di ∀i ∈ {1, 2, 3 · · ·n} ∧
+n
+�
+i=1
+Tij = d′
+j ∀j ∈ {1, 2, 3 · · ·n}
+(7)
+Therefore, we use a function that determines the distance between two texts as the cumulative
+sum of the minimum distance each word in one text must move in vector space to the closest word
+in the other text. Word embeddings derived from word2vec will be utilized for this work because
+of their capability to maintain critical aspects of the context in which a word is used.
+WMD is used quite frequently these days to calculate semantic distances, and this is one of
+the reasons why. The only problem is that the complexity of computing the constrained minimum
+cumulative cost in the worst case is O(p3 log u), where u is the number of unique words in the text
+(Skianis et al., 2020). Therefore, when working with texts that contain a large number of unique
+words, WMD may perform poorly. However, there are some techniques that improve performance.
+In the context of this work, we have used the library gensim3 implementation of the WMD fed
+by the word embeddings from word2vec (Mikolov et al., 2013). In this way, what was previously a
+rival synonym library now becomes an adversarial library that helps keep semantic distance under
+control.
+5.3. Illustrative examples
+Example 5 provides us with information about engineering that was taken from Wikipedia.
+We are interested in observing how ORUGA operates while attempting to minimize the FKGL
+readability score, the number of words that need to be replaced, and the semantic distance between
+the original text and the text that has been processed. We are replacing the synonyms with the
+help of WordNet, and we are moving forward with the MOO with the help of NSGA-II. The
+ultimate goal is to ensure a very low risk of distortion of the original message that wanted to be
+communicated.
+3https://pypi.org/project/gensim/
+24
+
+Example 5. Engineering
+Original text Source: Wikipedia
+“Big data refers to data sets that are too large or complex to be dealt with by
+traditional data-processing application software. Data with many fields (rows) of-
+fer greater statistical power, while data with higher complexity (more attributes or
+columns) may lead to a higher false discovery rate. Big data analysis challenges
+include capturing data, data storage, data analysis, search, sharing, transfer, visu-
+alization, querying, updating, information privacy, and data source. Big data was
+originally associated with three key concepts volume, variety, and velocity.
+The
+analysis of big data presents challenges in sampling, and thus previously allowing
+for only observations and sampling. Thus a fourth concept, veracity, refers to the
+quality or insightfulness of the data.” ARI score: 14.43.
+12.9
+13
+13.1
+13.2
+13.3
+10
+0.15
+0.2
+Readability score
+Words to be replaced
+Semantic distance
+ORUGA - Final
+Minimize ARI score with the least risk of distorting the original message
+Big data bring up to data sets that are too big or complex to be dealt with by
+traditional data-processing application software. Data with many fields (rows) offer
+greater statistical power, while data with higher complexity (more attributes or
+columns) may lead to a higher false finding rate. Big data analysis challenges include
+capturing data, data storage, data analysis, search, sharing, transfer, visualization,
+querying, updating, information privacy, and data source. Big data was originally
+tied in with three key concepts volume, variety, and velocity. The analysis of big
+data presents challenges in sampling, and thus previously let for only observations
+and sampling. Thus a fourth concept, veracity, refers to the quality or acumen of
+the data. ARI score: 13.33 - ▽ 7.62%
+25
+
+The example shows that we no longer operate as blindly as before. Now, we also minimize
+the words to be replaced and the semantic distance between the initial and the generated text.
+Therefore, the results are much more reasonable and can be relied upon to work in exploitation
+environments. The readability score optimization is less spectacular than before, but the risk of
+distorting the original message, both in form and content, is much more substantially reduced.
+5.4. Experimental study
+In this section, we have performed the empirical study to test this version of ORUGA. To do
+so, we designed the experiments through an experimental setup. We performed the experiments
+and collected the raw data. And finally, we proceeded with the analysis of the collected data.
+5.4.1. Experimental setup
+The accurate adjustment of the MOO strategy’s parameters is not the primary focus here, as
+was the case with the previous parts of this research. One configuration works quite well after a
+brief preliminary study based on a conventional parameter-setting strategy. This came about as a
+result of what was mentioned above. The following is a list of the parameters that we have used:
+• Population size {10, 15, 20}: 20
+• Number of parents mating {10, 15, 20}: 20
+• Number of genes: one per candidate word to be substituted by a synonym
+• Fitness function: Equations 1, 2, 3, 4, words to be replaced, and WMD (Kusner et al., 2015).
+• Stop condition: {300, 600, 900}: 900 generations
+5.4.2. Experiments
+We are going to proceed with the experiments concerning minimizing the impact on the meaning
+of the original message. We focus now on being able to produce a result that can be put (or be
+close to being put into exploitation). To do this, we will conduct experiments to see if we can
+control the difference in meaning between the original message’s content and the generated text.
+Figure 5 shows us the summary of all the results obtained for the ten use cases that we have
+been studying throughout this research work. As can be seen, each use case, no matter how topical
+or challenging, corresponds to a good number of solutions ranging from the most conservative (the
+one that has the least risk of distorting the original message) to the most aggressive (the one
+that reduces the readability score more conclusively at the risk of distorting the original message).
+According to previous experiments, each color represents a degree of difficulty.
+26
+
+12.6
+12.8
+13
+13.2
+13.4
+10
+0.1
+0.15
+Readabity score
+Words to be replaced
+Semantic distance
+Use Case #1 - science
+12.4
+12.6
+12.8
+13
+2
+4
+6
+2
+4
+6
+·10−2
+Readabity score
+Words to be replaced
+Semantic distance
+Use Case #2 - history
+15.8
+16
+16.2
+16.4
+16.6
+5
+10
+5 · 10−2
+0.1
+0.15
+Readabity score
+Words to be replaced
+Semantic distance
+Use Case #3 - geography
+10.5
+10.6
+10.7
+10.8
+10.9
+4
+6
+8
+6 · 10−2
+8 · 10−2
+0.1
+Readabity score
+Words to be replaced
+Semantic distance
+Use Case #4 - sports
+10.8
+11
+11.2
+11.4
+11.6
+11.8
+12
+10
+15
+0.1
+0.15
+0.2
+Readabity score
+Words to be replaced
+Semantic distance
+Use Case #5 - engineering
+9.6
+9.7
+9.8
+9.9
+10
+10.1 5
+10
+6 · 10−2
+8 · 10−2
+0.1
+0.12
+Readabity score
+Words to be replaced
+Semantic distance
+Use Case #6 - geography
+13.1
+13.2
+13.3
+13.4
+16
+18
+0.16
+0.18
+0.2
+0.22
+Readabity score
+Words to be replaced
+Semantic distance
+Use Case #7 - engineering
+14.1
+14.15
+14.2
+14.25
+14.3
+8
+10
+8 · 10−2
+9 · 10−2
+0.1
+0.11
+0.12
+Readabity score
+Words to be replaced
+Semantic distance
+Use Case #8 - history
+18.4
+18.6
+18.8
+19
+5
+10
+6 · 10−2
+8 · 10−2
+0.1
+0.12
+0.14
+Readabity score
+Words to be replaced
+Semantic distance
+Use Case #9 - sports
+8
+8.1
+8.2
+8.3
+12
+14
+0.1
+0.11
+0.12
+0.13
+0.14
+Readabity score
+Words to be replaced
+Semantic distance
+Use Case #10 - history
+Figure 5: Summary of the results obtained for the third (and final) version of ORUGA
+27
+
+5.4.3. Discussion
+We have seen how it is possible to build a solution that enhances the readability of texts of
+very different natures and readability levels, and we implemented that solution. In addition, this
+sort of optimization is conducted while paying attention to both the content and the structure of
+the texts in question to minimize the impact of replacing words with synonyms that are a better
+fit for the readability criteria of the various formulas we have researched.
+Throughout this research, we have seen how it is possible to use genetic algorithms to improve
+the readability of any text formulated in English. As we have explained earlier, optimization is
+bi-directional. That is, it can automatically increase or decrease the readability of the text. Once
+we established the basis for the optimization, we could control the number of words that could
+be replaced; we could also control that the semantic distance between the original text and the
+final text does not skyrocket. Therefore, our initial goals of building a solution that improves the
+readability of any text without significantly altering its form or content have been achieved.
+6. Conclusion
+In this research work, we have seen how ORUGA can automatically optimize the readability of
+a text by using genetic algorithms. We have shown that by automatically replacing some words of
+the text to be optimized by their synonyms, we can optimize the readability levels in the direction
+(minimize or maximize) we wish. Neither the content nor the form of the text is altered because
+a minimal impact on the transformation of the original text is sought through various additional
+MOO techniques. Although, in theory, analogous solutions could be built using neural language
+models, this approach has a significant advantage: it is unsupervised and requires no training.
+An exhaustive empirical study has shown that we have successfully performed all experiments.
+In the first instance, these experiments consisted of processing texts of different natures (history,
+geography, sports, nature, and science) using three synonym libraries and using different readability
+metrics. In the second instance: designing a MOO solution to control the number of words to be
+replaced. We have tested different texts to assess and compare the feasibility of this approach.
+Furthermore, in the third instance: measuring and controlling the semantic distance between the
+original text and the one that will finally be outputted. For this, we have used a novel technique
+that uses a library of synonyms to control the results obtained with another library of different
+synonyms in an adversarial process.
+The results demonstrate that our hypothesis about text readability optimization at the begin-
+ning of this paper is valid. Despite the success of this research, it is necessary to bear in mind
+that simple formulas are typically simpler to put into practice, which is a limitation of this body
+of work. These formulas have a fundamental inability to model the semantics of word usage in
+context, which is needed to capture richer ideas of text difficulty.
+28
+
+As future work, a possible approach (yet computationally expensive) would be using a model
+of contextual embeddings such as BERT on a large dataset. Then, it should be necessary to store
+pairs of words and corresponding contextual representations and then use the nearest neighbors
+approach to identify synonyms that include the context. In this way, the impact of text processing
+will be even more limited.
+Source code
+The source code of this approach is published under MIT license in the following Github
+repository: https://github.com/jorge-martinez-gil/oruga.
+Acknowledgments
+The research reported in this paper has been funded by the Federal Ministry for Climate
+Action, Environment, Energy, Mobility, Innovation, and Technology (BMK), the Federal Ministry
+for Digital and Economic Affairs (BMDW), and the State of Upper Austria in the frame of SCCH,
+a center in the COMET - Competence Centers for Excellent Technologies Programme managed
+by Austrian Research Promotion Agency FFG.
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+page_content=' martinez-gil@ scch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' at Abstract This research presents ORUGA, a method that tries to automatically optimize the readability of any text in English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The core idea behind the method is that certain factors affect the readability of a text, some of which are quantifiable (number of words, syllables, presence or absence of adverbs, and so on).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The nature of these factors allows us to implement a genetic learning strategy to replace some existing words with their most suitable synonyms to facilitate optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In addition, this research seeks to preserve both the original text’s content and form through multi-objective optimization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In this way, neither the text’s syntactic structure nor the semantic content of the original message is significantly distorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' An exhaustive study on a substantial number and diversity of texts confirms that our method was able to optimize the degree of readability in all cases without significantly altering their form or meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The source code of this approach is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='com/jorge-martinez-gil/oruga Keywords: Text readability, Text Optimization, Genetic Algorithms 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Introduction Readability is a measure that tells us how easy it is to read a text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' It corresponds to the level of literacy that is expected from the readers in the target audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In this way, readability is considered one of the most critical factors that facilitate the user experience when consuming in- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' It is crucial because it is key to establishing a trusting relationship between information producers and consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' It must be considered that some factors, such as complexity, legibility, or typography, contribute to making a text readable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' However, not all factors are quantifiable and cannot be optimized by automatic techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In this paper, we focus solely and exclusively on factors of a quantifiable nature, which always revolve around basic or advanced statistics associated with the text to be optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Therefore, text readability refers to how simple it is to read and comprehend a given text, depending on its unique characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' These characteristics are usually measurable through metrics like the number of syllables in a sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The diversity of words used to create a readability score can be used to gauge this measure (Collins-Thompson, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Therefore, ORUGA is intended to facilitate the reader’s ability to understand a text by optimizing its readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In the context of this work, a clear distinction between readability and quality should be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Quality emphasizes essential elements like grammar, spelling, and voice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' However, the goal of producing textual information as plainly as possible and better matching it with its audience is what is meant when a text is said to be readable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In this way, text readability partially overlaps with the notion of text difficulty, which is also an essential aspect of human language, and has an impact on the daily lives of most people who consume written information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Our research is based on the premise that readability can be measured by considering met- rics related to the text and then using a specific mathematical formula to calculate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Because different readability scores are calculated using different mathematical formulas, it is possible to design a strategy that replaces many of the terms in the text with synonyms using a global opti- mization scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Such optimization is about maximizing or minimizing the result yielded by such mathematical formulas at the user’s convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In practice, such a strategy can be implemented through a genetic algorithm, as shown throughout this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Therefore, our research offers a viewpoint from the field of computer science, concentrating on the fundamental text representations and metrics utilized by readability assessment methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In this way, the most significant contributions of this work to state-of-the-art are the following: We present, for the first time, a technique that can automatically optimize the readability of any text, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', we can minimize or maximize the degree of readability of a text automatically without substantive changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' We study which are the best sources of synonyms currently available for text readability opti- mization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Specifically, we have studied Wordnet, word2vec, and web scraping and established a classification around optimizing up to ten texts of different natures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' We present an additional method based on multi-objective optimization, whose mission is to ensure that the minimum number of words needed in the original text is replaced in such a way that the structure of the text is not impacted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Last but not least, we studied several strategies that allow us to measure (and therefore optimize) the semantic distance between the original text and the generated text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In this way, the impact on the original text’s meaning can be controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' This research work is structured as follows: Section 2 shows state-of-the-art methods and tools for improving text readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Section 3 presents the technical details of our proposal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' these technical details are based on the design of a genetic cutting strategy that allows us to explore a vast search space while consuming a reasonable amount of resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Section 4 explains how to minimize the impact on the form of the original text using a multi-objective optimization 2 technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Section 5 shows how to preserve the essence of the original text by ensuring that the distance between the original text and the text obtained is kept as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' It is necessary to remark that sections 3, 4, and 5 present information about the design of different experiments and the raw results obtained, and their subsequent analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Finally, we conclude with the main lessons that can be learned from this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' State-of-the-art Let us begin with a formal definition of readability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' for instance,(Chall & Dale, 1995) defines readability as the ”total number of elements in a given text that affect a reader’s success.” This reader’s success is a measure of how well a text that is read at an optimal speed can be understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' At the same time, (Mc Laughlin, 1969) defines readability as ”the level at which certain people find reading material convincing and understandable.” Beyond these definitions, we are particularly interested in the quantifiable aspects of readability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', what can be objectively measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Oth- erwise, it would not be easy to proceed to its optimization using a computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Let us see what the literature says about these quantifiable aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Text readability in the scientific literature Readability metrics usually use simple features to calculate the degree of readability of a text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Some commonly used features are the number of sentences or words, the ratio of unique words, the total number of syllables, the proportion of unique words in the text, the number of digits, the number of words with many syllables, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Although, at first, it may seem that these metrics are simplistic, they are very commonly used for two important reasons: they are much cheaper and faster to use than the alternatives consisting of human surveys, and according to experts, they usually give exceptionally reliable results that are in line with reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The goal of improving readability is to increase the chances that readers can understand the thoughts and ideas reflected in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' So that misunderstanding is minimized, information processing is facilitated without requiring much effort and energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' With this goal in mind, many sources can be found that advise how to improve a text’s readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' However, these are manually compiled protocols that a human operator must translate into reality by modifying the text manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' For example, for a given metric, it is better to shorten sentences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' for another, it is better to replace complex words with simpler ones, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' This is precisely where our contribution to the state-of-the-art lies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Optimizing texts automat- ically using a metric as a target means we do not have to concern about taking any manual action leading to altering the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The genetic algorithm will find a way to proceed automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Why is text readability important?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' There are several contexts and population groups for which readability is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Especially when it is necessary to convey a written message to an audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' For example, 3 Teachers need to be sure of the readability of a text before deciding whether it is appropriate for their students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' This is particularly important in language learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' With the method presented here, the text can be optimized for a certain niche of learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In the world of advertising, readability allows for building a trust relationship between ad- vertisers and potential consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Advertising goods or services using texts with high read- ability is usually not a good idea since the message might not reach an essential part of the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' This is even more important in the search engine optimization sector, as many search engines use readability metrics as a ranking factor when responding to user searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Readability is also relevant for professionals who work on websites (Pantula & Kuppusamy, 2022), news (Qin, 2021), or even educational materials (Ante, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In some countries, there is even a legal requirement that government agencies provide textual information with certain readability levels to reach the entire population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Readability metrics There are several metrics to quantify how readable a text is (Meade & Smith, 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Most metrics have been designed for the English language (Maqsood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', 2022), although works also explore readability in other languages (Madrazo Azpiazu & Pera, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Without being exhaus- tive, we can mention, in chronological order, some of the metrics that enjoy or have enjoyed more significant popularity when dealing with the English language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Text readability depends not only on the characteristics of the text but also on the educational background of the individuals interested in understanding the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' We will see this reflected below when measuring readability using formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' We will see how readability metrics take a text as input and calculate a numerical score that usually corresponds to the level of education required to understand the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Dale-Chall readability The Dale-Chall readability formula (Dale & Chall, 1948) requires a list of 3,000 words that fourth-grade U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' students could reliably understand, as shown in Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' DCRF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='1579 �difficult words total words × 100 � + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='0496 � total words total sentences � (1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' SMOG readability The SMOG readability level (Mc Laughlin, 1969) can be assessed through a formula originally used for checking health messages, as shown in Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' It corresponds to the years of education necessary to understand the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' SMOG = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='0430 � number of polysyllables × 30 number of sentences + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='1291 (2) 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' ARI readability The ARI assesses the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' grade level required to read a text (Senter & Smith, 1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In some ways, it is similar to other formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Its difference is that rather than counting syllables, it counts characters: the more characters, the more complex the word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' It also counts sentences as shown in Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' This sets it apart from some other formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' ARI = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='71 �total Characters total Words � + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='5 � total Words total Sentences � − 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='43 (3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Flesch Kincaid readability Flesch Kincaid’s readability score, as shown in Equation 4, is a metric based on grade levels that is used commonly in the insurance industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Grade levels made it much easier for people to understand (Kincaid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', 1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' A Flesch Kincaid Grade Level (FKGL) between 8 and 10 means that the text should be accessible to the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' FKGL remains the most widely-used formula today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' FKGL = 206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='835 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='015 � total words total sentences � − 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='6 �total syllables total words � (4) Over the past decade, several natural language processing (NLP) techniques have been pro- posed to determine the readability of a text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Thus, as opposed to the classical approach of using formulas that measure a limited set of text features, these new variants have attempted to mea- sure the difficulty of understanding sentences and words and even the complexity of the syntax (Martinc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Even some techniques based on predictors of readability, such as cohesion and coherence, have received considerable attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' However, so far, these approaches have yet to be able to predict the readability of a text better than the classical techniques discussed here (Todirascu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Semantic Similarity The field of semantic similarity measurement (Martinez-Gil, 2022) is one of the most ac- tive in several different research communities (information retrieval, database integration, nat- ural language processing, and so on) (Rus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' This is due to its significant implica- tions on many available frameworks, methods, and tools (Navigli & Martelli, 2019) both in in- dustry and academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The literature around this topic has skyrocketed in the last few years (Chandrasekaran & Mago, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' However, most research works focus on determining the similar- ity between words (Zhu & Iglesias, 2017), phrases, or documents (Martinez-Gil & Chaves-Gonzalez, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Martinez-Gil & Chaves-Gonzalez, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In this way, rarely the likelihood of effectively throwing out the most similar words to a given one has been discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' To this effect, there are synonym libraries that have been man- ually compiled and some word embedding techniques that do the job well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The most promi- 5 nent solutions in this direction are WordNet (Pedersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', 2004), word2vec (Mikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', 2013), or BERT (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' However, it must be taken into account that the re- source requirements for techniques based on the computation of word embeddings are very high (Martinez-Gil & Chaves-Gonzalez, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Contribution over the state-of-the-art Determining text readability based on a formal analysis of the structures and words used has been a recurring theme in the literature over the last few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' As a result, many metrics have been proposed to measure text readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' A common denominator of all these metrics is that a high score usually means the text is difficult to understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In other words, a higher degree of study is needed to understand it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' For this reason, many communication professionals often use tools to help them discern whether the text fits a given audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' However, none of these tools can do the professional’s job automatically, which is why our contribution is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' To the best of our knowledge, this is the first time anyone has proposed automatically improving the readability of text without significantly altering the content or the text’s form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In addition, we put this innovative method through its paces using a wide range of texts that varied in subject matter and level of readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Furthermore, we provide the source code of the first implementation for anyone interested in experimenting with or improving the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Part I: Design and Implementation of a Functional Solution In the following, we explain our proposal for text readability optimization using a method based on genetic algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In this section, we outline the technical preliminaries, discuss the implementation, show an illustrative example of how our approach works in practice, and conduct an experimental study using real data and use cases that can help us get an idea of the performance of this approach in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Technical Preliminaries Our hypothesis is that we can find a set of synonyms to replace some words in the original text so that the value returned by the readability formula can be optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' For example, if the readability formula rewards or penalizes long words, we have to find synonyms of less length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In reality, we only have to concern about understanding which formula best represents readability in our specific scenario and indicate it as a fitness function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The genetic algorithm will understand how to proceed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Thus, we are faced with a classical optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' We intend to act only on the vocabulary since it is one of the most critical parts of that language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' It is widely assumed that vocabulary is the essential part of a language because, without vocabulary, it is impossible to compose any message (Wilkins, 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' We do not act on proper nouns, prepositions, or other stop words to avoid distorting the original message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 6 While this problem can be solved by a brute-force search over the range of the words of a given text w0, w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', wn, the GA method scales very well when dealing with large texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In this case, a brute-force search would be prohibitively expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' We could act on other aspects, such as the structure, but then we would risk distorting the original text’s essence again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Our idea is to find the combination of words (which will be encoded in the form of an individual) that optimizes the desired objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The choice of the fitness function is effortless and has the advantage that the solution automatically identifies what kind of words lead to the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In this work, we work with three main sources of synonyms: WordNet (Miller, 1995), which is a thesaurus that has been manually compiled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' It is probably one of the most widely used dictionaries in information systems and will give us several alternatives to substitute each candidate word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' No surprises are to be expected in this library, except perhaps the substitution of words with a synonym that has a sense far from the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In any case, we have a solution to this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' wordvec (Mikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', 2013), which is an approach that calculates the vectors associated with each word according to a textual corpus of relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Once each vector has been computed, a computation process can be performed by which the N vectors most similar to a given one are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' This way, synonyms of relevance are obtained for the word in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Note that this process is computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Web scraping (Mitchell, 2018), which consists of obtaining the synonyms from some websites, usually specialized, so that we can shuffle several alternatives per candidate word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' This method obtains many high-quality word candidates, but it should be used responsibly because it can cause problems on the server side if many hundreds or thousands of requests are made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Therefore, the method is fine for experimentation, but it would be unreasonable to exploit it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Implementation The implementation of this novel approach is based on a classical optimization scheme using genetic algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Algorithm 1 briefly explains in pseudo-code how the whole process is performed by adapting the classical structure of the genetic algorithm (Forrest, 1996) where different operators capable of implementing selection, cross-over and mutation processes are considered in order to evolve a given population towards the desired objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Please note that before starting the evolution process, a pre-processing of the text must be done in order to identify the candidate words to be replaced by a synonym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' We propose that all words are candidates except: proper names, stop words, and prepositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The reason for this is that they are often words for which it is really difficult to find a synonym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='Algorithm 1 Optimizing Readability Using Genetic Algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='1: procedure ORUGA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='2: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='population ← generationRandomIndividual (population) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='3: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='calculateReadabilityScore (population) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='4: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='while (stop condition not reached) do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='5: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='parents ← selectionOfIndividuals (population) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='6: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='offspring ← Crossover (parents) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='7: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='offspring ← Mutation (offspring) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='8: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='offspring ← calculateReadabilityScore (offspring) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='9: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='population ← updatePopulation (offspring) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='10: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='endwhile ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='11: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='optimizedText ← optimizedIndividual (population) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='12: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='optimizedText ← correctErrorsIfNecessary (optimizedText) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='13: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='return optimizedText ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='In relation to the genetic algorithm itself,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' the parametric details of the solution will be discussed below,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' but it is possible to see how we implement it in the form of a classical evolutionary process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' This means that a population of individuals is selected randomly, and their readability score is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' We then proceed with an iterative process of selection, crossover, and mutation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' the best individuals are passed from generation to generation until one of the stopping criteria is met: the highest possible has been reached (unlikely), or the number of iterations has been exhausted (very likely).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Finally, at the end of the evolutionary process, we correct the text in case grammatical errors are produced by substituting a synonym that does not fit the tense and the form of the sentence in which it is framed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' This way, we obtain a corrected readability score, which may vary slightly from the one derived automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In return, we ensure that the results are usable, or at least close to being usable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Illustrative examples Example 1 shows us how ORUGA works with written material about science extracted from Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In fact, our aim is to observe how ORUGA behaves when trying to minimize the FKGL readability score using synonyms from the library WordNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Let us remember that FKGL (or one of its variants) is probably the most widely used metric and its optimization brings advantages in several fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Please note that the text to be treated can be of any length, but to facilitate the presentation to the reader, we have chosen (and will choose throughout this paper) one that contains only several sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 8 Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Science Original text Source: Wikipedia “The sea moderates the climate and has important roles in the water cycle, car- bon cycle, and nitrogen cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Humans harnessing and studying the sea have been recorded since ancient times, and evidenced well into prehistory, while its modern scientific study is called oceanography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The most abundant solid dissolved in sea- water is sodium chloride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The water also contains salts of magnesium, calcium, potassium, and mercury, amongst many other elements, some in minute concentra- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Salinity varies widely, being lower near the surface and the mouths of large rivers and higher in the depths of the ocean;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' however, the relative proportions of dissolved salts vary little across the oceans.” FKGL score: 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' ORUGA - minimizing the FKGL score - library Wordnet The sea moderates the climate and has important part in the H2O cycle, C cycle, and N cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Humans harnessing and studying the sea have been taped since ancient times, and attested well into prehistory, while its modern scientific study is called oceanography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The most abundant solid fade out in brine is Na chloride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The H2O too contains salts of magnesium, calcium, potassium, and mercury, amongst many other elements, some in min concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Salinity changes widely, being got down near the surface and the mouths of large rivers and higher in the depth of the ocean;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' however, the relative proportions of fade out salts change little across the oceans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' FKGL score: 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='85 As can be seen, ORUGA can optimize the textual input by first automatically identifying which words can be replaced by a synonym, and then undertaking a process of searching for synonyms that improve the results of the metric to be optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In this way, the impact on the initial message is minimal, although it is true that some synonyms can slightly distort the meaning of the text, and therefore final supervision by the user is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' However, there is no need to concern because this problem will be addressed later in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Let us look now at Example 2, which is a written text about the history of Austria that has been also extracted from Wikipedia, and we would like to to minimize the FKGL readability score using synonyms automatically obtained by web scraping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 9 Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' History Original text Source: Wikipedia “Austria emerged from the remnants of the Eastern and Hungarian March at the end of the first millennium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Originally a margraviate of Bavaria, it developed into a duchy of the Holy Roman Empire in 1156 and was later made an archduchy in 1453.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In the 16th century, Vienna began serving as the empire administrative capital and Austria thus became the heartland of the Habsburg monarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' After the dissolution of the Holy Roman Empire in 1806, Austria established its own empire, which became a great power and the dominant member of the German Confederation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The defeat in the Austro-Prussian War of 1866 led to the end of the Confederation and paved the way for the establishment of Austria-Hungary a year later.” FKGL score: 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='20 ORUGA - minimizing the FKGL score - web scraping Austria looms from the debris of the Eastern and Hungarian March at the end of the first millennium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Originally a margravate of Bavaria, it matured within a duchy of the Holy Roman Empire in 1156 and was next made an arch duchy in 1453.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In the 16th century, Vienna lead plate as the command departmental central and Austria thus come the heartland of the Habsburg monarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' After the divorce of the Holy Roman Empire in 1806, Austria settled its own empire, that come a great power and the dominant branch of the German Confederation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The defeat in the Austro-Prussian War of 1866 led to the end of the Confederation and brick the way for the founding of Austria-Hungary a year later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' FKGL score: 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='35 Once again, we can see how the genetic algorithm has acted intelligently to decrease the text readability score and therefore make the text accessible to more people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' It is clear that the suitability of some words may be subject to debate, but the first objective of this research, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', to optimize the readability score, has been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' As we mentioned before, we will be concerned to outline a final product later in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Finally, let us look at the opposite case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', let us see if we can make a text more difficult to read without losing its essence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In Example 3, we have a text about sports also extracted from Wikipedia, and we do not want to make the text accessible to as many people as possible, but on the contrary, we want to increase the level of readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' We will now try a different metric, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', ARI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 10 Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Sports Original text Source: Wikipedia “Real Madrid Club de Futbol, meaning Royal Madrid Football Club, commonly referred to as Real Madrid, is a Spanish professional football club based in Madrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Founded in 1902 as Madrid Football Club, the club has traditionally worn a white home kit since its inception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The honorific title real is Spanish for Royal and was bestowed to the club by King Alfonso XIII in 1920 together with the royal crown in the emblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Real Madrid have played their home matches in the Santiago Bernabeu Stadium in downtown Madrid since 1947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Unlike most European sporting entities, Real Madrid members (socios) have owned and operated the club throughout its history.” ARI score: 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' ORUGA - maximizing the ARI score - web scraping Real Madrid Club de Futbol, connotation Royal Madrid Football Club, frequently referred to as Real Madrid, is a Spanish professional football business established in Madrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Founded in 1902 as Madrid Football Club, the business has consis- tently timeworn an alabaster home kit since its inception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The appellation title real is Spanish for Royal and was entrusted to the business by King Alfonso XIII in 1920 together alongside the aristocratic culmination in the emblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Real Madrid have played their familiar matches in the Santiago Bernab´eu Stadium in downtown Madrid afterward 1947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Unlike most European sporting entities, Real Madrid as- semblage (socios) have owned and negotiated the business throughout its history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' ARI score: 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='53 As can be seen, after processing the fragment related to Real Madrid extracted from Wikipedia, the genetic algorithm selects synonyms that are much longer and more complicated to read to increase the ARI score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Although a use case consisting of making the text less accessible to people is hard to imagine, it may find application in some specific niches of learning, education, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Experimental study In this section, we explain the details of an empirical study that we have carried out to know the feasibility of our new method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' To do so, we first established the conditions of the experi- ments by setting up an experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Secondly, we have run and obtained the raw results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Furthermore, thirdly, and lastly, we have proceeded with the analysis of the data obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Experimental setup Adjusting the parameters of the genetic algorithm differs from the focus of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Therefore, we have chosen a classical parameter setting, which has been shown to work quite well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Standard tuning of the parameters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', through a grid search, is a possible line of future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In the meantime, the parameters we have used for our experiments are the following: Population size {10, 15, 20}: 20 Number of parents mating {10, 15, 20}: 10 Number of genes: one per candidate word to be substituted by a synonym Fitness function: the user can choose among Equations 1, 2, 3, 4 Stop condition: {100, 200, 300}: 300 generations To test whether our approach can optimize text readability, we semi-randomly selected ten texts extracted from Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' These texts are classified into several categories engineering, geography, history, science, and sports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Some metrics, such as SMOG, require at least 30 sentences to apply their formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' For cases where we do not reach 30 sentences, we will duplicate the text until we reach that number of sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' It should also be noted that depending on which readability category each use case represents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' For example, magenta represents the highest difficulty (FKGL between 15 and 18), equivalent to an academic paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Red represents medium-high difficulty (FKGL between 12 and 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The black color represents a medium difficulty (FKGL between 9 and 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Furthermore, the blue color represents a low-medium difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' It is estimated that up to 80% of the native English-speaking population could understand text with an FKGL between 6 and 9, which is what the blue value represents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Furthermore, every experiment was run on a standard computer with 32 GB of primary memory and an Intel Core i7-8700 processor running at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='20 GHz on Microsoft Windows 10 64-bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Most of the functionality has been implemented using the library PyGAD1, an open-source Python library for building genetic algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Experiments The first experiment is the one that is the most useful in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' It consists of trying to minimize the readability of the ten proposed texts so that the processed text can reach the largest possible audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' To do that, we want to use readability metrics which estimate the readability of a text based on simple aspects such as syllable and word counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' We use the FKGL score since this metric, or some of its variants, has been used for decades on traditional texts, and it is still one of the most common and widely used traditional readability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 1https://pypi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='org/project/pygad/ 12 01 02 03 04 05 06 07 08 09 10 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='5 #Use Case Improvement (GL) Figure 1: Results for the minimization of the FKGL score using WordNet Fig 1 shows us the results obtained when reducing the FKGL for the texts under consideration using the library WordNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The results shown are the summary of ten independent runs per use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Since we are dealing with cold start methods with random values, and there is even a component of randomness in the mutations, we will almost always get a different result, so it is essential to show the results in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Positive results have been obtained in all 100 experiments performed (ten runs for ten use cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In addition, minimum improvements, average improvements, and even maximum improvements have been achieved in the range between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='77 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='73 points on the FKGL scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Fig 2 shows us the results obtained using the synonym calculation method using word2vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' As it is possible to see, the variance of the results is enormous, which means that using this library will make the results not very predictable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' At the same time, as in the previous case, all 100 experiments were able to obtain readability improvements that ranged between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='42 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='90-grade levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Fig 3 shows us the results we have obtained by searching for synonyms with web scraping techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The results this time have a smaller variance than in the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In addition, we have again obtained favorable results in all 100 experiments performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The optimization achieved ranges between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='02 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='20-grade levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Table 1 shows the summary results of the 300 experiments performed (100 for each synonym library).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' For WordNet, a median optimization of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='63-grade levels is expected, very similar to 13 01 02 03 04 05 06 07 08 09 10 1 2 3 4 #Use Case Improvement (GL) Figure 2: Results for the minimization of the FKGL score using word2vec 01 02 03 04 05 06 07 08 09 10 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='5 4 #Use Case Improvement (GL) Figure 3: Results for the minimization of the FKGL score using Web Scraping 14 Library Minimum Median Maximum WordNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='77 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='63 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='73 word2vec 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='96 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='90 Web scraping 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='61 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='20 Table 1: Summary of the results obtained using the different libraries of synonyms Use case DCRF SMOG ARI FKGL 01/science 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
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+page_content='27 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
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+page_content='04 Table 2: Summary of the results obtained for the experiments performed w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The numerical values represent absolute improvements after using ORUGA that achieved by web scraping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The median optimization using word2vec is the worst of the three libraries considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Please note that we are not judging here the quality of the replacements, but the values to optimize the input text independently of the quality of the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Table 2 shows a summary of all the results we have obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The texts have been randomly obtained from Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The range of values shown indicates the minimum value that could be reached and the maximum value (for minimization and maximization problems respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' We analyze the four formulas that we consider most representative, but the analysis of other formulas would not be a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' When working with SMOG, we must be careful because the text to be analyzed requires at least 30 sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' We have duplicated the text for these experiments so that these 30 sentences can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Table 3 shows a summary of all the results we have obtained when maximizing the readability scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' As can be seen by comparing with the table above, it is much easier to increase the level of readability than to decrease it, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
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+page_content=', it is easier to make a text difficult to read than the other way around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
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+page_content=' Discussion We have seen how it is possible to build a solution that optimizes the readability of texts of different natures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Moreover, such optimization is done respecting the content and the form of such texts, trying to minimize the impact of word replacements by synonyms that better fit the readability criteria of the different formulas we have studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' We have seen how optimization can occur in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' First, it can be done in such a way as to reduce the readability score, which 15 Use case DCRF SMOG ARI FKGL 01/science 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
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+page_content='04 10/history 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='52 Table 3: Summary of the results obtained for the experiments performed w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The numerical values represent absolute improvements after using ORUGA will allow more people to understand the text perfectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' This is undoubtedly the most practical option in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Furthermore, secondly, it can be done to increase the readability score, allowing a smaller number of people to understand the processed text unambiguously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' This option has less practical utility than can be discerned at first glance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Based on the research we have carried out, possible improvements can be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' For example, a multi-objective optimization algorithm could simultaneously optimize the readability score by affecting as few words as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In this way, the impact of our method on the original text would be even more negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' A solution front could allow the human operator to decide the trade-off between the score modification and the replaced words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Finally, and as a limitation of our method, it can be observed that sometimes the processed text contains minor grammatical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' This is because we have yet to use techniques that allow, for example, to choose the appropriate verb tense for the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Here there are two alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' On the one hand, we can implement better methods for correcting grammatical errors to obtain a corrected readability score;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' on the other hand, we can give the user the option to edit or choose the word that best fits each moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Part II: Minimizing the impact on the form of the original text While in Section 3, we have seen that it is possible to design a functional solution to optimize the text readability, we have also seen that the approach can be intrusive at times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' That is, the replacement of many words by synonyms can lead to a distortion of the original message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' For this reason, in this section, we focus on minimizing the impact of ORUGA on the original text by replacing as few words as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' To do so, we will build a solution based on multi-objective optimization (MOO) that allows us to optimize readability and simultaneously minimize the num- ber of replacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' This section comprises the technical preliminaries, the implementation, some illustrative examples, and an empirical study to test several texts of different natures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Technical Preliminaries We have already seen how in the field of text readability, there has been a great effort to build straightforward formulas that can be understood by people and have a good correlation to how easily a text can be read from a human perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Now we go one step further to obtain a higher quality result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' MOO is a strategy in which two or more objectives are simultaneously optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' This is the situation we find ourselves in, given that we want: on the one hand, to improve the readability of a text, and on the other hand, we want to reduce as much as possible the number of words that need to be replaced to improve readability, and thus to minimize the impact that our approach has on the original text form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Furthermore, MOO is useful when decisions must be made despite potential trade-offs between more than one orthogonal objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Again, this is our situation because our goals of maximizing readability while simultaneously replacing the fewest possible words require us to pursue two completely different goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In situations like this, different solutions can simultaneously fulfill all objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' As a result, all optimal solutions ought to be regarded as equivalent merit without any external evaluation from a human operator (Martinez-Gil & Chaves-Gonzalez, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' More formally, we can model a MOO problem as expressed in Equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' min (s1(⃗x), s2(⃗x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' , sn(⃗x)) subject to ⃗x ∈ X (5) In MOO, no solution addresses all objective functions simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' As a result, the priority should be placed on finding solutions that cannot make any goals better without making at least one of the other goals worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Therefore, a solution ⃗x1 ∈ X is said to dominate another one ⃗x2 ∈ X if the conditions expressed in Equation 6 are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' si(⃗x1) ≤ si(⃗x2) ∀i ∈ {1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', n} sj(⃗x1) < sj(⃗x2) ∃j ∈ {1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', n} (6) In this way, a solution ⃗x ∈ X is optimal if no solution might dominate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' An element x is said to dominate another element y if x is not worse than y concerning all the goals and is strictly better than y for at least one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The elements of the search space that are not dominated give rise to a Pareto front, which represents the best possible solutions to the orthogonal objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Implementation There are several implementations for MOO strategies Kukkonen & Lampinen (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Zhang & Li (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' It is beyond the scope of this paper to consider them all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' However, we will look at one of the best ones, NSGA-II (Deb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' This strategy is summarized in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Its mode of operation is based on the concepts of fronts and crowding distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 17 NSGA-II adheres to the basic structure of a genetic algorithm but employs a different approach to mating and selection for survival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In the NSGA-II, the first step is to select individuals in a front-wise fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In this way, a situation will arise where it will be necessary to divide a front because it will not be possible for all individuals to survive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The solutions are chosen according to the crowding distance for this particular splitting front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Within the parameters of the objective space, the Manhattan Distance corresponds to the crowding distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' On the other hand, it is desired that the extreme points be maintained with each new generation, and as a result, an infinite crowding distance is assigned to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Algorithm 2 shows us a possible implementation of this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='Algorithm 2 MOO Technique for Optimizing Text Readability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='1: procedure ORUGA2-MOO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='2: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='population ← initializePopulation () ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='3: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='population ← generationRandomIndividual (population) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='4: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='calculateReadabilityScore (population) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='5: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='assignRankBasedOnPareto (population) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='6: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='auxiliarPopulation ← generationChildPopulation (population) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='7: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='while (stop condition not reached) do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='8: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='for (each individual in population and auxiliarPopulation) do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='9: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='solution ← calculateReadabilityScore (population) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='10: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='solution ← assignRankBasedOnPareto (population) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='11: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='solution ← generateNonDominateSolutions (population) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='12: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='solution ← determiningCrowdingDistance (population) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='13: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='for (each solution) do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='14: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='population ← addingSolutionsNextGeneration (population) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='15: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='16: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='17: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='population ← selectPointsLowFrontHighCrowdingDistance (population) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='18: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='population ← generationNextPopulation (population) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='19: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='end while ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='20: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='optimizedText ← optimizedIndividual (solution) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='21: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='optimizedText ← correctErrorsIfNecessary (optimizedText) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='22: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='return optimizedText ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='NSGA-II is an example of a genetic algorithm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' and it possesses the three characteristics listed below: It operates based on an elitist principle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' which states that only the most privileged mem- bers of a population are permitted to be passed down to subsequent generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In addition, it employs a mechanism specifically designed to preserve diversity (crowding distance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' As a direct consequence of this, it can identify non-dominated solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Illustrative examples Since we are trying to optimize two orthogonal objectives simultaneously, it is impossible to offer a single solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Nevertheless, we can put in the hands of the human operator a front of 18 solutions ranging from a total optimization by modifying the most significant number of words to a minor optimization by touching a minimum number of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' It is up to the human operator to decide which solution to choose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Example 4 shows a real trace that controls the number of words to be replaced using the MOO technique known as NSGA-II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The text on which it operates is about geography and has been extracted from Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Once again, we must insist that although, in theory, it would be possible to edit the words manually to satisfy the criteria of a given metric, this approach is transparent and works automatically for any desired metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Please note that the words in blue are candidates to be replaced by a synonym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Geography Source: Wikipedia Goal: Minimize the FKGL score by using Wordnet synonyms and minimize the words to be replaced using NSGA-II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' “Niagara Falls is a group of three waterfalls at the southern end of Niagara Gorge, spanning the border between the province of Ontario in Canada and the state of New York in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The largest of the three is Horseshoe Falls, which straddles the international border of the two countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' It is also known as the Canadian Falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The smaller American Falls and Bridal Veil Falls lie within the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Bridal Veil Falls is separated from Horseshoe Falls by Goat Island and from American Falls by Luna Island, with both islands situated in New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Formed by the Niagara River, which drains Lake Erie into Lake Ontario, the combined falls have the highest flow rate of any waterfall in North America that has a vertical drop of more than 50 m (160 ft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' During peak daytime tourist hours, more than 168,000 m3 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='9 million cu ft) of water goes over the crest of the falls every minute.” FKGL score: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Words to be replaced FKGL expected 5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='43 (▽ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='71%) 6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
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+page_content='10%) 7 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='13 (▽ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='50%) 8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
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+page_content='90%) 9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
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+page_content='56%) 10 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='84 (▽ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='21%) 11 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='76 (▽ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='96%) 12 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='68 (▽ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='70%) 13 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='61 (▽ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='35%) As it is possible to observe, minor changes can be made to the original text and still optimize readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' It is still an open question whether minor changes in form are not so minor in meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' But that open question will be addressed later in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Experimental study This section will explain the specifics of an empirical study we conducted to determine whether this novel approach is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' To accomplish this, we initially prepared an experimental setup so that we could determine the parameters of the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Second, we completed the test and collected the unprocessed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' We have moved forward with the analysis of the data that we have gathered, which brings us to our third and last point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Experimental setup As was the case in the preceding part, the meticulous fine-tuning of the MOO strategy’s pa- rameters will not be the primary focus of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' As a result, following a brief preliminary study based on a scheme of traditional parameter settings, one configuration works quite well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' As a direct consequence of this,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' the following is a list of the parameters that we used for our experiments: Population size {10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 20}: 20 Number of parents mating {10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 20}: 20 Number of genes: one per candidate word to be substituted by a synonym Fitness function: the user can choose among Equations 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' words to be replaced Stop condition: {300,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 600,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 900}: 900 generations Furthermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' every experiment was run on a standard computer with 32 GB of primary memory and an Intel Core i7-8700 processor running at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='20 GHz on Microsoft Windows 10 64-bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Most of the functionality has been implemented using the library jMetalPy2, an open-source Python library for designing and implementing MOO strategies (Ben´ıtez-Hidalgo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Experiments Now we will proceed with the experiments concerning minimizing the impact on the original message’s form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' While we focused previously on pure optimization, we focus here on minimizing the number of replaced words to affect how the message looks as little as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In Figure 4, we can see a summary of the results obtained after our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' What we have done is try to minimize the readability score (FKGL) at the same time as the number of words to be replaced in the ten use cases we are using throughout this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' As can be seen, we always obtain a Pareto front of solutions which indicates that the fewer words replaced, the less optimization is carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' However, it should also be noted that the fewer words replaced also means a more negligible impact on the form of the initial message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' It should be noted that different colors have been used for the Pareto fronts as in the previous experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Each color represents the degree of difficulty of each case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 2https://jmetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='io/jMetalPy/ 20 10 11 12 13 14 15 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
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+page_content='9 Words to be replaced Readability Score Use Case #1 - science FKGL 4 5 6 7 8 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
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+page_content='8 Words to be replaced Readability Score Use Case #2 - history FKGL 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
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+page_content='4 Words to be replaced Readability Score Use Case #3 - geography FKGL 2 3 4 5 6 7 8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
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+page_content='8 11 Words to be replaced Readability Score Use Case #4 - sports FKGL 7 8 9 10 11 12 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='6 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='8 12 Words to be replaced Readability Score Use Case #5 - engineering FKGL 8 9 10 11 12 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='9 10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='4 Words to be replaced Readability Score Use Case #6 - geography FKGL 10 11 12 13 14 15 13 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
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+page_content='6 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='8 Words to be replaced Readability Score Use Case #7 - engineering FKGL 6 7 8 9 10 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='3 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
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+page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='6 Words to be replaced Readability Score Use Case #8 - history FKGL 6 7 8 9 10 11 12 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='4 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='6 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='8 Words to be replaced Readability Score Use Case #9 - sports FKGL 13 14 15 16 17 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='9 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='3 Words to be replaced Readability Score Use Case #10 - history FKGL Figure 4: Non-dominated solutions for ten use cases obtained using NSGA-II 21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Discussion We have shown how it is possible to design a solution that enhances the readability of texts, and we implemented that solution through MOO techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In addition, this sort of optimization is carried out while paying attention to the form of the texts in question to minimize the impact of replacing words with synonyms that better fit the readability criteria of the various formulas we have researched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' As we explained earlier the paper, this kind of optimization can also occur in two distinct ways: it can be carried out to lower the readability score, making it possible for more people to comprehend the text altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Also, as a second possibility, it is possible to do so in order to improve the readability score, which will enable a lesser number of individuals to comprehend the processed text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Although the solutions will always be presented as a Pareto front for the human operator to choose his solution, it is also possible to define profiles: conservative, medium, and aggressive, which will opt for the conservative, medium, or more daring versions respectively without any question, without prejudice to the fact that it should be possible to manually edit a word that does not fit the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Part III: Preserving the essence of the original text In Section 3, we have seen how it is possible to design and implement a functional solution to optimize the readability of any text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Although such a solution seems to work quite well, it cannot be considered fully automatic since a human user still has to evaluate whether any word used to replace original terms is not out of place in the context of the original message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In Section 4, we have used a MOO strategy to ensure that we only replace a minimum number of words in the original text in order to preserve its form, and we even go a step further and allow a human operator to decide the degree of impact of ORUGA on the form of the original text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In Section 5, we will address a remaining problem, which is to keep under control the semantic distance between the original text and the generated text so that a short distance guarantees that the original text has not been distorted, while a long-distance means that we have been able to optimize readability by a large amount, but at a high cost in terms of altering the meaning of the original text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The importance of this third and final part is that success in our strategy is what can guarantee that the solution is entirely unsupervised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Technical Preliminaries The additional element we will add in this part is to measure the semantic distance between the original text and the text to be delivered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In this way, we can control any significant change in the meaning of the message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' This semantic distance will be the third objective in our MOO strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 22 In the literature, there are methods that allow us to determine the semantic distance between two pieces of text in a meaningful way, even when those pieces do not share any words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' To do that, words are embedded as vectors using this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' It has been demonstrated to perform better than many of the methods considered to be state-of-the-art in the k-nearest neighbor’s classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' With the help of Word Mover’s Distance (WMD) (Kusner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', 2015), and given pre-trained word embeddings, it is possible to automatically assess the semantic distance between two texts by computing the minimum distance that the embedded words of one text need to travel to reach the ones of another text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' So, for example, we can use the WordNet library to calculate synonyms that allow us to optimize text readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' At the same time, we can use word2vec to supervise that the semantic between the generated text and the original text is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' We could use as a metaphor that it is an adversarial process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The beauty of this approach is that the different synonym libraries now do not compete with each other but collaborate to try to measure (and therefore facilitate control) the semantic distance between the original text and the final text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Values close to zero will indicate that the meaning of the texts under consideration is practically equivalent, while distances approaching infinity indicate that the texts are incredibly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Moreover, we do not have to concern about whether the genetic algorithm replaces a word with one or more words (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', ’considering’ by ’taking into account’) since the WMD is prepared for this contiguity, as it assumes by design that the texts will not have the same length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Since when using WMD, each word is matched against all other words, but weighted by a flow matrix T that ensures the semantic distance will be symmetric, even when an unequal number of words must be matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Implementation In this work, we have decided to use WMD to facilitate the measurement of the semantic distance between the initial text and the text that will be delivered at the end of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' This ensures that the synonyms used to replace candidate words are coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' This choice is because WMD can measure the amount of semantic distance that separates two pieces of text by comparing the words that are important to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' This is true even if the two pieces of text do not share any words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In addition to that, the method makes use of a representation known as the bag-of-words representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The idea behind this method is that it should be possible to figure out how far apart two different texts are by figuring out the best way to move the distribution of the source text and the text being targeted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' We can formally define our strategy, so that let d and d′ be the embedding representation of two texts, and T ∈ Rn×n where Tij ≥ 0 means how much of word i in d travels to word j in d′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Furthermore, the distance between i and j might be c(i, j) = ∥xi − xj∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' By c(i, j), we denote the 23 cost of moving from one word to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In order to transform d into d′, it is necessary to be sure that the flow from i is equivalent to di so that � j Tij = di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In this way, the minimum cumulative cost of moving d to d′, given all these constraints, is provided by the solution shown in Equation 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' arg min n � i,j=1 Tijc(i, j) subject to n � j=1 Tij = di ∀i ∈ {1, 2, 3 · · ·n} ∧ n � i=1 Tij = d′ j ∀j ∈ {1, 2, 3 · · ·n} (7) Therefore, we use a function that determines the distance between two texts as the cumulative sum of the minimum distance each word in one text must move in vector space to the closest word in the other text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Word embeddings derived from word2vec will be utilized for this work because of their capability to maintain critical aspects of the context in which a word is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' WMD is used quite frequently these days to calculate semantic distances, and this is one of the reasons why.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The only problem is that the complexity of computing the constrained minimum cumulative cost in the worst case is O(p3 log u), where u is the number of unique words in the text (Skianis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Therefore, when working with texts that contain a large number of unique words, WMD may perform poorly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' However, there are some techniques that improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In the context of this work, we have used the library gensim3 implementation of the WMD fed by the word embeddings from word2vec (Mikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In this way, what was previously a rival synonym library now becomes an adversarial library that helps keep semantic distance under control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Illustrative examples Example 5 provides us with information about engineering that was taken from Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' We are interested in observing how ORUGA operates while attempting to minimize the FKGL readability score, the number of words that need to be replaced, and the semantic distance between the original text and the text that has been processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' We are replacing the synonyms with the help of WordNet, and we are moving forward with the MOO with the help of NSGA-II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The ultimate goal is to ensure a very low risk of distortion of the original message that wanted to be communicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 3https://pypi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='org/project/gensim/ 24 Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Engineering Original text Source: Wikipedia “Big data refers to data sets that are too large or complex to be dealt with by traditional data-processing application software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Data with many fields (rows) of- fer greater statistical power, while data with higher complexity (more attributes or columns) may lead to a higher false discovery rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Big data analysis challenges include capturing data, data storage, data analysis, search, sharing, transfer, visu- alization, querying, updating, information privacy, and data source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Big data was originally associated with three key concepts volume, variety, and velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The analysis of big data presents challenges in sampling, and thus previously allowing for only observations and sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Thus a fourth concept, veracity, refers to the quality or insightfulness of the data.” ARI score: 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
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+page_content='2 Readability score Words to be replaced Semantic distance ORUGA - Final Minimize ARI score with the least risk of distorting the original message Big data bring up to data sets that are too big or complex to be dealt with by traditional data-processing application software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Data with many fields (rows) offer greater statistical power, while data with higher complexity (more attributes or columns) may lead to a higher false finding rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Big data analysis challenges include capturing data, data storage, data analysis, search, sharing, transfer, visualization, querying, updating, information privacy, and data source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Big data was originally tied in with three key concepts volume, variety, and velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The analysis of big data presents challenges in sampling, and thus previously let for only observations and sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Thus a fourth concept, veracity, refers to the quality or acumen of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' ARI score: 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='33 - ▽ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='62% 25 The example shows that we no longer operate as blindly as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Now, we also minimize the words to be replaced and the semantic distance between the initial and the generated text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Therefore, the results are much more reasonable and can be relied upon to work in exploitation environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The readability score optimization is less spectacular than before, but the risk of distorting the original message, both in form and content, is much more substantially reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Experimental study In this section, we have performed the empirical study to test this version of ORUGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' To do so, we designed the experiments through an experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' We performed the experiments and collected the raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' And finally, we proceeded with the analysis of the collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Experimental setup The accurate adjustment of the MOO strategy’s parameters is not the primary focus here, as was the case with the previous parts of this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' One configuration works quite well after a brief preliminary study based on a conventional parameter-setting strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' This came about as a result of what was mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The following is a list of the parameters that we have used: Population size {10, 15, 20}: 20 Number of parents mating {10, 15, 20}: 20 Number of genes: one per candidate word to be substituted by a synonym Fitness function: Equations 1, 2, 3, 4, words to be replaced, and WMD (Kusner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Stop condition: {300, 600, 900}: 900 generations 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
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+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Experiments We are going to proceed with the experiments concerning minimizing the impact on the meaning of the original message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' We focus now on being able to produce a result that can be put (or be close to being put into exploitation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' To do this, we will conduct experiments to see if we can control the difference in meaning between the original message’s content and the generated text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Figure 5 shows us the summary of all the results obtained for the ten use cases that we have been studying throughout this research work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' As can be seen, each use case, no matter how topical or challenging, corresponds to a good number of solutions ranging from the most conservative (the one that has the least risk of distorting the original message) to the most aggressive (the one that reduces the readability score more conclusively at the risk of distorting the original message).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' According to previous experiments, each color represents a degree of difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
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+page_content='14 Readabity score Words to be replaced Semantic distance Use Case #10 - history Figure 5: Summary of the results obtained for the third (and final) version of ORUGA 27 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
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+page_content=' Discussion We have seen how it is possible to build a solution that enhances the readability of texts of very different natures and readability levels, and we implemented that solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In addition, this sort of optimization is conducted while paying attention to both the content and the structure of the texts in question to minimize the impact of replacing words with synonyms that are a better fit for the readability criteria of the various formulas we have researched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Throughout this research, we have seen how it is possible to use genetic algorithms to improve the readability of any text formulated in English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' As we have explained earlier, optimization is bi-directional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' That is, it can automatically increase or decrease the readability of the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Once we established the basis for the optimization, we could control the number of words that could be replaced;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' we could also control that the semantic distance between the original text and the final text does not skyrocket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Therefore, our initial goals of building a solution that improves the readability of any text without significantly altering its form or content have been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Conclusion In this research work, we have seen how ORUGA can automatically optimize the readability of a text by using genetic algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' We have shown that by automatically replacing some words of the text to be optimized by their synonyms, we can optimize the readability levels in the direction (minimize or maximize) we wish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Neither the content nor the form of the text is altered because a minimal impact on the transformation of the original text is sought through various additional MOO techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Although, in theory, analogous solutions could be built using neural language models, this approach has a significant advantage: it is unsupervised and requires no training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' An exhaustive empirical study has shown that we have successfully performed all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In the first instance, these experiments consisted of processing texts of different natures (history, geography, sports, nature, and science) using three synonym libraries and using different readability metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In the second instance: designing a MOO solution to control the number of words to be replaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' We have tested different texts to assess and compare the feasibility of this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Furthermore, in the third instance: measuring and controlling the semantic distance between the original text and the one that will finally be outputted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' For this, we have used a novel technique that uses a library of synonyms to control the results obtained with another library of different synonyms in an adversarial process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' The results demonstrate that our hypothesis about text readability optimization at the begin- ning of this paper is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Despite the success of this research, it is necessary to bear in mind that simple formulas are typically simpler to put into practice, which is a limitation of this body of work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' These formulas have a fundamental inability to model the semantics of word usage in context, which is needed to capture richer ideas of text difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' 28 As future work, a possible approach (yet computationally expensive) would be using a model of contextual embeddings such as BERT on a large dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Then, it should be necessary to store pairs of words and corresponding contextual representations and then use the nearest neighbors approach to identify synonyms that include the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' In this way, the impact of text processing will be even more limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Source code The source code of this approach is published under MIT license in the following Github repository: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='com/jorge-martinez-gil/oruga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Acknowledgments The research reported in this paper has been funded by the Federal Ministry for Climate Action, Environment, Energy, Mobility, Innovation, and Technology (BMK), the Federal Ministry for Digital and Economic Affairs (BMDW), and the State of Upper Austria in the frame of SCCH, a center in the COMET - Competence Centers for Excellent Technologies Programme managed by Austrian Research Promotion Agency FFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
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+page_content=' Linguistics in language teaching volume 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Edward Arnold London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Zhang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', & Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
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+page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
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+page_content=' URL: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='1109/TEVC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='892759.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='1109/TEVC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='892759.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' Zhu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=', & Iglesias, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
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+page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
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+page_content=' Data Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
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+page_content=' URL: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
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+page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content='2610428.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQfhPid/content/2301.00374v1.pdf'}
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+Draft version January 6, 2023
+Typeset using LATEX twocolumn style in AASTeX63
+The HETDEX Survey: Emission Line Exploration and Source Classification∗
+Dustin Davis,1 Karl Gebhardt,1 Erin Mentuch Cooper,1, 2 Robin Ciardullo,3, 4 Maximilian Fabricius,5, 6 Daniel J. Farrow,6, 5
+John J. Feldmeier,7 Steven L. Finkelstein,1 Eric Gawiser,8 Caryl Gronwall,3, 4 Gary J. Hill,2, 1 Ulrich Hopp,6, 5
+Lindsay R. House,1 Donghui Jeong,3, 4 Wolfram Kollatschny,9 Eiichiro Komatsu,10, 11 Martin Landriau,12 Chenxu Liu,1
+Shun Saito,13, 11 Sarah Tuttle,14 Isak G. B. Wold,15, 16, 17 Gregory R. Zeimann,18 and Yechi Zhang19, 20
+1Department of Astronomy, The University of Texas at Austin, 2515 Speedway Boulevard, Austin, TX 78712, USA
+2McDonald Observatory, The University of Texas at Austin, 2515 Speedway Boulevard, Austin, TX 78712, USA
+3Department of Astronomy & Astrophysics, The Pennsylvania State University, University Park, PA 16802, USA
+4Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA 16802, USA
+5Max-Planck Institut für extraterrestrische Physik, Giessenbachstrasse 1, 85748 Garching, Germany
+6Universitäts-Sternwarte, Fakultät für Physik, Ludwig-Maximilians Universität München, Scheinerstr. 1, 81679 München, Germany
+7Department of Physics, Astronomy, Geology, and Environmental Sciences, Youngstown State University Youngstown, OH 44555
+8Department of Physics and Astronomy, Rutgers, The State University of New Jersey, Piscataway, NJ 08854, USA
+9Institut für Astrophysik, Universität Göttingen, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany
+10Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany
+11Kavli Institute for the Physics and Mathematics of the Universe (WPI), Todai Institutes for Advanced Study, the University of Tokyo, Kashiwanoha, Kashiwa,
+Chiba 277-8583, Japan
+12Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA
+13Institute for Multi-messenger Astrophysics and Cosmology, Department of Physics, Missouri University of Science and Technology, 1315 N Pine St, Rolla, MO
+65409
+14Department of Astronomy, University of Washington, Seattle, 3910 15th Ave NE, Room C319, Seattle WA 98195-0002
+15Astrophysics Science Division, Goddard Space Flight Center, Greenbelt, MD 20771, USA
+16Department of Physics, The Catholic University of America, Washington, DC 20064, USA
+17Center for Research and Exploration in Space Science and Technology, NASA/GSFC, Greenbelt, MD 20771
+18Hobby-Eberly Telescope, University of Texas, Austin, Austin, TX, 78712, USA
+19Institute for Cosmic Ray Research, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan
+20Department of Astronomy, Graduate School of Science, the University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan
+ABSTRACT
+The Hobby-Eberly Telescope Dark Energy Experiment (HETDEX) is an untargeted spectroscopic survey that
+aims to measure the expansion rate of the Universe at 𝑧 ∼ 2.4 to 1% precision for both 𝐻(𝑧) and 𝐷 𝐴(𝑧). HETDEX
+is in the process of mapping in excess of one million Lyman-𝛼 emitting (LAE) galaxies and a similar number of
+lower-z galaxies as a tracer of the large-scale structure. The success of the measurement is predicated on the post-
+observation separation of galaxies with Ly𝛼 emission from the lower-𝑧 interloping galaxies, primarily [O II], with
+low contamination and high recovery rates. The Emission Line eXplorer (ELiXer) is the principal classification
+tool for HETDEX, providing a tunable balance between contamination and completeness as dictated by science
+needs. By combining multiple selection criteria, ELiXer improves upon the 20 Å rest-frame equivalent width cut
+commonly used to distinguish LAEs from lower-𝑧 [O II] emitting galaxies. Despite a spectral resolving power,
+R ∼ 800, that cannot resolve the [O ii] doublet, we demonstrate the ability to distinguish LAEs from foreground
+galaxies with 98.1% accuracy. We estimate a contamination rate of Ly𝛼 by [O II] of 1.2% and a Ly𝛼 recovery
+rate of 99.1% using the default ELiXer configuration. These rates meet the HETDEX science requirements.
+Keywords: Dark energy(351) – Emission line galaxies(459) – Lyman-alpha galaxies(978) – Redshift sur-
+veys(1378)
+∗ Based on observations obtained with the Hobby-Eberly Telescope, which
+is a joint project of the University of Texas at Austin, the Pennsylvania
+State University, Ludwig-Maximilians-Universität München, and Georg-
+August-Universität Göttingen. The HET is named in honor of its principal
+benefactors, William P. Hobby and Robert E. Eberly.
+1. INTRODUCTION
+It is generally acknowledged that the universe is expanding
+and that the expansion is accelerating. Though surprising
+at the time, the accelerated expansion has come to be the
+arXiv:2301.01799v1 [astro-ph.GA] 4 Jan 2023
+
+2
+Davis, et al.
+consensus understanding since the early work of Perlmutter
+et al. (1999) and Riess et al. (1998). Since then, many ob-
+servations have confirmed and refined the measures of this
+expansion with such increased precision that a possible ten-
+sion may have emerged in the results from the various broad
+measurement camps (Di Valentino et al. 2021; Aloni et al.
+2021, among others). Regardless, whether this tension is a
+consequence of real physics, as yet unidentified systematics,
+or some combination, we are essentially limited to only two
+anchor points, one from the recent past (∼ 72 km s−1 Mpc−1;
+Riess et al. 2009; Dhawan et al. 2018; Riess et al. 2021; Mort-
+sell et al. 2021, and others) and one from the Epoch of Re-
+combination (∼ 67 km s−1 Mpc−1; Alam et al. 2017; Planck
+Collaboration et al. 2020a; Aiola et al. 2020, and others),
+from which to constrain descriptions of dark energy. Further
+understanding requires additional data points from different
+epochs in the expansion history of the Universe. Multiple
+efforts are in progress to provide those data, including the fol-
+lowing, but far from exhaustive, list: the Dark Energy Survey
+(DES) (The Dark Energy Survey Collaboration 2005a), the
+Baryon Oscillation Spectroscopic Survey (BOSS) (Dawson
+et al. 2012), the extended Baryon Oscillation Spectroscopic
+Survey (eBOSS) (Alam et al. 2021), the Legacy Survey of
+Space and Time (LSST) (LSST Science Collaboration 2009),
+Euclid (Laureijs et al. 2011), the DESI Survey (DESI Col-
+laboration et al. 2016; Dey et al. 2019a), and, of course,
+the Hobby-Eberly Telescope Dark Energy Experiment (HET-
+DEX) (Ramsey et al. 1998; Gebhardt et al. 2021; Hill et al.
+2021).
+HETDEX is a multi-year untargeted spectroscopic survey
+designed to make new measurements of the Hubble Param-
+eter, 𝐻(𝑧), and the Angular Diameter Distance, 𝐷 𝐴(𝑧), at
+z∼2.4 to better than 1% accuracy in an effort to better charac-
+terize dark energy and look for possible evolution. HETDEX
+observations fall into two large, high galactic latitude fields.
+The ∼ 390 deg2 "Spring" field is centered near (RA,Dec)
+13h00m +53d00m and the ∼150 deg2 "Fall" field is centered
+near 1h30m +0d00m (Gebhardt et al. 2021). Functionally,
+HETDEX seeks to map the 3D positions of some 106 galaxies
+between 1.88 < 𝑧 < 3.52 and use their large scale clustering
+to derive 𝐻(𝑧) and 𝐷 𝐴(𝑧). More specifically, the galaxies
+HETDEX is using for large-scale structure are identified by
+their bright, conveniently red-shifted into the optical, Lyman-
+𝛼 emission lines. These Lyman-𝛼 Emitters (LAEs) are gen-
+erally small, blue, rapidly star-forming galaxies that, while
+uncommon in the local Universe, are present in large num-
+bers in the HETDEX redshift search window (Partridge &
+Peebles 1967; Gawiser et al. 2007; Nilsson 2007; Finkelstein
+2010, and many others).
+The HETDEX Visible Integral-Field Replicable Unit Spec-
+trographs (VIRUS; Hill et al. 2021) cover the wavelength
+range 3500-5500 Å with R∼750–900, and are optimized to
+detect Ly𝛼 flux down to ∼ 4 × 10−17 erg s−1 cm−2 (increasing
+to closer to 2 × 10−16 erg s−1 cm−2 at the extreme blue end
+of the range). This allows the detection of Ly𝛼 luminosities
+down to about 1042.3 erg s−1 for 𝑧 ∼ 2.4. Since it is of utmost
+importance to know the redshift of the observed galaxies, the
+emission must be correctly identified. However, the relatively
+narrow wavelength range often limits our ability to capture
+multiple emission lines and the low spectral resolving power
+prohibits most doublet splitting, making classifications dif-
+ficult. Around 95% of HETDEX emission line detections1
+are spectra containing only one, apparently single peaked
+(given the HETDEX spectral resolving power) emission line,
+and Ly𝛼 is not the only emission line to fall into this observed
+wavelength range. Neutral hydrogen (and dust) in each source
+galaxy’s Interstellar Medium (ISM) and in the Intergalactic
+Medium (IGM) along our line of sight effectively eliminate
+emission lines blueward of Ly𝛼 at higher redshifts (Haardt
+& Madau 1995; Meiksin 2006; Cowie & Hu 1998; Overzier
+et al. 2012; Vanzella et al. 2018), leaving low-𝑧 galaxies as
+the primary contaminate to be considered.
+In the relatively nearby universe, intrinsically small, line-
+emitting faint galaxies can be misidentified as their higher
+redshift cousins. In particular, at the low HETDEX spectral
+resolving power and with no strong lines in the wavelengths
+around it, the [O II] 3727Å emission line can be confused
+with Ly𝛼 1216Å which similarly appears unique in its spec-
+tral neighborhood. In a common case, HETDEX observa-
+tions detect only a single, fairly narrow, emission line and
+little or no continuum at the detection limits. Most likely
+the line is either Ly𝛼 and originates from a high-𝑧 galaxy,
+or [O II] from a low-redshift interloper, and unfortunately,
+these two primary cases occur in roughly equivalent num-
+bers (Adams et al. 2011; Gebhardt et al. 2021). Since the
+HETDEX 𝐻(𝑧) and 𝐷 𝐴(𝑧) measurements are sensitive to in-
+terloper clustering (Leung et al. 2017; Gebhardt et al. 2021;
+Farrow et al. 2021), contamination from [O II] in the LAE
+sample needs to be ≲ 2% (Gebhardt et al. 2021). Historically,
+a 20 Å equivalent width cut (using the rest-frame of Ly𝛼)
+has been used to segregate [O II] from Ly𝛼 (Gronwall et al.
+2007; Adams et al. 2011), and indeed, this criterion is quite
+effective. However, used by itself, the discriminant can still
+lead to >4% contamination and degrade the recovery of lower
+equivalent width Ly𝛼 lines (Acquaviva et al. 2014). Leung
+et al. (2017) improves on the 20 Å cut by taking a Bayesian
+approach and including information on the luminosity func-
+1 HDR3 is limited to emission line detections with SNR ≥ 4.8, of which 95%
+have only a single detected emission line. The fraction of detections with
+only a single line is partly a function of the SNR cut and other selection
+criteria used to define a sample. As in Mentuch Cooper (ApJ accepted),
+SNR ≥ 5.5 is commonly used as it is effectively free from noise detections
+(§5.4). For SNR ≥ 5.5, 70% of HETDEX spectra consist of only a single
+emission line and the entire sample is reduce by 60%.
+
+ELiXer
+3
+tions and equivalent width distributions of Ly𝛼 and [O II] .
+From their modeled data, they report an expected contami-
+nation by [O II] of between ∼ 0.5% and 3.0% at a cost of
+∼ 6.0% to 2.4% lost LAEs, depending on the methods used.
+This is a significant enhancement over the simpler 20 Å cut
+and, in this work, we are able to extend and improve on Leung
+et al. (2017) by (1) incorporating additional selection criteria,
+(2) considering other emission lines as contaminants, and (3)
+comparing directly against observational data.
+The HETDEX Emission Line eXplorer (ELiXer) software
+incorporates and extends these classification works, integrates
+supplemental data and additional classification criteria, and
+expands the analysis to consider more than two dozen other
+emission lines.
+Its primary objective is to classify every
+HETDEX emission line detection by assigning the correct
+redshift to the observed emission lines.
+In addition to its
+primary function as an emission line classifier, ELiXer also
+provides diagnostic and data integrity checking to supple-
+ment that of the HETDEX pipeline (Gebhardt et al. 2021),
+which is run prior to the ELiXer invocation and provides
+the detection coordinates, observation conditions, processed
+(calibrated, PSF weighted) spectra, emission line parameter
+measurements (flux, line width), and CCD information as
+ELiXer inputs. These features are useful for identifying and
+debugging some issues (e.g. errant sky subtraction, stuck/hot
+pixels, amplifier interference, etc) as well as in the manual
+inspection of individual detections.
+While ELiXer does classify all HETDEX detections re-
+gardless of magnitude, additional classification support is
+provided for continuum-bright sources via another software
+tool utilized by HETDEX called Diagnose, developed for
+the Hobby Eberly Telescope VIRUS Parallel Survey (HET-
+VIPS, Zeimann & et al. (in prep)). For a further description
+of source classification and redshift assignment of HETDEX
+sources please see Mentuch Cooper (ApJ accepted). Here,
+however, we focus only the bulk of the HETDEX detections,
+where ELiXer is the primary (or only) classifier. For this
+work, we reference ELiXer version 1.16 used in the gener-
+ation of the most recent HETDEX detections catalog, HET-
+DEX Data Release 3 (HDR3). This catalog contains more
+than 1.5 million entries and was released internally in April
+2022 with a public version to be released in the future. We
+report a projected HETDEX LAE contamination rate from
+[O II] of 1.2% (±0.1%) and an additional 0.8% (±0.1%) from
+all other sources, along with an LAE recovery rate of 99.1%
+(±3.3%) for the default classification configuration. ELiXer
+provides a tunable Ly𝛼 classifier, allowing the balancing of
+contamination vs. completeness as needed for specific science
+goals (see §4.4). ELiXer is a work in progress and contin-
+ues to evolve and improve as more data are collected, both
+from HETDEX and from other surveys, and as classification
+methods are added and refined.
+The remainder of this paper is organized as follows: Section
+2 provides an overview of the various photometric catalogs
+currently included in ELiXer. Section 3 describes the classi-
+fication methodologies and supporting functions. Section 4
+covers the selection of a Spectrocopic-z Assessment Sample
+(SzAS) providing spectroscopic redshifts from various imag-
+ing catalogs and the results of testing against that sample.
+Section 5 presents a discussion of the results and the science
+implications. Section 6 summarizes the work and future en-
+hancements. Example ELiXer detection reports are shown
+in Appendix-A with descriptions provided for the major fea-
+tures.
+Throughout the paper, the Planck 2018 cosmology (Planck
+Collaboration et al. 2020b) with ΩΛ= 0.69, Ωm= 0.31 and
+H0 = 67.7 km s−1 Mpc−1 is assumed. All magnitudes are in
+the AB system (Oke & Gunn 1983) and coordinates are J2000.
+2. IMAGING CATALOGS
+HETDEX is an untargeted spectroscopic survey, and the
+spectra alone provide most of the critical information for
+object classification. Coupled with the on-sky positions of
+the associated fibers, these data form the basis for the HET-
+DEX cosmology measurements. For the brighter detections,
+a source’s redshift and, to a lesser degree, its physical ex-
+tent and morphology can be determined securely from the
+spectra. However, for the fainter emission line detections,
+additional information from archival photometric imaging,
+including an object’s magnitude, color, angular/physical size,
+morphology, and even on-sky neighbors, can prove quite
+useful in ascertaining its identity. Even superimposing the
+HETDEX fiber positions on imaging data can provide diag-
+nostic checks on the astrometry and the reduction pipeline.
+Given these substantial benefits, ELiXer attempts to match all
+HETDEX observations with multi-band archival photometry
+at the highest angular resolution and imaging depth available.
+2.1. Individual Catalog Summaries
+At the time of writing, ELiXer references 11 separate imag-
+ing catalogs, most with their own associated object cata-
+log. These catalogs are of varying depth, resolution, band-
+coverage, and footprint. Additional catalogs can be added
+at any time and several new or expanded source lists are an-
+ticipated before the next HETDEX data release. With the
+exceptions of an 𝑟-band survey from the HyperSuprimeCam
+group (HSC-DEX) and a 𝑔-band survey from Kitt Peak Na-
+tional Observatory (KPNO;HETDEX-IM) that were specially
+designed and executed for HETDEX, all imaging and object
+catalogs are archival and publicly available. These catalogs
+are summarized in Table 1 and in the list below.
+
+4
+Davis, et al.
+Table 1. Summary of the imaging surveys incorporated into ELiXer
+Name
+HETDEX Field
+Overlap1
+Filters and Depth2
+PSF FWHM3
+Object Catalog4
+Canada-France-Hawaii
+Telescope Legacy Survey
+(CFHTLS)
+Spring
+4%
+Deep: 𝑢(26.3), 𝑔(26.0),
+𝑟(25.6), 𝑖(25.4), 𝑧(25.0)
+Wide: 𝑢(25.2), 𝑔(25.5),
+𝑟(25.0), 𝑖(24.8), 𝑧(23.9)
+0.6-1.0′′
+phot-𝑧
+𝐻𝑆𝑇 Cosmic Assembly
+Near-infrared Deep
+Extragalactic Legacy Survey
+(CANDELS) in the Extended
+Groth Strip (EGS)
+Spring
+<1%
+ACS/WFC: F606W, F814W
+WFC3: F105W, F125W,
+F140W, F160W
+0.08′′
+spec-𝑧, phot-𝑧
+𝐻𝑆𝑇 Cosmic Assembly
+Near-infrared Deep
+Extragalactic Legacy Survey
+(CANDELS) in the Great
+Observatories Origins Deep
+Survey, North (GOODS-N)
+Spring
+<1%
+ACS/WFC: F435W, F606W,
+F775W, F814W
+WFC3: F105W, F125W,
+F160W
+0.08′′
+spec-𝑧, phot-𝑧
+Hyper Suprime-Cam
+HETDEX Survey
+(HSC-DEX)
+Spring
+44%
+𝑟(25.5)
+0.6-1.0′′
+mag only
+Kitt Peak National
+Observatory HETDEX
+Imaging Survey (KPNO;
+HETDEX-IM)
+Spring
+20%
+𝑔(24.4)
+1.1-1.5′′
+mag only
+Cosmic Evolution Survey
+(COSMOS) with Dark
+Energy Camera (DECam)
+Fall
+2%
+𝑔(25.5), 𝑟(25.5)
+0.7-1.0′′
+(1) phot-𝑧 (Laigle+2015)
+(2) mag only
+Hyper Suprime-Cam Subaru
+Strategic Program
+(HSC-SSP)
+Fall
+29%
+Deep 𝑔(27.5), 𝑟(27.1),
+𝑖(26.8), 𝑧(26.3), 𝑦(25.3)
+Wide 𝑔(26.5), 𝑟(26.1),
+𝑖(25.9), 𝑧(25.1) ,𝑦(24.4)
+0.6-1.0′′
+mag only
+Spitzer/HETDEX
+Exploratory Large-Area
+(SHELA) with Dark Energy
+Camera (DECam)
+Fall
+25%
+𝑢(25.4), 𝑔(25.1), 𝑟(24.7),
+𝑖(24.0), 𝑧(23.7)
+0.7-1.0′′
+mag only
+Dark Energy Camera Legacy
+Survey (DECaLS)
+Spring & Fall
+17%
+𝑔(24.0), 𝑟(23.4), 𝑧(22.5)
+1.2′′
+No
+Panoramic Survey Telescope
+and Rapid Response System
+(Pan-STARRS)
+Spring & Fall
+<1%
+𝑔(23.3), 𝑟(23.2), 𝑖(23.1),
+𝑧(22.3), 𝑦(21.3)
+1.0-1.3′′
+No
+Sloan Digital Sky Survey
+(SDSS) DR16
+Spring & Fall
+<1%
+𝑢(22.0), 𝑔(23.1), 𝑟(22.7),
+𝑖(22.2), 𝑧(20.7)
+1.3′′
+spec-𝑧, phot-𝑧
+1Fraction of HETDEX Data Release 3 within each catalog footprint, except for DECaLS, Pan-STARRS, and SDSS which report only the
+fraction which does not also overlap with a previously listed catalog. Since multiple catalogs overlap, the column sums to > 100%.
+2Approximate average AB depth over the whole catalog as reported, typically for point sources and 2′′apertures. For some 𝑔 and 𝑟 filters and
+some image tiles, ELiXer uses its own estimated depths at 1′′and 2′′apertures. Not all surveys use the same SDSS ugriz filters, though for this
+purpose they are approximately similar. Only filters used by ELiXer are listed.
+3Typically in 𝑟-band
+4If not "No", also has an object catalog used by ELiXer with at least 𝑔 or 𝑟 magnitudes. Spec-𝑧 and/or phot-𝑧 redshifts are available where
+noted, but not necessarily for all object entries.
+
+ELiXer
+5
+• Canada-France-Hawaii
+Telescope
+Legacy
+Survey
+(CFHTLS): A multi-band (𝑢𝑔𝑟𝑖𝑧) imaging survey and
+joint venture of the National Research Council of
+Canada, the Institut National des Science de l’Univers
+of the Centre National de la Recherche Scientifique
+(CNRS) of France, and the University of Hawaii, uti-
+lizing the MegaPrime/MegaCam on the 3.6m Canada-
+France-Hawaii Telescope (CFHT) on Mauna Kea.
+ELiXer uses the deep and wide fields, D3/W3 cen-
+tered near RA 210◦, Dec +52◦. (Brimioulle et al. 2008;
+Cuillandre et al. 2012)
+• HST Cosmic Assembly Near-infrared Deep Extragalac-
+tic Legacy Survey (CANDELS) in the Extended Groth
+Strip (EGS): CANDELS is a deep 𝐻𝑆𝑇 survey (900+
+orbits) with multiple filters in the optical (using the Ad-
+vanced Camera for Surveys, ACS) and near-IR (using
+the Wide Field Camera 3, WFC3) studying on galaxy
+evolution with an emphasis on Cosmic Dawn and Cos-
+mic High Noon. The EGS is one of the five fields of
+CANDELS and is centered near RA 215◦, Dec +53◦.
+(Grogin et al. 2011; Koekemoer et al. 2011; Stefanon
+et al. 2017). The photometric redshifts used in ELiXer
+are provided by Andrews, B., et al, ApJ submitted.
+• HST Cosmic Assembly Near-infrared Deep Extragalac-
+tic Legacy Survey (CANDELS) in the Great Observa-
+tories Origins Deep Survey, North (GOODS-N): An-
+other of the 5 CANDELS fields (see previous bullet),
+GOODS-N is centered near RA 189◦, Dec +62◦ (Dick-
+inson et al. 2002; Grogin et al. 2011; Koekemoer et al.
+2011; Barro et al. 2019) Again, the photometric red-
+shifts used in ELiXer are provided by Andrews, B., et
+al, ApJ submitted.
+• Hyper Suprime-Cam HETDEX Survey (HSC-DEX):
+This survey consists of three nights of HSC 𝑟-band
+observations with the Subaru/HSC in 2015-2018 (PI:
+Andreas Schulze) and 2019-2020 (PI: Shiro Mukae)
+and covers the ∼ 250 deg2 area of the HETDEX Spring
+field. Data reduction and source detections were per-
+formed with version 6.7 of the HSC pipeline, hscPipe
+(Bosch et al. 2018), and produced 𝑟-band images with
+a 10𝜎 limit of 𝑟 = 25.1 mag in a 2′′ diameter circular
+aperture. These HSC 𝑟-band images are complemen-
+tary to the existing imaging data of the Kitt Peak 4-m
+Mosaic camera and the CFHT Wide-Field Legacy sur-
+vey.
+• Kitt Peak National Observatory HETDEX Imaging Sur-
+vey (KPNO; HETDEX-IM): A 𝑔-band survey with the
+Mosaic camera on the Mayall 4-m telescope at Kitt
+Peak National Observatory in 2011-2014 (PI: Robin
+Ciardullo).
+• Cosmic Evolution Survey (COSMOS) with Dark Energy
+Camera (DECam): The 3 deg2 ugriz-band COSMOS
+DECam catalog was generated with the same procedure
+used for the larger field of view SHELA DECam survey
+listed below (Wold et al. 2019). This also overlaps with
+Laigle et al. (2016).
+• Hyper Suprime-Cam Subaru Strategic Program (HSC-
+SSP): Multi-depth, multi-band, wide-field imaging sur-
+vey using the Hyper Suprime-Cam on the 8.2m Subaru
+at the Mauna Kea Observatories. For HETDEX Data
+Release 3, ELiXer uses HSC-SSP Public Data Release
+3 from August 2021. (Aihara et al. 2021)
+• Spitzer/HETDEX Exploratory Large-Area (SHELA)
+with Dark Energy Camera (DECam): This survey cov-
+ers 17.5 deg2 of the HETDEX Fall field within the
+Sloan Digital Sky Survey (SDSS) “Stripe 82” region.
+The ugriz-band DECam catalog is riz-band-selected
+and reaches a 5𝜎 depth of ∼ 24.5 AB mag for point
+sources (Wold et al. 2019).
+• Dark Energy Camera Legacy Survey (DECaLS): A
+multiband (𝑔𝑟𝑧) photometric survey, part of the Dark
+Energy Survey (The Dark Energy Survey Collabora-
+tion 2005b), based at the Cerro Tololo Inter-American
+Observatory using the Dark Energy Camera (DECam)
+on the 4m Blanco telescope. ELiXer uses Data Release
+9 which also includes observations from the Beijing-
+Arizona Sky Survey (BASS) and the Mayall z-band
+Legacy Survey (MzLS). (Dey et al. 2019b)
+• Panoramic Survey Telescope and Rapid Response Sys-
+tem (Pan-STARRS): Specifically, Pan-STARRS1, is a
+set of wide-field synoptic imaging surveys using the
+1.8m PS1 optical telescope at the Haleakala Observa-
+tories. PS1 collected data from 2010 through 2014.
+(Chambers et al. 2019)
+• Sloan Digital Sky Survey (SDSS): Multiband (𝑢𝑔𝑟𝑖𝑧)
+wide-field survey in operation since 2000 using a 2.5m
+optical telescope at the Apache Point Observatory.
+ELiXer uses Data Release 16 from SDSS Phase-IV.
+(Ahumada et al. 2019)
+2.2. ELiXer Aperture Photometry
+ELiXer directly uses the photometric imaging to gather
+aperture magnitudes for the HETDEX detections.
+While
+magnitudes are computed for each available filter, only 𝑔
+and 𝑟 magnitudes are used in the classification process (§3).
+For each HETDEX detection, ELiXer identifies the catalogs
+with overlapping imaging and gathers postage-stamp (9′′×9′′
+by default) imaging cutouts centered on the HETDEX detec-
+tion’s coordinates. Three sets of aperture magnitudes are then
+
+6
+Davis, et al.
+computed using the Python packages Astropy (Astropy Col-
+laboration et al. 2018a), Photutils (Bradley et al. 2020), and
+Source Extraction and Photometry (SEP) (Barbary 2016).
+The identified aperture(s) are used later to provide continuum
+estimates (§3.2) and size information (§3.5.1).
+First, ELiXer computes a magnitude for a dynamically sized
+circular aperture. We center the circular aperture on the HET-
+DEX coordinates, compute the magnitude within the aperture,
+and allow the aperture to grow until the magnitude stabilizes
+(e.g., Howell 1989). The initial size is set by a combination
+of the median seeing and pixel scale of the catalog+filter and
+is typically ∼ 1′′ in diameter. The magnitude within the aper-
+ture is computed, with the background determined from an
+annulus 2× to 3× the defined maximum allowed object aper-
+ture (6′′ diameter by default, for an annulus of 12′′ to 18′′).
+The aperture is then grown in steps of 0.′′1, with each measure-
+ment recorded, until the maximum diameter is reached. The
+smallest aperture size where the magnitude change to the next
+step up is less than 0.01 is assigned, and the corresponding
+magnitude is selected.
+Next, ELiXer uses SEP (Barbary 2016), which is based on
+the original Source Extractor (Bertin & Arnouts 1996), iter-
+ating over each cutout and records the magnitude, barycentric
+position, major and minor axes, and orientation of each iden-
+tified object. ELiXer also computes and records the angular
+separation from each barycenter to the HETDEX coordinates
+and the separation to the nearest point on the bounding ellipse
+if the HETDEX position lies outside that ellipse. The object
+with the nearest barycenter to the HETDEX position whose
+bounding ellipse includes the HETDEX position is consid-
+ered the best aperture match. If no object’s ellipse includes
+the HETDEX position, then the object with the nearest ellipse
+point to the HETDEX position but no more than 0.′′5 away is
+selected as the best match. If no object meets these criteria
+then no SEP found object is selected and the best circular
+aperture (see previous paragraph) is used for the aperture
+photometry.
+Lastly, at each SEP identified barycenter, ELiXer computes
+and records the background subtracted magnitude in a fixed,
+3.′′0 diameter circular aperture. These aperture magnitudes
+are intended for use in any fixed-aperture spectral energy
+distribution (SED)-fitting and color comparisons, but are not
+otherwise significantly used in the core ELiXer processing.
+2.3. Catalog Counterpart Matching
+ELiXer also attempts to match each HETDEX detection to
+one or more objects in each imaging catalog with a particular
+focus on 𝑔 and 𝑟 magnitudes, which can provide additional
+measures for use in other ELiXer functions. Object matching
+is based on a combination of barycenter position and agree-
+ment between the magnitudes reported by each catalog, the
+magnitudes computed within the ELiXer ellipses (§2.2), and
+the HETDEX spectrum estimated 𝑔-band magnitude.
+The nearest catalog object to the HETDEX position that
+falls within the selected best aperture (§2.2), or the near-
+est catalog object within 1.′′0 of the HETDEX position if
+no object falls within the best aperture, is identified as the
+catalog match object.
+If the candidate object’s reported
+magnitude is not compatible with the magnitude estimated
+from the HETDEX spectrum, then the next nearest object
+is evaluated until a match is found or the distance criteria
+are no longer satisfied.
+Compatibility with the HETDEX
+𝑔 magnitude (§3.2.1) is defined as an absolute difference
+of 0.5 magnitudes; if the HETDEX 𝑔 magnitude is fainter
+than the HETDEX magnitude limit (about 25𝐴𝐵), then no
+faint-side restriction is imposed. On the other end, if both the
+counterpart and the HETDEX magnitudes are brighter than
+22𝐴𝐵, they are considered compatible. For the purposes of
+this comparison, 𝑔 and 𝑟 are considered equivalent. There is
+at most one catalog match object per catalog+filter combi-
+nation. This object is later used for additional information,
+including spec-𝑧 and phot-𝑧 assignments if available, in the
+classification process.
+3. CLASSIFICATION
+Classifications in ELiXer are broadly interpreted as the
+identification of the redshifts of observed astrophysical ob-
+jects. This properly requires the additional steps of correctly
+associating an observed spectrum with a single host object
+and furthermore identifying or bounding what constitutes that
+"single object". More fundamentally, given a spectrum and
+a specified emission line in that spectrum, what we hereafter
+call the "anchor line", ELiXer attempts to determine the iden-
+tity, and thus the redshift, of that anchor line. Classification
+proceeds from the assumption that the anchor line is real and
+not spurious noise, an instrument or software artifact, or a
+misinterpretation of spectral data, such as the misidentifica-
+tion of continuum between two closely-separated absorption
+troughs. ELiXer initially assumes that the spectrum repre-
+sents a single object (single redshift), though later analysis
+explores the possibility that a HETDEX spectrum is a blend
+of spectra from discrete but immediately adjacent or over-
+lapping sources on sky (within a single, common detection
+aperture) at different redshifts.
+The focus of ELiXer’s classification is placed on distin-
+guishing Ly𝛼 from [O ii], by far the most common Ly𝛼 con-
+taminant in HETDEX data, and the bulk of the tests and
+conditions target that objective. Additional checks, described
+throughout this section, attempt to refine this bifurcated clas-
+sification and identify the spectral line(s) as any one of those
+listen in Table 2. As will be discussed in §5, these "Other"
+lines are encountered much less frequently than Ly𝛼 and
+
+ELiXer
+7
+[O ii] and, while they can be more challenging to identify,
+the HETDEX cosmology science is extremely robust against
+contamination from these misclassifications.
+The classification of HETDEX detections is organized to
+answer three increasingly general questions, with each an-
+swer incorporating the results of the previous question. First,
+closely following the work of Leung et al. (2017), we evaluate
+the relative likelihood that the target emission line is Ly𝛼 and
+rather than [O ii] (Adams et al. 2011; Gebhardt et al. 2021;
+Farrow et al. 2021). This is largely based on measurements
+of the emission line luminosity and equivalent width eval-
+uated against luminosity and equivalent width distributions
+of Ly𝛼 and [O ii] emitting galaxies from other publications
+interpolated at the redshift corresponding to the emission line
+wavelength (see §3.4). Second, we determine the confidence
+of the initial classification by performing checks against more
+than two dozen other emission lines. Here a weighted voting
+scheme is used with many independent (or semi-independent)
+rules applied to measured and derived features of the spec-
+trum and detection object. Third, we assign, with some rough
+measure of quality, the redshift and thus the specific identity
+of the emission line(s).
+This final step incorporates some
+additional rules and weights to combine all prior results.
+Broadly, ELiXer classifications build up evidence in a series
+of steps and then weighs the evidence to make a determina-
+tion. The high level steps are fairly serial and often largely
+independent, with their results only combined toward the end
+of the process. These major steps are described in more de-
+tail, and in roughly the same order, in the subsections that
+follow.
+1. Find, fit, and score all emission and absorption lines
+and set the anchor line
+2. Evaluate all combinations of found spectral lines for
+compatibility with redshifts, based on relative posi-
+tions, strengths, etc
+3. Collect additional (aperture) photometric imaging in-
+formation and any reported magnitude, spec-z, and
+phot-z measurements for the target object and its neigh-
+bors from non-HETDEX catalogs (Table 1)
+4. Evaluate spectra shape, lines, and imaging for consis-
+tency with known astrophysical objects (star, White
+Dwarf, AGN, meteor, low-z galaxy)
+5. Examine HETDEX data for corruption, pipeline arti-
+facts, and instrument issues.
+6. Test the compatibility of the anchor line with Ly𝛼
+7. Perform evaluations on the anchor line, including spec-
+tral and photometric information, to specifically distin-
+guish Ly𝛼 from [O ii]
+8. Perform separate evaluations on the anchor line, in-
+cluding spectral and photometric information, for con-
+sistency with lines other than Ly𝛼 and [O ii]
+9. Combine all evaluations to determine and rank likely
+redshifts and line classifications
+10. Re-evaluate redshift classification based on clustering
+with ELiXer results from the other neighboring HET-
+DEX detections
+The figures in this section illustrating some of the voting cri-
+teria and thresholds pull their data from the Spectroscopic-𝑧
+Assessment Sample (SzAS) whose selection and composition
+is described in Section 4.
+3.1. Line Finder
+Emission (and absorption) line detection is implemented
+as both a layered, untargeted search and a targeted line fit
+assuming an "anchor" line. More details will follow in the
+next subsections, but briefly put, the untargeted search scans
+the full width of the spectrum from blue to red, marks the lo-
+cations of possible emission line-centers, and attempts to fit a
+single Gaussian (in agreement with the measured instrumen-
+tal resolution; Hill et al.2021) to each position. The targeted
+search uses a single previously identified emission line (from
+the HETDEX input, user input, or the previous untargeted
+search) as an anchor and then assumes that anchor line is
+one of roughly two dozen potential emission lines (Table 2)
+and attempts to fit a Gaussian to the positions where other
+emission lines could be found, assuming that identify for
+the anchor line. The descriptions that follow are couched in
+terms of emission lines, as that is the primary use. A limited
+use of absorption lines is implemented and is described in
+§3.1.6.
+3.1.1. Untargeted Search
+The untargeted search scans the entire 1D HETDEX spec-
+trum to identify the positions and model the parameters of
+potential emission lines. It is used to (1) identify the strongest
+line as the reference or anchor line when no initial emission
+line is explicitly provided, (2) mark strong lines for con-
+sistency checks with redshift solutions and to help identify
+blended spectra, and (3) mark line positions for followup vi-
+sual inspection, without respect to the selected solution.
+Because Markov Chain Monte Carlo (MCMC) fits are rela-
+tively computationally expensive, and HETDEX spectra typ-
+ically have only one or very few emission lines, we do not
+want to perform such fits at each pixel along the spectrum.
+Instead, we first conduct a quick examination to narrow the
+potential locations of emission lines. We do this using two in-
+
+8
+Davis, et al.
+Table 2. Emission Line Candidates
+Name
+rest-𝜆 [Å]
+Name
+rest-𝜆 [Å]
+O VI
+1035
+H𝜂
+3835
+Ly𝛼
+1216
+[Ne III]
+3869
+N V
+1241
+H𝜁
+3889
+Si II
+1260
+(K) Ca II*
+3934
+Si IV
+1400
+[Ne III]
+3967
+C IV
+1549
+(H) Ca II*
+3968
+He II
+1640
+H𝜖
+3970
+C III]
+1909
+H𝛿
+4101
+C II]
+2326
+H𝛾
+4340
+Mg II
+2799
+H𝛽
+4861
+[Ne V]
+3346
+[O III]
+4959
+[Ne V]
+3426
+Na I
+4980
+[O II]
+3727
+[O III]
+5007
+Na I
+5153
+∗Fit as an absorption line
+Note— Possible identifications for spectral lines found
+in the HETDEX spectra.
+dependent algorithms and then combine the output positions
+into a single list for further examination.
+Two passes through the algorithms of this untargeted search
+are conducted. The first execution uses the native 2 Å binned
+HETDEX spectrum and focuses on identifying the common
+narrow spectral features. The second execution is performed
+after passing the original spectrum through a median filter
+(by default using a 5 pixel kernel), to smooth out some of
+the noise. This helps identify candidate emission lines that
+are wider than the ∼ 400 km s−1 resolution of the VIRUS
+spectrographs and may have small noise peaks within their
+overall broad shape.
+The first algorithm searches for the basic shape of an emis-
+sion feature, a general rise to a peak and then a decline. Due to
+the unavoidable noise in the data, the spectra are not smooth
+and the use of the first derivative to find zeros (and the second
+derivative to distinguish between an emission and absorption)
+results in more false detections than real spectral features. In-
+stead, we look for the general shape of the lines (a rise and fall
+in the flux of minimum height over a minimum width), based
+on the spectral resolution, flux limits, and noise of HETDEX.
+Sets of contiguous pixels that are sufficiently wide in the spec-
+tral direction and have the expected rise-peak-fall pattern are
+recorded as possible emission lines, and their line centers are
+recorded.
+The second algorithm counts contiguous pixels with flux
+values above some multiple of the corresponding noise (typi-
+cally SNR > 3, under the assumption that the flux uncertainty
+is distributed normally). Where the contiguous count of pix-
+els above this noise is greater than some count (here, typically
+3-5 pixels), the position of the highest flux value within that
+range is recorded as the possible emission line center. Es-
+sentially, this is just a SNR-cut over the spectrum. Unlike
+the first algorithm, the shape of the flux above the SNR-cut is
+irrelevant.
+The line centers from each algorithm are then passed to
+fitting (§3.1.3) and scoring routines (§3.1.4). When model fits
+to the flux at those positions are successful and the computed
+line score is sufficiently large, the feature is recorded to a list
+of potential spectral lines.
+After both the standard and broad line searches are con-
+ducted, the list of potential emission and absorption lines are
+merged into a single list, and any neighboring lines with line
+centers within in 4 Å of each other are combined into single
+entries by keeping only the feature with the largest line score.
+As a brief note: though this is not the normal operation of
+ELiXer under HETDEX, if no anchor line is specified for the
+spectrum to be classified, the line (emission or absorption)
+with the largest score (§3.1.4) found in this untargeted search
+is assumed as the anchor line. If the untargeted search fails
+to identify any spectral lines, the wavelength bin with the
+largest flux value is assigned as the anchor line position.
+3.1.2. Targeted Search
+Unlike the untargeted search described above, the targeted
+search does not scan for potential emission or absorption lines,
+but instead attempts to fit for an emission or absorption fea-
+ture at a specified position. Essentially, ELiXer attempts to fit
+spectral lines from a predefined list of common lines (Table
+2) at their expected observed wavelength positions given an
+assumed identity or redshift for the anchor line. The redshift
+assumptions come from alternately interpreting the anchor
+line as each of the common lines and from any matching spec-
+troscopic or photometric catalogs with a possible counterpart
+to the HETDEX detection. With each redshift assumption, all
+other lines in the subset that could occur within the HETDEX
+spectral window are fitted, allowing for some error in the sys-
+temic redshift (see Position Capture under §3.1.3). This is
+often redundant with the untargeted search in that, for higher
+signal-to-noise ratio (SNR) lines, the lines found in the tar-
+geted search are also found in the untargeted search. However
+lower SNR lines, [O iii] 𝜆4959 for example, can be missed in
+the initial sweep of the untargeted search. Fitting to a specific
+wavelength location helps avoids such misses.
+Each successfully fitted line for each assumed identity of
+the anchor line is scored (§3.1.4) and associated with the
+redshift solution (§3.3) for that identification.
+
+ELiXer
+9
+3.1.3. Line Fitting
+ELiXer uses a simple, 4-parameter (𝐴, 𝜇, 𝜎Line, 𝑦) single
+Gaussian as the model to fit emission and absorption features:
+𝐹(𝜆) =
+𝐴
+𝜎Line
+√
+2𝜋
+exp
+�
+− (𝜆 − 𝜇)2
+2𝜎2
+Line
+�
++ 𝑦,
+(1)
+where 𝐹(𝜆) is the flux per 2 Å wavelength bin, 𝐴 is the area
+under the curve or equivalently the integrated line-flux, 𝜇 is
+the line center, 𝜎Line is the measure of width, 𝑦 is the vertical
+offset, or flat continuum level, and 𝜆 is the wavelength (at the
+midpoint of a 2 Å wide wavelength bin).
+The flat continuum is a reasonable simplification, as no
+assumption is made as to the object type or its redshift, most
+HETDEX detections have continua at or below the survey’s
+continuum flux limit, and those objects with continua bright
+enough to have a shape typically have multiple emission lines
+or are too bright to support a Ly𝛼 classification. This con-
+tinuum estimate can be highly uncertain, especially for the
+noisier spectra, but as discussed later, multiple continuum es-
+timates are combined to improve the uncertainty and for the
+non-detections, the resulting equivalent width estimates are
+lower limits that favor a low contamination Ly𝛼 selection, at
+the cost of some completeness.
+Type I AGN may have broad lines that are not well fitted
+by a single Gaussian (Liu et al. 2022). Such detections are
+marked by ELiXer with warnings, but are not confused with
+the fainter, compact LAEs the software is designed to identify.
+We note, however, that it is possible that the simple emission
+line search can completely fail to find rare, extremely broad
+emission lines, as ∼ 3500 km s−1 is the maximum FWHM
+that ELiXer attempts to fit.
+More complex models, including the fitting of multiple
+emission and absorption lines within a single spectral fea-
+ture, have either proven to be unreliable, too computationally
+costly, and/or of limited utility for the main goal of simply
+identifying redshifts when the vast majority of line detections
+are well fit by the simple, single Gaussian model. Fitting
+for an emission line doublet would be useful in the effort to
+distinguish between Ly𝛼 and [O ii] however, given the low
+spectral resolving power of VIRUS, Δ𝜆/𝜆 ∼ 800 (Hill et al.
+2021), the [O II] doublet (3726, 3729 Å) is unresolved as
+are most other doublets (Mg II (2796, 2803 Å) is sometimes
+marginally resolved). The increased run time of fitting these
+extra parameters is not justified. For smaller data sets, such as
+for the case of AGN exploration, more complex fitting is war-
+ranted (Liu et al. 2022), but left to those specialized projects.
+For ELiXer’s classification needs, a description of the spectral
+feature that is limited to its position (wavelength), equivalent
+width (approximate integrated line flux and local continuum),
+and line width are sufficient. Additional parameters, such as
+the model’s skewness and kurtosis, and conditions combining
+those and other parameters have been explored but have not
+been found to improve the identification of real spectral fea-
+tures or aid in the classification, and are thus excluded from
+further discussion in this work.
+With the exception of the anchor line on which an MCMC fit
+is always performed, if a least square (LSQ) model fit passes
+its quality checks, no MCMC fit is conducted. This is due to
+the increased runtime cost of MCMC fitting weighed against
+the relatively modest needs for classification. In all MCMC
+cases however, an LSQ fit is performed first and its results are
+used as initial conditions (with appropriate randomization)
+for the MCMC algorithm.
+ELiXer uses the Python scipy
+package and its scipy.optimize.curve_fit (Virtanen et al. 2020)
+as the LSQ fitter; the MCMC fitter is from the Python emcee
+package (Foreman-Mackey et al. 2013). Uncertainties in the
+LSQ fit are estimated using the square root of the diagonal
+of the covariance matrix. Uncertainties in the MCMC fit are
+estimated using the 68% confidence interval in the parameter
+distribution.
+A series of loose checks evaluates the quality of each fit
+as minimally good, marginal, or poor. Poor fits are rejected;
+good fits are scored (see §3.1.4) in preparation for building
+solutions. Marginal solutions from the LSQ fitter are passed
+to the MCMC algorithm for improved optimization and re-
+evaluated. If the subsequent MCMC fit is good, the fit is
+scored and made eligible for inclusion in redshift solutions.
+If the MCMC fit is not sufficiently improved over the LSQ fit,
+it is rejected.
+The quality checks include following conditions:
+• Peak Capture: As a basic check, should the peak of
+the model fail to reproduce the most extreme measured
+data value near the line center within 50%, the fit is
+rejected. If the model is within 25% and 50% of the
+most extreme value, it is flagged for an MCMC fit.
+Should that MCMC fit fail to be within 25%, the fit is
+rejected and no line is assumed to be at that position.
+• Position Capture: If the fitted line center is greater
+than a configured maximum distance (in Å) from the
+local data extremum, the fit is rejected.
+The maxi-
+mum distance allowed can depend on the assumed line
+identification and its assumed position, with greater
+separations allowed for Ly𝛼 which can be significantly
+offset from the systemic redshift (Shapley et al. 2003;
+McLinden et al. 2011; Verhamme et al. 2018; Gurung-
+Ló pez et al. 2021, among others). During the untar-
+geted search, no variations are allowed and a default of
+8 Å (∼ 500 km s−1 in the HETDEX spectral range) is
+used.
+• Width Capture: If the fitted line width (here parame-
+terized as 𝜎) is less than 1.0 Å, i.e., significantly below
+
+10
+Davis, et al.
+the HETDEX spectral resolution of ∼ 2.0 Å (Hill et al.
+2021), or if the line width is greater than the config-
+ured maximum value of 17 Å (∼ 2700 km s−1 FWHM)
+or 25 Å (∼ 3500 km s−1 FWHM) for special, broad fit
+attempts, the fit is rejected.
+• Area Error: If the error on the line area (as estimated
+from the square root of the diagonal of the LSQ fit’s
+covariance matrix or the 68% confidence interval on
+the MCMC fit) is larger than the absolute value of the
+area (allowing for absorption or emission), the fit is
+rejected.
+• Local Uniqueness: This is used only in combination
+with other conditions. An emission or absorption line
+is considered unique if there is at most one other data
+extremum greater than 90% of this line’s peak between
+1× FWHM and 1× FWHM + 10 Å to either side of the
+line center.
+This is an alternate rough measure of local noise and is
+used primarily as a filter with low SNR lines.
+• SNR and 𝜒2: ELiXer uses the following definitions of
+SNR and 𝜒2:
+SNR =
+� √︁
+(𝐹(𝜆) − 𝑦)2
+√︁� (error2)
+,
+(2)
+𝜒2 =
+∑︁ �data − model
+error
+�2
+,
+(3)
+where the summations are over the wavelength bins
+within ±2𝜎 of the fit line center. 𝐹(𝜆) and 𝑦 are from
+Eqn 1. The model is the fitted flux evaluated at each
+corresponding wavelength bin for the data and the error
+is the uncertainty on the data.
+The uncertainty on the SNR is computed via standard
+error propagation using the MCMC or LSQ uncertain-
+ties on each of the model’s Gaussian parameters.
+If the LSQ fit is marginal given the previous conditions,
+it is rejected if (1) the SNR is less than 5.0 or (2) if the
+SNR is between 5.0 and 15.0 and the 𝜒2 is greater than
+2.0. These indicate poor fits to possibly noisy data and
+are generally not worth pursuing. Otherwise, the SNR
+and 𝜒2 are recorded for use in line scoring.
+3.1.4. Line Scoring
+Every successfully fitted emission and absorption line re-
+ceives a score based only on its own properties, without con-
+sideration to the position or properties of any other fitted
+emission or absorption lines. If that score exceeds a mini-
+mum threshold, the line, with its score, is accepted into a list
+of potential line candidates for later use in redshift solution
+finder (§3.3). The minimum threshold is configurable and
+is set, by default, to an empirically determined value based
+on the manual examination of many tens of thousands of ob-
+served spectra and a simulation of spectra drawn from median
+HETDEX noise properties (§3.1.5). Redshift solutions that
+fit multiple lines to the spectrum receive a separate "solution
+score" (§3.3) that is based, in part, on these individual "line
+scores".
+The line score attempts to capture and quantify features
+beyond just the signal-to-noise ratio, which is a less than ideal
+metric for broad emission lines fitted with a single Gaussian.
+The line score takes into account additional data including the
+magnitude of the integrated (fitted) line flux, the line position
+relative to expectations, and the uniqueness of the line within
+a local spectral region. The intent is to codify not just the
+presence of each potential emission line, but the consistency
+and significance of that line with respect to the spectrum at
+an assumed redshift.
+The line score calculation is defined as:
+𝑆𝐿 = 𝑆lim · 𝐴𝑁 · 𝑈𝑁 · 𝐹𝜆 · 𝑚𝜎 · 𝑚pix
+1+ | 𝛿𝑑𝑥0 |
+(4)
+where:
+• 𝑆𝐿 is the numerical line score. Noise peaks receive
+scores in the low single digits, typically less than 3.0.
+Weak emission lines (low SNR, low lineflux) typically
+receive scores in the 5.0 - 15.0 range. Extremely bright,
+high SNR lines can even exceed a score of 100.0, but
+are clipped to a maximum of 100.
+• 𝑆lim is the maximum allowed fitted SNR from a Gaus-
+sian fit, up to a configurable limit (20.0 by default).
+This helps scale the scoring by capping the maximum
+contribution of the SNR.
+• 𝐴𝑁 is the "Above Noise" factor, defined by the mea-
+sured flux value of the emission peak divided by a noise
+estimate at that position and normalized by a config-
+urable factor (by default, 5). The noise estimate used
+here is the standard deviation of the 3𝜎 clipped fluxes
+at the same wavelength over all (448) fibers on the
+detector. The value of 𝐴𝑁 is clipped to the range [0,3].
+• 𝑈𝑁 is an estimate of how unique the line is relative
+to the nearby spectrum (i.e., the presence of several
+similarly narrow, low flux peaks in the same wavelength
+range likely indicate noise in the spectrum). This is
+an encoding of the Local Uniqueness described in the
+previous subsection. If the candidate line is sufficiently
+broad, with a fit FWHM of greater than 6.5 Å or if fewer
+than 3 possible lines are found, the current candidate
+
+ELiXer
+11
+line is considered sufficiently unique and 𝑈𝑁 takes on
+a value of 1, otherwise it takes on a value of 1/2.
+• 𝐹𝜆 is the Gaussian fitted, continuum subtracted inte-
+grated line flux in units of 10−17 erg s−1 cm−2. There is
+no particular significance these units; they are simply
+used so that the value of the line score is generally in
+the range of 1-100.
+• 𝑚𝜎 encodes the minimum acceptable Gaussian fitted
+𝜎. Values of 𝜎 greater than 1 Å result in 𝑚𝜎 = 1, but
+values less than 1 Å receive a multiplicative penalty
+equal to the 𝜎 value as they are unlikely to have been fit
+to a real emission line. This is equivalent to min(𝜎, 1).
+• 𝑚pix encodes the minimum acceptable number of pixels
+(𝑁pix) over which the SNR of the line is calculated. If
+the number of pixels is less than 𝑁min (by default, 10
+pixels to either side of the wavelength bin containing the
+line center), there is a multiplicative penalty imposed
+equal to 𝑁pix / 𝑁min . Low numbers of pixels in the SNR
+measurement may be due to masked or invalid pixels or
+a line location near the edge of the wavelength range.
+This is equivalent to min�𝑁pix/𝑁min, 1�.
+• 𝛿𝑑𝑥0 is the offset, in Å, of the fit line center from the ex-
+pected location of the center line. For features found by
+the untargeted search (§3.1.1), this is the bin with the
+maximum (minimum, for absorption) flux within the
+spectrum slice being used to fit the line. For corrobo-
+rating features as part of the "Targeted Search" (§3.1.2),
+it is the expected position of the assumed feature for the
+given redshift.
+An adjustment is made to the 𝑆𝐿 if the fit SNR is less
+than 8.0 and the 𝜒2 is greater than 3.0. These are considered
+marginal fits that could have a large score due to the integrated
+line flux. In these cases, the score is reduced by a factor of
+(𝜒2 − 1).
+If the center of an emission line falls within a prominent
+sky line, specifically those centered at 3545 Å or 5462 Å, and
+if the FWHM does not extend past the sky line, the score is
+further reduced by a factor of 2, encoding the risk that the
+emission line is a relic of incomplete sky subtraction.
+For very broad lines (fit FWHM > 20 Å), the scoring is
+modified by rejecting the line (setting the 𝑆𝐿 to 0) if the fitted
+SNR is less than a minimum threshold (by default, 19) and
+the 𝜒2 of the Gaussian model is greater than a maximum (by
+default, 1.5). These fits tend to be poor, and caused either
+by artifacts in the data or the merging of multiple spectral
+features.
+Since the focus is on faint galaxies with continuum below
+the HETDEX sensitivity, absorption features do not factor
+strongly in classification for most HETDEX catalog objects.
+As such, their base scoring value is scaled by a factor of 1/2
+and optionally limited to a maximum value.
+3.1.5. Spectra Simulation and P(Noise)
+As part of the scoring and in an effort to quantify the
+probability that a fitted line is simply the product of noise, we
+use the line finding code to analyze simulated spectra, treating
+all identified emission lines as false positives. The procedure
+is applied only to emission lines, not absorption lines, but the
+results are applicable to both.
+As part of the configuration for ELiXer, we compute the
+PSF weighted spectral uncertainties versus wavelength from
+104 random, non-continuum detections from the entire HET-
+DEX catalog, and generate the median uncertainty for each
+wavelength bin.
+We then simulate 104 spectra, randomly
+drawing a flux for each wavelength bin (1036 random draws
+per spectrum over the range, 3470-5540 Å) according to the
+median uncertainty, and assuming a normal distribution about
+each uncertainty and no correlated noise between wavelength
+bins. Each simulated spectrum is passed through the line find-
+ing code and all identified emission lines are recorded with
+their line scores (§3.1.4). The line scores are binned in steps
+of 1.0 and normalized by the number of simulated spectra.
+This represents the simulated estimate of the probability that
+an emission line in a given scoring bin is the product of noise.
+This probability, 𝑃(Noise) monotonically decreases with in-
+creasing line score. Note that it is possible by this mechanism
+for a scoring bin to have a value of 𝑃(Noise) greater than
+1.0, and that is the case for the lowest scoring bins. For such
+cases, the probability is cropped to 1.0 and any emission line
+with a score that fall in those bins is considered to be noise.
+Higher scoring bins are cropped once the 𝑃(Noise) falls be-
+low 5 × 10−4, with that 𝑃(Noise) assumed for all emission
+lines with line scores above that value.
+When applied to line detections in real data, any line score
+below the lowest score for the bin is assumed to be noise
+and is rejected, and any line detection with a score above the
+highest score receives the 𝑃(Noise) of the highest score for the
+bin. These 𝑃(Noise) estimates factor in the Solution Scoring
+(§3.3), described later.
+Since the 𝑃(Noise) is based on the line scoring and on
+the uncertainties in the HETDEX PSF weighted spectra, any
+reformulation of the line scoring or any change to the HET-
+DEX pipeline that results in a change in flux uncertainties
+necessitates a re-computation of this mapping.
+3.1.6. Absorption Lines
+As called out by its name, ELiXer is primarily designed to
+identify and act on emission lines. Continuum bright HET-
+DEX detections (𝑔 < 22) are also analyzed with an indepen-
+dent software package (Diagnose, Zeimann & et al. (in prep)).
+
+12
+Davis, et al.
+Nevertheless, ELiXer does currently include a limited use of
+absorption lines, triggered either explicitly at its invocation
+or automatically for detections with continuum greater than
+2 × 10−17 erg s−1 cm−2 Å
+−1. The same untargeted search
+(§3.1.1) used for emission lines is executed for absorption
+lines, with the exception that the spectrum is first inverted
+by subtracting all the flux densities from the maximum flux
+density of the spectrum. This allows the fitter to treat the
+absorption lines as if they were emission lines, but only for
+purposes of line identification within the spectrum. The ac-
+tual fitting (§3.1.3) and initial scoring (§3.1.4) is performed
+on the original, non-inverted spectrum, with the appropriate
+sign changes to account for the different direction in the Gaus-
+sian model. And like the case for emission lines, the positions
+of absorption lines with scores above a configurable threshold
+are also marked in the 1D spectrum.
+While there are 26 emission lines checked by ELiXer, only
+the Ca ii (H&K) 3968,3934 Å absorption lines are explicitly
+fitted and used in spectral redshift identification. Addition-
+ally, these two lines are fit simultaneously and must appear
+together. If they occur at the edge of the spectral range, such
+that only one line could be found in the spectrum, the fit is
+not allowed. A simple assertion is made to the pair of lines,
+requiring them to be of similar flux and FWHM such that
+the difference in flux and FWHM must be with 50% of the
+mean of their mean values. If the assertion fails, the fit is
+rejected. If the assertion passes, the lines are both accepted
+and contribute to the solution scoring (§3.3).
+3.2. Continuum Estimates
+Much of the classification effort rests on an accurate mea-
+sure of the emission line equivalent width, so a robust es-
+timate of the continuum underlying the emission line is of
+major importance. There are several, independent and semi-
+independent estimates of the continuum which contribute to
+a single combined estimate.
+Since most of the independent estimates arise from pho-
+tometric imaging, we calibrate our continuum derived clas-
+sification properties (described later in this section) to the
+bandpass continuum estimates, all of which assume a flat
+spectrum over the bandpass with no emission or absorption
+line masking (see §3.2.1, §3.2.2, §3.2.3, and §3.2.4). This
+means we are slightly biased to overestimate the continuum
+level. This is more pronounced for objects such as AGN with
+strong, broad emission, but given the objective of accurate
+classification, this is a non-issue with these objects being a
+rare subset of HETDEX data and unlikely to be confused
+with the typical, continuum faint LAE. In the general case
+that ELiXer is designed to address, our objects have faint
+or undetected continuum and a single, faint emission line so
+the bandpass overestimate is minimal and serves as an upper
+limit.
+All continuum estimates from broadband photometry as-
+sume a flat spectrum point source over the bandpass and
+convert the magnitude to flux density at the emission line’s
+observed wavelength rather than the filter’s effective wave-
+length as:
+𝑓𝜆 = 𝑐 𝜆−2 × (3631 × 10−23) × 10−0.4𝑚
+(5)
+where 𝑓𝜆 is the flux density at the observed wavelength (in
+ergs cm−2 s−1 Å−1), 𝑐 is the speed of light in vacuum (Å s−1),
+𝜆 is the fitted, observed wavelength center (Å), and 𝑚 is
+the 𝑔 or 𝑟 magnitude.
+The literal constant is in units of
+ergs cm−2 s−1 Hz−1. As most of the HETDEX emission line
+detections have either only 𝑟 coverage or are undetected in
+the imaging even when multiple bands are available, a color
+correction to the photometric continuum estimate is rarely
+possible.
+In limited testing where photometric detections
+are made in both 𝑔 and 𝑟 no improvement in the classifica-
+tion performance and no change in the classification rates is
+found, and so no color correction is included in this version
+of ELiXer.
+3.2.1. HETDEX Spectrum
+The HETDEX spectrum covers the entire 𝑔 bandpass and
+therefore can be used to estimate an object’s 𝑔-band magni-
+tude without the use of external data. Sky and background
+subtraction is very good and the continuum level is consis-
+tently measurable ≲ 10−18 erg s−1 cm−2 Å
+−1(Gebhardt et al.
+2021). We use two methods to derive the 𝑔 magnitude from
+the HETDEX 1D spectrum. The first multiplies the HETDEX
+spectrum through the SDSS 𝑔 filter’s throughput curve using
+the Python speclite package (Kirkby 2020).
+ELiXer runs
+1000 realizations of the HETDEX spectrum, sampling over
+the flux errors, and assigns the biweight (Beers et al. 1990)
+of those realizations to define an estimated 𝑔-magnitude and
+its 68% confidence interval. The second method sums the
+total flux in the HETDEX spectrum, again with propagated
+errors, and uses the mean flux density and an 𝑓𝜆,eff of 4726 Å
+to set a continuum and the 𝑔-band magnitude. In both cases,
+the object is assumed to be a point-source. The combined
+continuum mean is converted into a 𝑔 magnitude for ease of
+use and comparison to other catalog reported magnitudes.
+While this estimate is reported as computed, it is used
+internally with an imposed flux density limit of 5.38 × 10−19
+erg s−1 cm−2 Å
+−1 (𝑔 = 25). When our measured HETDEX
+continuum flux density is at least 1.2× brighter than the limit,
+it receives the highest weight (4× standard) in the combined
+estimate (§3.2.4), as it is based on the same data that provides
+the line flux estimate. All other continuum estimates are from
+other data sources and matched by proximity. As the limit
+
+ELiXer
+13
+is approached, the weight rapidly drops to the standard vote
+weight and is considered a non-detection once the limit is
+reached.
+A second estimate of the continuum is obtained using the
+𝑦 offset from the Gaussian fit to the emission line (equation
+1). While this is the estimate nearest the emission line, it
+can also have a large uncertainty and the simple Gaussian
+model does not allow for asymmetric line flux or different
+continuum levels on either side of the line. When this esti-
+mate is brighter than the HETDEX limit, it receives a small,
+empirically set weight of 0.2× the standard vote, otherwise
+it receives zero weight and is not included in the combined
+continuum estimate.
+A third and final estimate is also recorded, but is not, by
+default, included in the combined continuum estimate. In this
+estimate, the continuum is still assumed to be flat in 𝑓𝜈, but
+all emission and absorption lines identified in the spectrum
+are masked at ±2𝜎 from the fitted line centers. The mean
+of the unmasked fluxes, with standard error propagation, is
+converted into a flux density and returned as the continuum
+estimate. With the exception of the continuum bright ob-
+jects with multiple, broad spectral lines mentioned earlier,
+this estimate is not significantly different from the speclite
+result and its inclusion in the combined estimate would be
+both redundant and somewhat inconsistent, given the other
+photometric estimates. It is, however, used internally in some
+diagnostic checks.
+3.2.2. Aperture Photometry
+The 𝑔 and 𝑟-bandpass continuum estimates come directly
+from run-time aperture photometry as described in section
+2.2. When an SEP aperture matches that of the HETDEX de-
+tection, its magnitude is used. If no SEP aperture is a match,
+then the smallest, stable ELiXer circular aperture provides the
+magnitude estimate. In either case, if the computed magni-
+tude is fainter than the imaging limit, that limit is used and
+the continuum value is flagged as a non-detected upper limit.
+Since the HETDEX emission lines appear in the 𝑔-band,
+an optional correction is allowed for translating an 𝑟-band
+continuum estimate to 𝑔-band, however this is not used by
+default, as an examination of 𝑔 and 𝑟 continuum estimates
+where both are available from the same instrument for the
+same objects shows no consistent trend. Additionally, Leung
+et al. (2017) finds no advantage in using 𝑔 over 𝑟 and their
+simulated data actually suggest that LAE/[O II] segregation
+is slightly improved with 𝑟, though this is not confirmed with
+the observed spectra in this work.
+If the measured aperture magnitude is brighter than the lim-
+iting magnitude of the image, it receives a full (1.0) weight in
+the final, combined estimate. If the measured aperture mag-
+nitude is fainter than the limit, it is treated as a non-detection
+and the limit is used in the combined estimate. When the limit
+is used for the aperture magnitude, the weight in the com-
+bined estimate is scaled down linearly from 1.0 to 0.0 as the
+limit grows brighter from 26𝐴𝐵 to 24𝐴𝐵 and a non-detection
+in that increasingly bright limit provides less and less useful
+information (noting that the HETDEX spectra has a mag-
+nitude limit near 𝑔 = 25). The 26𝐴𝐵 and 24𝐴𝐵 boundaries
+selected to roughly cover the the magnitude range of maxi-
+mal LAE and [O II] galaxy 𝑔 magnitude overlap in HETDEX.
+3.2.3. Catalog Counterpart
+Lastly, if a catalog counterpart can be matched to the
+HETDEX detection (§2.3), its reported bandpass magnitude
+(again, only 𝑔 or 𝑟) is added to the list of continuum esti-
+mates. A minimum 20% flux uncertainty is assumed, even if
+no uncertainty is reported by the catalog. All catalog reported
+values are assumed to be a proper detection and receive a full
+(1.0) weight.
+3.2.4. Combined Continuum
+The combined estimate is produced using the weighted
+mean of a subset of the individual continuum estimates, de-
+scribed in the immediately previous subsections, with less
+informative estimates and extreme outliers removed from con-
+sideration.
+At most, a single upper limit estimate is allowed in the sub-
+set and is selected as the deepest (faintest) upper limit. This
+is typically the limit from the deepest photometric imaging
+where there is no detection or where the aperture magnitude
+is fainter than the image’s limit. No upper limit is included if
+there exists a positive aperture detection. If there are three or
+more continuum estimates in the subset, a fairly aggressive
+clip is applied, which excludes the most extreme estimate(s)
+with values greater than 1.5× the weighted biweight scale
+(Davis et al. 2021) while retaining a minimum subset size
+of two. The final combined continuum estimate is then the
+weighted mean of the surviving continua in the subset:
+¯𝑓𝜆 =
+�
+𝑖
+� 𝑓𝜆𝑖 𝑤𝑖 𝜎−2
+𝑖
+�
+�
+𝑖 𝑤𝑖 𝜎−2
+𝑖
+,
+(6)
+Δ ¯𝑓𝜆 =
+√︄�
+𝑖
+�𝑤𝑖 𝜎2
+𝑖
+�
+�
+𝑖 𝑤𝑖
+,
+(7)
+where ¯𝑓𝜆 is the combined ("averaged") continuum estimate,
+𝑓𝜆𝑖 is an individual continuum estimate, 𝑤𝑖 is the associated
+weight, and 𝜎𝑖 is the associated standard deviation. The error,
+Δ ¯𝑓𝜆, is the square root of the weighed average of the variances.
+This defines the distribution over which the continuum is
+sampled for the P(LAE)/P(OII) classifier in the next subsec-
+tion.
+
+14
+Davis, et al.
+3.3. Redshift Solutions
+Distilled to its most basic functions, ELiXer’s raison d’être
+is to assign the correct redshift to every detection as the
+operative analog to the classification of the target emission
+line. The core approach to this objective is the testing and
+ranking (or scoring) of many possible redshift solutions.
+Clearly the most secure, and consequently the highest scor-
+ing, solutions are those with multiple identified spectral lines
+consistent with known rest-frame features at an assumed
+redshift. ELiXer’s initial set of redshift solutions is generated
+by iterating over the lines in Table 2 and assuming, in turn,
+that each one represents the target emission line identification
+(note that the H&K absorption lines are handled differently
+per §3.1.6). With each assumed redshift, ELiXer attempts
+to fit all in the list, and accumulates a total solution score
+based on the number and quality of the successes (§3.1.4).
+At this stage, only the relative line positions are considered,
+with flux ratios, required lines, and other criteria considered
+in later steps. The more lines that are found, the more ro-
+bust the solution. Unfortunately, only about 5% of ELiXer
+classifications are established with more than one identifi-
+able emission line, so additional methods must be applied to
+confidently identify the target emission lines and assign the
+corresponding redshift.
+3.3.1. Catalog Redshift Match
+When ELiXer matches a HETDEX detection to one (or
+more) catalog objects (§2.3) that have associated spectro-
+scopic and/or photometric redshift assignments, that in-
+formation is evaluated in the context of the emission and
+absorption lines identified in the HETDEX spectrum. The
+catalog supplied redshift, with its error, is applied to the
+target emission line and all other ELiXer identified lines and
+the resulting rest-frame wavelengths are checked for consis-
+tency with those in Table 2. If the catalog redshift results
+in rest-frame wavelength matches, it boosts any previously
+assigned ELiXer score (§3.3) for that redshift, with a larger
+weight given to spec-𝑧 (+100 to the redshift solution raw
+score, §3.3.5) than to phot-𝑧 (+5 to the redshift solution
+raw score). If an ELiXer redshift solution for that catalog
+redshift does not exist, one is created and scored in the same
+way. Approximately 0.1% of the HDR3 detections have a
+catalog matched spec-𝑧 counterpart and 1.5% have a phot-𝑧
+counterpart.
+3.3.2. Large Galaxy Mask
+In addition to matching redshift catalogs, ELiXer also
+compares the sky position and wavelength of each detection
+against an internal HETDEX catalog of large galaxies. We
+define this large galaxy catalog by searching the most recent
+versions of the RC3 (de Vaucouleurs et al. 1991)2 and the
+UGC (Nilson 1973)3 galaxy catalogs for objects larger than
+1 arcminute in diameter within our survey area.
+In total,
+we find 644 large galaxies in the Spring field, and 447 in
+the Fall field. For each system, we adopt the catalog’s basic
+parameters for position, position angle, ellipticity, and D25
+semi-major axis (i.e., the size of the galaxy defined by its
+𝐵-band isophote at 25.0 mag arcsec−2). Prior to inclusion in
+the large galaxy mask, each galaxy is manually inspected to
+confirm that these values are reasonable. Where values of
+these parameters are uncertain, they are corrected to values
+listed in the NASA/IPAC Extragalactic Database4 or through
+visual inspection of the galaxy in SDSS 𝑔-band images. Any
+HETDEX detection falling within 3× the D25 isophotal ra-
+dius of a large galaxy is tested against the spectral features
+expected for the system’s redshift.
+This matching is per-
+formed in exactly the same way as for the catalog matching
+in the previous section, except that the scoring is scaled in-
+versely by the distance in multiples of D25. The overall area
+of this large galaxy mask is dominated by a handful of nearby
+galaxies (NGC 5457 and NGC 4258 in the Spring field, and
+IC 1613 and NGC 474 in the Fall Field).
+3.3.3. Special Handling for [O III]
+The [O III] 5007 Å line can be problematic to identify by
+equivalent width based methods when other oxygen or Balmer
+lines are not detected as it can have a large equivalent width
+and appear similar to Ly𝛼.
+Low-𝑧 compact star forming
+galaxies, planetary nebulae (PNe), extragalactic H II regions,
+and the outer star forming regions of resolved galaxies could
+sometimes have detectable [O III] 5007 Å, but with [O III]
+4959 Å, [O II] 3727 Å, and H𝛽 that do not reach the threshold
+for a standard HETDEX detection. Such objects could be
+classified as Ly𝛼 by the base algorithms. To protect against
+such misclassifications, additional tests are needed.
+For observed emission lines redward of 5007 Å, but with-
+out any other nominally detected emission feature, a lower
+threshold for emission line detection is allowed at the ex-
+pected positions of [O III] 4959Å, [O II] 3727Å, and H𝛽. If
+one or more of those lines are detected at this reduced strin-
+gency, a redshift solution is created with a score of at least the
+minimum acceptable threshold, and a flag is set for followup
+manual inspection.
+If one or more of the above lines are found and there is no
+identified imaging counterpart, a flag is also set to indicate
+that this could be a planetary nebula, either in the Galaxy or
+in intergalactic space. Given the HETDEX lines of sight are
+2 available at: http://haroldcorwin.net/rc3/
+3 https://heasarc.gsfc.nasa.gov/W3Browse/galaxy-catalog/ugc.html
+4 http://ned.ipac.caltech.edu
+
+ELiXer
+15
+out of the plane of the Galaxy, the likelihood of encounter-
+ing Galactic planetary nebulae is reduced but is certainly not
+zero and several known Galactic planetaries are located in the
+HETDEX footprint. Given their physical proximity, most of
+these objects will have sizes of several arc-minutes, and we
+test for this by looking for large spatial clusterings of emis-
+sion at 5007 Å. When found, these regions are masked from
+use in HETDEX cosmology. A potentially more pernicious
+issue is planetary nebulae in the halos of nearby galaxies
+and intergalactic PNe within galaxies groups and clusters.
+These could be misinterpreted as background LAEs, though
+this risk is ameliorated via the check against the large galaxy
+mask (§3.3.2) and neighbor clustering (§3.7). Conversely,
+this comes at a (small) cost of the loss of some background
+LAEs with observed Ly𝛼 redshifted to match the [O iii] 5007
+Å line of on-sky adjacent foreground galaxies.
+We note that [O III] 5007 Å makes up only 1% of the SzAS
+detections and none are misidentified by ELiXer.
+3.3.4. Object Classifications Labels
+Based on combinations of spectral features (with examples
+given later in this subsection), some HETDEX detections are
+assigned classification labels. These labels indicate only that
+a detection is consistent with the class of object indicated by
+the label within the parameters defined for that class. Classi-
+fications are not mutually exclusive and are applied simply if
+the corresponding conditions are met. If none of the specific
+classification conditions are met, then no extra classification
+label is applied to the detection.
+The classification is not
+Boolean, but is scored, with the strength of the classifica-
+tion based on the number and quality of the conditions that
+are met. A negative classification can also be made if the
+failure to meet conditions is sufficiently extreme such that a
+classification is excluded (i.e., if the detection’s properties are
+grossly inconsistent with the given classification).
+Strongly consistent object classifications can be used to in-
+crease the score of the corresponding redshift solution, while
+strongly inconsistent classifications decrease the score of the
+corresponding solution. In this way, the object classification
+𝑐𝑎𝑛 modify the P(Ly𝛼) result (§3.5) by altering the score of a
+multi-line solution available to the P(Ly𝛼) routines. However,
+the conditions are relatively strict and the overall impact of
+labeling is small, with only ∼4% of detections actually meet
+the conditions to receive an object classification label.
+Additionally, a few generic labels are applied for ELiXer
+detections that are associated with unique object in a pho-
+tometric catalog (§2).
+These labels are only provided as
+suggestions and do not impact the scoring of the multi-line
+solutions.
+The ELiXer assigned labels are:
+• AGN ("agn") The "agn" label is set if a HETDEX
+spectrum contains (possibly broadened) emission lines
+consistent with those seen in AGN. These reference
+emission lines are:
+O VI (1035 Å), Ly𝛼 (1216 Å),
+N V (1241 Å), Si II (1260 Å), Si IV (1400 Å), C IV
+(1549 Å), He
+II (1640 Å), C
+III] (1909 Å), C
+II
+(2326 Å), Mg II (2799 Å), and [O II] (3727 Å). For
+some pairs of lines, bounds on relative line fluxes must
+be met and certain lines must be present to support the
+identification of other lines. For example, if a line as-
+sumed to be C IV is observed at 5000 Å, then a line for
+Ly𝛼 must also be found at 3295 Å and it should be at
+least as strong and have a similar FWHM as C IV. If
+no line is observed at 3295 Å or if the feature is much
+weaker than the assumed C IV line, then the identifica-
+tion is inconsistent with that of an AGN and the C IV
+solution receives a reduced score.
+• Low-𝑧 Galaxy ("lzg") The "lzg" logic is largely the same
+as the "agn" but with a different set of reference lines:
+[O II] (3727 Å), H𝜂 (3835 Å), H𝜁 (3889 Å), H𝜖/ionC2
+(3970 Å), H𝛿 (4101 Å), H𝛾 (4340 Å), H𝛽 (4861 Å),
+[O III] (4959 Å), and O III] (5007 Å). As with AGN,
+some bounds on line strengths must be met. For exam-
+ple, if a line assumed to be [O III] 5007 Å is observed at
+5300 Å, then another line at 5249 Å must be observed
+at one-third the strength. Similarly, for HETDEX de-
+tections with strong continuum, if an absorption line
+is assumed to be calcium H at 3968 Å, calcium K at
+3934 Å must also be present with at a similar equivalent
+width. If these criteria are satisfied, then the detection
+will be labeled "lzg". Moreover, an additional label of
+"o32" will be assigned to objects with an [O III] 5007
+Å to O II 3727 Å flux ratio greater than 5:1.
+• Meteor ("meteor")
+With any wide-field, long-term survey, meteor intru-
+sions on the extra-galactic observations are inevitable,
+and if not identified, they can be a significant nuisance
+source of emission (and sometimes of continuum) de-
+tections. A combination of methods are used to identify
+meteors in the detection catalog (Mentuch Cooper ApJ
+accepted).
+Since ELiXer processes only single detections in iso-
+lation, its meteor identification methodology focuses
+on the transient nature of the phenomenon and their
+fairly distinctive emission line signatures.
+To iden-
+tify a meteor emission, we divide a spectrum into
+9 non-overlapping, non-contiguous regions by wave-
+length (in Å) where meteor emission lines are common:
+[3570,3590], [3715,3745], [3824,3844], [3852,3864],
+[3926,3942], [3960,3976], [4210,4250], [4400,4450],
+and [5160,5220]. For the visually confirmed meteors in
+
+16
+Davis, et al.
+HETDEX, these regions often include bright features
+from Mg (3832, 3838, 5172, and 5183 Å) as well as
+typically fainter emission from Al, Ca, and Fe. Spec-
+tra that contain multiple emission lines that are within
+these ranges and are detected in only one of the three
+dithered exposures used for an observation are labeled
+as meteors.
+• White Dwarf ("wd") The white dwarf label logic is
+very basic and simply looks for the Hydrogen series
+absorption lines for DA and DAB types, the Helium
+series for DB types, and Carbon and Oxygen for DQ
+types. Additionally, to be classified as a white dwarf,
+the spectrum must have a blue spectral slope. Since
+the shape and width of the absorption features are not
+taken into account, norarethe presenceofother features
+(such as pronounced H and K (Ca ii) lines), it is possible
+to mislabel a main sequence star, particularly an A-type,
+as a white dwarf. However, given the high Galactic
+latitude of the HETDEX survey, we do not expect the
+set of HETDEX detections to contain many early-type
+stars.
+• Catalog Labels ("gal",
+"star",
+"agn") These are
+recorded as suggestions when matched to an exter-
+nal photometric catalog, but they do not influence any
+of the ELiXer logic. For example, an "agn" label from
+a photometric catalog matched to a HETDEX detection
+is considered separately from the ELiXer "agn" label
+logic described above and will appear in the classifica-
+tion labels even if the ELiXer spectral features analysis
+does not result in an "agn" label.
+3.3.5. Redshift Solution Scoring
+Each redshift solution receives three scores, a raw score, a
+(normalized) fractional score, and a scale score, so that the
+solutions can be rank ordered and assessed in terms of their
+viability. The raw score is the unweighted sum of the indi-
+vidual line scores (§3.1.4) of the spectral lines included in the
+solution, excluding the anchor line (which is common to all
+solutions), and including a multiplier based on the number
+of identified spectral lines and any multipliers from classifi-
+cation labels (§3.3.4), where they are strongly consistent or
+inconsistent. It is defined as:
+𝑟𝑠 =
+� 𝑛
+∑︁
+𝑖
+𝑙𝑠𝑖
+�
+× min
+�
+1, 1
+2
+�
+𝑛2 − 𝑛
+��
+× 𝑏,
+(8)
+where 𝑟𝑠 is the solution raw score, 𝑙𝑠 is a line score of an
+included spectral line, 𝑛 is the total number of spectral lines
+included in the solution not counting the anchor line, and
+𝑏 is any multiplier from the object classification label logic
+(typically 0.25 to 2.0).
+The raw score is normalized to produce the fractional score
+by dividing it by the sum of the raw scores of all redshift
+solutions.
+Lastly, a scale score is produced from the weighted sum
+of the probability that the solution is comprised of noise, the
+raw score, and the fractional score as:
+𝑠𝑠 =
+�
+1 −
+𝑛
+�
+𝑖
+𝑃(noise)𝑖
+�
+× 𝑤noise
++ min (1.0, 𝑟𝑠/𝐹) × 𝑤raw
++ 𝑓 𝑠 × 𝑤frac,
+(9)
+where 𝑠𝑠 is the scale score, 𝑃(noise)𝑖 is the probability that
+the included spectral line is noise (§3.1.5), 𝑤noise is the weight
+for this first term (by default, 0.40), 𝑟𝑠 is the raw score from
+Eqn (8), 𝐹 is the configured raw score scale factor (by default,
+50.0), 𝑤raw is the weight for this second term (by default,
+0.50), 𝑓 𝑠 is the fractional score, and 𝑤frac is the weight of
+this third term (by default, 0.10).
+3.4. P(LAE)/P(OII)
+P(LAE)/P(OII) (sometimes as PLAE/POII in other docu-
+mentation) represents the ratio of the relative probability that
+given a set of measured characteristics, an emission line is
+Ly𝛼 (representing an LAE) rather than [O ii]. These proba-
+bilities are based on the number of galaxies expected at the
+volume sampled by the redshift slices assuming the emission
+line is either Ly𝛼 or [O ii] given the measured line flux and
+equivalent width. The expected number of galaxies derives
+from the equivalent width distributions of Ly𝛼 and [O ii] con-
+ditioned on the luminosity functions found in Gronwall et al.
+(2014) and Ciardullo et al. (2013) respectively, interpolated
+or extrapolated as needed (see also Leung et al. (2017, Figure
+2)).
+This is an improvement on the commonly used 20 Å equiv-
+alent width cut (Gronwall et al. 2007; Adams et al. 2011) and
+is based largely on the analysis of Leung et al. (2017), and us-
+ing the specific translation and implementation described in
+(Farrow et al. 2021, primarily in Section 2). ELiXer slightly
+updates Farrow et al. (2021) by (1) using multiple independent
+or semi-independent estimates of the continuum (§3.2), (2)
+combining those estimates into a single, best-fit continuum
+value, and (3) sampling over the uncertainties in the measured
+line flux and continuum estimates to generate a (68%) con-
+fidence interval around each P(LAE)/P(OII) measurement.
+Partly for convenience and partly as a representation of the
+practical limits of this method, the ratio is cropped to values
+between 0.001 ≤ P(LAE)/P(OII) ≤ 1000.
+The interpretation of the P(LAE)/P(OII) value is not quite
+straightforward. While LAE evolution between 2 < 𝑧 < 4
+appears somewhat muted (Blanc et al. 2011; Santos et al.
+
+ELiXer
+17
+2021), there is more redshift evolution of the [O ii] systems
+(Gallego et al. 2002; Comparat et al. 2016; Saito et al. 2020;
+Park et al. 2015; Gao & Jing 2021) for 𝑧 < 0.5. This evolution
+may be underrepresented in the base P(LAE)/P(OII) code and
+lead to a deviation from the expectation that a ratio near 1
+should be interpreted as the likelihood of the emission line
+being Ly𝛼 or [O ii] is approximately equal. Building on the
+suggestion in Leung et al. (2017) of using different thresholds
+for the P(LAE)/P(OII) ratio at different observed wavelengths,
+ELiXer adopts an empirical threshold relation (§3.5.3).
+The overall combined P(LAE)/P(OII) value and its con-
+fidence interval factor significantly in the final automated
+classification of the emission line. It can frequently be the
+most influential (and sometimes the only) metric that is used
+in that classification (§3.5).
+3.5. P(LyA)
+Using some of the features/measurements already de-
+scribed, along with a set of additional features described
+below, ELiXer synthesizes an aggregate confidence in the
+classification of the anchor emission line as Ly𝛼 or not-Ly𝛼.
+For familiarity, this is couched in terms of a probability, la-
+beled as P(Ly𝛼) with values between 0 (definitely not Ly𝛼)
+and 1 (definitely Ly𝛼), but is not a true probability in the for-
+mal sense. P(Ly𝛼) is the result of a weighted voting system
+where each of the features described in this section provides
+a vote (typically 0 or 1, but can be in between) and that vote
+is given a weight based on the robustness or confidence of the
+measurement. With specifically noted exceptions, features
+that do not produce a clear preference are given zero or very
+little weight. The final P(Ly𝛼) value is then simply the sum
+over all votes multiplied by their respective weights:
+𝑃(Ly𝛼) =
+∑︁
+𝑖
+�vote𝑖 × weight𝑖
+�
+∑︁
+𝑖
+weight𝑖
+,
+(10)
+Note that the sum of the weights alone is not normalized and
+can exceed 1. In the relatively rare cases where the sum of all
+weights is less than 1, a special "uncertainty" vote is added
+with a value of 0.5 and a weight equal to 1−� weights, so that
+the weights do sum to 1. This helps capture the uncertainty
+in the classification and prevents one or two votes with very
+low weights from being dominant.
+The selection of voting criteria and the weights applied to
+the votes is the result of empirical analysis and trial-and-error
+testing and is discussed in Section 4. This is a little bit of
+the Central Limit Theorem and the Wisdom of the Crowd,
+even though the votes are not entirely independent as several
+incorporate similar elements and some are designed to handle
+edge cases not well covered by the others. No single vote is
+universally dominant, though each can be decisive under the
+right circumstances, such as the high weight of §3.5.2 when
+multiple emission lines are present or even a low weight vote
+from §3.5.5 for some moderate equivalent widths when the
+rest of the vote tally is near 0.5.
+As a word on the notation in this section; often [O ii] is
+used in place of "not-Ly𝛼" as [O ii] is the most common con-
+taminant. Votes "for [O ii]" are really votes for "not-Ly𝛼".
+Further, the figures in this subsection all show only those
+assessment sample detection emission lines that are Ly𝛼 or
+[O ii], so [O ii] is equivalent to "not-Ly𝛼".
+3.5.1. Object Size Vote
+In cases where a counterpart is identified and resolved in
+the 𝑔- or 𝑟-band imaging, the angular and physical extent of
+the counterpart contributes a vote. For this purpose, an ob-
+ject is considered resolved if the angular major diameter is
+greater than 1.1× the seeing FWHM. This includes artifi-
+cially enlarged footprints in the imaging due to the "bloom-
+ing" of bright sources that have saturated the detector. The
+proper physical diameter is computed assuming the redshift
+of [O ii], as larger objects tend to be more evolved and at lower
+redshift. The emission line FWHM is used to help break the
+size degeneracy between larger, lower-𝑧 objects and saturated,
+higher-redshift sources, via the assumption that the latter are
+AGN with a large emission line FWHM.
+The parameter thresholds are set from a manual partition-
+ing of classifications in scatter plots of angular and physical
+diameter versus the observed wavelength of the anchor emis-
+sion line, as shown in Figure 1. The conditions and their
+associated votes are summarized in Table 3.
+The specific
+limiting values of the FWHM help distinguish possible AGN
+with a broadened emission line, from lower redshift galaxies.
+It is reasonable for an AGN to receive a vote for Ly𝛼, but
+an angularly large object with a more narrow emission line
+is more likely an [O ii] emitter. The gap between the condi-
+tions avoids a vote where it is unclear. The angular diameter
+threshold (in arcseconds), 𝜃𝜆, is a piece-wise linear function:
+𝜃𝜆 =
+����
+����
+2.8,
+3727Å < 𝜆 ≤ 4000Å
+− 0.0018𝜆 + 10.0,
+4000Å < 𝜆 ≤ 5000Å
+1.0,
+𝜆 > 5000Å
+(11)
+The object size criteria results in a cast vote for 70% of the
+SzAS (down-selected to only contain Ly𝛼 and [O ii]), where
+the separation of [O ii] from Ly𝛼 is effective, with a Ly𝛼
+contamination rate of 4% in those votes.
+3.5.2. Multi-line Redshift Solutions Votes
+
+18
+Davis, et al.
+Table 3. Angular and Physical Diameter Votes
+Condition
+Vote
+Weight
+𝑑𝑝 < 3.0 kpc or 𝜃 < 𝜃𝜆
+1.0
+0.25
+𝑑𝑝 < 4.5 kpc
+1.0
+0.10
+𝜃 < 2.′′5 and FWHM > 1000 km s−1
+1.0
+0.25
+𝜃 > 2.′′5 and FWHM < 800km s−1
+0.0
+0.25
+else no vote
+NA
+0.00
+Note— Summary of angular and physical size votes. The
+conditions are ordered such that the logical evaluation
+results in at most one unique vote. If no conditions are
+met, there is no vote.
+𝑑𝑝 is the proper diameter in kpc.
+𝜃 is the angular diameter in arcsec.
+𝜃𝜆 is the minimum expected angular size for an [O ii]
+galaxy for the observed anchor emission line wavelength.
+𝐹𝑊𝐻𝑀 refers to the emission line.
+This criterion can generate multiple votes, one for each
+potential redshift solution (§3.3) based on the positions and
+fluxes of the fitted spectral lines.There must be two or more
+found spectral lines, with the scores based largely on the
+number of lines and their strengths (see §3.3.5). However,
+as is shown later in this subsection, solutions incorporating
+three or more lines receive an increased voting weight. At
+most, there will be a single Ly𝛼 (1.0) vote if there is a solution
+that supports the classification of the anchor line as Ly𝛼. All
+other redshift solutions necessarily require the anchor line to
+be something other than Ly𝛼, and therefore cast a not-Ly𝛼
+(0.0) vote. The weight each vote receives depends on the
+scaled solution score assigned multiplied through a sigmoid:
+𝑤0 = 𝑠𝑠/(1 + exp(0.75𝑚 − 𝑟𝑠))
+(12)
+where 𝑤0 is the initial voting weight, 𝑠𝑠 is the redshift solution
+scale score (Eqn 9), 𝑚 is the minimum acceptable score (25,
+by default), and 𝑟𝑠 is the redshift solution raw score (Eqn 8).
+An additional multiplier is applied for exceptionally strong
+redshift solutions with 3 or more contributing spectral lines:
+𝑤 = 𝑤0 × min(𝑟𝑠/𝑚, 10)
+(13)
+where 𝑤 is the modified voting weight, 𝑤0 is the original
+weight (Eqn 12), 𝑟𝑠 is the raw solution score, and 𝑚 is the
+minimum acceptable score. This multiplier is always greater
+than 1 since, by definition, a qualifying redshift solution must
+have a raw solution score greater than minimum acceptable
+value. The maximum value of 𝑤 is limited to 10× the original
+number, but that allows this vote to dominate with a high
+confidence redshift solution comprised of multiple, strong
+spectral lines.
+3500
+3750
+4000
+4250
+4500
+4750
+5000
+5250
+5500
+0
+2
+4
+6
+8
+10
+Diameter [arcsec]
+Threshold
+Ly
+[O II]
+3500
+3750
+4000
+4250
+4500
+4750
+5000
+5250
+5500
+ Observed [Å]
+0
+5
+10
+15
+20
+25
+Diameter [kpc]
+Figure 1. The separation of Ly𝛼 from [O ii] in the assessment sample
+(SzAS, §4) based on the angular (upper panel) and physical (lower
+panel) diameters. Errors are ∼ 0.′′2. The dashed line corresponds to
+the thresholds defined in Table3. There are no points blue-ward of
+3727 Å in the lower figure since the physical diameter is computed
+based on the assumption that the emission line is [O ii]. The lower
+panel is cropped to a maximum of 25 kpc for readability and shows
+two horizontal thresholds at 3.0 and 4.5 kpc, corresponding to the
+first two conditions in Table 3. This generates a vote for 70% of the
+SzAS with a 4% contamination of Ly𝛼 in those votes.
+This criteria does not often trigger a vote, casting one for
+only 7% of the SzAS, down-selected to only contain Ly𝛼 and
+[O ii], and 12% for the entire SzAS, but has no contamination
+of Ly𝛼 for those votes. Due to the bright skew in SzAS (see
+§4 and §5.1), this voting rate is exaggerated and is only cast
+for 2% of the 𝑔 >22 detections in HETDEX.
+3.5.3. P(LAE)/P(OII) Vote
+As most HETDEX detections are faint, single emission
+lines, the above criteria rarely produce strong redshift so-
+lutions, and the P(LAE)/P(OII) computation is often the
+most significant vote.
+The value (0 or 1) of the vote de-
+pends on which side of a wavelength dependent midpoint the
+P(LAE)/P(OII) ratio falls, and the weight of the vote increases
+
+ELiXer
+19
+with the distance of the ratio from that midpoint. The mid-
+point value, which separates the [O ii] (0) and Ly𝛼 (1) vote,
+is a modification of the binary condition suggested in Leung
+et al. (2017),
+𝜇 =
+�
+1.38,
+𝜆 ≤ 4255Å
+10.3,
+𝜆 > 4255Å
+(14)
+and is defined as
+𝜇 =
+�����
+����
+1.0,
+𝜆 ≤ 4000Å
+0.018𝜆 − 71,
+4000Å < 𝜆 ≤ 4500Å
+10.0,
+𝜆 > 4500Å
+(15)
+where 𝜇 is the midpoint or vote threshold and 𝜆 is the wave-
+length of the anchor emission line. Ratios nearer the midpoint
+suggest an increasingly equal likelihood that the source emis-
+sion line is [O ii] or Ly𝛼 and, as such, add little evidence for
+a classification. This is reflected in a low voting weight (𝑤)
+built from a Gaussian,
+𝑤 = 1 − exp
+�
+−
+� 𝑃 − 𝜇
+√
+2 𝜎
+�2�
+× (1 − 𝑖),
+(16)
+where
+𝑃 =
+�
+P(LAE)/P(OII),
+for P(LAE)/P(OII) ≥ 1
+P(OII)/P(LAE),
+for P(LAE)/P(OII) < 1
+(17)
+and 𝜇 is the midpoint and 𝜎 is the usual Gaussian width (here
+set to 5.0, which is tuned by hand to give balanced voting
+weights). The parameter 𝑖 is an ersatz standard deviation from
+the scaled 68% confidence interval around the P(LAE)/P(OII)
+(§3.4) and is defined as:
+𝑖 = 1
+2 ×
+�
+𝑈
+𝑈 + 1 −
+𝐿
+𝐿 + 1
+�
+(18)
+where 𝑈 is the upper bound of the confidence interval and 𝐿
+is the lower bound. As the P(LAE)/P(OII) ratio moves farther
+from the midpoint in either direction, the weight of the vote
+increases and rapidly asymptotes to 1.
+Alone, the P(LAE)/P(OII) vote is effective, with a 4% Ly𝛼
+contamination rate (by [O ii]) in the SzAS (down-selected to
+only contain Ly𝛼 and [O ii]), voting 90% of the time. As with
+the other equivalent width based votes, though, it struggles to
+identify Ly𝛼 emission lines when originating from non-LAE
+(i.e. low-EW Ly𝛼 emitting galaxies) (see also §5.3). As the
+P(LAE)/P(OII) computation includes the volumes sampled
+by the two assumed redshifts, it can become a less effective
+discriminator as the observed wavelengths approach the rest
+wavelength of [O ii] and that volume shrinks (§3.4 and Leung
+et al. (2017); Farrow et al. (2021)). The other votes, including
+two more based partly on the emission line equivalent width,
+§3.5.5 in particular, help compensate.
+3500
+3750
+4000
+4250
+4500
+4750
+5000
+5250
+5500
+ Observed [Å]
+3
+2
+1
+0
+1
+2
+3
+log(PLAE/POII)
+Threshold
+Ly
+[O II]
+Figure 2. P(LAE)/P(OII) distribution (clipped to 10±3) in the as-
+sessment sample (SzAS, §4) shown without the 68% confidence
+intervals (§3.5.3). The dashed line is the midpoint of the segrega-
+tion threshold (Eqns 15 - 18) with points above the line receiving a
+vote for Ly𝛼 and those below for not-Ly𝛼 with weights based on the
+distance from the threshold. This vote has a 4% contamination rate
+of Ly𝛼 by [O ii] in the SzAS.
+3.5.4. Line FWHM Vote
+This is logically one of the simplest votes. If the emission
+line FWHM is larger than 10.5 Å, as seen in Figure 3, the line
+receives a Ly𝛼 vote (1) with a weight as high as 1.0 using
+𝑤 = min(FWHM/10.5 − 1.0, 1.0),
+(19)
+where 𝑤 is the assigned weight of the line and FWHM is
+line’s fitted full-width at half-maximum. As the contamina-
+tion rate decreases with larger FWHM thresholds, the voting
+weight increases. If the lower uncertainty bound of the fitted
+FWHM, here defined as the fitted FWHM minus the uncer-
+tainty derived from standard error propagation, exceeds a
+configurable minimum (15.3 Å by default), the vote weight
+is set to the 1.0 maximum value, as [O ii] emission lines are
+rarely that broad. Also, as a consequence of the increasing
+FWHM threshold, these higher weighted votes tend to favor
+AGN selection and thereby helps reduce the confusion caused
+by lower AGN emission line equivalent widths. In short, it
+helps improve the recovery of Ly𝛼 (and decrease the misclas-
+sification as [O ii]) from AGN that can fail the other voting
+criteria based on equivalent width (§3.5.3, §3.5.5), bandpass
+magnitude (§3.5.7), and angular size (§3.5.1).
+This criteria casts a vote for 23% of the down-sampled
+SzAS (containing only Ly𝛼 and [O ii]) with a total Ly𝛼 con-
+tamination of 11%. This drops to 3% when considering votes
+
+20
+Davis, et al.
+with weights above 0.3 (received by 18% of the down-selected
+SzAS) and is contamination free for votes with weights above
+0.7 (received by 12% of the down-selected SzAS).
+We note that while this particular vote is a good discrim-
+inator against [O ii], it can confuse Ly𝛼 with other broad
+AGN lines, such as C iii] or C iv. We largely address this
+issue using multi-line redshift solutions (§3.3) and clustering
+(§3.7).
+3500
+3750
+4000
+4250
+4500
+4750
+5000
+5250
+5500
+ Observed [Å]
+5
+10
+15
+20
+25
+30
+Line FWHM [Å]
+Threshold
+Ly
+[O II]
+Figure 3. Ly𝛼 and [O ii] separation in the assessment sample (SzAS,
+§4) based on the emission line FWHM. The data points are shown
+without their uncertainties (∼ 14% ). The horizontal dashed line rep-
+resent the minimum threshold to receive a vote for Ly𝛼 as described
+by Eqn 19.
+3.5.5. Simplified Equivalent Width Vote
+This
+vote
+is
+somewhat
+redundant
+with
+the
+full
+P(LAE)/P(OII) vote (§3.5.3), but does not consider the red-
+shift based population distributions or observed wavelength
+variations. It slightly moderates the P(LAE)/P(OII) vote and
+can help push away from an uncertain classification where the
+P(LAE)/P(OII) vote has a low weight. It can also push toward
+an uncertain classification if the P(LAE)/P(OII) vote and this
+vote have similar weights, but different votes, allowing other
+voting criteria to have more influence. These two votes agree
+95% of the time and this simplified equivalent width vote is
+only important in these boundary cases.
+This simplified vote uses EW𝐿𝑦𝛼, which is defined by the
+Gaussian fitted line flux (§3.1.3) and the combined continuum
+estimate (§3.2). For EW𝐿𝑦𝛼 much greater or much less than
+20 Å, this reinforces the P(LAE)/P(OII) vote and helps nudge
+the solution away from the P(LAE)/P(OII) midpoint. If the
+EW𝐿𝑦𝛼 is greater than 30 Å, then the vote is for Ly𝛼 (1); if
+the EW𝐿𝑦𝛼 is less than 20 Å, the vote is for [O ii] (0). All
+other EW𝐿𝑦𝛼 values do not generate a vote.
+The assigned voting weights are based on the EW𝐿𝑦𝛼 lower
+(EW−
+𝐿𝑦𝛼) and upper (EW+
+𝐿𝑦𝛼) bounds and increase with con-
+ditions where the contamination is reduced. The maximum
+weight is limited to 0.5 so that the P(LAE)/P(OII) vote is
+dominant when both votes approach their maximum weights.
+In the pro-Ly𝛼 case, the weight is either 0 or between 0.1
+and 0.5 as:
+𝑤 =
+����
+����
+0.0,
+𝑟− ≤ 0.0
+0.1,
+0.0 < 𝑟− < 1.0
+max(0.1, min(0.5, 𝑟− − 1.0)),
+𝑟− ≥ 1.0
+(20)
+where 𝑤 is the assigned weight and 𝑟− = 1
+25 × 𝐸𝑊−
+𝐿𝑦𝛼.
+In the pro-[O ii] case, the weight is between 0.1 and 0.5 as:
+𝑤 =
+�
+0.1,
+𝑟+ < 1.0
+min(0.5, max(0.1, 𝑓 )),
+𝑟+ ≥ 1.0
+(21)
+𝑓 = −0.04 × 𝐸𝑊+
+𝐿𝑦𝛼 + 0.9
+(22)
+where 𝑤 is the assigned weight and 𝑟+ = 20 / 𝐸𝑊+
+𝐿𝑦𝛼.
+Figure 4 shows the Ly𝛼 and [O II] SzAS detections with
+rest-Ly𝛼 EW less than 100 Å (this includes all SzAS [O II]
+emission lines) with the voting thresholds marked.
+This criteria votes in 80% of the down-selected SzAS (con-
+taining only Ly𝛼 and [O ii]) with a Ly𝛼 contamination rate
+of 2%. Superficially, this is superior Ly𝛼/[O ii] segregation
+compared to the P(LAE)/P(OII) vote (§3.5.3, but by design,
+avoids voting in the difficult EW transition region (shaded
+region in Figure 4).
+3.5.6. Catalog Photometric Redshift Vote
+The photometric redshifts fits from the various included
+catalogs (§2) are often too broad to confidently pin down a
+tight redshift constraint. However, they can be sufficient to
+distinguish between low-𝑧 (𝑧 ≲ 0.7) and high-𝑧 (1.7 ≲ 𝑧 ≲ 3.7)
+objects and thus help separate [O ii] from Ly𝛼. If there are
+any photometric redshifts for a HETDEX detection, this vote
+simply takes the arithmetic mean of all phot-𝑧 measurements
+of the matched catalog counterpart from all contributing cat-
+alogs and compares it to the low-𝑧 and high-𝑧 ranges quoted
+above. If the mean falls within either range and is within a
+redshift distance of 0.5 of [O ii] or Ly𝛼 respectively, then the
+corresponding vote is cast with a weight of 0.5. If the mean
+falls outside of both ranges or if the redshift separation be-
+tween the mean and an assumption of [O ii] or Ly𝛼 is greater
+than 0.5, then no vote is cast.
+Only ∼ 1.5% of HDR3 sources have at least one phot-𝑧
+catalog counterpart match, so this vote rarely contributes to
+the P(Ly𝛼) logic. For the SzAS testing, since contributions
+from catalog phot-𝑧 and spec-𝑧 are necessarily turned off,
+this vote is never cast.
+
+ELiXer
+21
+3500
+3750
+4000
+4250
+4500
+4750
+5000
+5250
+5500
+ Observed [Å]
+0
+20
+40
+60
+80
+100
+rest-Ly EW [Å]
+Thresholds
+Ly
+[O II]
+Figure 4. Simplified rest-Ly𝛼 equivalent width vote applied to the
+assessment sample (SzAS, §4). The figure is cropped to a maximum
+EW of 100 Å for readability and plotted without the ∼ 16% errors.
+The SzAS contains no spectroscopically confirmed [O II] emission
+lines with rest-Ly𝛼 EW > 80 Å. Detections with EWs falling in the
+gray shaded region between 20 and 30 Å receive no vote while those
+above receive a Ly𝛼 vote and those below an [O II] vote with the
+weight of the vote modulated by the distance to the nearest threshold
+(§3.5.5).
+3.5.7. Apparent Magnitude and Equivalent Width Vote
+This vote is largely predicated on the observation that the
+HETDEX LAEs tend to be fainter than [O ii] galaxies. How-
+ever, there certainly exist bright LAEs (including AGN) and
+faint [O ii] galaxies, so the EW𝐿𝑦𝛼 is also incorporated into
+the decision to moderate it.
+The apparent magnitude used in this vote is the 𝑔-band
+magnitude derived from the HETDEXspectrum (§3.2), which
+has a limiting magnitude of ∼25𝐴𝐵. The magnitude threshold
+between votes for [O ii] and for Ly𝛼 are defined by a pair of
+lines whose parameters are set to optimize the segregation
+of those two samples.
+Objects with 𝑔 magnitudes fainter
+than the upper line of Figure 5 are more likely to be Ly𝛼,
+while those brighter than the lower line of Figure 5 are more
+likely to be [O ii]. The classification of objects, defined by
+their SzAS spectroscopic redshifts, lying between these two
+regimes is uncertain. The optimization over the slope and
+intercept parameters of these lines was performed using a
+simple grid search that maximizes the Ly𝛼 accuracy in one
+case and the [O ii] accuracy in the other. While an MCMC
+fit could be more precise, given the uncertainties in the data
+features and the desire to avoid over-fitting to the specific test
+set, the grid search is preferred. The accuracy is defined as:
+accuracy = 1 − 𝜉 + 𝜖
+Ω
+(23)
+where 𝜉 is the number of "true" Ly𝛼 ([O ii]) detections (here
+as the spec-z counterparts in the SzAS test sample), that are
+not identified by the selection, 𝜖 is the number of incorrectly
+classified Ly𝛼 ([O ii]) detections, and Ω is the total number
+of Ly𝛼 ([O ii]) classified detections. Here, "true" is assumed
+as the catalog based spectroscopic redshifts (§4).
+The two lines are defined as:
+𝑔+
+𝑇 = 1.10 × 10−3𝜆 + 18.0, (𝑙𝑜𝑤𝑒𝑟𝑙𝑖𝑛𝑒, 𝐹𝑖𝑔.5)
+(24)
+𝑔−
+𝑇 = 1.26 × 10−3𝜆 + 18.1, (𝑢𝑝𝑝𝑒𝑟𝑙𝑖𝑛𝑒, 𝐹𝑖𝑔.5)
+(25)
+where 𝑔+
+𝑇 is the faint magnitude threshold, 𝑔−
+𝑇 is the bright
+magnitude threshold, and 𝜆 is the wavelength (Å) of the an-
+chor emission line.
+We also define upper (faint) and lower (bright) bounds for
+the measured 𝑔 magnitude of each detection (𝑔+ and 𝑔−,
+respectively) based on the propagated errors of the HETDEX
+spectroscopically-measured 𝑔-band magnitude.
+The votes and their weights for this criterion are summa-
+rized in Table 4 with Figure 5 showing the segregation of Ly𝛼
+and [O II] with Eqns 24 and 25. As the 𝑔 magnitude becomes
+brighter, the voting weights for Ly𝛼 decrease and those for
+[O ii] increase. Large anchor line EW𝐿𝑦𝛼 favor Ly𝛼 and small
+EW𝐿𝑦𝛼 favor [O ii]. With the exception of spectra associated
+with objects having faint 𝑔 magnitudes, those spectra with an-
+chor line EW𝐿𝑦𝛼 (with error) between 15 Å and 30 Å receive
+no vote either way. Contamination of Ly𝛼 by [O II] for the
+down-selected SzAS is low with Ly𝛼 comprising 97% of the
+detections above the Neutral region in Figure 5). Conversely,
+Ly𝛼 represents only 44% within the Neutral region, where no
+vote is cast, and 14% below it, where the vote is cast for [O II].
+3.5.8. Disqualifications
+Disqualification conditions are a set of special classifica-
+tions and data integrity issues that can either contribute ad-
+ditional weighted votes against a Ly𝛼 classification or, in
+extreme cases, completely override the P(Ly𝛼) results.
+• Meteor: If the detections has a possible classification
+as a meteor (§3.3.4, a vote against Ly𝛼 is added with a
+weight equal to the strength of the meteor classification
+(0.0 - 5.0). Given its potentially large weight, this vote
+can be dominant. Regardless of the final result of the
+vote, the "meteor" label is attached to the detection.
+• Bad Pixel Flat: If a bad pixel flat is indicated by pixel-
+to-pixel variations or pixel flux values outside the ac-
+ceptable range for an emission-line on that part of the
+CCD, then the emission line may be entirely due to, or
+at least enhanced by, this artifact. The detection will
+thus receive a vote against Ly𝛼 with a weight equal to
+1.0 plus the sum of the relative weights of those fibers
+contributing to the spectrum that have a bad pixel flat.
+
+22
+Davis, et al.
+3500
+3750
+4000
+4250
+4500
+4750
+5000
+5250
+5500
+ Observed [Å]
+22.0
+22.0
+22.5
+22.5
+23.0
+23.0
+23.5
+23.5
+24.0
+24.0
+24.5
+24.5
+25.0
+25.0
+g
+g Limit
+[O II]
+Ly
+Neutral
+Figure 5.
+The apparent magnitude (error ∼ 0.1) and equivalent
+width vote, by itself, is highly effective at segregating Ly𝛼 from
+[O ii] against the assessment sample here (SzAS, §4). The Neutral
+region is defined by the lines of Eqns (24) and (25) as the lower and
+upper bounds respectively, and extends from 3727Å to the red edge
+of the HETDEX spectral window. Ly𝛼 emitters represent 97% of
+the down-selected SzAS above the Neutral region, 44% inside the
+Neutral region, and 14% below the Neutral region.
+The total weight for this vote is between 1.0 and 2.0.
+However, if the sum of the fiber weights exceeds a
+threshold, 0.50 by default, the entire P(Ly𝛼) vote is
+disqualified. Independent of the vote, the bad pixel flat
+flag is associated with the detection and shown on the
+ELiXer report.
+• Duplicate Fibers: If duplicated fibers (identified by
+repeated fiber identifiers or identical flux and error data
+vectors) appear in the detection spectra, the P(Ly𝛼) vote
+is disqualified. This is an indication of a data reduction
+problem.
+• Grossly Negative Spectrum: If less than 10% of the
+wavelength bins contain non-negative integrated flux
+values, the spectrum is considered "grossly negative"
+and suggests some issue in the reduction. In this case,
+the detection and the P(Ly𝛼) vote is disqualified.
+• Poor Observation: If the seeing FWHM is worse than
+a threshold (3′′ by default) or the throughput response,
+as defined by (Gebhardt et al. 2021), is less than a
+threshold (0.08, by default), the input observation is
+considered too poor to make a meaningful classification
+attempt and the vote is disqualified.
+• Bad Dither Norm: If the dither-to-dither normaliza-
+tion (Gebhardt et al. 2021) for the detection is above
+a threshold (3.0× by default), a potentially severe ob-
+Table 4. Apparent Magnitude and EW Votes
+Condition
+Vote
+Weight
+𝑔− > 𝑔+
+𝑇
+1.0
+0.50
+𝑔−
+𝑇 < 𝑔− < 𝑔+
+𝑇 < 𝑔+
+and 𝐸𝑊− > 80
+1.0
+0.50
+𝑔−
+𝑇 < 𝑔− < 𝑔+
+𝑇 < 𝑔+
+and 𝐸𝑊− > 30
+1.0
+0.30
+𝑔−
+𝑇 < 𝑔− < 𝑔+
+𝑇 < 𝑔+
+and 𝐸𝑊+ ≤ 15
+0.0
+0.25
+𝑔− < 𝑔−
+𝑇 < 𝑔+ < 𝑔+
+𝑇
+and 𝐸𝑊− > 80
+1.0
+0.30
+𝑔− < 𝑔−
+𝑇 < 𝑔+ < 𝑔+
+𝑇
+and 𝐸𝑊− > 30
+1.0
+0.15
+𝑔− < 𝑔−
+𝑇 < 𝑔+ < 𝑔+
+𝑇 and 𝐸𝑊+ ≤ 15
+0.0
+0.40
+𝑔+ < 𝑔−
+𝑇
+and 𝐸𝑊− > 80
+1.0
+0.25
+𝑔+ < 𝑔−
+𝑇
+and 𝐸𝑊− > 30
+1.0
+0.10
+𝑔+ < 𝑔−
+𝑇
+and 𝐸𝑊+ ≤ 15
+0.0
+0.50
+else no vote
+NA
+0.00
+Note— Summary of apparent magnitude and equivalent
+width votes.
+The conditions are ordered such that the
+logical evaluation results in at most one unique vote. If
+no conditions are met, there is no vote. The apparent 𝑔
+magnitude becomes brighter moving down the table.
+𝑔+
+𝑇 is the upper (faint) 𝑔 threshold as a function of 𝜆.
+𝑔−
+𝑇 is the lower (bright) 𝑔 threshold as a function of 𝜆.
+𝑔+ is the upper bound (faint) 𝑔 for the detection.
+𝑔− is the lower bound (bright) 𝑔 for the detection.
+𝐸𝑊+ is the upper bound restframe EW in Å, assuming
+Ly𝛼.
+𝐸𝑊− is the lower bound restframe EW in Å, assuming Ly𝛼.
+servation or reduction issue is indicated and the vote is
+disqualified.
+3.6. Best-z and Q(z)
+Unless there is a serious error or a disqualification (§3.5.8),
+ELiXer assigns a single, best guess redshift, "Best-𝑧", along
+with a quality score, "𝑄(𝑧)", as an indication of the confidence
+in that redshift. The assignment of the Best-𝑧 incorporates
+all prior information and analysis including the P(Ly𝛼), cat-
+alog spec-𝑧 and phot-𝑧, and any multi-line redshift solutions
+(§3.3). The 𝑄(𝑧) value takes on a continuous value between
+0 and 1, with 1 meaning "full confidence" and 0 meaning "no
+confidence" (i.e., the redshift is effectively a guess). Where
+the P(Ly𝛼) analysis is limited only to a determination as to
+whether the emission line is Ly𝛼 the Best-𝑧 logic attempts to
+fully specify the redshift. In the ideal scenario, there are mul-
+tiple high-SNR emission lines within the HETDEX spectrum,
+each corresponding to a known line at a consistent redshift.
+In such a case, the Best-𝑧 is clear and the corresponding 𝑄(𝑧)
+is 1.0. Such objects are rather rare, but they do define the
+starting benchmark.
+The Best-𝑧 is set as (1) the redshift from a qualified multi-
+line spec-𝑧 solution, (2) the Ly𝛼 redshift when there is no
+
+ELiXer
+23
+spec-𝑧 solution but P(Ly𝛼) favors Ly𝛼, or typically, (3) the
+[O II] redshift. In the last case, the redshift can be set to C III]
+or Mg II when the line is broad and occurs at a wavelength
+within the HETDEX spectral window where no other strong
+feature is expected to be found.
+The 𝑄(𝑧) confidence value is set based on the Best-𝑧 se-
+lection condition. It is primarily a function of the P(Ly𝛼)
+value and the multi-line solution score. 𝑄(𝑧) is maximized
+by P(Ly𝛼) when P(Ly𝛼) is near 0 or 1 and minimized when
+P(Ly𝛼) is near 0.5. The effect of the multi-line solution score,
+on the other hand, is a monotonic increase with the multi-line
+solution score. 𝑄(𝑧) may also have penalties and caps im-
+posed on it based on specific circumstances and flags, such
+as the detection being near a spatially extended, bright ob-
+ject or if the various continuum estimates (§3.2) disagree.
+If the multi-line solution and P(Ly𝛼) agree, the 𝑄(𝑧) score
+increases; if the two measures disagree, the 𝑄(𝑧) score is
+decreased based on the relative difference between the multi-
+line solution and P(Ly𝛼) strengths. The selection logic and
+𝑄(𝑧) assignment is summarized in Table 5.
+Since the majority of HETDEX objects are faint, with a
+single detected emission line, most (∼ 80%) receive a 𝑄(𝑧)
+score less than 0.5 with ∼ 35% in the lowest 𝑄(𝑧) bin (0-0.1).
+These are still usually correctly classified as is shown in
+Sections 4.4 and 5, but rely on less evidence and thus have a
+low 𝑄(𝑧) value.
+3.7. Clustering/Neighbor Redshift Matching
+In the low-surface brightness outer regions of spatially re-
+solved galaxies, HETDEX detections with low, PSF-weighted
+line fluxes (commonly arising from faint H II regions and
+planetary nebulae) may be incorrectly classified by ELiXer
+as Ly𝛼. To address this issue, ELiXer can optionally com-
+pare a detection against other nearby HETDEX detections and
+look for consistencies. When invoked, ELiXer examines all
+HETDEX emission line detections within 15′′ (by default) of
+the current detection under consideration, and tests for 𝑔-band
+magnitudes brighter than 23𝐴𝐵 with matching observed emis-
+sion line(s) of higher line score (§3.1.4). The presumption,
+which is borne out in testing, is that the brighter, higher-
+scoring detections are (1) better centered on the object and
+(2) more likely to receive the correct classification. The re-
+quirement to match the observed emission line wavelength(s)
+in addition to the on-sky proximity helps preserve the clas-
+sification of background objects with lines of sight passing
+near the brighter, foreground source. When more than one
+match is found, the highest scoring redshift solution is se-
+lected and if the selected object is brighter and higher scoring
+than the current detection’s solution, that neighbor’s classi-
+fication is used as a replacement. In other words, faint, low
+scoring detections can be assigned the more secure redshift of
+Table 5. Best-𝑧, 𝑄(𝑧) Summary
+Condition
+Best-𝑧
+𝑄(𝑧)
+Strong, multi-line spec-𝑧
+solution consistent with P(Ly𝛼)
+multi-line
+spec-𝑧
+4-5★
+Strong, multi-line spec-𝑧
+solution not consistent with
+P(Ly𝛼)
+multi-line
+spec-𝑧
+0-3★
+Weak, multi-line spec-𝑧
+solution consistent with P(Ly𝛼)
+multi-line
+spec-𝑧
+2-4★
+Weak, multi-line spec-𝑧
+solution not consistent with
+P(Ly𝛼)
+multi-line
+spec-𝑧
+1-3★
+P(Ly𝛼) only, ≳ 0.7
+Ly𝛼
+3-4★
+P(Ly𝛼) only, ≳ 0.5
+Ly𝛼
+0-2★
+P(Ly𝛼) ≲ 0.5 with single,
+broad emission line
+[O ii]
+Mg II,
+C III]
+0-1★
+P(Ly𝛼) only, ≲ 0.5
+[O ii]
+0-1★
+P(Ly𝛼) only, ≲ 0.3
+[O ii]
+0-2★
+Note— Summary of the Best-𝑧 and 𝑄(𝑧) logic. Specific values (0.0-1.0)
+of the 𝑄(𝑧) are not shown as they depend on details omitted, but are
+expressed as these qualitative descriptors: 5★ (∼1.0), 4★ (∼0.80), 3★
+(∼0.50), 2★ (∼0.35), 1★ (∼0.25), 0★ (∼0).
+an immediately adjacent, brighter, higher scoring "neighbor"
+detection when they share matching observed-frame emission
+lines and are assumed to represent different detections of the
+same object. When this update is made, the altered detection
+is marked with a flag and the detection ID number of the
+matching neighbor detection.
+This clustering has a relatively small effect, modifying less
+than 0.5% of all HETDEX emission line detections.
+The
+algorithm does not link nor otherwise combine the individual
+detections; all detections remain uniquely reported.
+4. TESTING AND RESULTS
+All the effort made toward classification is effectively mean-
+ingless without appropriate testing and a selection of a rea-
+sonable spec-𝑧 assessment sample (SzAS) against which to
+test. As HETDEX is a large and unique survey with no pre-
+selection of targets, it is impossible to collect an overlapping
+observational dataset of known redshifts of even remotely
+similar size (in terms of numbers of unique astrophysical ob-
+jects) and continuum depth. Beyond polling experts for clas-
+sifications based on visual inspection, and comparing ELiXer
+results against those of simulated objects, the best we can
+do is match HETDEX sources against spectroscopic redshift
+catalogs produced by other surveys.
+
+24
+Davis, et al.
+The assessment sample for this work is a composite of
+matched HETDEX detections from the public, archival cat-
+alogs described in Section 2 and in Mentuch Cooper (ApJ
+accepted). In all cases, these are spectroscopic redshifts only;
+no photo-𝑧 estimates are used in this assessment sample. For
+the catalog provided redshifts, source matching to HETDEX
+is based on sky position and apparent magnitude. The catalog
+source position must be inside or within 0.′′5 of the edge of
+the SEP aperture associated with the HETDEX detection if
+an aperture match is made (§2.2), or within 0.′′75 of the HET-
+DEX position if the object is fainter than 𝑔 = 24.5 and no
+SEP aperture is matched. The catalog matched spectroscopic
+redshifts are accepted as true.
+The assessment sample is down-selected to only those
+detections fainter than 𝑔 = 22 with redshifts that match any
+of the emission lines in Table 2 to within ±4 Å. Though the
+magnitude distribution still significantly skews to brighter
+objects, this filtering helps refine the selection to better align
+with the more common, fainter HETDEX detections. The
+result is a dataset consisting of 834 [O II] emission lines,
+384 Ly𝛼 lines, and 402 "Other" lines, including C IV, C III],
+Mg II, and H𝛽 as reported in the SzAS. Each redshift cor-
+responds to a unique HETDEX detection, however, these
+are not necessarily unique galaxies. For brighter, extended
+galaxies there can be more than one overlapping HETDEX
+emission line detection, and where there are multiple obser-
+vations covering the same position, the same galaxy may be
+detected more than once.
+Since ELiXer operates on each
+HETDEX detection individually, this is as intended.
+4.1. Definitions
+For the remainder of this work, we make the following
+definitions:
+• Accuracy: The number of agreements between the
+ELiXer assigned classification and the SzAS classi-
+fication divided by the number of ELiXer detections
+of that classification. A match is counted if the rest-
+frame wavelengths from the HETDEX observed wave-
+length and the SzAS and ELiXer assigned redshifts
+agree within ±4 Å.
+• Recovery: A fraction roughly equivalent to complete-
+ness, but with no correction made for survey biases.
+Here we refer to the number of detections of a partic-
+ular emission line identified by the ELiXer software
+that are matched 1:1 to that of the SzAS divided by the
+number of those emission lines in the SzAS.
+• Contamination: The fraction of detections within some
+defined range that are incorrectly classified. This may
+be further refined to the fraction of misclassifications
+by a particular emission line. For example, we will
+discuss the contamination in the Ly𝛼 sample by [O II]
+as a function of 𝑔-magnitude.
+Accuracy can be slightly under reported for broad, noisy lines
+where the fitted line center can be offset from the true center
+and where winds and radiative transfer effects can create a
+significant velocity offset from the systemic redshift.
+The
+±4 Å allowance covers all but the most extreme cases so the
+impact is minimal. Accuracy and contamination are direct
+inverses and, for any given emission line, they necessarily
+sum to unity. Accuracy and recovery are similar, but differ by
+the base divisor. For the recovery of detections, any contam-
+ination of one emission line comes at the direct cost to the
+recovery of another emission line. Conversely, the recovery
+counts of an emission line is also one minus the sum of the
+contaminations of all other emission line types. Notice that
+the relationship does not directly hold for recovery and con-
+tamination rates, as each of those rates have different divisors.
+4.2. Calibration
+Testing and calibration are combined in a highly iterative
+process.
+ELiXer is executed on the detections of the test
+dataset, but with catalog matching spec-𝑧 and phot-𝑧 turned
+off. That is, for the test runs, ELiXer does not include or con-
+sider the catalog reported spectroscopic redshifts that would,
+in a standard run, factor into the classification. The ELiXer
+output, specifically the P(Ly𝛼) values and the redshifts, are
+then compared to the test sample and checked for contami-
+nation, recovery, and accuracy. Disagreements between the
+ELiXer results and the assessment sample are examined,
+and manual adjustments to the individual votes and voting
+weights (§3, and §3.5 in particular), are made as warranted.
+Considerations against over-tuning and potentially incorrect
+test sample redshifts are addressed with deliberately loose
+fitting, low-order segmentation thresholds and by varying
+the composition of the test sample by creating random and
+targeted (in apparent magnitude, line FWHM, observation
+field, etc.) subsets. The process is repeated until there is good
+agreement (generally, matching 90-95% or better) between
+the ELiXer assigned redshifts and the test sample redshifts.
+With the focus on P(Ly𝛼) as the primary classification metric
+and with its flexible threshold selection, what constitutes
+"good" agreement is somewhat subjective but is also highly
+adaptable to the specific scientific needs. For example, the
+stacking of spectra to measure Lyman Continuum in Davis
+et al. (2021) is very sensitive to contamination but does not
+specifically require a highly complete sample and so utilizes
+a P(Ly𝛼) selection of 0.8 and greater. On the other hand, the
+𝐻(𝑧) and 𝐷 𝐴(𝑧) precision goals for the primary HETDEX
+science is less sensitive to contamination but needs to be
+largely complete (Gebhardt et al. 2021; Farrow et al. 2021)
+and a P(Ly𝛼) threshold of 0.5, or even lower, is more appro-
+
+ELiXer
+25
+priate.
+4.3. Additional Testing
+To supplement the catalog spec-𝑧 testing, several other test-
+ing and feedback efforts are actively used. Though the me-
+chanics vary, all provide checks on the ELiXer classifications
+with targeted detection subsets. As with the SzAS, the detec-
+tions where these alternate methods and ELiXer disagree are
+manually inspected and adjustments to the ELiXer classifica-
+tion algorithm(s) are made as warranted.
+These supplementary efforts fall into two categories. The
+first are automated machine learning classifiers, both super-
+vised and (sometimes) unsupervised. These are all in early
+development and explore various classification frameworks,
+with both T-distributed Stochastic Neighborhood Embedding
+(tSNE) (van der Maaten & Hinton 2008) and Autoencoder
+Neural Network (Wang et al. 2014) techniques showing good
+promise.
+The second category relies on manual, visual vetting. The
+first efforts focused on HETDEX collaboration experts and
+university students (after receiving training). A more recent
+science outreach effort has opened classification and gen-
+eral exploration to the public in a citizen science project
+on Zooniverse (https://www.zooniverse.org/).
+One work-
+flow of the Dark Energy Explorers (https://www.zooniverse.
+org/projects/erinmc/dark-energy-explorers) project (House &
+et al. in prep) tasks its citizen scientists to classify HETDEX
+detections as either being at low-𝑧 ("Nearby Galaxy or Star")
+or possibly high-𝑧 ("Distant Galaxy or nothing") using a re-
+duced ELiXer report that contains only sections of 2D fiber
+cutouts, single band (𝑔 or𝑟) photometric imaging, and a Gaus-
+sian fit to the emission line. Each detection receives 15 re-
+sponses with the aggregate classification reported as the mean
+of those responses. Even with this reduced information, these
+broad categories match with the ELiXer classification more
+than 92% of the time with House, et al, estimating 7.7% con-
+tamination and 90.7% recovery of high-𝑧 galaxies. As with
+the other methods, select disagreements between ELiXer and
+Zooniverse are reviewed for potential classification failures
+by ELiXer.
+4.4. Results Summary
+A comparison of the ELiXer classification/redshift assign-
+ments with those of the SzAS are summarized in Figure 6
+and in Table 6. Figure 6 breaks out the contamination and
+recovery rates by 𝑔-magnitude, with the counts of each type
+shown as a reference in the bottom panel. When there are very
+few classifications of a given type, such as faint [O II] and
+"Other" lines, the accuracy and recovery rates are not mean-
+ingful. Against the SzAS, ELiXer performs very well on Ly𝛼
+and [O II] classifications, but is challenged by the "Other"
+emission lines. As will be discussed later, the elevated con-
+tamination in the Ly𝛼 detections at bright magnitudes is a
+function of the biases in the SzAS as compared to the HET-
+DEX survey.
+Table 6 summarizes the cumulative performance of several
+different Ly𝛼/[O II] segregation methods against the SzAS
+identifications of the Ly𝛼 or [O II] line. This down-selection
+is made so that the comparisons of the ELiXer P(Ly𝛼) method
+(§3.5) at several selection thresholds is equitable, as 20 Å
+equivalent width cut and the P(LAE)/P(OII) method do not
+classify lines other than Ly𝛼 and [O II] .
+It is clear that
+each method is an effective classifier. Except at the extreme
+thresholds, the P(Ly𝛼) methods produce the lowest contami-
+nation and highest recovery rates, with P(Ly𝛼) > 0.5 yielding
+a good balance of contamination and recovery fraction. This
+is the default input for the ELiXer Best-𝑧 assignment (§3.6).
+Given the biases in the SzAS for bright objects and AGN,
+though, these results cannot be directly applied to the whole
+of HETDEX. However, a correction for these biases is made
+and discussed later in §5.1. We also caution that the detec-
+tions in the SzAS factor significantly in the calibration of the
+votes and weights of the P(Ly𝛼) metric. Although efforts
+are made to avoid over-fitting, these results could still be less
+reflective of HDR3 in general.
+The contamination rate of Ly𝛼 by [O II] is effectively flat
+as a function of the observed wavelength of the emission line.
+However, the recovery rate of Ly𝛼 sources trends lower as the
+observed wavelength moves redward. At the blue end of the
+HETDEX spectral range, 𝜆obs ≲ 4200Å, the recovery rate is
+∼97%; in the middle range, 4200 ≲ 𝜆obs ≲ 4800, the rate is
+∼91%; and at the red end, 4800 ≲ 𝜆obs the rate is ∼81%. This
+is an effect of larger numbers of faint [O II] emitting galaxies
+and fewer numbers of LAEs in their respective higher redshift
+regions. These [O II] galaxies are more similar in appearance
+to LAEs based on several of the metrics used in ELiXer, 𝑔
+and 𝑟 magnitudes, angular size, and even EW and line width
+to a lesser extent (see Sections 3.5.1, 3.5.4, 3.5.5, and 3.5.7
+and their figures). The observed emission line wavelength
+factors in the related votes help keep the Ly𝛼 contamination
+rate flat and low, but at the cost of the loss of some LAEs to
+[O II] classifications. As shown in Table 6, this can be tuned
+to improve the Ly𝛼 recovery rate at the expense of a higher
+contamination rate as dictated by particular science needs.
+5. DISCUSSION
+As can be seen from Figure 7, the sample we use for spectro-
+scopic assessment, SzAS, is highly biased to brighter detec-
+tions, somewhat biased to broader lines, and contains an over
+representation of emission lines other than Ly𝛼 and [O II],
+as compared to HETDEX as a whole. At its bright end, the
+sample is under-abundant in [O II] and over-abundant in Ly𝛼
+
+26
+Davis, et al.
+Table 6. Ly𝛼 vs [O II] Segregation on Assessment Sample
+Method
+Ly𝛼 Contamination
+Ly𝛼 Recovery
+Ly𝛼 rest EW > 20Å
+0.084
+0.708
+P(LAE)/P(OII) default 1
+0.090
+0.763
+P(LAE)/P(OII) optimized 1
+0.056
+0.724
+P(LAE)/P(OII) ELiXer 2
+0.042
+0.705
+P(Ly𝛼) > 0.7
+0.005
+0.752
+P(Ly𝛼) > 0.6
+0.007
+0.797
+P(Ly𝛼) > 0.5 3
+0.010
+0.903
+P(Ly𝛼) > 0.4
+0.027
+0.926
+P(Ly𝛼) > 0.3
+0.056
+0.940
+Note—The cumulative performance of various methods against the
+SzAS down-selected to only include [O II] (834 detections) and Ly𝛼
+(384 detections). This allows a fairer comparison of P(Ly𝛼) (§3.5) to
+the first three methods, which do not consider other lines. The SzAS is
+biased to bright objects, with an over representation of AGN, so these
+results do not directly translate to the larger population of HETDEX
+detections. An adjustment for these biases are made and discussed later
+in §5.1. Additionally, though efforts are made to avoid over-fitting to
+the SzAS, its detections significantly contribute to the determination
+of the votes and weights of the P(Ly𝛼) metric, so these results may not
+be as representative when considering all HETDEX detections.
+1Leung et al. (2017)
+2Modified P(LAE)/P(OII) optimized used in ELiXer (§3.5.3)
+3Default input to Best-𝑧 logic (§3.6)
+with the reverse at the faint end. Since these spectroscopic
+redshifts come from existing archival surveys (§4) and spec-
+troscopy is historically expensive, it stands to reason that the
+available spectra would favor brighter, rarer objects. An ex-
+pansion of the SzAS is underway in collaboration with DESI
+((Jelinsky et al. 2018; Levi et al. 2019)) which will provide
+higher spectral resolving power (R∼2000-5000) and a redder
+wavelength coverage (3600-9800Å) to selected HETDEX de-
+tections. This will increase the number of faint (𝑔 > 25) spec-
+tra in future assessment samples and bring their distributions
+more in line with HETDEX.
+While not completely devoid of faint objects, the SzAS
+contains a smaller fraction of its detections in the faintest bins
+compared to the full HETDEX sample. This is not unexpected
+and is not a significant issue. Given the methodology of the
+classification, ELiXer is likely to classify anything fainter
+than 𝑔 ∼25 as an LAE in the 1.9 < 𝑧 < 3.5 redshift range.
+While there are certainly [O II] emission-line galaxies with
+𝑧 < 0.5 and 𝑔 > 25, if we assume that this emission has
+a rest-frame equivalent width of less than 20 Å, then [O II]
+can be expected to be, at most, ∼ 3 × 10−17 erg s−1 cm−2.
+This maximum value is ∼2× fainter than the 50% flux limits
+22.5
+23.0
+23.5
+24.0
+24.5
+25.0
+0.00
+0.25
+0.50
+0.75
+1.00
+Contamination
+Ly
+[O II]
+Other
+22.5
+23.0
+23.5
+24.0
+24.5
+25.0
+0.00
+0.25
+0.50
+0.75
+1.00
+Recovery
+22.5
+23.0
+23.5
+24.0
+24.5
+25.0
+g
+0
+100
+200
+300
+400
+Number in Bin
+Figure 6. Performance summary of ELiXer classification and red-
+shift assignment vs. the SzAS in 𝑔-magnitude bins. ELiXer does
+very well with Ly𝛼 and [O II], as intended, but struggles with the
+"Other" lines, such as C IV 𝜆1550, C III] 𝜆1909, and Mg II 𝜆2800
+Note that the results for the faintest bin for Ly𝛼, the faintest 2 bins
+for [O II] and the faintest 4 bins for Other lines, denoted with open
+markers and dotted lines, have too few SzAS counts to be meaning-
+ful. The high contamination rate in Ly𝛼 at brighter magnitudes is a
+result of the biases in the SzAS and is discussed in section 5.
+for HETDEX (Gebhardt et al. 2021), making it unlikely that
+HETDEX would even detect an [O II] emission line from
+such a galaxy. Thus the reduced fraction of 𝑔 ≳ 25 objects in
+the SzAS, compared to HDR3, is largely moot.
+Nevertheless, the other biases cannot be ignored. While
+an uncorrected assessment sample can serve as a develop-
+ment test set and provide reasonable limits on the expected
+contamination, recovery, and accuracy rates for ELiXer clas-
+sifications, a correction is needed to extrapolate to the entire
+HETDEX emission line sample.
+5.1. Bias Correction to the Full HETDEX Catalog
+Given the clearly biased distribution of the assessment sam-
+ple as compared to the full HETDEX catalog, it is prudent to
+apply some measure of correction before extending the results
+from the SzAS to the full catalog. The correction chosen is
+
+ELiXer
+27
+23
+24
+25
+g
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Ly Fraction in Bin
+HDR3 ELiXer
+SzAS ELiXer
+SzAS Reported
+23
+24
+25
+g
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+OII Fraction in Bin
+HDR3 ELiXer
+SzAS ELiXer
+SzAS Reported
+23
+24
+25
+g
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Other Fraction in Bin
+HDR3 ELiXer
+SzAS ELiXer
+SzAS Reported
+18
+20
+22
+24
+g
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Fraction of Total
+HDR3 Bright (Excluded)
+HDR3
+g Limit = 25
+SzAS
+10
+20
+30
+Line FWHM [Å]
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+Fraction of g > 22
+HDR3
+SzAS
+0
+1
+2
+3
+z
+0.00
+0.05
+0.10
+0.15
+0.20
+Fraction of g > 22
+HDR3 ELiXer
+SzAS ELiXer
+SzAS Reported
+Figure 7. Summary of the ∼ 1600 emission line detections in the Spec-𝑧 Assessment Sample (SzAS) compared to the ∼ 1.5×106 detections in
+the HETDEX Data Release 3 (HDR3). The top panels show the relative fraction of Ly𝛼, [O II], and Other emission line detections as a function
+of 𝑔-magnitude, as classified by ELiXer and as reported by archival spec-𝑧 measurements in the SzAS. The ELiXer reported classifications
+represent more of an "apples to apples" comparison, as it is clear that the SzAS is skewed towards brighter magnitudes and is significantly
+overabundant in Other emission line detections. The Ly𝛼 and [O II] distributions are very similar fainter than about 23.5𝐴𝐵, but diverge at the
+brighter end. The lower left panel illustrates the bright bias. The lower-center panel shows an excess in the SzAS for broad emission lines;
+though not explicitly shown here, these broad lines are predominantly Ly𝛼, C IV 1549 Å, and C III] 1909 Å and originate from brighter, probably
+AGN, objects. The lower right panel echoes the over abundance of the Other emission lines, showing an increase in the fraction of 1.0 ≲ 𝑧 ≲ 2.0
+detections, likely AGN, compared to HDR3.
+relatively simplistic and, as will be shown a little later, has
+effectively no impact on the overall sample results.
+As seen earlier, the SzAS dataset is subdivided into Ly𝛼,
+[O II], and Other emission line detections, and each subset
+is binned by 𝑔 magnitude from 22𝐴𝐵 to 25𝐴𝐵 in steps of
+0.5, with the last bin containing all detections fainter than the
+25𝐴𝐵 flux limit. The contamination (by type) for each of the
+three classifications is computed against the SzAS in each 𝑔
+bin as defined in §4.1.
+To correct for the population biases in the SzAS compared
+to the full HDR3 sample, we consider the contamination rates
+in the SzAS to be functions of the per bin fractions of the
+contaminant, and the target type as classified by ELiXer. This
+allows us to use the same ELiXer classification rates in the
+full HDR3 sample as a correction to the SzAS rates. The
+applied correction to the SzAS values then is:
+𝐶′
+𝑖, 𝑗 =
+�∑︁
+𝑘
+𝐶𝑖, 𝑗,𝑘 × 𝐸𝐻, 𝑗,𝑘
+𝐸𝑆, 𝑗,𝑘
+× 𝑁𝐻,𝑖,𝑘
+�
+∑︁
+𝑘
+𝑁𝐻,𝑖,𝑘
+(26)
+where:
+• 𝐶′
+𝑖, 𝑗 is the corrected contamination rate of the target
+type 𝑖 (Ly𝛼, [O II], or Other) by contaminant 𝑗, such
+that 𝑖 ≠ 𝑗.
+• 𝐶𝑖, 𝑗,𝑘 is the directly computed contamination rate in
+the SzAS per 𝑔-magnitude bin, 𝑘 (matching the bins in
+Figure 6).
+
+28
+Davis, et al.
+• 𝐸𝐻,𝑖,𝑘 is the ELiXer classification fraction of the target
+type in HDR3 per 𝑔-magnitude bin.
+• 𝐸𝑆, 𝑗,𝑘 is the ELiXer classification fraction of the con-
+tamination type per 𝑔-magnitude bin in the SzAS.
+• 𝑁𝐻,𝑖,𝑘 is the number of target ELiXer classifications in
+HDR3 per 𝑔-magnitude bin.
+An additional simple correction is also applied to help ac-
+count for false positive (FPN) detections caused by noise
+interpreted as an emission line by the HETDEX line-finding
+algorithm (Gebhardt et al. 2021). These are random fluctua-
+tions in the PSF weighted spectrum from thermal electrons in
+the CCDs, stray photons, read noise, etc, that happen to scatter
+up and pass the various filtering thresholds in the line-finding
+code and masquerade as low SNR emission lines. They do
+not represent real astrophysical sources but when interpreted
+as such, they map to random locations in (RA, Dec, z)-space.
+As the candidate emission line SNR increases toward 5.5,
+the incidence of these FPN rapidly approaches zero. As dis-
+cussed later in §5.4, this has only a minimal impact on the
+HETDEX cosmological measurements. As an approximate
+correction, the ELiXer classification ratios in Eqn (26) for
+HDR3 are modified by assuming 30% of all detections with
+SNR < 5.0 and 15% of all detections with 5.0 ≤ SNR < 5.5 are
+false positives and simply removing those from all summed
+counts. Early indications are that the true FPN rates may be
+significantly less than this,
+(Mentuch Cooper ApJ accepted), so we believe the as-
+sumed FPN rates are overestimates.
+22.5
+23.0
+23.5
+24.0
+24.5
+25.0
+g
+0.000
+0.005
+0.010
+0.015
+0.020
+Cumulative [O II] Cont. Frac.
+Bias + FPN Corrected HDR3
+SzAS
+g Limit
+Figure 8. Cumulative (bright to faint) contamination of Ly𝛼 by
+[O II] as a function of 𝑔 magnitude using the default ELiXer con-
+figuration.
+The Bias + FPN Corrected HDR3 curve attempts to
+compensate for the biases in the SzAS (compared to all of HDR3)
+and account for false positives in the low-SNR regime (§5.1).
+22.5
+23.0
+23.5
+24.0
+24.5
+25.0
+g
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+Cumulative Other Cont. Frac.
+Bias + FPN Corrected HDR3
+SzAS
+g Limit
+Figure 9. Cumulative (bright to faint) contamination of Ly𝛼 by
+emission lines other than [O II] for 𝑔 > 22 using the default ELiXer
+configuration. The FPN + Bias Corrected HDR3 curve attempts to
+compensate for the biases in the SzAS and account for false positives
+due to random noise in the low-SNR regime (§5.1). The much larger
+contamination rate in the SzAS is largely driven by confusion of Ly𝛼
+vs. C III] and C IV, where the AGN population is significantly over
+represented (see Figure 7, §5.2 and §5.3).
+5.2. Performance
+Figures 8 and 9 show the cumulative (bright to faint) con-
+tamination fraction of Ly𝛼 by [O II] and all "Other" lines
+respectively, both for the SzAS and for the 𝑔 > 22 HDR3
+dataset. Table 7 reports the cumulative contamination rates
+from those two figures (highlighted by bold type face), pro-
+vides summary information on the contamination in [O II]
+and the "Other" lines, and gives the accuracy and recovery
+rates for all discussed line types. Note that the values for the
+SzAS corresponding to Table 6 are slightly different, since
+the detections for that table are down selected to only in-
+clude Ly𝛼 and [O II] . Overall, ELiXer performs extremely
+well in mitigating the contamination in the Ly𝛼 classification,
+and excels at the faint end against the primary contaminant,
+[O II]. This is what ELiXer is tuned to do. At brighter magni-
+tudes, non-[O II] contaminants are more problematic, though
+they represent only a small fraction of the total HETDEX
+dataset (Figure 7). For the HETDEX data releases, the final
+classification of these objects is assisted by the supplemental
+program, Diagnose (Zeimann & et al. in prep) (see also §5.3).
+The cumulative fractional contamination from [O II] has a
+peak between 𝑔 ∼ 23.0 and 𝑔 ∼ 24.0, where the numbers of
+[O II] and Ly𝛼 emitters are most similar. The total contami-
+nation rate sits at only 1.3% for the SzAS even with the [O ii]
+emitters outnumbering LAEs in that sample by more than
+2:1. For HDR3, when corrected for the SzAS distribution
+bias and predicted false positives from noise, the predicted
+contamination rate is 1.2%.
+While this already meets the
+HETDEX requirements, planned ELiXer enhancements, in-
+
+ELiXer
+29
+cluding updated Ly𝛼 and [O II] luminosity functions for the
+P(LAE)/P(OII) analysis (§3.4) and run-time phot-𝑧 fitting,
+should further decrease the contamination rate and improve
+overall accuracy.
+The cumulative fractional contamination of Ly𝛼 from all
+other lines in the SzAS is substantial at 26.4%. This, how-
+ever, is significantly inflated due to the over representation of
+AGN and C III] and C IV emission lines in the SzAS (Fig-
+ure 6, upper right panel). When projected onto the HDR3
+distribution and corrected for the SzAS distribution bias and
+noise driven false positives, this cumulative contamination
+fraction falls to a predicted 0.8% for the full HDR3 dataset.
+This is even better than the [O ii] contamination. However,
+given the large correction from the SzAS results (Figure 9),
+it is prudent to estimate a worst case contamination by these
+other lines by alternate means. These misclassifications in the
+SzAS are dominated by C iii] and C iv and are characterized
+by bright magnitudes and large line widths – median 𝑔 = 22.5
+± 0.5 and median emission line FWHM = 22 ± 8 Å. Using
+these properties as a guide, we select the fraction of HDR3
+detections with emission line FWHM > 14 Å and 𝑔 < 23,
+yielding 5.8% of HDR3 detections, of which we assume 1/3
+are misclassified as Ly𝛼. With 47% of detections classified
+as Ly𝛼, we then estimate the worst case contamination rate
+by Other lines at 4% (e.g.: 1
+3 · 0.058 / 0.47 = 0.04).
+While this is 5× the Bias + FPN Corrected contamination
+rate of 0.8%, this is still relatively small and the impact is
+far less than that of [O ii] contamination. The small scale
+clustering of [O ii] emitters projects to large scale clustering
+when misinterpreted as higher-z Ly𝛼. This is greatly dimin-
+ished with C iii] and C iv as the contamination sources shift to
+higher redshift and scales proportionally to the square of the
+ratio of the co-moving angular diameter distances (Gebhardt
+et al. 2019; Farrow et al. 2021). This means the HETDEX
+cosmology is some 6.5× less sensitive to C iii] contamination
+than [O ii] contamination and can tolerate ∼13% (or ∼16%
+for C iv) at the desired uncertainty. So, even the worst case
+contamination is well within the required tolerances. Addi-
+tionally, the focus for ELiXer has been on the largest con-
+taminant, [O II], as the contamination rate of other lines is
+expected to decrease with future improvements targeting their
+identification.
+Overall, the ELiXer accuracy is good in the HDR3 dataset,
+while that in SzAS is poorer. The weaker performance in the
+SzAS set is due to the bright-magnitude and broad-emission
+line biases in the SzAS; this is where ELiXer does not per-
+form as well. The stronger (estimated) accuracy in the full
+HETDEX population is bolstered by the large numbers of
+faint end detections that are highly biased towards being Ly𝛼.
+The results for ELiXer recovery rates are similarly mixed.
+The numbers are good for Ly𝛼 and [O II], which are, by far,
+the most common emission lines found by HETDEX. The
+recovery of all other emission lines is rather poor, and is
+largely an issue of the default behavior of the classification
+algorithms. When there is only a single line in a HETDEX
+spectrum, ELiXer heavily weights the various Ly𝛼 / [O II]
+segregation methods which, as stated above, assume no con-
+tamination other than [O II] . In this case, ELiXer delivers
+a binary result, Ly𝛼 vs not-Ly𝛼, at the expense of all other
+emission lines. Moreover, when analyzing particularly broad
+lines, ELiXer favors Ly𝛼 (often suggestive of an AGN) over
+[O II]; this also leads to the enhanced contamination of Ly𝛼
+by such "Other" lines. Additional identification metrics such
+as limited run-time phot-𝑧, spectral slope, and multi-Gaussian
+fits, could help improve these rates and will be explored in
+future versions.
+A preliminary evaluation of an assessment sample ex-
+panded with ∼ 1000 DESI provided spectroscopic redshifts,
+3/4 of which are for 𝑔 > 24 objects, is consistent with the
+HETDEX classification results of this work. The resulting
+assessment sample more closely matches the HETDEX mag-
+nitude and emission line distributions. After the observations
+are complete, the full, detailed results will be presented in
+Landriau et al, in preparation.
+5.3. Missing AGN and LBGs
+Since ELiXer largely relies on equivalent width to clas-
+sify most single-line spectra, the program currently does not
+perform well with Ly𝛼 emitting objects that are not classi-
+cal LAEs, i.e., broad-line AGN and Lyman-break galaxies
+(LBGs) which may have small Ly𝛼 equivalent widths (e.g.,
+Shapley et al. 2003). Moreover, ELiXer can also fail to find
+some of the broad emission lines associated with the AGN,
+which can result in misclassifications that would otherwise be
+correctly assigned by the multi-line redshift solutions (§3.3).
+This is particularly noticeable in the bright end of the SzAS
+(Figure 9), which has a disproportionately large number of
+AGN.
+Moreover, in AGN, ELiXer can confuse Ly𝛼 with
+C III] when C III] is the only significant emission line in the
+HETDEX spectral window (0.96 ≲ 𝑧 ≲ 1.25) or with C IV
+when the line fit to C III] fails. Other approaches are taken to
+identify and recover AGN missed or misclassified by ELiXer
+(Liu et al. 2022) and future updates to ELiXer should improve
+upon its classification performance with these emission lines.
+ELiXer also struggles to classify low Ly𝛼 EW LBGs. On
+the whole, given their name-defining detection methodology
+(Guhathakurta et al. 1990; Madau et al. 1996; Steidel et al.
+1996), LBGs tend to be more massive and more evolved
+than the typical LAE (Stark et al. 2010; Kornei et al. 2010;
+Jose et al. 2013; Vargas et al. 2014; Steidel et al. 2018) and,
+consequently, may contain more dust to inhibit the escape
+of Ly𝛼. While some LBGs also meet the definition of an
+LAE and are likely to be detected and correctly identified
+
+30
+Davis, et al.
+Table 7. Cumulative Classification Performance for HDR3
+Metric
+SzAS
+Bias + FPN Corrected
+Ly𝛼 Accuracy
+0.723
+0.981 ±0.034
+Ly𝛼 Recovery
+0.892
+0.991 ±0.033
+Ly𝛼 Contamination by [O II]
+0.013
+0.012 ±0.001
+Ly𝛼 Contamination by Other
+0.264
+0.008 ±0.001*
+[O II] Accuracy
+0.890
+0.965 ±0.034
+[O II] Recovery
+0.972
+0.970 ±0.034
+[O II] Contamination by Ly𝛼
+0.039
+0.021 ±0.001
+[O II] Contamination by Other
+0.071
+0.014 ±0.001
+Other Accuracy
+0.892
+0.916 ±0.032
+Other Recovery
+0.509
+0.294 ±0.010
+Other Contamination by Ly𝛼
+0.027
+0.006 ±0.001
+Other Contamination by [O II]
+0.081
+0.078 ±0.003
+Note—The cumulative performance of the ELiXer classifications on
+the SzAS and predictions for the full HDR3 dataset for detections
+with 𝑔 > 22 and using the default ELiXer configuration. The Bias
++ FNP Corrected column corrects for the sample biases in the SzAS
+dataset and for false positives in the full HDR3 dataset, assuming
+30% false positive rate below emission line SNR of 5.0 and 15% rate
+between 5.0 < SNR < 5.5. The values in the first column are slightly
+different than those in Table 6 since that table is down selected to
+only consider Ly𝛼 and [O II] detections. The bold type face rows
+correspond to the cumulative data points in the right-most (faintest)
+bins in Figure 8 and Figure 9.
+∗0.04 worst case estimate. See §5.2 for a discussion.
+as such by ELiXer, the more massive objects may often be
+confused with low-𝑧 [O II] emitters or even overlooked com-
+pletely if they exhibit weak Ly𝛼 emission or Ly𝛼 absorption.
+While relatively few in number compared to LAEs, the more
+massive LBGs do represent a highly biased mass tracer and
+are of value to HETDEX, so it is desirable to recover and
+correctly identify as many of them as possible. This means
+using methods that do not use equivalent width as their pri-
+mary discriminant.
+To that end, several machine learning
+approaches (both supervised and unsupervised) are being ex-
+plored, as are direct enhancements to ELiXer that incorporate
+additional classification methods, such as run-time photo-𝑧
+estimation.
+5.4. Contamination from Noise
+As stated earlier, ELiXer assumes an emission line detec-
+tion is real, and not the result of noise or an artifact of the
+data reduction. As the SNR of an emission line detection
+decreases, it does become more likely that the feature is the
+result of noise. However, unlike real, incorrectly classified
+emission lines, false positives from noise are not expected to
+cluster (they occur in random spectra at random wavelengths
+and thus map to random sky positions at random redshifts)
+and should only increase the uncertainty in the HETDEX cos-
+mological measurements and not introduce a bias. As such, it
+is of lesser concern than misclassifications. Nevertheless, as
+described earlier, a (likely overly) aggressive false positives
+correction (§5.1) is used for Figures 8 and 9 and for Table
+7 to better estimate the classification performance of ELiXer
+against the full HETDEX dataset.
+Separate efforts to identify the noise driven false positive
+rate include repeat observations of low SNR sources (based
+on the premise that random noise will not cause a repeat de-
+tection at the same position and wavelength; Mentuch Cooper
+(ApJ accepted)) and various machine learning techniques.
+Their goal is to allow a more accurate model of contamination
+from noise.
+5.5. Uncertainties
+The performance of ELiXer presented in the prior sections
+are shown without statistical uncertainties, though some un-
+certainty is implicit in its predictions for the whole of HET-
+DEX Data Release 3.
+For the SzAS results in this work, the ELiXer classifications
+have been taken as absolute, as the quality of the classifica-
+tions has not yet been calibrated to a proper probability. (This
+is a planned enhancement.) Since classifications are based on
+votes and weights, some of which have an MCMC element
+with a weak dependency on the initial random seed vectors,
+individual executions can occasionally result in a different
+classification due to conditions falling just to either side of a
+threshold (though the quality score (𝑄(𝑧)) is generally unaf-
+fected; see §3.6). Similarly the catalog reported spec-𝑧 values
+are taken as truth, and matching against the reported values is
+done as described in §4, with a ±4 Å allowance, independent
+of the uncertainties in the spec-𝑧 or the fitted emission line
+center (§3.1.3). Nevertheless, many realizations of ELiXer
+classification runs compared against the SzAS have shown
+the results to be highly stable and repeatable.
+In projecting the SzAS results onto the full HDR3 dataset,
+a few additional sources of uncertainty arise, such as the as-
+sumed false positive rate, which is binned only as a function
+of SNR. However, as with the SzAS, we assume the ELiXer
+classifications to be strictly categorical and the reported frac-
+tions subject only to rounding error. Anticipated expansion
+and improvements to the SzAS, including better matching
+to the HETDEX magnitude and emission line width distribu-
+tions, will help address the systematics between the SzAS and
+the full HETDEX sample beyond the simplified corrections
+of §5.1.
+As rough estimate on the uncertainties in the accuracy,
+recovery, and contamination rates reported for HDR3, we
+
+ELiXer
+31
+use the fraction of detections that are most susceptible to
+classification changes as described in this subsection. This is
+effectively captured by the largest factor in the classifications,
+P(Ly𝛼), where P(Ly𝛼) is least certain and least stable against
+change due to randomness in sampling (i.e., near 0.5). As 7%
+of HDR3 detections have 0.4 < P(Ly𝛼) < 0.6, we assume a
+±3.5% uncertainty on those rates.
+6. SUMMARY
+As the primary emission line classifier for HETDEX,
+ELiXer must produce quality redshift identifications that are
+highly accurate, complete, and with minimal contamination.
+With a resolving power ranging from 750–950, HETDEX
+cannot split the [O II] doublet, so object classification must
+rely heavily on continuum information combined with equiv-
+alent width distributions. By incorporating improvements to
+established Ly𝛼/[O II] separation mechanics, from the 20 Å
+equivalent width cut (Gronwall et al. 2007; Adams et al. 2011)
+to the P(LAE)/P(OII) ratio (Leung et al. 2017), and by com-
+bining additional partitioning techniques, ELiXer produces
+classifications that outperform the HETDEX science require-
+ments for Ly𝛼 contamination by its principle low-𝑧 interloper,
+[O II] 3727 Å, while providing a good recovery rate (Table 7).
+The lower than required 1.2% contamination of Ly𝛼 by [O II]
+affords the option to loosen the project’s strict classification
+thresholds in exchange for gains in the Ly𝛼 recovery fraction
+or completeness.
+Though they occupy a small fraction of HETDEX emission
+line detections, lines other than [O II] 3727Å, such C III]
+1909 Å, and C IV 1549 Å represent a larger source of Ly𝛼
+contamination in the biased SzAS. However, as described in
+§5.2, these lines are not expected to produce a significant
+clustering signal or bias in the 𝑧 = 2.4 measures of 𝐻(𝑧) and
+𝐷 𝐴(𝑧). Regardless, planned enhancements to ELiXer and a
+larger spectroscopic redshift test sample (more aligned with
+the HETDEX distribution) will improve these classifications
+and further reduce Ly𝛼 contamination.
+The HETDEX project is continuing to work towards re-
+ducing the rate of false positive detections as a function of
+the emission line signal-to-noise ratio (Mentuch Cooper ApJ
+accepted). Early indications suggest the contamination from
+noise is small above the 4.8-5.0 SNR acceptance threshold
+for detections. Regardless, these noise driven false positives
+should only add white noise to the LAE cluster signal. Al-
+though this increases the uncertainty in the HETDEX mea-
+surements, it should not introduce specific features in the
+galaxy power spectrum.
+ELiXer continues to evolve.
+Future enhancements and
+revised voting criteria will be tested against expanded as-
+sessment samples drawn from forthcoming data releases.
+This will improve the current classification capabilities, en-
+abling new and higher precision science. Although ELiXer
+is designed for and calibrated to HETDEX, the methodology
+developed in this work can be adapted to other low-resolution,
+narrow wavelength range spectroscopic surveys.
+ACKNOWLEDGMENTS
+The authors thank the anonymous reviewer for the helpful
+feedback which assisted in improving this manuscript.
+HETDEX is led by the University of Texas at Austin
+McDonald Observatory and Department of Astronomy
+with participation from the Ludwig-Maximilians-Universität
+München, Max-Planck-Institut für Extraterrestrische Physik
+(MPE), Leibniz-Institut für Astrophysik Potsdam (AIP),
+Texas A&M University, The Pennsylvania State University,
+Institut für Astrophysik Göttingen, The University of Oxford,
+Max-Planck-Institut für Astrophysik (MPA), The University
+of Tokyo, and Missouri University of Science and Technology.
+In addition to Institutional support, HETDEX is funded by the
+National Science Foundation (grant AST-0926815), the State
+of Texas, the US Air Force (AFRL FA9451-04-2-0355), and
+generous support from private individuals and foundations.
+Observations were obtained with the Hobby-Eberly Tele-
+scope (HET), which is a joint project of the Univer-
+sity of Texas at Austin, the Pennsylvania State Univer-
+sity, Ludwig-Maximilians-Universität München, and Georg-
+August-Universität Göttingen. The HET is named in honor
+of its principal benefactors, William P. Hobby and Robert E.
+Eberly.
+VIRUS is a joint project of the University of Texas at
+Austin, Leibniz-Institut für Astrophysik Potsdam (AIP), Texas
+A&M University (TAMU), Max-Planck-Institut für Extrater-
+restrische Physik (MPE), Ludwig-Maximilians-Universität
+Muenchen, Pennsylvania State University, Institut fur Astro-
+physik Göttingen, University of Oxford, and the Max-Planck-
+Institut für Astrophysik (MPA). In addition to Institutional
+support, VIRUS was partially funded by the National Science
+Foundation, the State of Texas, and generous support from
+private individuals and foundations.
+The authors acknowledge the Texas Advanced Comput-
+ing Center (TACC) at The University of Texas at Austin for
+providing high performance computing, visualization, and
+storage resources that have contributed to the research results
+reported within this paper. URL:http://www.tacc.utexas.edu
+The Institute for Gravitation and the Cosmos is supported by
+the Eberly College of Science and the Office of the Senior Vice
+President for Research at the Pennsylvania State University.
+KG acknowledges support from NSF-2008793.
+This research benefits from the open-source projects Python
+(Van Rossum & Drake 2009), astropy (Astropy Collaboration
+et al. 2018b), numpy (Harris et al. 2020), photutils (Bradley
+
+32
+Davis, et al.
+et al. 2020), and others in the open-source community.
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+ELiXer
+35
+APPENDIX
+A. EXAMPLE ELIXER DETECTION REPORTS
+In this Appendix, we include two ELiXer detection reports as examples of those used for visual inspection and diagnostics. The
+first, Figure 10, is a somewhat unusual HETDEX LAE: it has a very high emission line SNR, it is matched to a source contained
+in multiple catalogs (§2), and has several photometric and spectroscopic redshift determinations, and it lies in an area of sky with
+deep HST imaging. It is presented to illustrate the various sections within an ELiXer report. The second, Figure 11, is more
+representative of the typical HETDEX LAE and is shown here to that end.
+1
+2
+3
+4
+5
+6
+7
+8
+9
+11
+10
+12
+13
+Figure 10. Example ELiXer detection report. This is a somewhat uncommon example selected to illustrate elements that are not always
+present for an individual detection, such as the classification label, warning flags, multiple catalog references, and photometric redshift PDFs.
+Descriptions of the bulleted features are provided below.
+1. Summary - From left to right: (1) computed Equivalent Width of the emission line in the rest-frame of Ly𝛼, the combined
+continuum estimate (§3.2.4), (2) P(LAE)/P(OII) (§3.4) and 68% confidence interval using the combined continuum estimate,
+(3) P(Ly𝛼) score (§3.5), (4) Quality score for the Best-𝑧 redshift (§3.6), (5) Best-𝑧 redshift, (6) Classification labels (§3.3.4)
+if any, (7) Error/Warning Flags5 if any; in this example, there is a warning flag indicating a small disagreement in the
+𝑔-magnitudes calculated from the spectrum.
+5 Flags are not explicitly described in this work but are part of data release
+documentation
+
+EW: 196.8±13.5AP(LAE)/P(0II): 1000 1800
+P(Lyα): 0.999
+Q(z): 0.61
+z: 2.2688 Lyα Flags:0x00002000
+2022-03-16 19:23:49
+Version 1.16.5
+ID:3007744560(3007744560.pdf)
+2DSpe
+Pixel Flat
+Smoothee
+With Sky
+0bs: 20200513v014_3007744560
+x, y: 252, 39
+Primary Spec_Slot_IFU_AMP: 418_057_064_RU
+12 Je-17x2A
+F=1.4"
+T=0.147 N=1.41 A=0.88
+10
+RA,Dec (214.776810,52.825974)
+入 = 3973.74A
+FWHM = 9.8(±0.5)A
+LineFlux = 5.90(±0.23)e-16
+Cont(n) = 1.50(±i.00)e-18
+8
+Cont(w) = 5.00(±1.50)e-19 (gmag 25.07 24:3)
+R
+EWr = 120.00(±80.00) (w: 360.00(±110.00))A
+S/N = 25.0(±0.6)
+x2 = 1.1(±0.2)
+P(LAE)/P(01I): 1000 188 (w: 1000 1888)
+18
+28
+3920
+3940
+0965
+086
+4000
+4020
+LyA z = 2.2688
+3 0II Z = 0.0660
+CII {
+0%
+0
+SilV
+0
+e-17x2A
+10
+3500
+3600
+3700
+3800
+3900
+4000
+4100
+4200
+4300
+4400
+4500
+4600
+4700
+4800
+4900
+5000
+5100
+5200
+5300
+5400
+5500
+CANDELS/EGS : Possible Matches = 4 (within +/- 3
+3")
+P(LAE)/P(0II): 1000 188 (f606w)
+Fiber Positions
+Lineflux Map
+CFHTLS/Meg(27.3) u
+CFHTLS/Meg(27.3) g
+ACS WFC(30.0) f606w
+ACS WFC(30.0) f814w
+CFHTLS/Meg(27.3) z
+4
+7.
+2
+2
+2
+2
+2
+0
+0-
+0
+2
+-2
+-4
+2
+4
+0
+4
+0
+-4
+0
+arcsecs
+s/b: 18.20 +/- 0.148
+m:24.2 re:1.4"s:0.2'
+m:24.7 re:1.3" s:0.3"
+m:24.8 re:0.4" s:0.3"
+m:24.1 re:1.0" s:0.3
+EWr:205.PLAE:1000
+EWr: 297. PLAE: 1000
+Phot z PDF
+0.282259"
+Separation
+0.312691"
+2.63637"
+0.997
+0.997
+Match score
+0.847
+214.776848, 52.825899
+214.776846, 52.825890
+214.776664, 52.825246
+RA, Dec
+2.27416
+N/A
+N/A
+Spec z
+2.101
+N/A
+2.886
+Photo z
+280.00(±15.00)A
+350.00(±15.00)A
+4100.00(±720.00)A
+Est LyA rest-Ew
+24.42(-0.05,0.05)f606w
+24.49(24.47,24.52) r
+27.32(-0.21,0.26)f606w
+mag
+1000 1888
+1000 18
+1000 1888
+P(LAE)/P(OII)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+--. Oll z(virus) = 0.0659746
+-: LyA z (VIRUS) = 2.2687736
+Davis, et al.
+2. Timestamp + Version - Displays the date and time of the creation of this report and the ELiXer version number.
+3. Detection Details - A block of information about the HETDEX observation and the emission line detection. From top to
+bottom: (1) Detection ID number and file name, (2) Observation ID, (3) IFU+Amp address of the fiber nearest the detection
+center, (4) ’F’ = seeing FWHM in arcsecs, ’T’ = effective throughput at 4540 Å, ’N’ = dither to dither normalization, ’A’
+= aperture correction (divisor), (5) J2000 equatorial coordinates of the PSF weighted detection center in decimal degrees,
+(6) emission line wavelength center and FWHM, (7) integrated emission line flux, (8) continuum estimate (§3.2) from the
+spectrum within ±40 Å of the line center, (9) continuum estimate and 𝑔-magnitude from the full width of the spectrum,
+(10) equivalent width in Ly𝛼 rest-frame with the continuum estimates from (8) and (9) respectively, (10) signal-to-noise
+ratio and 𝜒2 of the emission line fit, (11) P(LAE)/P(OII) using the continuum estimates from (8) and (9) respectively, (12)
+redshifts assuming Ly𝛼 and [O ii], (13) multi-line emission line identification (§3.3), if one is selected, with its quality
+score, name, rest-wavelength, redshift, and equivalent width in its own restframe using the continuum estimate in (9).
+4. 2D Fiber Cutouts - 5 × 3 grid of cutouts within ±40 Å of the detection line center in the spectral direction and ±1 fiber
+in the CCD direction6. The left most column is the pre-smoothing cutout with all rectifications and sky subtraction. The
+center column is the pixel flat, with any significant deviations marked in red (none in this example). The right most column
+is the same as the left most column but smoothed with a 2 × 2 Gaussian filter. The top row (highlighed in black) is the
+weighted sum of all contributing fibers. The rows below (blue, green, orange, red) are the highest four fibers as weighted
+by PSF modeled flux. The values (in very small print) to the left of the grid represent (from top to bottom): the normalized
+fiber weight in the PSF, the 𝜒2 of the fit to the fiber profile, and the fiber number on the CCD. The values (in very small
+print) to the right of the grid represent (from top to bottom): the fiber center distance to the detection center (in arcsecs),
+the CCD pixel coordinate of the fiber center, the exposure date, the observation number and exposure number for that date,
+and the IFU spectrograph ID, amplifier ID, and fiber number on that amplifier.
+5. Key CCD Region - ±10 fibers in the CCD direction and ±40 Å in the spectral direction around the detection center for the
+fiber nearest the detection, shown before and after sky subtraction.
+6. 1D Line Fit - the 1D emission line fit to the data. This matches the gold highlighted section in the full 1D spectrum. Values
+are integrated fluxes in 2 Å wide bins.
+7. 1D Spectrum - the full 1D spectrum as integrated fluxes in 2 Å wide bins. The gray background gives the estimated. The
+two vertical gray-hashed bars point out the two strongest sky-lines. The gold highlighted region is the anchor emission line.
+Any other colored regions, if present, highlight other spectral lines that support the selected multi-line redshift solution. The
+other red labels ("NV","SiII","SiIV","CIV","HeII") mark the positions of other possible lines in the spectrum, assuming the
+anchor line is Ly𝛼; in this spectrum, none of these confirming lines are detectable. The colored labels above the spectrum
+represent the positions of other common lines if the anchor emission line were one of the features listed below the spectrum
+with the matching color.
+8. Main Catalog Summary - displays the name of the catalog with the deepest imaging used in the report, along with the
+number of potential catalog counterparts (if any) and the P(LAE)/P(OII) found from the continuum estimate of the listed
+filter.
+9. Fiber Positions - the footprint of all fibers contributing to the detection plotted over a stacked image from the catalog with
+the deepest imaging. The four colored fibers match those in the 5 × 3 grid in (4). Fibers with a dashed outer ring are at the
+edge of the detector. The PSF weighted center of the detection is marked with a red cross.
+10. Lineflux Map - wavelength collapsed flux intensity map summing over ±3𝜎 from the emission line center. The values under
+the image are an estimate of significance based on the flux inside a 1′′ radius aperture and the standard deviation of flux
+inside a 5′′ to 7′′annulus, corrected for area. The lower section of the Lineflux Map in this example is blank as that region
+happens to fall off the edge of the CCD.
+11. Imaging Stamps - postage stamp cutouts of the deepest imaging available to ELiXer, shown in increasing order from blue
+(left) to red (right). Only the bluest five filters are shown, though more may be available. Overplotted are colored 1′′ per
+6 Fibers adjacent on the CCD are not necessarily adjacent on sky
+
+ELiXer
+37
+side squares corresponding to the positions of possible catalog counterparts. The top three (see (12) are shown in blue, red,
+and green, with all others displayed in white. In this example, the blue and red squares overlap, so only the red is obviously
+visible, but they mark the same object. The overplotted ellipses are SEP identified sources (§2.2). A gold ellipse marks the
+object selected by ELiXer as the most likely counterpart, while all other objects are marked in white. If the bounding ellipse
+is dashed, then it has been expanded to be a 1′′ radius circle for visibility. The text above each cutout indicates the catalog
+name, and the approximate imaging depth and the filter. The values under the cutouts correspond to the gold aperture and
+are: ’m’ = aperture magnitude, ’re’ = the effective radius of the ellipse in arcsecs, ’s’ = separation between the center of the
+aperture and the HETDEX PSF weighted center in arcsecs,"EWr" - the equivalent width in the Ly𝛼 rest-frame using the
+aperture magnitude as the continuum estimate, "PLAE" - P(LAE)/P(OII) using the aperture magnitude as the continuum
+estimate. All values are computed for 𝑔 and 𝑟 (or equivalent) filters, but not always for other bands.
+12. Catalog Counterparts - basic information on up to the top three most likely catalog counterparts, based on magnitude
+and distance, which correspond to colored squares on the Imaging Stamps. In this example, the blue, red, and green
+objects are actually the same source, but reported from different catalogs, and their corresponding squares in the Imaging
+Stamps overlap. The "Separation" is the distance in arcsec between the HETDEX detection position and the catalog
+reported position. This offset can sometimes be sizeable, especially for extended objects where the catalog reports a surface
+brightness center and the HETDEX detection is more toward the object’s edge. The reported P(LAE)/P(OII) value uses the
+catalog’s reported bandpass magnitude as the continuum estimate, not the aperture magnitude from the Imaging Stamps.
+13. Catalog z PDFs - if available, shows the photometric redshift PDFs, color coded to match the top three catalog counterparts.
+In this example, there is no PDF for the red counterpart (from the CFHTLS catalog), so only blue and green PDFs are
+shown. Circles, again with a matching color, mark the reported spectroscopic redshift, if available. The green dashed line
+represents the redshift if the emission line is [O II], while the red dashed line shows the same for Ly𝛼. Since the anchor
+line in this example is Ly𝛼, there is a precise match with the spec-𝑧 and a close match with the phot-𝑧 for the object marked
+in blue.
+
+38
+Davis, et al.
+Figure 11. The ELiXer report of a typical HETDEX LAE. Note that this region of sky has fewer and shallower imaging data, and more limited
+catalog data compared to Figure 10. It is included here as a counter to the more illustrative, but less common example of Figure 10.
+
+EW: 109.8±24.2A P(LAE)/P(0II): 1000 1000
+P(Lyα): 0.999
+Q(z): 0.36
+ z: 2.3484 Lyα
+2022-03-15 17:20:04
+Version 1.16.5
+ID: 3013462701 (3013462701.pdf)
+Pixel Flat
+Smooth
+With Sky
+0bs: 20210729v012_3013462701
+x, y: 290, 948
+Primary Spec_Slot_IFU_AMP: 410_024_039_LL
+e-17x2A
+F=2.2"
+T=0.129 N=1.45 A=0.89
+RA,Dec (220.022415,52.361256)
+^ = 4070.56A FWHM = 12.4(±2.3)A
+LineFlux = 2.20(±0.32)e-16
+Cont(n) =
+-3.00(±1.10)e-18
+Cont(w) = 5.70(±1.90)e-19 (gmag 24.82 25:24 *)
+EWr = 110.00(±42.00) (w: 110.00(±42.00))A
+S/N = 5.3(±0.5)
+x2 = 1.0(±0.2)
+P(LAE)/P(0II): 1000 188
+4020
+4040
+4060
+4080
+4100
+4120
+LyA z = 2.3484
+4 0II Z = 0.0919
+CII
+人
+ silv
+Silv
+5.0
+2.5
+3500
+3600
+3700
+3800
+3900
+4000
+4100
+4300
+4400
+4700
+4800
+4900
+5000
+5100
+5200
+5300
+5400
+5500
+HSC-DEX : Possible Matches = 1 (within +/- 3")
+P(LAE)/P(0II): 1000 1888 (r)
+Fiber Positions
+Lineflux Map
+KPNO(24.7) g
+HSC(26.2) r
+-4
+-4
+0
+0
+4
+-2
+arcsecs
+s/b: 3.11 +/- 0.146
+m:24.7 rc:1.6"s:0.2"
+m:25.0 re:0.9" $:0.6"
+EWr: 76. PLAE: 1000
+EWr:156.PLAE:1000
+Separation
+0.631354"
+0.994
+Match score
+220.022128, 52.361252
+RA, Dec
+N/A
+Phot z plot not available.
+Spec z
+N/A
+Photo z
+120.00(±30.00)A
+Est LyA rest-Ew
+24.39(24.14,24.72)R
+mag
+P(LAE)/P(OII)
+1000 1888
\ No newline at end of file
diff --git a/V9AzT4oBgHgl3EQf1f5A/content/tmp_files/load_file.txt b/V9AzT4oBgHgl3EQf1f5A/content/tmp_files/load_file.txt
new file mode 100644
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+page_content=' The Catholic University of America,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
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+page_content=' USA 19Institute for Cosmic Ray Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
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+page_content=' Graduate School of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' the University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 7-3-1 Hongo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Bunkyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Tokyo 113-0033,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Japan ABSTRACT The Hobby-Eberly Telescope Dark Energy Experiment (HETDEX) is an untargeted spectroscopic survey that aims to measure the expansion rate of the Universe at 𝑧 ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4 to 1% precision for both 𝐻(𝑧) and 𝐷 𝐴(𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' HETDEX is in the process of mapping in excess of one million Lyman-𝛼 emitting (LAE) galaxies and a similar number of lower-z galaxies as a tracer of the large-scale structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The success of the measurement is predicated on the post- observation separation of galaxies with Ly𝛼 emission from the lower-𝑧 interloping galaxies, primarily [O II], with low contamination and high recovery rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The Emission Line eXplorer (ELiXer) is the principal classification tool for HETDEX, providing a tunable balance between contamination and completeness as dictated by science needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' By combining multiple selection criteria, ELiXer improves upon the 20 Å rest-frame equivalent width cut commonly used to distinguish LAEs from lower-𝑧 [O II] emitting galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Despite a spectral resolving power, R ∼ 800, that cannot resolve the [O ii] doublet, we demonstrate the ability to distinguish LAEs from foreground galaxies with 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' We estimate a contamination rate of Ly𝛼 by [O II] of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2% and a Ly𝛼 recovery rate of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1% using the default ELiXer configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' These rates meet the HETDEX science requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Keywords: Dark energy(351) – Emission line galaxies(459) – Lyman-alpha galaxies(978) – Redshift sur- veys(1378) ∗ Based on observations obtained with the Hobby-Eberly Telescope, which is a joint project of the University of Texas at Austin, the Pennsylvania State University, Ludwig-Maximilians-Universität München, and Georg- August-Universität Göttingen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The HET is named in honor of its principal benefactors, William P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Hobby and Robert E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Eberly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' INTRODUCTION It is generally acknowledged that the universe is expanding and that the expansion is accelerating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Though surprising at the time, the accelerated expansion has come to be the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='01799v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='GA] 4 Jan 2023 2 Davis, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' consensus understanding since the early work of Perlmutter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (1999) and Riess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Since then, many ob- servations have confirmed and refined the measures of this expansion with such increased precision that a possible ten- sion may have emerged in the results from the various broad measurement camps (Di Valentino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Aloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021, among others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Regardless, whether this tension is a consequence of real physics, as yet unidentified systematics, or some combination, we are essentially limited to only two anchor points, one from the recent past (∼ 72 km s−1 Mpc−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Riess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Dhawan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Riess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Mort- sell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021, and others) and one from the Epoch of Re- combination (∼ 67 km s−1 Mpc−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Alam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Aiola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2020, and others), from which to constrain descriptions of dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Further understanding requires additional data points from different epochs in the expansion history of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Multiple efforts are in progress to provide those data, including the fol- lowing, but far from exhaustive, list: the Dark Energy Survey (DES) (The Dark Energy Survey Collaboration 2005a), the Baryon Oscillation Spectroscopic Survey (BOSS) (Dawson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2012), the extended Baryon Oscillation Spectroscopic Survey (eBOSS) (Alam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021), the Legacy Survey of Space and Time (LSST) (LSST Science Collaboration 2009), Euclid (Laureijs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2011), the DESI Survey (DESI Col- laboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2019a), and, of course, the Hobby-Eberly Telescope Dark Energy Experiment (HET- DEX) (Ramsey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' HETDEX is a multi-year untargeted spectroscopic survey designed to make new measurements of the Hubble Param- eter, 𝐻(𝑧), and the Angular Diameter Distance, 𝐷 𝐴(𝑧), at z∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4 to better than 1% accuracy in an effort to better charac- terize dark energy and look for possible evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' HETDEX observations fall into two large, high galactic latitude fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The ∼ 390 deg2 "Spring" field is centered near (RA,Dec) 13h00m +53d00m and the ∼150 deg2 "Fall" field is centered near 1h30m +0d00m (Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Functionally, HETDEX seeks to map the 3D positions of some 106 galaxies between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='88 < 𝑧 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='52 and use their large scale clustering to derive 𝐻(𝑧) and 𝐷 𝐴(𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' More specifically, the galaxies HETDEX is using for large-scale structure are identified by their bright, conveniently red-shifted into the optical, Lyman- 𝛼 emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' These Lyman-𝛼 Emitters (LAEs) are gen- erally small, blue, rapidly star-forming galaxies that, while uncommon in the local Universe, are present in large num- bers in the HETDEX redshift search window (Partridge & Peebles 1967;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Gawiser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Nilsson 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Finkelstein 2010, and many others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The HETDEX Visible Integral-Field Replicable Unit Spec- trographs (VIRUS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021) cover the wavelength range 3500-5500 Å with R∼750–900, and are optimized to detect Ly𝛼 flux down to ∼ 4 × 10−17 erg s−1 cm−2 (increasing to closer to 2 × 10−16 erg s−1 cm−2 at the extreme blue end of the range).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This allows the detection of Ly𝛼 luminosities down to about 1042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3 erg s−1 for 𝑧 ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Since it is of utmost importance to know the redshift of the observed galaxies, the emission must be correctly identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' However, the relatively narrow wavelength range often limits our ability to capture multiple emission lines and the low spectral resolving power prohibits most doublet splitting, making classifications dif- ficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Around 95% of HETDEX emission line detections1 are spectra containing only one, apparently single peaked (given the HETDEX spectral resolving power) emission line, and Ly𝛼 is not the only emission line to fall into this observed wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Neutral hydrogen (and dust) in each source galaxy’s Interstellar Medium (ISM) and in the Intergalactic Medium (IGM) along our line of sight effectively eliminate emission lines blueward of Ly𝛼 at higher redshifts (Haardt & Madau 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Meiksin 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Cowie & Hu 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Overzier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Vanzella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2018), leaving low-𝑧 galaxies as the primary contaminate to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In the relatively nearby universe, intrinsically small, line- emitting faint galaxies can be misidentified as their higher redshift cousins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In particular, at the low HETDEX spectral resolving power and with no strong lines in the wavelengths around it, the [O II] 3727Å emission line can be confused with Ly𝛼 1216Å which similarly appears unique in its spec- tral neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In a common case, HETDEX observa- tions detect only a single, fairly narrow, emission line and little or no continuum at the detection limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Most likely the line is either Ly𝛼 and originates from a high-𝑧 galaxy, or [O II] from a low-redshift interloper, and unfortunately, these two primary cases occur in roughly equivalent num- bers (Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Since the HETDEX 𝐻(𝑧) and 𝐷 𝐴(𝑧) measurements are sensitive to in- terloper clustering (Leung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Farrow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021), contamination from [O II] in the LAE sample needs to be ≲ 2% (Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Historically, a 20 Å equivalent width cut (using the rest-frame of Ly𝛼) has been used to segregate [O II] from Ly𝛼 (Gronwall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2011), and indeed, this criterion is quite effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' However, used by itself, the discriminant can still lead to >4% contamination and degrade the recovery of lower equivalent width Ly𝛼 lines (Acquaviva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Leung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (2017) improves on the 20 Å cut by taking a Bayesian approach and including information on the luminosity func- 1 HDR3 is limited to emission line detections with SNR ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='8, of which 95% have only a single detected emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The fraction of detections with only a single line is partly a function of the SNR cut and other selection criteria used to define a sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As in Mentuch Cooper (ApJ accepted), SNR ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 is commonly used as it is effectively free from noise detections (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For SNR ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5, 70% of HETDEX spectra consist of only a single emission line and the entire sample is reduce by 60%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' ELiXer 3 tions and equivalent width distributions of Ly𝛼 and [O II] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' From their modeled data, they report an expected contami- nation by [O II] of between ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0% at a cost of ∼ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0% to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4% lost LAEs, depending on the methods used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This is a significant enhancement over the simpler 20 Å cut and, in this work, we are able to extend and improve on Leung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (2017) by (1) incorporating additional selection criteria, (2) considering other emission lines as contaminants, and (3) comparing directly against observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The HETDEX Emission Line eXplorer (ELiXer) software incorporates and extends these classification works, integrates supplemental data and additional classification criteria, and expands the analysis to consider more than two dozen other emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Its primary objective is to classify every HETDEX emission line detection by assigning the correct redshift to the observed emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In addition to its primary function as an emission line classifier, ELiXer also provides diagnostic and data integrity checking to supple- ment that of the HETDEX pipeline (Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021), which is run prior to the ELiXer invocation and provides the detection coordinates, observation conditions, processed (calibrated, PSF weighted) spectra, emission line parameter measurements (flux, line width), and CCD information as ELiXer inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' These features are useful for identifying and debugging some issues (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' errant sky subtraction, stuck/hot pixels, amplifier interference, etc) as well as in the manual inspection of individual detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' While ELiXer does classify all HETDEX detections re- gardless of magnitude, additional classification support is provided for continuum-bright sources via another software tool utilized by HETDEX called Diagnose, developed for the Hobby Eberly Telescope VIRUS Parallel Survey (HET- VIPS, Zeimann & et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (in prep)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For a further description of source classification and redshift assignment of HETDEX sources please see Mentuch Cooper (ApJ accepted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Here, however, we focus only the bulk of the HETDEX detections, where ELiXer is the primary (or only) classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For this work, we reference ELiXer version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='16 used in the gener- ation of the most recent HETDEX detections catalog, HET- DEX Data Release 3 (HDR3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This catalog contains more than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 million entries and was released internally in April 2022 with a public version to be released in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' We report a projected HETDEX LAE contamination rate from [O II] of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2% (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1%) and an additional 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='8% (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1%) from all other sources, along with an LAE recovery rate of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1% (±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3%) for the default classification configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' ELiXer provides a tunable Ly𝛼 classifier, allowing the balancing of contamination vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' completeness as needed for specific science goals (see §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' ELiXer is a work in progress and contin- ues to evolve and improve as more data are collected, both from HETDEX and from other surveys, and as classification methods are added and refined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The remainder of this paper is organized as follows: Section 2 provides an overview of the various photometric catalogs currently included in ELiXer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Section 3 describes the classi- fication methodologies and supporting functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Section 4 covers the selection of a Spectrocopic-z Assessment Sample (SzAS) providing spectroscopic redshifts from various imag- ing catalogs and the results of testing against that sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Section 5 presents a discussion of the results and the science implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Section 6 summarizes the work and future en- hancements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Example ELiXer detection reports are shown in Appendix-A with descriptions provided for the major fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Throughout the paper, the Planck 2018 cosmology (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2020b) with ΩΛ= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='69, Ωm= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='31 and H0 = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='7 km s−1 Mpc−1 is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' All magnitudes are in the AB system (Oke & Gunn 1983) and coordinates are J2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' IMAGING CATALOGS HETDEX is an untargeted spectroscopic survey, and the spectra alone provide most of the critical information for object classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Coupled with the on-sky positions of the associated fibers, these data form the basis for the HET- DEX cosmology measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For the brighter detections, a source’s redshift and, to a lesser degree, its physical ex- tent and morphology can be determined securely from the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' However, for the fainter emission line detections, additional information from archival photometric imaging, including an object’s magnitude, color, angular/physical size, morphology, and even on-sky neighbors, can prove quite useful in ascertaining its identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Even superimposing the HETDEX fiber positions on imaging data can provide diag- nostic checks on the astrometry and the reduction pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Given these substantial benefits, ELiXer attempts to match all HETDEX observations with multi-band archival photometry at the highest angular resolution and imaging depth available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Individual Catalog Summaries At the time of writing, ELiXer references 11 separate imag- ing catalogs, most with their own associated object cata- log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' These catalogs are of varying depth, resolution, band- coverage, and footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Additional catalogs can be added at any time and several new or expanded source lists are an- ticipated before the next HETDEX data release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' With the exceptions of an 𝑟-band survey from the HyperSuprimeCam group (HSC-DEX) and a 𝑔-band survey from Kitt Peak Na- tional Observatory (KPNO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='HETDEX-IM) that were specially designed and executed for HETDEX, all imaging and object catalogs are archival and publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' These catalogs are summarized in Table 1 and in the list below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 4 Davis, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Summary of the imaging surveys incorporated into ELiXer Name HETDEX Field Overlap1 Filters and Depth2 PSF FWHM3 Object Catalog4 Canada-France-Hawaii Telescope Legacy Survey (CFHTLS) Spring 4% Deep: 𝑢(26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3), 𝑔(26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0), 𝑟(25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='6), 𝑖(25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4), 𝑧(25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0) Wide: 𝑢(25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2), 𝑔(25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5), 𝑟(25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0), 𝑖(24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='8), 𝑧(23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='9) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='6-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0′′ phot-𝑧 𝐻𝑆𝑇 Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS) in the Extended Groth Strip (EGS) Spring <1% ACS/WFC: F606W, F814W WFC3: F105W, F125W, F140W, F160W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='08′′ spec-𝑧, phot-𝑧 𝐻𝑆𝑇 Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS) in the Great Observatories Origins Deep Survey, North (GOODS-N) Spring <1% ACS/WFC: F435W, F606W, F775W, F814W WFC3: F105W, F125W, F160W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='08′′ spec-𝑧, phot-𝑧 Hyper Suprime-Cam HETDEX Survey (HSC-DEX) Spring 44% 𝑟(25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='6-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0′′ mag only Kitt Peak National Observatory HETDEX Imaging Survey (KPNO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' HETDEX-IM) Spring 20% 𝑔(24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5′′ mag only Cosmic Evolution Survey (COSMOS) with Dark Energy Camera (DECam) Fall 2% 𝑔(25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5), 𝑟(25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='7-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0′′ (1) phot-𝑧 (Laigle+2015) (2) mag only Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP) Fall 29% Deep 𝑔(27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5), 𝑟(27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1), 𝑖(26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='8), 𝑧(26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3), 𝑦(25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3) Wide 𝑔(26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5), 𝑟(26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1), 𝑖(25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='9), 𝑧(25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1) ,𝑦(24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='6-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0′′ mag only Spitzer/HETDEX Exploratory Large-Area (SHELA) with Dark Energy Camera (DECam) Fall 25% 𝑢(25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4), 𝑔(25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1), 𝑟(24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='7), 𝑖(24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0), 𝑧(23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='7) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='7-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0′′ mag only Dark Energy Camera Legacy Survey (DECaLS) Spring & Fall 17% 𝑔(24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0), 𝑟(23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4), 𝑧(22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2′′ No Panoramic Survey Telescope and Rapid Response System (Pan-STARRS) Spring & Fall <1% 𝑔(23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3), 𝑟(23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2), 𝑖(23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1), 𝑧(22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3), 𝑦(21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3′′ No Sloan Digital Sky Survey (SDSS) DR16 Spring & Fall <1% 𝑢(22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0), 𝑔(23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1), 𝑟(22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='7), 𝑖(22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2), 𝑧(20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='7) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3′′ spec-𝑧, phot-𝑧 1Fraction of HETDEX Data Release 3 within each catalog footprint, except for DECaLS, Pan-STARRS, and SDSS which report only the fraction which does not also overlap with a previously listed catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Since multiple catalogs overlap, the column sums to > 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2Approximate average AB depth over the whole catalog as reported, typically for point sources and 2′′apertures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For some 𝑔 and 𝑟 filters and some image tiles, ELiXer uses its own estimated depths at 1′′and 2′′apertures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Not all surveys use the same SDSS ugriz filters, though for this purpose they are approximately similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Only filters used by ELiXer are listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3Typically in 𝑟-band 4If not "No", also has an object catalog used by ELiXer with at least 𝑔 or 𝑟 magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Spec-𝑧 and/or phot-𝑧 redshifts are available where noted, but not necessarily for all object entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' ELiXer 5 Canada-France-Hawaii Telescope Legacy Survey (CFHTLS): A multi-band (𝑢𝑔𝑟𝑖𝑧) imaging survey and joint venture of the National Research Council of Canada, the Institut National des Science de l’Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii, uti- lizing the MegaPrime/MegaCam on the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='6m Canada- France-Hawaii Telescope (CFHT) on Mauna Kea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' ELiXer uses the deep and wide fields, D3/W3 cen- tered near RA 210◦, Dec +52◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (Brimioulle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Cuillandre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2012) HST Cosmic Assembly Near-infrared Deep Extragalac- tic Legacy Survey (CANDELS) in the Extended Groth Strip (EGS): CANDELS is a deep 𝐻𝑆𝑇 survey (900+ orbits) with multiple filters in the optical (using the Ad- vanced Camera for Surveys, ACS) and near-IR (using the Wide Field Camera 3, WFC3) studying on galaxy evolution with an emphasis on Cosmic Dawn and Cos- mic High Noon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The EGS is one of the five fields of CANDELS and is centered near RA 215◦, Dec +53◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (Grogin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Koekemoer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Stefanon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The photometric redshifts used in ELiXer are provided by Andrews, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=', et al, ApJ submitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' HST Cosmic Assembly Near-infrared Deep Extragalac- tic Legacy Survey (CANDELS) in the Great Observa- tories Origins Deep Survey, North (GOODS-N): An- other of the 5 CANDELS fields (see previous bullet), GOODS-N is centered near RA 189◦, Dec +62◦ (Dick- inson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Grogin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Koekemoer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Barro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2019) Again, the photometric red- shifts used in ELiXer are provided by Andrews, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=', et al, ApJ submitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Hyper Suprime-Cam HETDEX Survey (HSC-DEX): This survey consists of three nights of HSC 𝑟-band observations with the Subaru/HSC in 2015-2018 (PI: Andreas Schulze) and 2019-2020 (PI: Shiro Mukae) and covers the ∼ 250 deg2 area of the HETDEX Spring field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Data reduction and source detections were per- formed with version 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='7 of the HSC pipeline, hscPipe (Bosch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2018), and produced 𝑟-band images with a 10𝜎 limit of 𝑟 = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1 mag in a 2′′ diameter circular aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' These HSC 𝑟-band images are complemen- tary to the existing imaging data of the Kitt Peak 4-m Mosaic camera and the CFHT Wide-Field Legacy sur- vey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Kitt Peak National Observatory HETDEX Imaging Sur- vey (KPNO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' HETDEX-IM): A 𝑔-band survey with the Mosaic camera on the Mayall 4-m telescope at Kitt Peak National Observatory in 2011-2014 (PI: Robin Ciardullo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Cosmic Evolution Survey (COSMOS) with Dark Energy Camera (DECam): The 3 deg2 ugriz-band COSMOS DECam catalog was generated with the same procedure used for the larger field of view SHELA DECam survey listed below (Wold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This also overlaps with Laigle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Hyper Suprime-Cam Subaru Strategic Program (HSC- SSP): Multi-depth, multi-band, wide-field imaging sur- vey using the Hyper Suprime-Cam on the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2m Subaru at the Mauna Kea Observatories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For HETDEX Data Release 3, ELiXer uses HSC-SSP Public Data Release 3 from August 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (Aihara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021) Spitzer/HETDEX Exploratory Large-Area (SHELA) with Dark Energy Camera (DECam): This survey cov- ers 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 deg2 of the HETDEX Fall field within the Sloan Digital Sky Survey (SDSS) “Stripe 82” region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The ugriz-band DECam catalog is riz-band-selected and reaches a 5𝜎 depth of ∼ 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 AB mag for point sources (Wold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Dark Energy Camera Legacy Survey (DECaLS): A multiband (𝑔𝑟𝑧) photometric survey, part of the Dark Energy Survey (The Dark Energy Survey Collabora- tion 2005b), based at the Cerro Tololo Inter-American Observatory using the Dark Energy Camera (DECam) on the 4m Blanco telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' ELiXer uses Data Release 9 which also includes observations from the Beijing- Arizona Sky Survey (BASS) and the Mayall z-band Legacy Survey (MzLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2019b) Panoramic Survey Telescope and Rapid Response Sys- tem (Pan-STARRS): Specifically, Pan-STARRS1, is a set of wide-field synoptic imaging surveys using the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='8m PS1 optical telescope at the Haleakala Observa- tories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' PS1 collected data from 2010 through 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (Chambers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2019) Sloan Digital Sky Survey (SDSS): Multiband (𝑢𝑔𝑟𝑖𝑧) wide-field survey in operation since 2000 using a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5m optical telescope at the Apache Point Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' ELiXer uses Data Release 16 from SDSS Phase-IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (Ahumada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2019) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' ELiXer Aperture Photometry ELiXer directly uses the photometric imaging to gather aperture magnitudes for the HETDEX detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' While magnitudes are computed for each available filter, only 𝑔 and 𝑟 magnitudes are used in the classification process (§3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For each HETDEX detection, ELiXer identifies the catalogs with overlapping imaging and gathers postage-stamp (9′′×9′′ by default) imaging cutouts centered on the HETDEX detec- tion’s coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Three sets of aperture magnitudes are then 6 Davis, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' computed using the Python packages Astropy (Astropy Col- laboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2018a), Photutils (Bradley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2020), and Source Extraction and Photometry (SEP) (Barbary 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The identified aperture(s) are used later to provide continuum estimates (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2) and size information (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' First, ELiXer computes a magnitude for a dynamically sized circular aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' We center the circular aperture on the HET- DEX coordinates, compute the magnitude within the aperture, and allow the aperture to grow until the magnitude stabilizes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=', Howell 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The initial size is set by a combination of the median seeing and pixel scale of the catalog+filter and is typically ∼ 1′′ in diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The magnitude within the aper- ture is computed, with the background determined from an annulus 2× to 3× the defined maximum allowed object aper- ture (6′′ diameter by default, for an annulus of 12′′ to 18′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The aperture is then grown in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='′′1, with each measure- ment recorded, until the maximum diameter is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The smallest aperture size where the magnitude change to the next step up is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='01 is assigned, and the corresponding magnitude is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Next, ELiXer uses SEP (Barbary 2016), which is based on the original Source Extractor (Bertin & Arnouts 1996), iter- ating over each cutout and records the magnitude, barycentric position, major and minor axes, and orientation of each iden- tified object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' ELiXer also computes and records the angular separation from each barycenter to the HETDEX coordinates and the separation to the nearest point on the bounding ellipse if the HETDEX position lies outside that ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The object with the nearest barycenter to the HETDEX position whose bounding ellipse includes the HETDEX position is consid- ered the best aperture match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If no object’s ellipse includes the HETDEX position, then the object with the nearest ellipse point to the HETDEX position but no more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='′′5 away is selected as the best match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If no object meets these criteria then no SEP found object is selected and the best circular aperture (see previous paragraph) is used for the aperture photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Lastly, at each SEP identified barycenter, ELiXer computes and records the background subtracted magnitude in a fixed, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='′′0 diameter circular aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' These aperture magnitudes are intended for use in any fixed-aperture spectral energy distribution (SED)-fitting and color comparisons, but are not otherwise significantly used in the core ELiXer processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Catalog Counterpart Matching ELiXer also attempts to match each HETDEX detection to one or more objects in each imaging catalog with a particular focus on 𝑔 and 𝑟 magnitudes, which can provide additional measures for use in other ELiXer functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Object matching is based on a combination of barycenter position and agree- ment between the magnitudes reported by each catalog, the magnitudes computed within the ELiXer ellipses (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2), and the HETDEX spectrum estimated 𝑔-band magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The nearest catalog object to the HETDEX position that falls within the selected best aperture (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2), or the near- est catalog object within 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='′′0 of the HETDEX position if no object falls within the best aperture, is identified as the catalog match object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If the candidate object’s reported magnitude is not compatible with the magnitude estimated from the HETDEX spectrum, then the next nearest object is evaluated until a match is found or the distance criteria are no longer satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Compatibility with the HETDEX 𝑔 magnitude (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1) is defined as an absolute difference of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 magnitudes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' if the HETDEX 𝑔 magnitude is fainter than the HETDEX magnitude limit (about 25𝐴𝐵), then no faint-side restriction is imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' On the other end, if both the counterpart and the HETDEX magnitudes are brighter than 22𝐴𝐵, they are considered compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For the purposes of this comparison, 𝑔 and 𝑟 are considered equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' There is at most one catalog match object per catalog+filter combi- nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This object is later used for additional information, including spec-𝑧 and phot-𝑧 assignments if available, in the classification process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' CLASSIFICATION Classifications in ELiXer are broadly interpreted as the identification of the redshifts of observed astrophysical ob- jects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This properly requires the additional steps of correctly associating an observed spectrum with a single host object and furthermore identifying or bounding what constitutes that "single object".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' More fundamentally, given a spectrum and a specified emission line in that spectrum, what we hereafter call the "anchor line", ELiXer attempts to determine the iden- tity, and thus the redshift, of that anchor line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Classification proceeds from the assumption that the anchor line is real and not spurious noise, an instrument or software artifact, or a misinterpretation of spectral data, such as the misidentifica- tion of continuum between two closely-separated absorption troughs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' ELiXer initially assumes that the spectrum repre- sents a single object (single redshift), though later analysis explores the possibility that a HETDEX spectrum is a blend of spectra from discrete but immediately adjacent or over- lapping sources on sky (within a single, common detection aperture) at different redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The focus of ELiXer’s classification is placed on distin- guishing Ly𝛼 from [O ii], by far the most common Ly𝛼 con- taminant in HETDEX data, and the bulk of the tests and conditions target that objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Additional checks, described throughout this section, attempt to refine this bifurcated clas- sification and identify the spectral line(s) as any one of those listen in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As will be discussed in §5, these "Other" lines are encountered much less frequently than Ly𝛼 and ELiXer 7 [O ii] and, while they can be more challenging to identify, the HETDEX cosmology science is extremely robust against contamination from these misclassifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The classification of HETDEX detections is organized to answer three increasingly general questions, with each an- swer incorporating the results of the previous question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' First, closely following the work of Leung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (2017), we evaluate the relative likelihood that the target emission line is Ly𝛼 and rather than [O ii] (Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Farrow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This is largely based on measurements of the emission line luminosity and equivalent width eval- uated against luminosity and equivalent width distributions of Ly𝛼 and [O ii] emitting galaxies from other publications interpolated at the redshift corresponding to the emission line wavelength (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Second, we determine the confidence of the initial classification by performing checks against more than two dozen other emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Here a weighted voting scheme is used with many independent (or semi-independent) rules applied to measured and derived features of the spec- trum and detection object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Third, we assign, with some rough measure of quality, the redshift and thus the specific identity of the emission line(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This final step incorporates some additional rules and weights to combine all prior results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Broadly, ELiXer classifications build up evidence in a series of steps and then weighs the evidence to make a determina- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The high level steps are fairly serial and often largely independent, with their results only combined toward the end of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' These major steps are described in more de- tail, and in roughly the same order, in the subsections that follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Find, fit, and score all emission and absorption lines and set the anchor line 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Evaluate all combinations of found spectral lines for compatibility with redshifts, based on relative posi- tions, strengths, etc 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Collect additional (aperture) photometric imaging in- formation and any reported magnitude, spec-z, and phot-z measurements for the target object and its neigh- bors from non-HETDEX catalogs (Table 1) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Evaluate spectra shape, lines, and imaging for consis- tency with known astrophysical objects (star, White Dwarf, AGN, meteor, low-z galaxy) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Examine HETDEX data for corruption, pipeline arti- facts, and instrument issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Test the compatibility of the anchor line with Ly𝛼 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Perform evaluations on the anchor line, including spec- tral and photometric information, to specifically distin- guish Ly𝛼 from [O ii] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Perform separate evaluations on the anchor line, in- cluding spectral and photometric information, for con- sistency with lines other than Ly𝛼 and [O ii] 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Combine all evaluations to determine and rank likely redshifts and line classifications 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Re-evaluate redshift classification based on clustering with ELiXer results from the other neighboring HET- DEX detections The figures in this section illustrating some of the voting cri- teria and thresholds pull their data from the Spectroscopic-𝑧 Assessment Sample (SzAS) whose selection and composition is described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Line Finder Emission (and absorption) line detection is implemented as both a layered, untargeted search and a targeted line fit assuming an "anchor" line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' More details will follow in the next subsections, but briefly put, the untargeted search scans the full width of the spectrum from blue to red, marks the lo- cations of possible emission line-centers, and attempts to fit a single Gaussian (in agreement with the measured instrumen- tal resolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2021) to each position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The targeted search uses a single previously identified emission line (from the HETDEX input, user input, or the previous untargeted search) as an anchor and then assumes that anchor line is one of roughly two dozen potential emission lines (Table 2) and attempts to fit a Gaussian to the positions where other emission lines could be found, assuming that identify for the anchor line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The descriptions that follow are couched in terms of emission lines, as that is the primary use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' A limited use of absorption lines is implemented and is described in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Untargeted Search The untargeted search scans the entire 1D HETDEX spec- trum to identify the positions and model the parameters of potential emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' It is used to (1) identify the strongest line as the reference or anchor line when no initial emission line is explicitly provided, (2) mark strong lines for con- sistency checks with redshift solutions and to help identify blended spectra, and (3) mark line positions for followup vi- sual inspection, without respect to the selected solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Because Markov Chain Monte Carlo (MCMC) fits are rela- tively computationally expensive, and HETDEX spectra typ- ically have only one or very few emission lines, we do not want to perform such fits at each pixel along the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Instead, we first conduct a quick examination to narrow the potential locations of emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' We do this using two in- 8 Davis, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Emission Line Candidates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='Name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='rest-𝜆 [Å] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='Name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='rest-𝜆 [Å] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='O VI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1035 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='H𝜂 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3835 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='Ly𝛼 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1216 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='[Ne III] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3869 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='N V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1241 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='H𝜁 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3889 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='Si II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1260 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='(K) Ca II* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3934 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='Si IV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='[Ne III] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3967 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='C IV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1549 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='(H) Ca II* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3968 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='He II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1640 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='H𝜖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3970 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='C III] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1909 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='H𝛿 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='C II] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='H𝛾 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4340 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='Mg II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2799 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='H𝛽 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4861 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='[Ne V] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3346 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='[O III] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4959 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='[Ne V] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3426 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='Na I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4980 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='[O II] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='[O III] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5007 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='Na I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5153 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='∗Fit as an absorption line ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='Note— Possible identifications for spectral lines found ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='in the HETDEX spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' dependent algorithms and then combine the output positions into a single list for further examination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Two passes through the algorithms of this untargeted search are conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The first execution uses the native 2 Å binned HETDEX spectrum and focuses on identifying the common narrow spectral features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The second execution is performed after passing the original spectrum through a median filter (by default using a 5 pixel kernel), to smooth out some of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This helps identify candidate emission lines that are wider than the ∼ 400 km s−1 resolution of the VIRUS spectrographs and may have small noise peaks within their overall broad shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The first algorithm searches for the basic shape of an emis- sion feature, a general rise to a peak and then a decline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Due to the unavoidable noise in the data, the spectra are not smooth and the use of the first derivative to find zeros (and the second derivative to distinguish between an emission and absorption) results in more false detections than real spectral features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In- stead, we look for the general shape of the lines (a rise and fall in the flux of minimum height over a minimum width), based on the spectral resolution, flux limits, and noise of HETDEX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Sets of contiguous pixels that are sufficiently wide in the spec- tral direction and have the expected rise-peak-fall pattern are recorded as possible emission lines, and their line centers are recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The second algorithm counts contiguous pixels with flux values above some multiple of the corresponding noise (typi- cally SNR > 3, under the assumption that the flux uncertainty is distributed normally).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Where the contiguous count of pix- els above this noise is greater than some count (here, typically 3-5 pixels), the position of the highest flux value within that range is recorded as the possible emission line center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Es- sentially, this is just a SNR-cut over the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Unlike the first algorithm, the shape of the flux above the SNR-cut is irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The line centers from each algorithm are then passed to fitting (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3) and scoring routines (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' When model fits to the flux at those positions are successful and the computed line score is sufficiently large, the feature is recorded to a list of potential spectral lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' After both the standard and broad line searches are con- ducted, the list of potential emission and absorption lines are merged into a single list, and any neighboring lines with line centers within in 4 Å of each other are combined into single entries by keeping only the feature with the largest line score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As a brief note: though this is not the normal operation of ELiXer under HETDEX, if no anchor line is specified for the spectrum to be classified, the line (emission or absorption) with the largest score (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4) found in this untargeted search is assumed as the anchor line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If the untargeted search fails to identify any spectral lines, the wavelength bin with the largest flux value is assigned as the anchor line position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Targeted Search Unlike the untargeted search described above, the targeted search does not scan for potential emission or absorption lines, but instead attempts to fit for an emission or absorption fea- ture at a specified position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Essentially, ELiXer attempts to fit spectral lines from a predefined list of common lines (Table 2) at their expected observed wavelength positions given an assumed identity or redshift for the anchor line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The redshift assumptions come from alternately interpreting the anchor line as each of the common lines and from any matching spec- troscopic or photometric catalogs with a possible counterpart to the HETDEX detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' With each redshift assumption, all other lines in the subset that could occur within the HETDEX spectral window are fitted, allowing for some error in the sys- temic redshift (see Position Capture under §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This is often redundant with the untargeted search in that, for higher signal-to-noise ratio (SNR) lines, the lines found in the tar- geted search are also found in the untargeted search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' However lower SNR lines, [O iii] 𝜆4959 for example, can be missed in the initial sweep of the untargeted search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Fitting to a specific wavelength location helps avoids such misses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Each successfully fitted line for each assumed identity of the anchor line is scored (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4) and associated with the redshift solution (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3) for that identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' ELiXer 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Line Fitting ELiXer uses a simple,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 4-parameter (𝐴,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝜇,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝜎Line,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝑦) single Gaussian as the model to fit emission and absorption features: 𝐹(𝜆) = 𝐴 𝜎Line √ 2𝜋 exp � − (𝜆 − 𝜇)2 2𝜎2 Line � + 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (1) where 𝐹(𝜆) is the flux per 2 Å wavelength bin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝐴 is the area under the curve or equivalently the integrated line-flux,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝜇 is the line center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝜎Line is the measure of width,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝑦 is the vertical offset,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' or flat continuum level,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' and 𝜆 is the wavelength (at the midpoint of a 2 Å wide wavelength bin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The flat continuum is a reasonable simplification, as no assumption is made as to the object type or its redshift, most HETDEX detections have continua at or below the survey’s continuum flux limit, and those objects with continua bright enough to have a shape typically have multiple emission lines or are too bright to support a Ly𝛼 classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This con- tinuum estimate can be highly uncertain, especially for the noisier spectra, but as discussed later, multiple continuum es- timates are combined to improve the uncertainty and for the non-detections, the resulting equivalent width estimates are lower limits that favor a low contamination Ly𝛼 selection, at the cost of some completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Type I AGN may have broad lines that are not well fitted by a single Gaussian (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Such detections are marked by ELiXer with warnings, but are not confused with the fainter, compact LAEs the software is designed to identify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' We note, however, that it is possible that the simple emission line search can completely fail to find rare, extremely broad emission lines, as ∼ 3500 km s−1 is the maximum FWHM that ELiXer attempts to fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' More complex models, including the fitting of multiple emission and absorption lines within a single spectral fea- ture, have either proven to be unreliable, too computationally costly, and/or of limited utility for the main goal of simply identifying redshifts when the vast majority of line detections are well fit by the simple, single Gaussian model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Fitting for an emission line doublet would be useful in the effort to distinguish between Ly𝛼 and [O ii] however, given the low spectral resolving power of VIRUS, Δ𝜆/𝜆 ∼ 800 (Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021), the [O II] doublet (3726, 3729 Å) is unresolved as are most other doublets (Mg II (2796, 2803 Å) is sometimes marginally resolved).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The increased run time of fitting these extra parameters is not justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For smaller data sets, such as for the case of AGN exploration, more complex fitting is war- ranted (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2022), but left to those specialized projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For ELiXer’s classification needs, a description of the spectral feature that is limited to its position (wavelength), equivalent width (approximate integrated line flux and local continuum), and line width are sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Additional parameters, such as the model’s skewness and kurtosis, and conditions combining those and other parameters have been explored but have not been found to improve the identification of real spectral fea- tures or aid in the classification, and are thus excluded from further discussion in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' With the exception of the anchor line on which an MCMC fit is always performed, if a least square (LSQ) model fit passes its quality checks, no MCMC fit is conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This is due to the increased runtime cost of MCMC fitting weighed against the relatively modest needs for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In all MCMC cases however, an LSQ fit is performed first and its results are used as initial conditions (with appropriate randomization) for the MCMC algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' ELiXer uses the Python scipy package and its scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='curve_fit (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2020) as the LSQ fitter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' the MCMC fitter is from the Python emcee package (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Uncertainties in the LSQ fit are estimated using the square root of the diagonal of the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Uncertainties in the MCMC fit are estimated using the 68% confidence interval in the parameter distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' A series of loose checks evaluates the quality of each fit as minimally good, marginal, or poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Poor fits are rejected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' good fits are scored (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4) in preparation for building solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Marginal solutions from the LSQ fitter are passed to the MCMC algorithm for improved optimization and re- evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If the subsequent MCMC fit is good, the fit is scored and made eligible for inclusion in redshift solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If the MCMC fit is not sufficiently improved over the LSQ fit, it is rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The quality checks include following conditions: Peak Capture: As a basic check, should the peak of the model fail to reproduce the most extreme measured data value near the line center within 50%, the fit is rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If the model is within 25% and 50% of the most extreme value, it is flagged for an MCMC fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Should that MCMC fit fail to be within 25%, the fit is rejected and no line is assumed to be at that position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Position Capture: If the fitted line center is greater than a configured maximum distance (in Å) from the local data extremum, the fit is rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The maxi- mum distance allowed can depend on the assumed line identification and its assumed position, with greater separations allowed for Ly𝛼 which can be significantly offset from the systemic redshift (Shapley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' McLinden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Verhamme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Gurung- Ló pez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021, among others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' During the untar- geted search, no variations are allowed and a default of 8 Å (∼ 500 km s−1 in the HETDEX spectral range) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Width Capture: If the fitted line width (here parame- terized as 𝜎) is less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 Å, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=', significantly below 10 Davis, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' the HETDEX spectral resolution of ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 Å (Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021), or if the line width is greater than the config- ured maximum value of 17 Å (∼ 2700 km s−1 FWHM) or 25 Å (∼ 3500 km s−1 FWHM) for special, broad fit attempts, the fit is rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Area Error: If the error on the line area (as estimated from the square root of the diagonal of the LSQ fit’s covariance matrix or the 68% confidence interval on the MCMC fit) is larger than the absolute value of the area (allowing for absorption or emission), the fit is rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Local Uniqueness: This is used only in combination with other conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' An emission or absorption line is considered unique if there is at most one other data extremum greater than 90% of this line’s peak between 1× FWHM and 1× FWHM + 10 Å to either side of the line center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This is an alternate rough measure of local noise and is used primarily as a filter with low SNR lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' SNR and 𝜒2: ELiXer uses the following definitions of SNR and 𝜒2: SNR = � √︁ (𝐹(𝜆) − 𝑦)2 √︁� (error2) , (2) 𝜒2 = ∑︁ �data − model error �2 , (3) where the summations are over the wavelength bins within ±2𝜎 of the fit line center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝐹(𝜆) and 𝑦 are from Eqn 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The model is the fitted flux evaluated at each corresponding wavelength bin for the data and the error is the uncertainty on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The uncertainty on the SNR is computed via standard error propagation using the MCMC or LSQ uncertain- ties on each of the model’s Gaussian parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If the LSQ fit is marginal given the previous conditions, it is rejected if (1) the SNR is less than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 or (2) if the SNR is between 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 and the 𝜒2 is greater than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' These indicate poor fits to possibly noisy data and are generally not worth pursuing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Otherwise, the SNR and 𝜒2 are recorded for use in line scoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Line Scoring Every successfully fitted emission and absorption line re- ceives a score based only on its own properties, without con- sideration to the position or properties of any other fitted emission or absorption lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If that score exceeds a mini- mum threshold, the line, with its score, is accepted into a list of potential line candidates for later use in redshift solution finder (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The minimum threshold is configurable and is set, by default, to an empirically determined value based on the manual examination of many tens of thousands of ob- served spectra and a simulation of spectra drawn from median HETDEX noise properties (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Redshift solutions that fit multiple lines to the spectrum receive a separate "solution score" (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3) that is based, in part, on these individual "line scores".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The line score attempts to capture and quantify features beyond just the signal-to-noise ratio, which is a less than ideal metric for broad emission lines fitted with a single Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The line score takes into account additional data including the magnitude of the integrated (fitted) line flux, the line position relative to expectations, and the uniqueness of the line within a local spectral region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The intent is to codify not just the presence of each potential emission line, but the consistency and significance of that line with respect to the spectrum at an assumed redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The line score calculation is defined as: 𝑆𝐿 = 𝑆lim · 𝐴𝑁 · 𝑈𝑁 · 𝐹𝜆 · 𝑚𝜎 · 𝑚pix 1+ | 𝛿𝑑𝑥0 | (4) where: 𝑆𝐿 is the numerical line score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Noise peaks receive scores in the low single digits, typically less than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Weak emission lines (low SNR, low lineflux) typically receive scores in the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 - 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Extremely bright, high SNR lines can even exceed a score of 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0, but are clipped to a maximum of 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝑆lim is the maximum allowed fitted SNR from a Gaus- sian fit, up to a configurable limit (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 by default).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This helps scale the scoring by capping the maximum contribution of the SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝐴𝑁 is the "Above Noise" factor, defined by the mea- sured flux value of the emission peak divided by a noise estimate at that position and normalized by a config- urable factor (by default, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The noise estimate used here is the standard deviation of the 3𝜎 clipped fluxes at the same wavelength over all (448) fibers on the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The value of 𝐴𝑁 is clipped to the range [0,3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝑈𝑁 is an estimate of how unique the line is relative to the nearby spectrum (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=', the presence of several similarly narrow, low flux peaks in the same wavelength range likely indicate noise in the spectrum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This is an encoding of the Local Uniqueness described in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If the candidate line is sufficiently broad, with a fit FWHM of greater than 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 Å or if fewer than 3 possible lines are found, the current candidate ELiXer 11 line is considered sufficiently unique and 𝑈𝑁 takes on a value of 1, otherwise it takes on a value of 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝐹𝜆 is the Gaussian fitted, continuum subtracted inte- grated line flux in units of 10−17 erg s−1 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' There is no particular significance these units;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' they are simply used so that the value of the line score is generally in the range of 1-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝑚𝜎 encodes the minimum acceptable Gaussian fitted 𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Values of 𝜎 greater than 1 Å result in 𝑚𝜎 = 1, but values less than 1 Å receive a multiplicative penalty equal to the 𝜎 value as they are unlikely to have been fit to a real emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This is equivalent to min(𝜎, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝑚pix encodes the minimum acceptable number of pixels (𝑁pix) over which the SNR of the line is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If the number of pixels is less than 𝑁min (by default, 10 pixels to either side of the wavelength bin containing the line center), there is a multiplicative penalty imposed equal to 𝑁pix / 𝑁min .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Low numbers of pixels in the SNR measurement may be due to masked or invalid pixels or a line location near the edge of the wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This is equivalent to min�𝑁pix/𝑁min, 1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝛿𝑑𝑥0 is the offset, in Å, of the fit line center from the ex- pected location of the center line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For features found by the untargeted search (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1), this is the bin with the maximum (minimum, for absorption) flux within the spectrum slice being used to fit the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For corrobo- rating features as part of the "Targeted Search" (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2), it is the expected position of the assumed feature for the given redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' An adjustment is made to the 𝑆𝐿 if the fit SNR is less than 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 and the 𝜒2 is greater than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' These are considered marginal fits that could have a large score due to the integrated line flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In these cases, the score is reduced by a factor of (𝜒2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If the center of an emission line falls within a prominent sky line, specifically those centered at 3545 Å or 5462 Å, and if the FWHM does not extend past the sky line, the score is further reduced by a factor of 2, encoding the risk that the emission line is a relic of incomplete sky subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For very broad lines (fit FWHM > 20 Å), the scoring is modified by rejecting the line (setting the 𝑆𝐿 to 0) if the fitted SNR is less than a minimum threshold (by default, 19) and the 𝜒2 of the Gaussian model is greater than a maximum (by default, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' These fits tend to be poor, and caused either by artifacts in the data or the merging of multiple spectral features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Since the focus is on faint galaxies with continuum below the HETDEX sensitivity, absorption features do not factor strongly in classification for most HETDEX catalog objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As such, their base scoring value is scaled by a factor of 1/2 and optionally limited to a maximum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Spectra Simulation and P(Noise) As part of the scoring and in an effort to quantify the probability that a fitted line is simply the product of noise, we use the line finding code to analyze simulated spectra, treating all identified emission lines as false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The procedure is applied only to emission lines, not absorption lines, but the results are applicable to both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As part of the configuration for ELiXer, we compute the PSF weighted spectral uncertainties versus wavelength from 104 random, non-continuum detections from the entire HET- DEX catalog, and generate the median uncertainty for each wavelength bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' We then simulate 104 spectra, randomly drawing a flux for each wavelength bin (1036 random draws per spectrum over the range, 3470-5540 Å) according to the median uncertainty, and assuming a normal distribution about each uncertainty and no correlated noise between wavelength bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Each simulated spectrum is passed through the line find- ing code and all identified emission lines are recorded with their line scores (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The line scores are binned in steps of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 and normalized by the number of simulated spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This represents the simulated estimate of the probability that an emission line in a given scoring bin is the product of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This probability, 𝑃(Noise) monotonically decreases with in- creasing line score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Note that it is possible by this mechanism for a scoring bin to have a value of 𝑃(Noise) greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0, and that is the case for the lowest scoring bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For such cases, the probability is cropped to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 and any emission line with a score that fall in those bins is considered to be noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Higher scoring bins are cropped once the 𝑃(Noise) falls be- low 5 × 10−4, with that 𝑃(Noise) assumed for all emission lines with line scores above that value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' When applied to line detections in real data, any line score below the lowest score for the bin is assumed to be noise and is rejected, and any line detection with a score above the highest score receives the 𝑃(Noise) of the highest score for the bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' These 𝑃(Noise) estimates factor in the Solution Scoring (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3), described later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Since the 𝑃(Noise) is based on the line scoring and on the uncertainties in the HETDEX PSF weighted spectra, any reformulation of the line scoring or any change to the HET- DEX pipeline that results in a change in flux uncertainties necessitates a re-computation of this mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Absorption Lines As called out by its name, ELiXer is primarily designed to identify and act on emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Continuum bright HET- DEX detections (𝑔 < 22) are also analyzed with an indepen- dent software package (Diagnose, Zeimann & et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (in prep)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 12 Davis, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Nevertheless, ELiXer does currently include a limited use of absorption lines, triggered either explicitly at its invocation or automatically for detections with continuum greater than 2 × 10−17 erg s−1 cm−2 Å −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The same untargeted search (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1) used for emission lines is executed for absorption lines, with the exception that the spectrum is first inverted by subtracting all the flux densities from the maximum flux density of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This allows the fitter to treat the absorption lines as if they were emission lines, but only for purposes of line identification within the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The ac- tual fitting (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3) and initial scoring (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4) is performed on the original, non-inverted spectrum, with the appropriate sign changes to account for the different direction in the Gaus- sian model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' And like the case for emission lines, the positions of absorption lines with scores above a configurable threshold are also marked in the 1D spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' While there are 26 emission lines checked by ELiXer, only the Ca ii (H&K) 3968,3934 Å absorption lines are explicitly fitted and used in spectral redshift identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Addition- ally, these two lines are fit simultaneously and must appear together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If they occur at the edge of the spectral range, such that only one line could be found in the spectrum, the fit is not allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' A simple assertion is made to the pair of lines, requiring them to be of similar flux and FWHM such that the difference in flux and FWHM must be with 50% of the mean of their mean values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If the assertion fails, the fit is rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If the assertion passes, the lines are both accepted and contribute to the solution scoring (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Continuum Estimates Much of the classification effort rests on an accurate mea- sure of the emission line equivalent width, so a robust es- timate of the continuum underlying the emission line is of major importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' There are several, independent and semi- independent estimates of the continuum which contribute to a single combined estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Since most of the independent estimates arise from pho- tometric imaging, we calibrate our continuum derived clas- sification properties (described later in this section) to the bandpass continuum estimates, all of which assume a flat spectrum over the bandpass with no emission or absorption line masking (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3, and §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This means we are slightly biased to overestimate the continuum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This is more pronounced for objects such as AGN with strong, broad emission, but given the objective of accurate classification, this is a non-issue with these objects being a rare subset of HETDEX data and unlikely to be confused with the typical, continuum faint LAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In the general case that ELiXer is designed to address, our objects have faint or undetected continuum and a single, faint emission line so the bandpass overestimate is minimal and serves as an upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' All continuum estimates from broadband photometry as- sume a flat spectrum point source over the bandpass and convert the magnitude to flux density at the emission line’s observed wavelength rather than the filter’s effective wave- length as: 𝑓𝜆 = 𝑐 𝜆−2 × (3631 × 10−23) × 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4𝑚 (5) where 𝑓𝜆 is the flux density at the observed wavelength (in ergs cm−2 s−1 Å−1), 𝑐 is the speed of light in vacuum (Å s−1), 𝜆 is the fitted, observed wavelength center (Å), and 𝑚 is the 𝑔 or 𝑟 magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The literal constant is in units of ergs cm−2 s−1 Hz−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As most of the HETDEX emission line detections have either only 𝑟 coverage or are undetected in the imaging even when multiple bands are available, a color correction to the photometric continuum estimate is rarely possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In limited testing where photometric detections are made in both 𝑔 and 𝑟 no improvement in the classifica- tion performance and no change in the classification rates is found, and so no color correction is included in this version of ELiXer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' HETDEX Spectrum The HETDEX spectrum covers the entire 𝑔 bandpass and therefore can be used to estimate an object’s 𝑔-band magni- tude without the use of external data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Sky and background subtraction is very good and the continuum level is consis- tently measurable ≲ 10−18 erg s−1 cm−2 Å −1(Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' We use two methods to derive the 𝑔 magnitude from the HETDEX 1D spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The first multiplies the HETDEX spectrum through the SDSS 𝑔 filter’s throughput curve using the Python speclite package (Kirkby 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' ELiXer runs 1000 realizations of the HETDEX spectrum, sampling over the flux errors, and assigns the biweight (Beers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 1990) of those realizations to define an estimated 𝑔-magnitude and its 68% confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The second method sums the total flux in the HETDEX spectrum, again with propagated errors, and uses the mean flux density and an 𝑓𝜆,eff of 4726 Å to set a continuum and the 𝑔-band magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In both cases, the object is assumed to be a point-source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The combined continuum mean is converted into a 𝑔 magnitude for ease of use and comparison to other catalog reported magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' While this estimate is reported as computed, it is used internally with an imposed flux density limit of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='38 × 10−19 erg s−1 cm−2 Å −1 (𝑔 = 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' When our measured HETDEX continuum flux density is at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2× brighter than the limit, it receives the highest weight (4× standard) in the combined estimate (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4), as it is based on the same data that provides the line flux estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' All other continuum estimates are from other data sources and matched by proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As the limit ELiXer 13 is approached, the weight rapidly drops to the standard vote weight and is considered a non-detection once the limit is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' A second estimate of the continuum is obtained using the 𝑦 offset from the Gaussian fit to the emission line (equation 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' While this is the estimate nearest the emission line, it can also have a large uncertainty and the simple Gaussian model does not allow for asymmetric line flux or different continuum levels on either side of the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' When this esti- mate is brighter than the HETDEX limit, it receives a small, empirically set weight of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2× the standard vote, otherwise it receives zero weight and is not included in the combined continuum estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' A third and final estimate is also recorded, but is not, by default, included in the combined continuum estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In this estimate, the continuum is still assumed to be flat in 𝑓𝜈, but all emission and absorption lines identified in the spectrum are masked at ±2𝜎 from the fitted line centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The mean of the unmasked fluxes, with standard error propagation, is converted into a flux density and returned as the continuum estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' With the exception of the continuum bright ob- jects with multiple, broad spectral lines mentioned earlier, this estimate is not significantly different from the speclite result and its inclusion in the combined estimate would be both redundant and somewhat inconsistent, given the other photometric estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' It is, however, used internally in some diagnostic checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Aperture Photometry The 𝑔 and 𝑟-bandpass continuum estimates come directly from run-time aperture photometry as described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' When an SEP aperture matches that of the HETDEX de- tection, its magnitude is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If no SEP aperture is a match, then the smallest, stable ELiXer circular aperture provides the magnitude estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In either case, if the computed magni- tude is fainter than the imaging limit, that limit is used and the continuum value is flagged as a non-detected upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Since the HETDEX emission lines appear in the 𝑔-band, an optional correction is allowed for translating an 𝑟-band continuum estimate to 𝑔-band, however this is not used by default, as an examination of 𝑔 and 𝑟 continuum estimates where both are available from the same instrument for the same objects shows no consistent trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Additionally, Leung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (2017) finds no advantage in using 𝑔 over 𝑟 and their simulated data actually suggest that LAE/[O II] segregation is slightly improved with 𝑟, though this is not confirmed with the observed spectra in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If the measured aperture magnitude is brighter than the lim- iting magnitude of the image, it receives a full (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0) weight in the final, combined estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If the measured aperture mag- nitude is fainter than the limit, it is treated as a non-detection and the limit is used in the combined estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' When the limit is used for the aperture magnitude, the weight in the com- bined estimate is scaled down linearly from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 as the limit grows brighter from 26𝐴𝐵 to 24𝐴𝐵 and a non-detection in that increasingly bright limit provides less and less useful information (noting that the HETDEX spectra has a mag- nitude limit near 𝑔 = 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The 26𝐴𝐵 and 24𝐴𝐵 boundaries selected to roughly cover the the magnitude range of maxi- mal LAE and [O II] galaxy 𝑔 magnitude overlap in HETDEX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Catalog Counterpart Lastly, if a catalog counterpart can be matched to the HETDEX detection (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3), its reported bandpass magnitude (again, only 𝑔 or 𝑟) is added to the list of continuum esti- mates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' A minimum 20% flux uncertainty is assumed, even if no uncertainty is reported by the catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' All catalog reported values are assumed to be a proper detection and receive a full (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0) weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Combined Continuum The combined estimate is produced using the weighted mean of a subset of the individual continuum estimates, de- scribed in the immediately previous subsections, with less informative estimates and extreme outliers removed from con- sideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' At most, a single upper limit estimate is allowed in the sub- set and is selected as the deepest (faintest) upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This is typically the limit from the deepest photometric imaging where there is no detection or where the aperture magnitude is fainter than the image’s limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' No upper limit is included if there exists a positive aperture detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If there are three or more continuum estimates in the subset, a fairly aggressive clip is applied, which excludes the most extreme estimate(s) with values greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5× the weighted biweight scale (Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021) while retaining a minimum subset size of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The final combined continuum estimate is then the weighted mean of the surviving continua in the subset: ¯𝑓𝜆 = � 𝑖 � 𝑓𝜆𝑖 𝑤𝑖 𝜎−2 𝑖 � � 𝑖 𝑤𝑖 𝜎−2 𝑖 , (6) Δ ¯𝑓𝜆 = √︄� 𝑖 �𝑤𝑖 𝜎2 𝑖 � � 𝑖 𝑤𝑖 , (7) where ¯𝑓𝜆 is the combined ("averaged") continuum estimate, 𝑓𝜆𝑖 is an individual continuum estimate, 𝑤𝑖 is the associated weight, and 𝜎𝑖 is the associated standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The error, Δ ¯𝑓𝜆, is the square root of the weighed average of the variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This defines the distribution over which the continuum is sampled for the P(LAE)/P(OII) classifier in the next subsec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 14 Davis, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Redshift Solutions Distilled to its most basic functions, ELiXer’s raison d’être is to assign the correct redshift to every detection as the operative analog to the classification of the target emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The core approach to this objective is the testing and ranking (or scoring) of many possible redshift solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Clearly the most secure, and consequently the highest scor- ing, solutions are those with multiple identified spectral lines consistent with known rest-frame features at an assumed redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' ELiXer’s initial set of redshift solutions is generated by iterating over the lines in Table 2 and assuming, in turn, that each one represents the target emission line identification (note that the H&K absorption lines are handled differently per §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' With each assumed redshift, ELiXer attempts to fit all in the list, and accumulates a total solution score based on the number and quality of the successes (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' At this stage, only the relative line positions are considered, with flux ratios, required lines, and other criteria considered in later steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The more lines that are found, the more ro- bust the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Unfortunately, only about 5% of ELiXer classifications are established with more than one identifi- able emission line, so additional methods must be applied to confidently identify the target emission lines and assign the corresponding redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Catalog Redshift Match When ELiXer matches a HETDEX detection to one (or more) catalog objects (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3) that have associated spectro- scopic and/or photometric redshift assignments, that in- formation is evaluated in the context of the emission and absorption lines identified in the HETDEX spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The catalog supplied redshift, with its error, is applied to the target emission line and all other ELiXer identified lines and the resulting rest-frame wavelengths are checked for consis- tency with those in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If the catalog redshift results in rest-frame wavelength matches, it boosts any previously assigned ELiXer score (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3) for that redshift, with a larger weight given to spec-𝑧 (+100 to the redshift solution raw score, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5) than to phot-𝑧 (+5 to the redshift solution raw score).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If an ELiXer redshift solution for that catalog redshift does not exist, one is created and scored in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1% of the HDR3 detections have a catalog matched spec-𝑧 counterpart and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5% have a phot-𝑧 counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Large Galaxy Mask In addition to matching redshift catalogs, ELiXer also compares the sky position and wavelength of each detection against an internal HETDEX catalog of large galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' We define this large galaxy catalog by searching the most recent versions of the RC3 (de Vaucouleurs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 1991)2 and the UGC (Nilson 1973)3 galaxy catalogs for objects larger than 1 arcminute in diameter within our survey area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In total, we find 644 large galaxies in the Spring field, and 447 in the Fall field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For each system, we adopt the catalog’s basic parameters for position, position angle, ellipticity, and D25 semi-major axis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=', the size of the galaxy defined by its 𝐵-band isophote at 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 mag arcsec−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Prior to inclusion in the large galaxy mask, each galaxy is manually inspected to confirm that these values are reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Where values of these parameters are uncertain, they are corrected to values listed in the NASA/IPAC Extragalactic Database4 or through visual inspection of the galaxy in SDSS 𝑔-band images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Any HETDEX detection falling within 3× the D25 isophotal ra- dius of a large galaxy is tested against the spectral features expected for the system’s redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This matching is per- formed in exactly the same way as for the catalog matching in the previous section, except that the scoring is scaled in- versely by the distance in multiples of D25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The overall area of this large galaxy mask is dominated by a handful of nearby galaxies (NGC 5457 and NGC 4258 in the Spring field, and IC 1613 and NGC 474 in the Fall Field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Special Handling for [O III] The [O III] 5007 Å line can be problematic to identify by equivalent width based methods when other oxygen or Balmer lines are not detected as it can have a large equivalent width and appear similar to Ly𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Low-𝑧 compact star forming galaxies, planetary nebulae (PNe), extragalactic H II regions, and the outer star forming regions of resolved galaxies could sometimes have detectable [O III] 5007 Å, but with [O III] 4959 Å, [O II] 3727 Å, and H𝛽 that do not reach the threshold for a standard HETDEX detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Such objects could be classified as Ly𝛼 by the base algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' To protect against such misclassifications, additional tests are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For observed emission lines redward of 5007 Å, but with- out any other nominally detected emission feature, a lower threshold for emission line detection is allowed at the ex- pected positions of [O III] 4959Å, [O II] 3727Å, and H𝛽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If one or more of those lines are detected at this reduced strin- gency, a redshift solution is created with a score of at least the minimum acceptable threshold, and a flag is set for followup manual inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If one or more of the above lines are found and there is no identified imaging counterpart, a flag is also set to indicate that this could be a planetary nebula, either in the Galaxy or in intergalactic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Given the HETDEX lines of sight are 2 available at: http://haroldcorwin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='net/rc3/ 3 https://heasarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='gov/W3Browse/galaxy-catalog/ugc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='html 4 http://ned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='ipac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='edu ELiXer 15 out of the plane of the Galaxy, the likelihood of encounter- ing Galactic planetary nebulae is reduced but is certainly not zero and several known Galactic planetaries are located in the HETDEX footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Given their physical proximity, most of these objects will have sizes of several arc-minutes, and we test for this by looking for large spatial clusterings of emis- sion at 5007 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' When found, these regions are masked from use in HETDEX cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' A potentially more pernicious issue is planetary nebulae in the halos of nearby galaxies and intergalactic PNe within galaxies groups and clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' These could be misinterpreted as background LAEs, though this risk is ameliorated via the check against the large galaxy mask (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2) and neighbor clustering (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Conversely, this comes at a (small) cost of the loss of some background LAEs with observed Ly𝛼 redshifted to match the [O iii] 5007 Å line of on-sky adjacent foreground galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' We note that [O III] 5007 Å makes up only 1% of the SzAS detections and none are misidentified by ELiXer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Object Classifications Labels Based on combinations of spectral features (with examples given later in this subsection), some HETDEX detections are assigned classification labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' These labels indicate only that a detection is consistent with the class of object indicated by the label within the parameters defined for that class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Classi- fications are not mutually exclusive and are applied simply if the corresponding conditions are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If none of the specific classification conditions are met, then no extra classification label is applied to the detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The classification is not Boolean, but is scored, with the strength of the classifica- tion based on the number and quality of the conditions that are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' A negative classification can also be made if the failure to meet conditions is sufficiently extreme such that a classification is excluded (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=', if the detection’s properties are grossly inconsistent with the given classification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Strongly consistent object classifications can be used to in- crease the score of the corresponding redshift solution, while strongly inconsistent classifications decrease the score of the corresponding solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In this way, the object classification 𝑐𝑎𝑛 modify the P(Ly𝛼) result (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5) by altering the score of a multi-line solution available to the P(Ly𝛼) routines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' However, the conditions are relatively strict and the overall impact of labeling is small, with only ∼4% of detections actually meet the conditions to receive an object classification label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Additionally, a few generic labels are applied for ELiXer detections that are associated with unique object in a pho- tometric catalog (§2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' These labels are only provided as suggestions and do not impact the scoring of the multi-line solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The ELiXer assigned labels are: AGN ("agn") The "agn" label is set if a HETDEX spectrum contains (possibly broadened) emission lines consistent with those seen in AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' These reference emission lines are: O VI (1035 Å), Ly𝛼 (1216 Å), N V (1241 Å), Si II (1260 Å), Si IV (1400 Å), C IV (1549 Å), He II (1640 Å), C III] (1909 Å), C II (2326 Å), Mg II (2799 Å), and [O II] (3727 Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For some pairs of lines, bounds on relative line fluxes must be met and certain lines must be present to support the identification of other lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For example, if a line as- sumed to be C IV is observed at 5000 Å, then a line for Ly𝛼 must also be found at 3295 Å and it should be at least as strong and have a similar FWHM as C IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If no line is observed at 3295 Å or if the feature is much weaker than the assumed C IV line, then the identifica- tion is inconsistent with that of an AGN and the C IV solution receives a reduced score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Low-𝑧 Galaxy ("lzg") The "lzg" logic is largely the same as the "agn" but with a different set of reference lines: [O II] (3727 Å), H𝜂 (3835 Å), H𝜁 (3889 Å), H𝜖/ionC2 (3970 Å), H𝛿 (4101 Å), H𝛾 (4340 Å), H𝛽 (4861 Å), [O III] (4959 Å), and O III] (5007 Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As with AGN, some bounds on line strengths must be met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For exam- ple, if a line assumed to be [O III] 5007 Å is observed at 5300 Å, then another line at 5249 Å must be observed at one-third the strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Similarly, for HETDEX de- tections with strong continuum, if an absorption line is assumed to be calcium H at 3968 Å, calcium K at 3934 Å must also be present with at a similar equivalent width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If these criteria are satisfied, then the detection will be labeled "lzg".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Moreover, an additional label of "o32" will be assigned to objects with an [O III] 5007 Å to O II 3727 Å flux ratio greater than 5:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Meteor ("meteor") With any wide-field, long-term survey, meteor intru- sions on the extra-galactic observations are inevitable, and if not identified, they can be a significant nuisance source of emission (and sometimes of continuum) de- tections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' A combination of methods are used to identify meteors in the detection catalog (Mentuch Cooper ApJ accepted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Since ELiXer processes only single detections in iso- lation, its meteor identification methodology focuses on the transient nature of the phenomenon and their fairly distinctive emission line signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' To iden- tify a meteor emission, we divide a spectrum into 9 non-overlapping, non-contiguous regions by wave- length (in Å) where meteor emission lines are common: [3570,3590], [3715,3745], [3824,3844], [3852,3864], [3926,3942], [3960,3976], [4210,4250], [4400,4450], and [5160,5220].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For the visually confirmed meteors in 16 Davis, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' HETDEX, these regions often include bright features from Mg (3832, 3838, 5172, and 5183 Å) as well as typically fainter emission from Al, Ca, and Fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Spec- tra that contain multiple emission lines that are within these ranges and are detected in only one of the three dithered exposures used for an observation are labeled as meteors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' White Dwarf ("wd") The white dwarf label logic is very basic and simply looks for the Hydrogen series absorption lines for DA and DAB types, the Helium series for DB types, and Carbon and Oxygen for DQ types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Additionally, to be classified as a white dwarf, the spectrum must have a blue spectral slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Since the shape and width of the absorption features are not taken into account, norarethe presenceofother features (such as pronounced H and K (Ca ii) lines), it is possible to mislabel a main sequence star, particularly an A-type, as a white dwarf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' However, given the high Galactic latitude of the HETDEX survey, we do not expect the set of HETDEX detections to contain many early-type stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Catalog Labels ("gal", "star", "agn") These are recorded as suggestions when matched to an exter- nal photometric catalog, but they do not influence any of the ELiXer logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For example, an "agn" label from a photometric catalog matched to a HETDEX detection is considered separately from the ELiXer "agn" label logic described above and will appear in the classifica- tion labels even if the ELiXer spectral features analysis does not result in an "agn" label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Redshift Solution Scoring Each redshift solution receives three scores, a raw score, a (normalized) fractional score, and a scale score, so that the solutions can be rank ordered and assessed in terms of their viability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The raw score is the unweighted sum of the indi- vidual line scores (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4) of the spectral lines included in the solution, excluding the anchor line (which is common to all solutions), and including a multiplier based on the number of identified spectral lines and any multipliers from classifi- cation labels (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4), where they are strongly consistent or inconsistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' It is defined as: 𝑟𝑠 = � 𝑛 ∑︁ 𝑖 𝑙𝑠𝑖 � × min � 1, 1 2 � 𝑛2 − 𝑛 �� × 𝑏, (8) where 𝑟𝑠 is the solution raw score, 𝑙𝑠 is a line score of an included spectral line, 𝑛 is the total number of spectral lines included in the solution not counting the anchor line, and 𝑏 is any multiplier from the object classification label logic (typically 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='25 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The raw score is normalized to produce the fractional score by dividing it by the sum of the raw scores of all redshift solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Lastly, a scale score is produced from the weighted sum of the probability that the solution is comprised of noise, the raw score, and the fractional score as: 𝑠𝑠 = � 1 − 𝑛 � 𝑖 𝑃(noise)𝑖 � × 𝑤noise + min (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0, 𝑟𝑠/𝐹) × 𝑤raw + 𝑓 𝑠 × 𝑤frac, (9) where 𝑠𝑠 is the scale score, 𝑃(noise)𝑖 is the probability that the included spectral line is noise (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5), 𝑤noise is the weight for this first term (by default, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='40), 𝑟𝑠 is the raw score from Eqn (8), 𝐹 is the configured raw score scale factor (by default, 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0), 𝑤raw is the weight for this second term (by default, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='50), 𝑓 𝑠 is the fractional score, and 𝑤frac is the weight of this third term (by default, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' P(LAE)/P(OII) P(LAE)/P(OII) (sometimes as PLAE/POII in other docu- mentation) represents the ratio of the relative probability that given a set of measured characteristics, an emission line is Ly𝛼 (representing an LAE) rather than [O ii].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' These proba- bilities are based on the number of galaxies expected at the volume sampled by the redshift slices assuming the emission line is either Ly𝛼 or [O ii] given the measured line flux and equivalent width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The expected number of galaxies derives from the equivalent width distributions of Ly𝛼 and [O ii] con- ditioned on the luminosity functions found in Gronwall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (2014) and Ciardullo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (2013) respectively, interpolated or extrapolated as needed (see also Leung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (2017, Figure 2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This is an improvement on the commonly used 20 Å equiv- alent width cut (Gronwall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2011) and is based largely on the analysis of Leung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (2017), and us- ing the specific translation and implementation described in (Farrow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021, primarily in Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' ELiXer slightly updates Farrow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (2021) by (1) using multiple independent or semi-independent estimates of the continuum (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2), (2) combining those estimates into a single, best-fit continuum value, and (3) sampling over the uncertainties in the measured line flux and continuum estimates to generate a (68%) con- fidence interval around each P(LAE)/P(OII) measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Partly for convenience and partly as a representation of the practical limits of this method, the ratio is cropped to values between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='001 ≤ P(LAE)/P(OII) ≤ 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The interpretation of the P(LAE)/P(OII) value is not quite straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' While LAE evolution between 2 < 𝑧 < 4 appears somewhat muted (Blanc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' ELiXer 17 2021), there is more redshift evolution of the [O ii] systems (Gallego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Comparat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Gao & Jing 2021) for 𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This evolution may be underrepresented in the base P(LAE)/P(OII) code and lead to a deviation from the expectation that a ratio near 1 should be interpreted as the likelihood of the emission line being Ly𝛼 or [O ii] is approximately equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Building on the suggestion in Leung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (2017) of using different thresholds for the P(LAE)/P(OII) ratio at different observed wavelengths, ELiXer adopts an empirical threshold relation (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The overall combined P(LAE)/P(OII) value and its con- fidence interval factor significantly in the final automated classification of the emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' It can frequently be the most influential (and sometimes the only) metric that is used in that classification (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' P(LyA) Using some of the features/measurements already de- scribed, along with a set of additional features described below, ELiXer synthesizes an aggregate confidence in the classification of the anchor emission line as Ly𝛼 or not-Ly𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For familiarity, this is couched in terms of a probability, la- beled as P(Ly𝛼) with values between 0 (definitely not Ly𝛼) and 1 (definitely Ly𝛼), but is not a true probability in the for- mal sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' P(Ly𝛼) is the result of a weighted voting system where each of the features described in this section provides a vote (typically 0 or 1, but can be in between) and that vote is given a weight based on the robustness or confidence of the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' With specifically noted exceptions, features that do not produce a clear preference are given zero or very little weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The final P(Ly𝛼) value is then simply the sum over all votes multiplied by their respective weights: 𝑃(Ly𝛼) = ∑︁ 𝑖 �vote𝑖 × weight𝑖 � ∑︁ 𝑖 weight𝑖 , (10) Note that the sum of the weights alone is not normalized and can exceed 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In the relatively rare cases where the sum of all weights is less than 1, a special "uncertainty" vote is added with a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 and a weight equal to 1−� weights, so that the weights do sum to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This helps capture the uncertainty in the classification and prevents one or two votes with very low weights from being dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The selection of voting criteria and the weights applied to the votes is the result of empirical analysis and trial-and-error testing and is discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This is a little bit of the Central Limit Theorem and the Wisdom of the Crowd, even though the votes are not entirely independent as several incorporate similar elements and some are designed to handle edge cases not well covered by the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' No single vote is universally dominant, though each can be decisive under the right circumstances, such as the high weight of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2 when multiple emission lines are present or even a low weight vote from §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 for some moderate equivalent widths when the rest of the vote tally is near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As a word on the notation in this section;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' often [O ii] is used in place of "not-Ly𝛼" as [O ii] is the most common con- taminant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Votes "for [O ii]" are really votes for "not-Ly𝛼".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Further, the figures in this subsection all show only those assessment sample detection emission lines that are Ly𝛼 or [O ii], so [O ii] is equivalent to "not-Ly𝛼".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Object Size Vote In cases where a counterpart is identified and resolved in the 𝑔- or 𝑟-band imaging, the angular and physical extent of the counterpart contributes a vote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For this purpose, an ob- ject is considered resolved if the angular major diameter is greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1× the seeing FWHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This includes artifi- cially enlarged footprints in the imaging due to the "bloom- ing" of bright sources that have saturated the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The proper physical diameter is computed assuming the redshift of [O ii], as larger objects tend to be more evolved and at lower redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The emission line FWHM is used to help break the size degeneracy between larger, lower-𝑧 objects and saturated, higher-redshift sources, via the assumption that the latter are AGN with a large emission line FWHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The parameter thresholds are set from a manual partition- ing of classifications in scatter plots of angular and physical diameter versus the observed wavelength of the anchor emis- sion line, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The conditions and their associated votes are summarized in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The specific limiting values of the FWHM help distinguish possible AGN with a broadened emission line, from lower redshift galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' It is reasonable for an AGN to receive a vote for Ly𝛼, but an angularly large object with a more narrow emission line is more likely an [O ii] emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The gap between the condi- tions avoids a vote where it is unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The angular diameter threshold (in arcseconds), 𝜃𝜆, is a piece-wise linear function: 𝜃𝜆 = ���� ���� 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='8, 3727Å < 𝜆 ≤ 4000Å − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0018𝜆 + 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0, 4000Å < 𝜆 ≤ 5000Å 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0, 𝜆 > 5000Å (11) The object size criteria results in a cast vote for 70% of the SzAS (down-selected to only contain Ly𝛼 and [O ii]), where the separation of [O ii] from Ly𝛼 is effective, with a Ly𝛼 contamination rate of 4% in those votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Multi-line Redshift Solutions Votes 18 Davis, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Angular and Physical Diameter Votes Condition Vote Weight 𝑑𝑝 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 kpc or 𝜃 < 𝜃𝜆 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='25 𝑑𝑝 < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 kpc 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='10 𝜃 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='′′5 and FWHM > 1000 km s−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='25 𝜃 > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='′′5 and FWHM < 800km s−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='25 else no vote NA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='00 Note— Summary of angular and physical size votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The conditions are ordered such that the logical evaluation results in at most one unique vote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If no conditions are met, there is no vote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝑑𝑝 is the proper diameter in kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝜃 is the angular diameter in arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝜃𝜆 is the minimum expected angular size for an [O ii] galaxy for the observed anchor emission line wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝐹𝑊𝐻𝑀 refers to the emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This criterion can generate multiple votes, one for each potential redshift solution (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3) based on the positions and fluxes of the fitted spectral lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='There must be two or more found spectral lines, with the scores based largely on the number of lines and their strengths (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' However, as is shown later in this subsection, solutions incorporating three or more lines receive an increased voting weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' At most, there will be a single Ly𝛼 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0) vote if there is a solution that supports the classification of the anchor line as Ly𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' All other redshift solutions necessarily require the anchor line to be something other than Ly𝛼, and therefore cast a not-Ly𝛼 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0) vote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The weight each vote receives depends on the scaled solution score assigned multiplied through a sigmoid: 𝑤0 = 𝑠𝑠/(1 + exp(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='75𝑚 − 𝑟𝑠)) (12) where 𝑤0 is the initial voting weight, 𝑠𝑠 is the redshift solution scale score (Eqn 9), 𝑚 is the minimum acceptable score (25, by default), and 𝑟𝑠 is the redshift solution raw score (Eqn 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' An additional multiplier is applied for exceptionally strong redshift solutions with 3 or more contributing spectral lines: 𝑤 = 𝑤0 × min(𝑟𝑠/𝑚, 10) (13) where 𝑤 is the modified voting weight, 𝑤0 is the original weight (Eqn 12), 𝑟𝑠 is the raw solution score, and 𝑚 is the minimum acceptable score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This multiplier is always greater than 1 since, by definition, a qualifying redshift solution must have a raw solution score greater than minimum acceptable value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The maximum value of 𝑤 is limited to 10× the original number, but that allows this vote to dominate with a high confidence redshift solution comprised of multiple, strong spectral lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3500 3750 4000 4250 4500 4750 5000 5250 5500 0 2 4 6 8 10 Diameter [arcsec] Threshold Ly [O II] 3500 3750 4000 4250 4500 4750 5000 5250 5500 Observed [Å] 0 5 10 15 20 25 Diameter [kpc] Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The separation of Ly𝛼 from [O ii] in the assessment sample (SzAS, §4) based on the angular (upper panel) and physical (lower panel) diameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Errors are ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='′′2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The dashed line corresponds to the thresholds defined in Table3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' There are no points blue-ward of 3727 Å in the lower figure since the physical diameter is computed based on the assumption that the emission line is [O ii].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The lower panel is cropped to a maximum of 25 kpc for readability and shows two horizontal thresholds at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 kpc, corresponding to the first two conditions in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This generates a vote for 70% of the SzAS with a 4% contamination of Ly𝛼 in those votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This criteria does not often trigger a vote, casting one for only 7% of the SzAS, down-selected to only contain Ly𝛼 and [O ii], and 12% for the entire SzAS, but has no contamination of Ly𝛼 for those votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Due to the bright skew in SzAS (see §4 and §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1), this voting rate is exaggerated and is only cast for 2% of the 𝑔 >22 detections in HETDEX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' P(LAE)/P(OII) Vote As most HETDEX detections are faint, single emission lines, the above criteria rarely produce strong redshift so- lutions, and the P(LAE)/P(OII) computation is often the most significant vote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The value (0 or 1) of the vote de- pends on which side of a wavelength dependent midpoint the P(LAE)/P(OII) ratio falls, and the weight of the vote increases ELiXer 19 with the distance of the ratio from that midpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The mid- point value, which separates the [O ii] (0) and Ly𝛼 (1) vote, is a modification of the binary condition suggested in Leung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (2017), 𝜇 = � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='38, 𝜆 ≤ 4255Å 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3, 𝜆 > 4255Å (14) and is defined as 𝜇 = ����� ���� 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0, 𝜆 ≤ 4000Å 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='018𝜆 − 71, 4000Å < 𝜆 ≤ 4500Å 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0, 𝜆 > 4500Å (15) where 𝜇 is the midpoint or vote threshold and 𝜆 is the wave- length of the anchor emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Ratios nearer the midpoint suggest an increasingly equal likelihood that the source emis- sion line is [O ii] or Ly𝛼 and, as such, add little evidence for a classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This is reflected in a low voting weight (𝑤) built from a Gaussian, 𝑤 = 1 − exp � − � 𝑃 − 𝜇 √ 2 𝜎 �2� × (1 − 𝑖), (16) where 𝑃 = � P(LAE)/P(OII), for P(LAE)/P(OII) ≥ 1 P(OII)/P(LAE), for P(LAE)/P(OII) < 1 (17) and 𝜇 is the midpoint and 𝜎 is the usual Gaussian width (here set to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0, which is tuned by hand to give balanced voting weights).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The parameter 𝑖 is an ersatz standard deviation from the scaled 68% confidence interval around the P(LAE)/P(OII) (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4) and is defined as: 𝑖 = 1 2 × � 𝑈 𝑈 + 1 − 𝐿 𝐿 + 1 � (18) where 𝑈 is the upper bound of the confidence interval and 𝐿 is the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As the P(LAE)/P(OII) ratio moves farther from the midpoint in either direction, the weight of the vote increases and rapidly asymptotes to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Alone, the P(LAE)/P(OII) vote is effective, with a 4% Ly𝛼 contamination rate (by [O ii]) in the SzAS (down-selected to only contain Ly𝛼 and [O ii]), voting 90% of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As with the other equivalent width based votes, though, it struggles to identify Ly𝛼 emission lines when originating from non-LAE (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' low-EW Ly𝛼 emitting galaxies) (see also §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As the P(LAE)/P(OII) computation includes the volumes sampled by the two assumed redshifts, it can become a less effective discriminator as the observed wavelengths approach the rest wavelength of [O ii] and that volume shrinks (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4 and Leung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Farrow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The other votes, including two more based partly on the emission line equivalent width, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 in particular, help compensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3500 3750 4000 4250 4500 4750 5000 5250 5500 Observed [Å] 3 2 1 0 1 2 3 log(PLAE/POII) Threshold Ly [O II] Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' P(LAE)/P(OII) distribution (clipped to 10±3) in the as- sessment sample (SzAS, §4) shown without the 68% confidence intervals (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The dashed line is the midpoint of the segrega- tion threshold (Eqns 15 - 18) with points above the line receiving a vote for Ly𝛼 and those below for not-Ly𝛼 with weights based on the distance from the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This vote has a 4% contamination rate of Ly𝛼 by [O ii] in the SzAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Line FWHM Vote This is logically one of the simplest votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If the emission line FWHM is larger than 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 Å, as seen in Figure 3, the line receives a Ly𝛼 vote (1) with a weight as high as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 using 𝑤 = min(FWHM/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0), (19) where 𝑤 is the assigned weight of the line and FWHM is line’s fitted full-width at half-maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As the contamina- tion rate decreases with larger FWHM thresholds, the voting weight increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If the lower uncertainty bound of the fitted FWHM, here defined as the fitted FWHM minus the uncer- tainty derived from standard error propagation, exceeds a configurable minimum (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3 Å by default), the vote weight is set to the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 maximum value, as [O ii] emission lines are rarely that broad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Also, as a consequence of the increasing FWHM threshold, these higher weighted votes tend to favor AGN selection and thereby helps reduce the confusion caused by lower AGN emission line equivalent widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In short, it helps improve the recovery of Ly𝛼 (and decrease the misclas- sification as [O ii]) from AGN that can fail the other voting criteria based on equivalent width (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5), bandpass magnitude (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='7), and angular size (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This criteria casts a vote for 23% of the down-sampled SzAS (containing only Ly𝛼 and [O ii]) with a total Ly𝛼 con- tamination of 11%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This drops to 3% when considering votes 20 Davis, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' with weights above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3 (received by 18% of the down-selected SzAS) and is contamination free for votes with weights above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='7 (received by 12% of the down-selected SzAS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' We note that while this particular vote is a good discrim- inator against [O ii], it can confuse Ly𝛼 with other broad AGN lines, such as C iii] or C iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' We largely address this issue using multi-line redshift solutions (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3) and clustering (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3500 3750 4000 4250 4500 4750 5000 5250 5500 Observed [Å] 5 10 15 20 25 30 Line FWHM [Å] Threshold Ly [O II] Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Ly𝛼 and [O ii] separation in the assessment sample (SzAS, §4) based on the emission line FWHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The data points are shown without their uncertainties (∼ 14% ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The horizontal dashed line rep- resent the minimum threshold to receive a vote for Ly𝛼 as described by Eqn 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Simplified Equivalent Width Vote This vote is somewhat redundant with the full P(LAE)/P(OII) vote (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3), but does not consider the red- shift based population distributions or observed wavelength variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' It slightly moderates the P(LAE)/P(OII) vote and can help push away from an uncertain classification where the P(LAE)/P(OII) vote has a low weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' It can also push toward an uncertain classification if the P(LAE)/P(OII) vote and this vote have similar weights, but different votes, allowing other voting criteria to have more influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' These two votes agree 95% of the time and this simplified equivalent width vote is only important in these boundary cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This simplified vote uses EW𝐿𝑦𝛼, which is defined by the Gaussian fitted line flux (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3) and the combined continuum estimate (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For EW𝐿𝑦𝛼 much greater or much less than 20 Å, this reinforces the P(LAE)/P(OII) vote and helps nudge the solution away from the P(LAE)/P(OII) midpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If the EW𝐿𝑦𝛼 is greater than 30 Å, then the vote is for Ly𝛼 (1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' if the EW𝐿𝑦𝛼 is less than 20 Å, the vote is for [O ii] (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' All other EW𝐿𝑦𝛼 values do not generate a vote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The assigned voting weights are based on the EW𝐿𝑦𝛼 lower (EW− 𝐿𝑦𝛼) and upper (EW+ 𝐿𝑦𝛼) bounds and increase with con- ditions where the contamination is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The maximum weight is limited to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 so that the P(LAE)/P(OII) vote is dominant when both votes approach their maximum weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In the pro-Ly𝛼 case, the weight is either 0 or between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 as: 𝑤 = ���� ���� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0, 𝑟− ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 < 𝑟− < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 max(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1, min(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5, 𝑟− − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0)), 𝑟− ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 (20) where 𝑤 is the assigned weight and 𝑟− = 1 25 × 𝐸𝑊− 𝐿𝑦𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In the pro-[O ii] case, the weight is between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 as: 𝑤 = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1, 𝑟+ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 min(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5, max(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1, 𝑓 )), 𝑟+ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 (21) 𝑓 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='04 × 𝐸𝑊+ 𝐿𝑦𝛼 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='9 (22) where 𝑤 is the assigned weight and 𝑟+ = 20 / 𝐸𝑊+ 𝐿𝑦𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Figure 4 shows the Ly𝛼 and [O II] SzAS detections with rest-Ly𝛼 EW less than 100 Å (this includes all SzAS [O II] emission lines) with the voting thresholds marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This criteria votes in 80% of the down-selected SzAS (con- taining only Ly𝛼 and [O ii]) with a Ly𝛼 contamination rate of 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Superficially, this is superior Ly𝛼/[O ii] segregation compared to the P(LAE)/P(OII) vote (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3, but by design, avoids voting in the difficult EW transition region (shaded region in Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Catalog Photometric Redshift Vote The photometric redshifts fits from the various included catalogs (§2) are often too broad to confidently pin down a tight redshift constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' However, they can be sufficient to distinguish between low-𝑧 (𝑧 ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='7) and high-𝑧 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='7 ≲ 𝑧 ≲ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='7) objects and thus help separate [O ii] from Ly𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If there are any photometric redshifts for a HETDEX detection, this vote simply takes the arithmetic mean of all phot-𝑧 measurements of the matched catalog counterpart from all contributing cat- alogs and compares it to the low-𝑧 and high-𝑧 ranges quoted above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If the mean falls within either range and is within a redshift distance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 of [O ii] or Ly𝛼 respectively, then the corresponding vote is cast with a weight of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If the mean falls outside of both ranges or if the redshift separation be- tween the mean and an assumption of [O ii] or Ly𝛼 is greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5, then no vote is cast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Only ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5% of HDR3 sources have at least one phot-𝑧 catalog counterpart match, so this vote rarely contributes to the P(Ly𝛼) logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For the SzAS testing, since contributions from catalog phot-𝑧 and spec-𝑧 are necessarily turned off, this vote is never cast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' ELiXer 21 3500 3750 4000 4250 4500 4750 5000 5250 5500 Observed [Å] 0 20 40 60 80 100 rest-Ly EW [Å] Thresholds Ly [O II] Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Simplified rest-Ly𝛼 equivalent width vote applied to the assessment sample (SzAS, §4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The figure is cropped to a maximum EW of 100 Å for readability and plotted without the ∼ 16% errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The SzAS contains no spectroscopically confirmed [O II] emission lines with rest-Ly𝛼 EW > 80 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Detections with EWs falling in the gray shaded region between 20 and 30 Å receive no vote while those above receive a Ly𝛼 vote and those below an [O II] vote with the weight of the vote modulated by the distance to the nearest threshold (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Apparent Magnitude and Equivalent Width Vote This vote is largely predicated on the observation that the HETDEX LAEs tend to be fainter than [O ii] galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' How- ever, there certainly exist bright LAEs (including AGN) and faint [O ii] galaxies, so the EW𝐿𝑦𝛼 is also incorporated into the decision to moderate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The apparent magnitude used in this vote is the 𝑔-band magnitude derived from the HETDEXspectrum (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2), which has a limiting magnitude of ∼25𝐴𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The magnitude threshold between votes for [O ii] and for Ly𝛼 are defined by a pair of lines whose parameters are set to optimize the segregation of those two samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Objects with 𝑔 magnitudes fainter than the upper line of Figure 5 are more likely to be Ly𝛼, while those brighter than the lower line of Figure 5 are more likely to be [O ii].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The classification of objects, defined by their SzAS spectroscopic redshifts, lying between these two regimes is uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The optimization over the slope and intercept parameters of these lines was performed using a simple grid search that maximizes the Ly𝛼 accuracy in one case and the [O ii] accuracy in the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' While an MCMC fit could be more precise, given the uncertainties in the data features and the desire to avoid over-fitting to the specific test set, the grid search is preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The accuracy is defined as: accuracy = 1 − 𝜉 + 𝜖 Ω (23) where 𝜉 is the number of "true" Ly𝛼 ([O ii]) detections (here as the spec-z counterparts in the SzAS test sample), that are not identified by the selection, 𝜖 is the number of incorrectly classified Ly𝛼 ([O ii]) detections, and Ω is the total number of Ly𝛼 ([O ii]) classified detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Here, "true" is assumed as the catalog based spectroscopic redshifts (§4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The two lines are defined as: 𝑔+ 𝑇 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='10 × 10−3𝜆 + 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0, (𝑙𝑜𝑤𝑒𝑟𝑙𝑖𝑛𝑒, 𝐹𝑖𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5) (24) 𝑔− 𝑇 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='26 × 10−3𝜆 + 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1, (𝑢𝑝𝑝𝑒𝑟𝑙𝑖𝑛𝑒, 𝐹𝑖𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5) (25) where 𝑔+ 𝑇 is the faint magnitude threshold, 𝑔− 𝑇 is the bright magnitude threshold, and 𝜆 is the wavelength (Å) of the an- chor emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' We also define upper (faint) and lower (bright) bounds for the measured 𝑔 magnitude of each detection (𝑔+ and 𝑔−, respectively) based on the propagated errors of the HETDEX spectroscopically-measured 𝑔-band magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The votes and their weights for this criterion are summa- rized in Table 4 with Figure 5 showing the segregation of Ly𝛼 and [O II] with Eqns 24 and 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As the 𝑔 magnitude becomes brighter, the voting weights for Ly𝛼 decrease and those for [O ii] increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Large anchor line EW𝐿𝑦𝛼 favor Ly𝛼 and small EW𝐿𝑦𝛼 favor [O ii].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' With the exception of spectra associated with objects having faint 𝑔 magnitudes, those spectra with an- chor line EW𝐿𝑦𝛼 (with error) between 15 Å and 30 Å receive no vote either way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Contamination of Ly𝛼 by [O II] for the down-selected SzAS is low with Ly𝛼 comprising 97% of the detections above the Neutral region in Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Conversely, Ly𝛼 represents only 44% within the Neutral region, where no vote is cast, and 14% below it, where the vote is cast for [O II].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Disqualifications Disqualification conditions are a set of special classifica- tions and data integrity issues that can either contribute ad- ditional weighted votes against a Ly𝛼 classification or, in extreme cases, completely override the P(Ly𝛼) results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Meteor: If the detections has a possible classification as a meteor (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4, a vote against Ly𝛼 is added with a weight equal to the strength of the meteor classification (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 - 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Given its potentially large weight, this vote can be dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Regardless of the final result of the vote, the "meteor" label is attached to the detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Bad Pixel Flat: If a bad pixel flat is indicated by pixel- to-pixel variations or pixel flux values outside the ac- ceptable range for an emission-line on that part of the CCD, then the emission line may be entirely due to, or at least enhanced by, this artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The detection will thus receive a vote against Ly𝛼 with a weight equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 plus the sum of the relative weights of those fibers contributing to the spectrum that have a bad pixel flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 22 Davis, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3500 3750 4000 4250 4500 4750 5000 5250 5500 Observed [Å] 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 g g Limit [O II] Ly Neutral Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The apparent magnitude (error ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1) and equivalent width vote, by itself, is highly effective at segregating Ly𝛼 from [O ii] against the assessment sample here (SzAS, §4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The Neutral region is defined by the lines of Eqns (24) and (25) as the lower and upper bounds respectively, and extends from 3727Å to the red edge of the HETDEX spectral window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Ly𝛼 emitters represent 97% of the down-selected SzAS above the Neutral region, 44% inside the Neutral region, and 14% below the Neutral region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The total weight for this vote is between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' However, if the sum of the fiber weights exceeds a threshold, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='50 by default, the entire P(Ly𝛼) vote is disqualified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Independent of the vote, the bad pixel flat flag is associated with the detection and shown on the ELiXer report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Duplicate Fibers: If duplicated fibers (identified by repeated fiber identifiers or identical flux and error data vectors) appear in the detection spectra, the P(Ly𝛼) vote is disqualified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This is an indication of a data reduction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Grossly Negative Spectrum: If less than 10% of the wavelength bins contain non-negative integrated flux values, the spectrum is considered "grossly negative" and suggests some issue in the reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In this case, the detection and the P(Ly𝛼) vote is disqualified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Poor Observation: If the seeing FWHM is worse than a threshold (3′′ by default) or the throughput response, as defined by (Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021), is less than a threshold (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='08, by default), the input observation is considered too poor to make a meaningful classification attempt and the vote is disqualified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Bad Dither Norm: If the dither-to-dither normaliza- tion (Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021) for the detection is above a threshold (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0× by default), a potentially severe ob- Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Apparent Magnitude and EW Votes Condition Vote Weight 𝑔− > 𝑔+ 𝑇 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='50 𝑔− 𝑇 < 𝑔− < 𝑔+ 𝑇 < 𝑔+ and 𝐸𝑊− > 80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='50 𝑔− 𝑇 < 𝑔− < 𝑔+ 𝑇 < 𝑔+ and 𝐸𝑊− > 30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='30 𝑔− 𝑇 < 𝑔− < 𝑔+ 𝑇 < 𝑔+ and 𝐸𝑊+ ≤ 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='25 𝑔− < 𝑔− 𝑇 < 𝑔+ < 𝑔+ 𝑇 and 𝐸𝑊− > 80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='30 𝑔− < 𝑔− 𝑇 < 𝑔+ < 𝑔+ 𝑇 and 𝐸𝑊− > 30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='15 𝑔− < 𝑔− 𝑇 < 𝑔+ < 𝑔+ 𝑇 and 𝐸𝑊+ ≤ 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='40 𝑔+ < 𝑔− 𝑇 and 𝐸𝑊− > 80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='25 𝑔+ < 𝑔− 𝑇 and 𝐸𝑊− > 30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='10 𝑔+ < 𝑔− 𝑇 and 𝐸𝑊+ ≤ 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='50 else no vote NA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='00 Note— Summary of apparent magnitude and equivalent width votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The conditions are ordered such that the logical evaluation results in at most one unique vote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If no conditions are met, there is no vote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The apparent 𝑔 magnitude becomes brighter moving down the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝑔+ 𝑇 is the upper (faint) 𝑔 threshold as a function of 𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝑔− 𝑇 is the lower (bright) 𝑔 threshold as a function of 𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝑔+ is the upper bound (faint) 𝑔 for the detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝑔− is the lower bound (bright) 𝑔 for the detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝐸𝑊+ is the upper bound restframe EW in Å, assuming Ly𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝐸𝑊− is the lower bound restframe EW in Å, assuming Ly𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' servation or reduction issue is indicated and the vote is disqualified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Best-z and Q(z) Unless there is a serious error or a disqualification (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='8), ELiXer assigns a single, best guess redshift, "Best-𝑧", along with a quality score, "𝑄(𝑧)", as an indication of the confidence in that redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The assignment of the Best-𝑧 incorporates all prior information and analysis including the P(Ly𝛼), cat- alog spec-𝑧 and phot-𝑧, and any multi-line redshift solutions (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The 𝑄(𝑧) value takes on a continuous value between 0 and 1, with 1 meaning "full confidence" and 0 meaning "no confidence" (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=', the redshift is effectively a guess).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Where the P(Ly𝛼) analysis is limited only to a determination as to whether the emission line is Ly𝛼 the Best-𝑧 logic attempts to fully specify the redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In the ideal scenario, there are mul- tiple high-SNR emission lines within the HETDEX spectrum, each corresponding to a known line at a consistent redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In such a case, the Best-𝑧 is clear and the corresponding 𝑄(𝑧) is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Such objects are rather rare, but they do define the starting benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The Best-𝑧 is set as (1) the redshift from a qualified multi- line spec-𝑧 solution, (2) the Ly𝛼 redshift when there is no ELiXer 23 spec-𝑧 solution but P(Ly𝛼) favors Ly𝛼, or typically, (3) the [O II] redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In the last case, the redshift can be set to C III] or Mg II when the line is broad and occurs at a wavelength within the HETDEX spectral window where no other strong feature is expected to be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The 𝑄(𝑧) confidence value is set based on the Best-𝑧 se- lection condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' It is primarily a function of the P(Ly𝛼) value and the multi-line solution score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝑄(𝑧) is maximized by P(Ly𝛼) when P(Ly𝛼) is near 0 or 1 and minimized when P(Ly𝛼) is near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The effect of the multi-line solution score, on the other hand, is a monotonic increase with the multi-line solution score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝑄(𝑧) may also have penalties and caps im- posed on it based on specific circumstances and flags, such as the detection being near a spatially extended, bright ob- ject or if the various continuum estimates (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2) disagree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If the multi-line solution and P(Ly𝛼) agree, the 𝑄(𝑧) score increases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' if the two measures disagree, the 𝑄(𝑧) score is decreased based on the relative difference between the multi- line solution and P(Ly𝛼) strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The selection logic and 𝑄(𝑧) assignment is summarized in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Since the majority of HETDEX objects are faint, with a single detected emission line, most (∼ 80%) receive a 𝑄(𝑧) score less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 with ∼ 35% in the lowest 𝑄(𝑧) bin (0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' These are still usually correctly classified as is shown in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4 and 5, but rely on less evidence and thus have a low 𝑄(𝑧) value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Clustering/Neighbor Redshift Matching In the low-surface brightness outer regions of spatially re- solved galaxies, HETDEX detections with low, PSF-weighted line fluxes (commonly arising from faint H II regions and planetary nebulae) may be incorrectly classified by ELiXer as Ly𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' To address this issue, ELiXer can optionally com- pare a detection against other nearby HETDEX detections and look for consistencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' When invoked, ELiXer examines all HETDEX emission line detections within 15′′ (by default) of the current detection under consideration, and tests for 𝑔-band magnitudes brighter than 23𝐴𝐵 with matching observed emis- sion line(s) of higher line score (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The presumption, which is borne out in testing, is that the brighter, higher- scoring detections are (1) better centered on the object and (2) more likely to receive the correct classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The re- quirement to match the observed emission line wavelength(s) in addition to the on-sky proximity helps preserve the clas- sification of background objects with lines of sight passing near the brighter, foreground source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' When more than one match is found, the highest scoring redshift solution is se- lected and if the selected object is brighter and higher scoring than the current detection’s solution, that neighbor’s classi- fication is used as a replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In other words, faint, low scoring detections can be assigned the more secure redshift of Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Best-𝑧, 𝑄(𝑧) Summary Condition Best-𝑧 𝑄(𝑧) Strong, multi-line spec-𝑧 solution consistent with P(Ly𝛼) multi-line spec-𝑧 4-5★ Strong, multi-line spec-𝑧 solution not consistent with P(Ly𝛼) multi-line spec-𝑧 0-3★ Weak, multi-line spec-𝑧 solution consistent with P(Ly𝛼) multi-line spec-𝑧 2-4★ Weak, multi-line spec-𝑧 solution not consistent with P(Ly𝛼) multi-line spec-𝑧 1-3★ P(Ly𝛼) only, ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='7 Ly𝛼 3-4★ P(Ly𝛼) only, ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 Ly𝛼 0-2★ P(Ly𝛼) ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 with single, broad emission line [O ii] Mg II, C III] 0-1★ P(Ly𝛼) only, ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 [O ii] 0-1★ P(Ly𝛼) only, ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3 [O ii] 0-2★ Note— Summary of the Best-𝑧 and 𝑄(𝑧) logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Specific values (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0) of the 𝑄(𝑧) are not shown as they depend on details omitted, but are expressed as these qualitative descriptors: 5★ (∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0), 4★ (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='80), 3★ (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='50), 2★ (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='35), 1★ (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='25), 0★ (∼0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' an immediately adjacent, brighter, higher scoring "neighbor" detection when they share matching observed-frame emission lines and are assumed to represent different detections of the same object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' When this update is made, the altered detection is marked with a flag and the detection ID number of the matching neighbor detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This clustering has a relatively small effect, modifying less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5% of all HETDEX emission line detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The algorithm does not link nor otherwise combine the individual detections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' all detections remain uniquely reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' TESTING AND RESULTS All the effort made toward classification is effectively mean- ingless without appropriate testing and a selection of a rea- sonable spec-𝑧 assessment sample (SzAS) against which to test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As HETDEX is a large and unique survey with no pre- selection of targets, it is impossible to collect an overlapping observational dataset of known redshifts of even remotely similar size (in terms of numbers of unique astrophysical ob- jects) and continuum depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Beyond polling experts for clas- sifications based on visual inspection, and comparing ELiXer results against those of simulated objects, the best we can do is match HETDEX sources against spectroscopic redshift catalogs produced by other surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 24 Davis, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The assessment sample for this work is a composite of matched HETDEX detections from the public, archival cat- alogs described in Section 2 and in Mentuch Cooper (ApJ accepted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In all cases, these are spectroscopic redshifts only;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' no photo-𝑧 estimates are used in this assessment sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For the catalog provided redshifts, source matching to HETDEX is based on sky position and apparent magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The catalog source position must be inside or within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='′′5 of the edge of the SEP aperture associated with the HETDEX detection if an aperture match is made (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2), or within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='′′75 of the HET- DEX position if the object is fainter than 𝑔 = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 and no SEP aperture is matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The catalog matched spectroscopic redshifts are accepted as true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The assessment sample is down-selected to only those detections fainter than 𝑔 = 22 with redshifts that match any of the emission lines in Table 2 to within ±4 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Though the magnitude distribution still significantly skews to brighter objects, this filtering helps refine the selection to better align with the more common, fainter HETDEX detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The result is a dataset consisting of 834 [O II] emission lines, 384 Ly𝛼 lines, and 402 "Other" lines, including C IV, C III], Mg II, and H𝛽 as reported in the SzAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Each redshift cor- responds to a unique HETDEX detection, however, these are not necessarily unique galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For brighter, extended galaxies there can be more than one overlapping HETDEX emission line detection, and where there are multiple obser- vations covering the same position, the same galaxy may be detected more than once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Since ELiXer operates on each HETDEX detection individually, this is as intended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Definitions For the remainder of this work, we make the following definitions: Accuracy: The number of agreements between the ELiXer assigned classification and the SzAS classi- fication divided by the number of ELiXer detections of that classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' A match is counted if the rest- frame wavelengths from the HETDEX observed wave- length and the SzAS and ELiXer assigned redshifts agree within ±4 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Recovery: A fraction roughly equivalent to complete- ness, but with no correction made for survey biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Here we refer to the number of detections of a partic- ular emission line identified by the ELiXer software that are matched 1:1 to that of the SzAS divided by the number of those emission lines in the SzAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Contamination: The fraction of detections within some defined range that are incorrectly classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This may be further refined to the fraction of misclassifications by a particular emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For example, we will discuss the contamination in the Ly𝛼 sample by [O II] as a function of 𝑔-magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Accuracy can be slightly under reported for broad, noisy lines where the fitted line center can be offset from the true center and where winds and radiative transfer effects can create a significant velocity offset from the systemic redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The ±4 Å allowance covers all but the most extreme cases so the impact is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Accuracy and contamination are direct inverses and, for any given emission line, they necessarily sum to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Accuracy and recovery are similar, but differ by the base divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For the recovery of detections, any contam- ination of one emission line comes at the direct cost to the recovery of another emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Conversely, the recovery counts of an emission line is also one minus the sum of the contaminations of all other emission line types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Notice that the relationship does not directly hold for recovery and con- tamination rates, as each of those rates have different divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Calibration Testing and calibration are combined in a highly iterative process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' ELiXer is executed on the detections of the test dataset, but with catalog matching spec-𝑧 and phot-𝑧 turned off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' That is, for the test runs, ELiXer does not include or con- sider the catalog reported spectroscopic redshifts that would, in a standard run, factor into the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The ELiXer output, specifically the P(Ly𝛼) values and the redshifts, are then compared to the test sample and checked for contami- nation, recovery, and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Disagreements between the ELiXer results and the assessment sample are examined, and manual adjustments to the individual votes and voting weights (§3, and §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 in particular), are made as warranted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Considerations against over-tuning and potentially incorrect test sample redshifts are addressed with deliberately loose fitting, low-order segmentation thresholds and by varying the composition of the test sample by creating random and targeted (in apparent magnitude, line FWHM, observation field, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=') subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The process is repeated until there is good agreement (generally, matching 90-95% or better) between the ELiXer assigned redshifts and the test sample redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' With the focus on P(Ly𝛼) as the primary classification metric and with its flexible threshold selection, what constitutes "good" agreement is somewhat subjective but is also highly adaptable to the specific scientific needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For example, the stacking of spectra to measure Lyman Continuum in Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (2021) is very sensitive to contamination but does not specifically require a highly complete sample and so utilizes a P(Ly𝛼) selection of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='8 and greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' On the other hand, the 𝐻(𝑧) and 𝐷 𝐴(𝑧) precision goals for the primary HETDEX science is less sensitive to contamination but needs to be largely complete (Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Farrow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021) and a P(Ly𝛼) threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5, or even lower, is more appro- ELiXer 25 priate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Additional Testing To supplement the catalog spec-𝑧 testing, several other test- ing and feedback efforts are actively used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Though the me- chanics vary, all provide checks on the ELiXer classifications with targeted detection subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As with the SzAS, the detec- tions where these alternate methods and ELiXer disagree are manually inspected and adjustments to the ELiXer classifica- tion algorithm(s) are made as warranted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' These supplementary efforts fall into two categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The first are automated machine learning classifiers, both super- vised and (sometimes) unsupervised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' These are all in early development and explore various classification frameworks, with both T-distributed Stochastic Neighborhood Embedding (tSNE) (van der Maaten & Hinton 2008) and Autoencoder Neural Network (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2014) techniques showing good promise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The second category relies on manual, visual vetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The first efforts focused on HETDEX collaboration experts and university students (after receiving training).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' A more recent science outreach effort has opened classification and gen- eral exploration to the public in a citizen science project on Zooniverse (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='zooniverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='org/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' One work- flow of the Dark Energy Explorers (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='zooniverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' org/projects/erinmc/dark-energy-explorers) project (House & et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' in prep) tasks its citizen scientists to classify HETDEX detections as either being at low-𝑧 ("Nearby Galaxy or Star") or possibly high-𝑧 ("Distant Galaxy or nothing") using a re- duced ELiXer report that contains only sections of 2D fiber cutouts, single band (𝑔 or𝑟) photometric imaging, and a Gaus- sian fit to the emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Each detection receives 15 re- sponses with the aggregate classification reported as the mean of those responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Even with this reduced information, these broad categories match with the ELiXer classification more than 92% of the time with House, et al, estimating 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='7% con- tamination and 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='7% recovery of high-𝑧 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As with the other methods, select disagreements between ELiXer and Zooniverse are reviewed for potential classification failures by ELiXer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Results Summary A comparison of the ELiXer classification/redshift assign- ments with those of the SzAS are summarized in Figure 6 and in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Figure 6 breaks out the contamination and recovery rates by 𝑔-magnitude, with the counts of each type shown as a reference in the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' When there are very few classifications of a given type, such as faint [O II] and "Other" lines, the accuracy and recovery rates are not mean- ingful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Against the SzAS, ELiXer performs very well on Ly𝛼 and [O II] classifications, but is challenged by the "Other" emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As will be discussed later, the elevated con- tamination in the Ly𝛼 detections at bright magnitudes is a function of the biases in the SzAS as compared to the HET- DEX survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Table 6 summarizes the cumulative performance of several different Ly𝛼/[O II] segregation methods against the SzAS identifications of the Ly𝛼 or [O II] line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This down-selection is made so that the comparisons of the ELiXer P(Ly𝛼) method (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5) at several selection thresholds is equitable, as 20 Å equivalent width cut and the P(LAE)/P(OII) method do not classify lines other than Ly𝛼 and [O II] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' It is clear that each method is an effective classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Except at the extreme thresholds, the P(Ly𝛼) methods produce the lowest contami- nation and highest recovery rates, with P(Ly𝛼) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 yielding a good balance of contamination and recovery fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This is the default input for the ELiXer Best-𝑧 assignment (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Given the biases in the SzAS for bright objects and AGN, though, these results cannot be directly applied to the whole of HETDEX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' However, a correction for these biases is made and discussed later in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' We also caution that the detec- tions in the SzAS factor significantly in the calibration of the votes and weights of the P(Ly𝛼) metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Although efforts are made to avoid over-fitting, these results could still be less reflective of HDR3 in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The contamination rate of Ly𝛼 by [O II] is effectively flat as a function of the observed wavelength of the emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' However, the recovery rate of Ly𝛼 sources trends lower as the observed wavelength moves redward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' At the blue end of the HETDEX spectral range, 𝜆obs ≲ 4200Å, the recovery rate is ∼97%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' in the middle range, 4200 ≲ 𝜆obs ≲ 4800, the rate is ∼91%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' and at the red end, 4800 ≲ 𝜆obs the rate is ∼81%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This is an effect of larger numbers of faint [O II] emitting galaxies and fewer numbers of LAEs in their respective higher redshift regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' These [O II] galaxies are more similar in appearance to LAEs based on several of the metrics used in ELiXer, 𝑔 and 𝑟 magnitudes, angular size, and even EW and line width to a lesser extent (see Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='7 and their figures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The observed emission line wavelength factors in the related votes help keep the Ly𝛼 contamination rate flat and low, but at the cost of the loss of some LAEs to [O II] classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As shown in Table 6, this can be tuned to improve the Ly𝛼 recovery rate at the expense of a higher contamination rate as dictated by particular science needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' DISCUSSION As can be seen from Figure 7, the sample we use for spectro- scopic assessment, SzAS, is highly biased to brighter detec- tions, somewhat biased to broader lines, and contains an over representation of emission lines other than Ly𝛼 and [O II], as compared to HETDEX as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' At its bright end, the sample is under-abundant in [O II] and over-abundant in Ly𝛼 26 Davis, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Ly𝛼 vs [O II] Segregation on Assessment Sample Method Ly𝛼 Contamination Ly𝛼 Recovery Ly𝛼 rest EW > 20Å 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='708 P(LAE)/P(OII) default 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='763 P(LAE)/P(OII) optimized 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='724 P(LAE)/P(OII) ELiXer 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='705 P(Ly𝛼) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='752 P(Ly𝛼) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='797 P(Ly𝛼) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='903 P(Ly𝛼) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='926 P(Ly𝛼) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='940 Note—The cumulative performance of various methods against the SzAS down-selected to only include [O II] (834 detections) and Ly𝛼 (384 detections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This allows a fairer comparison of P(Ly𝛼) (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5) to the first three methods, which do not consider other lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The SzAS is biased to bright objects, with an over representation of AGN, so these results do not directly translate to the larger population of HETDEX detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' An adjustment for these biases are made and discussed later in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Additionally, though efforts are made to avoid over-fitting to the SzAS, its detections significantly contribute to the determination of the votes and weights of the P(Ly𝛼) metric, so these results may not be as representative when considering all HETDEX detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 1Leung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (2017) 2Modified P(LAE)/P(OII) optimized used in ELiXer (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3) 3Default input to Best-𝑧 logic (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='6) with the reverse at the faint end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Since these spectroscopic redshifts come from existing archival surveys (§4) and spec- troscopy is historically expensive, it stands to reason that the available spectra would favor brighter, rarer objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' An ex- pansion of the SzAS is underway in collaboration with DESI ((Jelinsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Levi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2019)) which will provide higher spectral resolving power (R∼2000-5000) and a redder wavelength coverage (3600-9800Å) to selected HETDEX de- tections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This will increase the number of faint (𝑔 > 25) spec- tra in future assessment samples and bring their distributions more in line with HETDEX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' While not completely devoid of faint objects, the SzAS contains a smaller fraction of its detections in the faintest bins compared to the full HETDEX sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This is not unexpected and is not a significant issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Given the methodology of the classification, ELiXer is likely to classify anything fainter than 𝑔 ∼25 as an LAE in the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='9 < 𝑧 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 redshift range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' While there are certainly [O II] emission-line galaxies with 𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 and 𝑔 > 25, if we assume that this emission has a rest-frame equivalent width of less than 20 Å, then [O II] can be expected to be, at most, ∼ 3 × 10−17 erg s−1 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This maximum value is ∼2× fainter than the 50% flux limits 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
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+page_content='00 Contamination Ly [O II] Other 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
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+page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='00 Recovery 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
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+page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 g 0 100 200 300 400 Number in Bin Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Performance summary of ELiXer classification and red- shift assignment vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' the SzAS in 𝑔-magnitude bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' ELiXer does very well with Ly𝛼 and [O II], as intended, but struggles with the "Other" lines, such as C IV 𝜆1550, C III] 𝜆1909, and Mg II 𝜆2800 Note that the results for the faintest bin for Ly𝛼, the faintest 2 bins for [O II] and the faintest 4 bins for Other lines, denoted with open markers and dotted lines, have too few SzAS counts to be meaning- ful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The high contamination rate in Ly𝛼 at brighter magnitudes is a result of the biases in the SzAS and is discussed in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' for HETDEX (Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021), making it unlikely that HETDEX would even detect an [O II] emission line from such a galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Thus the reduced fraction of 𝑔 ≳ 25 objects in the SzAS, compared to HDR3, is largely moot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Nevertheless, the other biases cannot be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' While an uncorrected assessment sample can serve as a develop- ment test set and provide reasonable limits on the expected contamination, recovery, and accuracy rates for ELiXer clas- sifications, a correction is needed to extrapolate to the entire HETDEX emission line sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Bias Correction to the Full HETDEX Catalog Given the clearly biased distribution of the assessment sam- ple as compared to the full HETDEX catalog, it is prudent to apply some measure of correction before extending the results from the SzAS to the full catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The correction chosen is ELiXer 27 23 24 25 g 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 Ly Fraction in Bin HDR3 ELiXer SzAS ELiXer SzAS Reported 23 24 25 g 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
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+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 OII Fraction in Bin HDR3 ELiXer SzAS ELiXer SzAS Reported 23 24 25 g 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 Other Fraction in Bin HDR3 ELiXer SzAS ELiXer SzAS Reported 18 20 22 24 g 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='6 Fraction of Total HDR3 Bright (Excluded) HDR3 g Limit = 25 SzAS 10 20 30 Line FWHM [Å] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='40 Fraction of g > 22 HDR3 SzAS 0 1 2 3 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='20 Fraction of g > 22 HDR3 ELiXer SzAS ELiXer SzAS Reported Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Summary of the ∼ 1600 emission line detections in the Spec-𝑧 Assessment Sample (SzAS) compared to the ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5×106 detections in the HETDEX Data Release 3 (HDR3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The top panels show the relative fraction of Ly𝛼, [O II], and Other emission line detections as a function of 𝑔-magnitude, as classified by ELiXer and as reported by archival spec-𝑧 measurements in the SzAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The ELiXer reported classifications represent more of an "apples to apples" comparison, as it is clear that the SzAS is skewed towards brighter magnitudes and is significantly overabundant in Other emission line detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The Ly𝛼 and [O II] distributions are very similar fainter than about 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5𝐴𝐵, but diverge at the brighter end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The lower left panel illustrates the bright bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The lower-center panel shows an excess in the SzAS for broad emission lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' though not explicitly shown here, these broad lines are predominantly Ly𝛼, C IV 1549 Å, and C III] 1909 Å and originate from brighter, probably AGN, objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The lower right panel echoes the over abundance of the Other emission lines, showing an increase in the fraction of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 ≲ 𝑧 ≲ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 detections, likely AGN, compared to HDR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' relatively simplistic and, as will be shown a little later, has effectively no impact on the overall sample results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As seen earlier, the SzAS dataset is subdivided into Ly𝛼, [O II], and Other emission line detections, and each subset is binned by 𝑔 magnitude from 22𝐴𝐵 to 25𝐴𝐵 in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5, with the last bin containing all detections fainter than the 25𝐴𝐵 flux limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The contamination (by type) for each of the three classifications is computed against the SzAS in each 𝑔 bin as defined in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' To correct for the population biases in the SzAS compared to the full HDR3 sample, we consider the contamination rates in the SzAS to be functions of the per bin fractions of the contaminant, and the target type as classified by ELiXer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This allows us to use the same ELiXer classification rates in the full HDR3 sample as a correction to the SzAS rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The applied correction to the SzAS values then is: 𝐶′ 𝑖, 𝑗 = �∑︁ 𝑘 𝐶𝑖, 𝑗,𝑘 × 𝐸𝐻, 𝑗,𝑘 𝐸𝑆, 𝑗,𝑘 × 𝑁𝐻,𝑖,𝑘 � ∑︁ 𝑘 𝑁𝐻,𝑖,𝑘 (26) where: 𝐶′ 𝑖, 𝑗 is the corrected contamination rate of the target type 𝑖 (Ly𝛼, [O II], or Other) by contaminant 𝑗, such that 𝑖 ≠ 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝐶𝑖, 𝑗,𝑘 is the directly computed contamination rate in the SzAS per 𝑔-magnitude bin, 𝑘 (matching the bins in Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 28 Davis, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝐸𝐻,𝑖,𝑘 is the ELiXer classification fraction of the target type in HDR3 per 𝑔-magnitude bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝐸𝑆, 𝑗,𝑘 is the ELiXer classification fraction of the con- tamination type per 𝑔-magnitude bin in the SzAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 𝑁𝐻,𝑖,𝑘 is the number of target ELiXer classifications in HDR3 per 𝑔-magnitude bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' An additional simple correction is also applied to help ac- count for false positive (FPN) detections caused by noise interpreted as an emission line by the HETDEX line-finding algorithm (Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' These are random fluctua- tions in the PSF weighted spectrum from thermal electrons in the CCDs, stray photons, read noise, etc, that happen to scatter up and pass the various filtering thresholds in the line-finding code and masquerade as low SNR emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' They do not represent real astrophysical sources but when interpreted as such, they map to random locations in (RA, Dec, z)-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As the candidate emission line SNR increases toward 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5, the incidence of these FPN rapidly approaches zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As dis- cussed later in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4, this has only a minimal impact on the HETDEX cosmological measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As an approximate correction, the ELiXer classification ratios in Eqn (26) for HDR3 are modified by assuming 30% of all detections with SNR < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 and 15% of all detections with 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 ≤ SNR < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 are false positives and simply removing those from all summed counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Early indications are that the true FPN rates may be significantly less than this, (Mentuch Cooper ApJ accepted), so we believe the as- sumed FPN rates are overestimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 g 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='020 Cumulative [O II] Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Frac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Bias + FPN Corrected HDR3 SzAS g Limit Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Cumulative (bright to faint) contamination of Ly𝛼 by [O II] as a function of 𝑔 magnitude using the default ELiXer con- figuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The Bias + FPN Corrected HDR3 curve attempts to compensate for the biases in the SzAS (compared to all of HDR3) and account for false positives in the low-SNR regime (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 g 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='35 Cumulative Other Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Frac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Bias + FPN Corrected HDR3 SzAS g Limit Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Cumulative (bright to faint) contamination of Ly𝛼 by emission lines other than [O II] for 𝑔 > 22 using the default ELiXer configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The FPN + Bias Corrected HDR3 curve attempts to compensate for the biases in the SzAS and account for false positives due to random noise in the low-SNR regime (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The much larger contamination rate in the SzAS is largely driven by confusion of Ly𝛼 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' C III] and C IV, where the AGN population is significantly over represented (see Figure 7, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2 and §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Performance Figures 8 and 9 show the cumulative (bright to faint) con- tamination fraction of Ly𝛼 by [O II] and all "Other" lines respectively, both for the SzAS and for the 𝑔 > 22 HDR3 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Table 7 reports the cumulative contamination rates from those two figures (highlighted by bold type face), pro- vides summary information on the contamination in [O II] and the "Other" lines, and gives the accuracy and recovery rates for all discussed line types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Note that the values for the SzAS corresponding to Table 6 are slightly different, since the detections for that table are down selected to only in- clude Ly𝛼 and [O II] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Overall, ELiXer performs extremely well in mitigating the contamination in the Ly𝛼 classification, and excels at the faint end against the primary contaminant, [O II].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This is what ELiXer is tuned to do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' At brighter magni- tudes, non-[O II] contaminants are more problematic, though they represent only a small fraction of the total HETDEX dataset (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For the HETDEX data releases, the final classification of these objects is assisted by the supplemental program, Diagnose (Zeimann & et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' in prep) (see also §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The cumulative fractional contamination from [O II] has a peak between 𝑔 ∼ 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 and 𝑔 ∼ 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0, where the numbers of [O II] and Ly𝛼 emitters are most similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The total contami- nation rate sits at only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3% for the SzAS even with the [O ii] emitters outnumbering LAEs in that sample by more than 2:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For HDR3, when corrected for the SzAS distribution bias and predicted false positives from noise, the predicted contamination rate is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' While this already meets the HETDEX requirements, planned ELiXer enhancements, in- ELiXer 29 cluding updated Ly𝛼 and [O II] luminosity functions for the P(LAE)/P(OII) analysis (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4) and run-time phot-𝑧 fitting, should further decrease the contamination rate and improve overall accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The cumulative fractional contamination of Ly𝛼 from all other lines in the SzAS is substantial at 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This, how- ever, is significantly inflated due to the over representation of AGN and C III] and C IV emission lines in the SzAS (Fig- ure 6, upper right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' When projected onto the HDR3 distribution and corrected for the SzAS distribution bias and noise driven false positives, this cumulative contamination fraction falls to a predicted 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='8% for the full HDR3 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This is even better than the [O ii] contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' However, given the large correction from the SzAS results (Figure 9), it is prudent to estimate a worst case contamination by these other lines by alternate means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' These misclassifications in the SzAS are dominated by C iii] and C iv and are characterized by bright magnitudes and large line widths – median 𝑔 = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 and median emission line FWHM = 22 ± 8 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Using these properties as a guide, we select the fraction of HDR3 detections with emission line FWHM > 14 Å and 𝑔 < 23, yielding 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='8% of HDR3 detections, of which we assume 1/3 are misclassified as Ly𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' With 47% of detections classified as Ly𝛼, we then estimate the worst case contamination rate by Other lines at 4% (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' : 1 3 · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='058 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='47 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='04).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' While this is 5× the Bias + FPN Corrected contamination rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='8%, this is still relatively small and the impact is far less than that of [O ii] contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The small scale clustering of [O ii] emitters projects to large scale clustering when misinterpreted as higher-z Ly𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This is greatly dimin- ished with C iii] and C iv as the contamination sources shift to higher redshift and scales proportionally to the square of the ratio of the co-moving angular diameter distances (Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Farrow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This means the HETDEX cosmology is some 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5× less sensitive to C iii] contamination than [O ii] contamination and can tolerate ∼13% (or ∼16% for C iv) at the desired uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' So, even the worst case contamination is well within the required tolerances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Addi- tionally, the focus for ELiXer has been on the largest con- taminant, [O II], as the contamination rate of other lines is expected to decrease with future improvements targeting their identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Overall, the ELiXer accuracy is good in the HDR3 dataset, while that in SzAS is poorer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The weaker performance in the SzAS set is due to the bright-magnitude and broad-emission line biases in the SzAS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' this is where ELiXer does not per- form as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The stronger (estimated) accuracy in the full HETDEX population is bolstered by the large numbers of faint end detections that are highly biased towards being Ly𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The results for ELiXer recovery rates are similarly mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The numbers are good for Ly𝛼 and [O II], which are, by far, the most common emission lines found by HETDEX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The recovery of all other emission lines is rather poor, and is largely an issue of the default behavior of the classification algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' When there is only a single line in a HETDEX spectrum, ELiXer heavily weights the various Ly𝛼 / [O II] segregation methods which, as stated above, assume no con- tamination other than [O II] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In this case, ELiXer delivers a binary result, Ly𝛼 vs not-Ly𝛼, at the expense of all other emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Moreover, when analyzing particularly broad lines, ELiXer favors Ly𝛼 (often suggestive of an AGN) over [O II];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' this also leads to the enhanced contamination of Ly𝛼 by such "Other" lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Additional identification metrics such as limited run-time phot-𝑧, spectral slope, and multi-Gaussian fits, could help improve these rates and will be explored in future versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' A preliminary evaluation of an assessment sample ex- panded with ∼ 1000 DESI provided spectroscopic redshifts, 3/4 of which are for 𝑔 > 24 objects, is consistent with the HETDEX classification results of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The resulting assessment sample more closely matches the HETDEX mag- nitude and emission line distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' After the observations are complete, the full, detailed results will be presented in Landriau et al, in preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Missing AGN and LBGs Since ELiXer largely relies on equivalent width to clas- sify most single-line spectra, the program currently does not perform well with Ly𝛼 emitting objects that are not classi- cal LAEs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=', broad-line AGN and Lyman-break galaxies (LBGs) which may have small Ly𝛼 equivalent widths (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=', Shapley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Moreover, ELiXer can also fail to find some of the broad emission lines associated with the AGN, which can result in misclassifications that would otherwise be correctly assigned by the multi-line redshift solutions (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This is particularly noticeable in the bright end of the SzAS (Figure 9), which has a disproportionately large number of AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Moreover, in AGN, ELiXer can confuse Ly𝛼 with C III] when C III] is the only significant emission line in the HETDEX spectral window (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='96 ≲ 𝑧 ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='25) or with C IV when the line fit to C III] fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Other approaches are taken to identify and recover AGN missed or misclassified by ELiXer (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2022) and future updates to ELiXer should improve upon its classification performance with these emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' ELiXer also struggles to classify low Ly𝛼 EW LBGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' On the whole, given their name-defining detection methodology (Guhathakurta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Madau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Steidel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 1996), LBGs tend to be more massive and more evolved than the typical LAE (Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Kornei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Jose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Vargas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Steidel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2018) and, consequently, may contain more dust to inhibit the escape of Ly𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' While some LBGs also meet the definition of an LAE and are likely to be detected and correctly identified 30 Davis, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Cumulative Classification Performance for HDR3 Metric SzAS Bias + FPN Corrected Ly𝛼 Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='723 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='981 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='034 Ly𝛼 Recovery 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='892 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='991 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='033 Ly𝛼 Contamination by [O II] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='012 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='001 Ly𝛼 Contamination by Other 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='264 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='008 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='001* [O II] Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='890 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='965 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='034 [O II] Recovery 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='972 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='970 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='034 [O II] Contamination by Ly𝛼 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='021 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='001 [O II] Contamination by Other 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='071 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='014 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='001 Other Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='892 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='916 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='032 Other Recovery 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='509 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='294 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='010 Other Contamination by Ly𝛼 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='006 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='001 Other Contamination by [O II] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='081 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='078 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='003 Note—The cumulative performance of the ELiXer classifications on the SzAS and predictions for the full HDR3 dataset for detections with 𝑔 > 22 and using the default ELiXer configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The Bias + FNP Corrected column corrects for the sample biases in the SzAS dataset and for false positives in the full HDR3 dataset, assuming 30% false positive rate below emission line SNR of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 and 15% rate between 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 < SNR < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The values in the first column are slightly different than those in Table 6 since that table is down selected to only consider Ly𝛼 and [O II] detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The bold type face rows correspond to the cumulative data points in the right-most (faintest) bins in Figure 8 and Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' ∗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='04 worst case estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' See §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2 for a discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' as such by ELiXer, the more massive objects may often be confused with low-𝑧 [O II] emitters or even overlooked com- pletely if they exhibit weak Ly𝛼 emission or Ly𝛼 absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' While relatively few in number compared to LAEs, the more massive LBGs do represent a highly biased mass tracer and are of value to HETDEX, so it is desirable to recover and correctly identify as many of them as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This means using methods that do not use equivalent width as their pri- mary discriminant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' To that end, several machine learning approaches (both supervised and unsupervised) are being ex- plored, as are direct enhancements to ELiXer that incorporate additional classification methods, such as run-time photo-𝑧 estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Contamination from Noise As stated earlier, ELiXer assumes an emission line detec- tion is real, and not the result of noise or an artifact of the data reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As the SNR of an emission line detection decreases, it does become more likely that the feature is the result of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' However, unlike real, incorrectly classified emission lines, false positives from noise are not expected to cluster (they occur in random spectra at random wavelengths and thus map to random sky positions at random redshifts) and should only increase the uncertainty in the HETDEX cos- mological measurements and not introduce a bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As such, it is of lesser concern than misclassifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Nevertheless, as described earlier, a (likely overly) aggressive false positives correction (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1) is used for Figures 8 and 9 and for Table 7 to better estimate the classification performance of ELiXer against the full HETDEX dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Separate efforts to identify the noise driven false positive rate include repeat observations of low SNR sources (based on the premise that random noise will not cause a repeat de- tection at the same position and wavelength;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Mentuch Cooper (ApJ accepted)) and various machine learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Their goal is to allow a more accurate model of contamination from noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Uncertainties The performance of ELiXer presented in the prior sections are shown without statistical uncertainties, though some un- certainty is implicit in its predictions for the whole of HET- DEX Data Release 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' For the SzAS results in this work, the ELiXer classifications have been taken as absolute, as the quality of the classifica- tions has not yet been calibrated to a proper probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' (This is a planned enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=') Since classifications are based on votes and weights, some of which have an MCMC element with a weak dependency on the initial random seed vectors, individual executions can occasionally result in a different classification due to conditions falling just to either side of a threshold (though the quality score (𝑄(𝑧)) is generally unaf- fected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Similarly the catalog reported spec-𝑧 values are taken as truth, and matching against the reported values is done as described in §4, with a ±4 Å allowance, independent of the uncertainties in the spec-𝑧 or the fitted emission line center (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Nevertheless, many realizations of ELiXer classification runs compared against the SzAS have shown the results to be highly stable and repeatable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In projecting the SzAS results onto the full HDR3 dataset, a few additional sources of uncertainty arise, such as the as- sumed false positive rate, which is binned only as a function of SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' However, as with the SzAS, we assume the ELiXer classifications to be strictly categorical and the reported frac- tions subject only to rounding error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Anticipated expansion and improvements to the SzAS, including better matching to the HETDEX magnitude and emission line width distribu- tions, will help address the systematics between the SzAS and the full HETDEX sample beyond the simplified corrections of §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As rough estimate on the uncertainties in the accuracy, recovery, and contamination rates reported for HDR3, we ELiXer 31 use the fraction of detections that are most susceptible to classification changes as described in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This is effectively captured by the largest factor in the classifications, P(Ly𝛼), where P(Ly𝛼) is least certain and least stable against change due to randomness in sampling (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=', near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' As 7% of HDR3 detections have 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4 < P(Ly𝛼) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='6, we assume a ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5% uncertainty on those rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' SUMMARY As the primary emission line classifier for HETDEX, ELiXer must produce quality redshift identifications that are highly accurate, complete, and with minimal contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' With a resolving power ranging from 750–950, HETDEX cannot split the [O II] doublet, so object classification must rely heavily on continuum information combined with equiv- alent width distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' By incorporating improvements to established Ly𝛼/[O II] separation mechanics, from the 20 Å equivalent width cut (Gronwall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2011) to the P(LAE)/P(OII) ratio (Leung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2017), and by com- bining additional partitioning techniques, ELiXer produces classifications that outperform the HETDEX science require- ments for Ly𝛼 contamination by its principle low-𝑧 interloper, [O II] 3727 Å, while providing a good recovery rate (Table 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The lower than required 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2% contamination of Ly𝛼 by [O II] affords the option to loosen the project’s strict classification thresholds in exchange for gains in the Ly𝛼 recovery fraction or completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Though they occupy a small fraction of HETDEX emission line detections, lines other than [O II] 3727Å, such C III] 1909 Å, and C IV 1549 Å represent a larger source of Ly𝛼 contamination in the biased SzAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' However, as described in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2, these lines are not expected to produce a significant clustering signal or bias in the 𝑧 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4 measures of 𝐻(𝑧) and 𝐷 𝐴(𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Regardless, planned enhancements to ELiXer and a larger spectroscopic redshift test sample (more aligned with the HETDEX distribution) will improve these classifications and further reduce Ly𝛼 contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The HETDEX project is continuing to work towards re- ducing the rate of false positive detections as a function of the emission line signal-to-noise ratio (Mentuch Cooper ApJ accepted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Early indications suggest the contamination from noise is small above the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='8-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 SNR acceptance threshold for detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Regardless, these noise driven false positives should only add white noise to the LAE cluster signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Al- though this increases the uncertainty in the HETDEX mea- surements, it should not introduce specific features in the galaxy power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' ELiXer continues to evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Future enhancements and revised voting criteria will be tested against expanded as- sessment samples drawn from forthcoming data releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This will improve the current classification capabilities, en- abling new and higher precision science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Although ELiXer is designed for and calibrated to HETDEX, the methodology developed in this work can be adapted to other low-resolution, narrow wavelength range spectroscopic surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' ACKNOWLEDGMENTS The authors thank the anonymous reviewer for the helpful feedback which assisted in improving this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' HETDEX is led by the University of Texas at Austin McDonald Observatory and Department of Astronomy with participation from the Ludwig-Maximilians-Universität München,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Max-Planck-Institut für Extraterrestrische Physik (MPE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Leibniz-Institut für Astrophysik Potsdam (AIP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Texas A&M University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The Pennsylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Institut für Astrophysik Göttingen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The University of Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Max-Planck-Institut für Astrophysik (MPA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' and Missouri University of Science and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In addition to Institutional support, HETDEX is funded by the National Science Foundation (grant AST-0926815), the State of Texas, the US Air Force (AFRL FA9451-04-2-0355), and generous support from private individuals and foundations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Observations were obtained with the Hobby-Eberly Tele- scope (HET), which is a joint project of the Univer- sity of Texas at Austin, the Pennsylvania State Univer- sity, Ludwig-Maximilians-Universität München, and Georg- August-Universität Göttingen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The HET is named in honor of its principal benefactors, William P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Hobby and Robert E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Eberly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' VIRUS is a joint project of the University of Texas at Austin, Leibniz-Institut für Astrophysik Potsdam (AIP), Texas A&M University (TAMU), Max-Planck-Institut für Extrater- restrische Physik (MPE), Ludwig-Maximilians-Universität Muenchen, Pennsylvania State University, Institut fur Astro- physik Göttingen, University of Oxford, and the Max-Planck- Institut für Astrophysik (MPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In addition to Institutional support, VIRUS was partially funded by the National Science Foundation, the State of Texas, and generous support from private individuals and foundations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The authors acknowledge the Texas Advanced Comput- ing Center (TACC) at The University of Texas at Austin for providing high performance computing, visualization, and storage resources that have contributed to the research results reported within this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' URL:http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='tacc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='edu The Institute for Gravitation and the Cosmos is supported by the Eberly College of Science and the Office of the Senior Vice President for Research at the Pennsylvania State University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' KG acknowledges support from NSF-2008793.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This research benefits from the open-source projects Python (Van Rossum & Drake 2009), astropy (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2018b), numpy (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
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+page_content=', & et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' in prep, n ELiXer 35 APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' EXAMPLE ELIXER DETECTION REPORTS In this Appendix, we include two ELiXer detection reports as examples of those used for visual inspection and diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The first, Figure 10, is a somewhat unusual HETDEX LAE: it has a very high emission line SNR, it is matched to a source contained in multiple catalogs (§2), and has several photometric and spectroscopic redshift determinations, and it lies in an area of sky with deep HST imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' It is presented to illustrate the various sections within an ELiXer report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The second, Figure 11, is more representative of the typical HETDEX LAE and is shown here to that end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 1 2 3 4 5 6 7 8 9 11 10 12 13 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Example ELiXer detection report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This is a somewhat uncommon example selected to illustrate elements that are not always present for an individual detection, such as the classification label, warning flags, multiple catalog references, and photometric redshift PDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Descriptions of the bulleted features are provided below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Summary - From left to right: (1) computed Equivalent Width of the emission line in the rest-frame of Ly𝛼, the combined continuum estimate (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4), (2) P(LAE)/P(OII) (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4) and 68% confidence interval using the combined continuum estimate, (3) P(Ly𝛼) score (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5), (4) Quality score for the Best-𝑧 redshift (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='6), (5) Best-𝑧 redshift, (6) Classification labels (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4) if any, (7) Error/Warning Flags5 if any;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' in this example, there is a warning flag indicating a small disagreement in the 𝑔-magnitudes calculated from the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 5 Flags are not explicitly described in this work but are part of data release documentation EW: 196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='8±13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5AP(LAE)/P(0II): 1000 1800 P(Lyα): 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='999 Q(z): 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='61 z: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2688 Lyα Flags:0x00002000 2022-03-16 19:23:49 Version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 ID:3007744560(3007744560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='pdf) 2DSpe Pixel Flat Smoothee With Sky 0bs: 20200513v014_3007744560 x, y: 252, 39 Primary Spec_Slot_IFU_AMP: 418_057_064_RU 12 Je-17x2A F=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4" T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='147 N=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='41 A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='88 10 RA,Dec (214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='776810,52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='825974) 入 = 3973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='74A FWHM = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='8(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5)A LineFlux = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='90(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='23)e-16 Cont(n) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='50(±i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='00)e-18 8 Cont(w) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='00(±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='50)e-19 (gmag 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='07 24:3) R EWr = 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='00(±80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='00) (w: 360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='00(±110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='00))A S/N = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='6) x2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2) P(LAE)/P(01I): 1000 188 (w: 1000 1888) 18 28 3920 3940 0965 086 4000 4020 LyA z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2688 3 0II Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0660 CII { 0% 0 SilV 0 e-17x2A 10 3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000 5100 5200 5300 5400 5500 CANDELS/EGS : Possible Matches = 4 (within +/- 3 3") P(LAE)/P(0II): 1000 188 (f606w) Fiber Positions Lineflux Map CFHTLS/Meg(27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3) u CFHTLS/Meg(27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3) g ACS WFC(30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0) f606w ACS WFC(30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0) f814w CFHTLS/Meg(27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3) z 4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2 2 2 2 2 0 0- 0 2 2 4 2 4 0 4 0 4 0 arcsecs s/b: 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='20 +/- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='148 m:24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2 re:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4"s:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content="2' m:24." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='7 re:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3" s:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3" m:24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='8 re:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4" s:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3" m:24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='1 re:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0" s:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3 EWr:205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='PLAE:1000 EWr: 297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' PLAE: 1000 Phot z PDF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='282259" Separation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='312691" 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='63637" 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='997 Match score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='847 214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='776848, 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='825899 214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='776846, 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='825890 214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='776664, 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='825246 RA, Dec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='27416 N/A N/A Spec z 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='101 N/A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='886 Photo z 280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='00(±15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='00)A 350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='00(±15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='00)A 4100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='00(±720.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='00)A Est LyA rest-Ew 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='42(-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='05,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='05)f606w 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='49(24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='47,24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='52) r 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='32(-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='21,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='26)f606w mag 1000 1888 1000 18 1000 1888 P(LAE)/P(OII) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 --.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Oll z(virus) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0659746 : LyA z (VIRUS) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2687736 Davis, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Timestamp + Version - Displays the date and time of the creation of this report and the ELiXer version number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Detection Details - A block of information about the HETDEX observation and the emission line detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' From top to bottom: (1) Detection ID number and file name, (2) Observation ID, (3) IFU+Amp address of the fiber nearest the detection center, (4) ’F’ = seeing FWHM in arcsecs, ’T’ = effective throughput at 4540 Å, ’N’ = dither to dither normalization, ’A’ = aperture correction (divisor), (5) J2000 equatorial coordinates of the PSF weighted detection center in decimal degrees, (6) emission line wavelength center and FWHM, (7) integrated emission line flux, (8) continuum estimate (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2) from the spectrum within ±40 Å of the line center, (9) continuum estimate and 𝑔-magnitude from the full width of the spectrum, (10) equivalent width in Ly𝛼 rest-frame with the continuum estimates from (8) and (9) respectively, (10) signal-to-noise ratio and 𝜒2 of the emission line fit, (11) P(LAE)/P(OII) using the continuum estimates from (8) and (9) respectively, (12) redshifts assuming Ly𝛼 and [O ii], (13) multi-line emission line identification (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3), if one is selected, with its quality score, name, rest-wavelength, redshift, and equivalent width in its own restframe using the continuum estimate in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 2D Fiber Cutouts - 5 × 3 grid of cutouts within ±40 Å of the detection line center in the spectral direction and ±1 fiber in the CCD direction6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The left most column is the pre-smoothing cutout with all rectifications and sky subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The center column is the pixel flat, with any significant deviations marked in red (none in this example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The right most column is the same as the left most column but smoothed with a 2 × 2 Gaussian filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The top row (highlighed in black) is the weighted sum of all contributing fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The rows below (blue, green, orange, red) are the highest four fibers as weighted by PSF modeled flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The values (in very small print) to the left of the grid represent (from top to bottom): the normalized fiber weight in the PSF, the 𝜒2 of the fit to the fiber profile, and the fiber number on the CCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The values (in very small print) to the right of the grid represent (from top to bottom): the fiber center distance to the detection center (in arcsecs), the CCD pixel coordinate of the fiber center, the exposure date, the observation number and exposure number for that date, and the IFU spectrograph ID, amplifier ID, and fiber number on that amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Key CCD Region - ±10 fibers in the CCD direction and ±40 Å in the spectral direction around the detection center for the fiber nearest the detection, shown before and after sky subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 1D Line Fit - the 1D emission line fit to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This matches the gold highlighted section in the full 1D spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Values are integrated fluxes in 2 Å wide bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 1D Spectrum - the full 1D spectrum as integrated fluxes in 2 Å wide bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The gray background gives the estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The two vertical gray-hashed bars point out the two strongest sky-lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The gold highlighted region is the anchor emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Any other colored regions, if present, highlight other spectral lines that support the selected multi-line redshift solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The other red labels ("NV","SiII","SiIV","CIV","HeII") mark the positions of other possible lines in the spectrum, assuming the anchor line is Ly𝛼;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' in this spectrum, none of these confirming lines are detectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The colored labels above the spectrum represent the positions of other common lines if the anchor emission line were one of the features listed below the spectrum with the matching color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Main Catalog Summary - displays the name of the catalog with the deepest imaging used in the report, along with the number of potential catalog counterparts (if any) and the P(LAE)/P(OII) found from the continuum estimate of the listed filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Fiber Positions - the footprint of all fibers contributing to the detection plotted over a stacked image from the catalog with the deepest imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The four colored fibers match those in the 5 × 3 grid in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Fibers with a dashed outer ring are at the edge of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The PSF weighted center of the detection is marked with a red cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Lineflux Map - wavelength collapsed flux intensity map summing over ±3𝜎 from the emission line center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The values under the image are an estimate of significance based on the flux inside a 1′′ radius aperture and the standard deviation of flux inside a 5′′ to 7′′annulus, corrected for area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The lower section of the Lineflux Map in this example is blank as that region happens to fall off the edge of the CCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Imaging Stamps - postage stamp cutouts of the deepest imaging available to ELiXer, shown in increasing order from blue (left) to red (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Only the bluest five filters are shown, though more may be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Overplotted are colored 1′′ per 6 Fibers adjacent on the CCD are not necessarily adjacent on sky ELiXer 37 side squares corresponding to the positions of possible catalog counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The top three (see (12) are shown in blue, red, and green, with all others displayed in white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In this example, the blue and red squares overlap, so only the red is obviously visible, but they mark the same object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The overplotted ellipses are SEP identified sources (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' A gold ellipse marks the object selected by ELiXer as the most likely counterpart, while all other objects are marked in white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' If the bounding ellipse is dashed, then it has been expanded to be a 1′′ radius circle for visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The text above each cutout indicates the catalog name, and the approximate imaging depth and the filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The values under the cutouts correspond to the gold aperture and are: ’m’ = aperture magnitude, ’re’ = the effective radius of the ellipse in arcsecs, ’s’ = separation between the center of the aperture and the HETDEX PSF weighted center in arcsecs,"EWr" - the equivalent width in the Ly𝛼 rest-frame using the aperture magnitude as the continuum estimate, "PLAE" - P(LAE)/P(OII) using the aperture magnitude as the continuum estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' All values are computed for 𝑔 and 𝑟 (or equivalent) filters, but not always for other bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Catalog Counterparts - basic information on up to the top three most likely catalog counterparts, based on magnitude and distance, which correspond to colored squares on the Imaging Stamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In this example, the blue, red, and green objects are actually the same source, but reported from different catalogs, and their corresponding squares in the Imaging Stamps overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The "Separation" is the distance in arcsec between the HETDEX detection position and the catalog reported position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' This offset can sometimes be sizeable, especially for extended objects where the catalog reports a surface brightness center and the HETDEX detection is more toward the object’s edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The reported P(LAE)/P(OII) value uses the catalog’s reported bandpass magnitude as the continuum estimate, not the aperture magnitude from the Imaging Stamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Catalog z PDFs - if available, shows the photometric redshift PDFs, color coded to match the top three catalog counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' In this example, there is no PDF for the red counterpart (from the CFHTLS catalog), so only blue and green PDFs are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Circles, again with a matching color, mark the reported spectroscopic redshift, if available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The green dashed line represents the redshift if the emission line is [O II], while the red dashed line shows the same for Ly𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Since the anchor line in this example is Ly𝛼, there is a precise match with the spec-𝑧 and a close match with the phot-𝑧 for the object marked in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' 38 Davis, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' The ELiXer report of a typical HETDEX LAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Note that this region of sky has fewer and shallower imaging data, and more limited catalog data compared to Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' It is included here as a counter to the more illustrative, but less common example of Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' EW: 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='8±24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2A P(LAE)/P(0II): 1000 1000 P(Lyα): 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='999 Q(z): 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='36 z: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3484 Lyα 2022-03-15 17:20:04 Version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='5 ID: 3013462701 (3013462701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='pdf) Pixel Flat Smooth With Sky 0bs: 20210729v012_3013462701 x, y: 290, 948 Primary Spec_Slot_IFU_AMP: 410_024_039_LL e-17x2A F=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2" T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='129 N=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='45 A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='89 RA,Dec (220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='022415,52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='361256) ^ = 4070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='56A FWHM = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='4(±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='3)A LineFlux = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='20(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='32)e-16 Cont(n) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
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+page_content='5) x2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
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+page_content='3484 4 0II Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0919 CII 人 silv Silv 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
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+page_content='7) g HSC(26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2) r 4 4 0 0 4 2 arcsecs s/b: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
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+page_content='146 m:24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='7 rc:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='6"s:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='2" m:25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='0 re:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='9" $:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='6" EWr: 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' PLAE: 1000 EWr:156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='PLAE:1000 Separation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='631354" 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='994 Match score 220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='022128, 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='361252 RA, Dec N/A Phot z plot not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content=' Spec z N/A Photo z 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='00(±30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='00)A Est LyA rest-Ew 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='39(24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
+page_content='14,24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQf1f5A/content/2301.01799v1.pdf'}
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+Restarts subject to approximate sharpness:
+A parameter-free and optimal scheme for first-order methods
+Ben Adcock, Matthew J. Colbrook, Maksym Neyra-Nesterenko
+January 9, 2023
+Abstract
+Sharpness is an almost generic assumption in continuous optimization that bounds the dis-
+tance from minima by objective function suboptimality. It leads to the acceleration of first-order
+methods via restarts. However, sharpness involves problem-specific constants that are typically
+unknown, and previous restart schemes reduce convergence rates. Moreover, such schemes are
+challenging to apply in the presence of noise or approximate model classes (e.g., in compressive
+imaging or learning problems), and typically assume that the first-order method used produces
+feasible iterates.
+We consider the assumption of approximate sharpness, a generalization of
+sharpness that incorporates an unknown constant perturbation to the objective function er-
+ror. This constant offers greater robustness (e.g., with respect to noise or relaxation of model
+classes) for finding approximate minimizers. By employing a new type of search over the un-
+known constants, we design a restart scheme that applies to general first-order methods and
+does not require the first-order method to produce feasible iterates. Our scheme maintains the
+same convergence rate as when assuming knowledge of the constants. The rates of convergence
+we obtain for various first-order methods either match the optimal rates or improve on previ-
+ously established rates for a wide range of problems. We showcase our restart scheme on several
+examples and point to future applications and developments of our framework and theory.
+Keywords: First-order methods, Restarting and acceleration, Approximate sharpness, Convex
+optimization, Convergence rates, Inverse problems
+Mathematics Subject Classification: 65K0, 65B99, 68Q25, 90C25, 90C60
+1
+Introduction
+First-order methods are the workhorse of much of modern continuous optimization [6, 10, 24, 59].
+They are widely used to solve large-scale problems because of their excellent scalability and easiness
+of implementation. However, standard first-order methods often converge slowly, for instance, when
+applied to nonsmooth objective functions or functions lacking strong convexity. This has motivated
+a large amount of work on speeding up such methods [11,30,48,55,60,64,65].
+Recently there has been significant interest in using restarts to accelerate the convergence of
+first-order methods [1,13,27,33,34,37,39,44,46,47,49,52,57,61,62,66,68,69,71]. A restart scheme
+repeatedly takes the output of an optimization algorithm instance as the initial point of a new
+instance or “restart”, and additionally may reselect the algorithm parameters before executing the
+2Corresponding author: m.colbrook@damtp.cam.ac.uk
+DAMTP, Centre for Mathematical Sciences, University of Cambridge, UK
+1
+arXiv:2301.02268v1 [math.OC] 5 Jan 2023
+
+new instance. Under the right conditions, the objective error and feasibility gap decay faster for
+the restarted scheme than for the underlying (unrestarted) first-order method.
+However, as discussed below, existing restart schemes either require somewhat restrictive as-
+sumptions in which various constants are known, or attain suboptimal convergence rates. This
+paper overcomes these limitations. We introduce a general restart scheme that applies to a broad
+class of convex optimization problems, generalizes and improves upon various existing schemes, and
+leads to optimal complexity bounds for a wide range of problems.
+1.1
+The problem
+We consider the general convex optimization problem
+min
+x∈Q f(x),
+(1.1)
+where f : D → R is a proper, closed convex function with non-empty effective domain D ⊆ Cn, and
+Q ⊆ Cn is a closed, convex set with Q ⊂ D. Let ˆf denote the optimal value of (1.1) and �
+X ⊂ Q
+denote its set of minimizers, where we assume that �
+X is non-empty.
+Our key assumption is that f satisfies the following approximate sharpness condition
+d(x, �
+X) ≤
+�
+f(x) − ˆf + gQ(x) + η
+α
+�1/β
+,
+∀x ∈ D,
+(1.2)
+for a metric d on Cn and some constants α > 0, β ≥ 1, η ≥ 0. We slightly abuse notation by
+defining d(x, S) := infz∈S d(x, z) for a set S ⊆ Cn. Here, gQ : D → R+ is a function satisfying
+gQ(x) = 0 ⇐⇒ x ∈ Q
+and for any sequence {xm} ⊂ D, d(xm, Q) → 0 implies g(xm) → 0. In this paper, we assume that
+the function gQ is known, but that the constants η, α and β (or a subset thereof) are unknown.
+We refer to gQ as the feasibility gap function and f − ˆf as the objective (function) error.
+To formulate a restart scheme that accelerates an optimization algorithm solving (1.1), we
+assume that f satisfies (1.2), and that we have access to an optimization algorithm Γ : R++ ×
+R++ × D → D that defines a map (δ, ϵ, x0) �→ x, with the property that
+d(x0, �
+X) ≤ δ
+=⇒
+f(x) − ˆf + gQ(x) ≤ ϵ, where x = Γ(δ, ϵ, x0).
+(1.3)
+In essence, for an initial value x0 within distance δ of an optimal solution, the algorithm produces
+an output x that is ϵ-suboptimal, i.e., f(x) − ˆf ≤ ϵ, and ϵ-feasible, i.e., gQ(x) ≤ ϵ, for (1.1).
+Assumption (1.3) is a generic condition that appear in typical convergence analysis of first-order
+methods. In Section 4, we describe various examples of first-order optimization methods that yield
+algorithms satisfying this assumption. See also [66].
+The algorithms Γ we consider in this paper are iterative. We define the cost function CΓ :
+R++ × R++ → N, where CΓ(δ, ϵ) represents an upper bound on the number of iterations Γ needs
+to compute x = Γ(δ, ϵ, x0) for any starting value x0 satisfying d(x0, �
+X) ≤ δ. One can generalize
+this framework to also consider cost in terms of floating point operations or other measures of time
+complexity. It is assumed that CΓ is nondecreasing in its first argument and nonincreasing in its
+second argument. Examples are given in Section 4 for various first-order methods.
+2
+
+1.2
+Motivations
+The assumption (1.2) is much weaker than typical assumptions for acceleration, such as strong
+convexity. It can be considered an approximate version of the sharpness condition considered in [69]
+(see (1.6)). We discuss its links to other error bounds in Section 1.4. There are two key differences
+between (1.2) and sharpness. First, we do not assume that the sharpness condition is exact, i.e.,
+we have an additional η ≥ 0 term that controls the approximation. This is very important in many
+applications and for noisy data, and provides greater robustness of our results. For example, when
+considering sparse recovery, (1.2) covers both noisy measurements and approximately sparse vectors
+[27], which is more realistic than exact sparse recovery from noiseless measurements. Second, we do
+not require iterates of our algorithm to be feasible, and this is captured by the additional feasibility
+gap function gQ. This adds further flexibility and efficiency when selecting the first-order method
+for the restart scheme (e.g., the primal-dual algorithm considered in Section 4.5).
+The other key motivation for this work is that we do not assume knowledge of the constants
+α, β, and η. When these parameters are known, it is relatively straightforward to derive a restart
+scheme. However, the constants are rarely known in practice. For example, sharpness holds for
+general subanalytic convex functions [17], but the proof of this result uses topological arguments
+that are far from constructive. As another example, in a sparse recovery problem, η depends on the
+noise level and the sparsity level of the unknown vector, neither of which are typically known. In
+some applications, one may have bounds for one or more of these constants. Nevertheless, if such
+bounds are loose – for instance, global bounds may be highly pessimistic near minimizers – this
+can lead to inefficient schemes. Our method obviates the need for such bounds. However, it also
+allows the user to input such prior information (e.g., exact values of or ranges for the constants) if
+these are available.
+1.3
+Contributions
+The following theorem, which follows directly from the results presented in Section 3, summarizes
+our main convergence rates result.
+Theorem 1.1. Let α, β and η be (unknown) approximate sharpness constants of f in (1.2).
+Consider Algorithm 2 for fixed a, b > 1, r < 1, α0 > 0, β0 ≥ 1 and the choices of schedule criterion
+and assignment functions described in Section 3.2. Then running Algorithm 2 with
+t ≳ K(ε),
+ε → 0+,
+(total inner) iterations, where K(ε) is given in (3.3), implies that
+f(x(t)) − ˆf + gQ(x(t)) ≤ max{η, ε}.
+Let β∗ = b⌈logb(β/β0)⌉β0. If, in addition, CΓ satisfies
+CΓ(δ, ϵ) ≤ Cδd1/ϵd2 + 1,
+C, d1, d2 > 0,
+(1.4)
+for all δ, ϵ > 0, then we have
+K(ε) ≤ ˆCεd1/β∗−d2 ·
+�
+⌈log(1/ε)⌉ ,
+if d2 ≤ d1/β∗,
+1,
+if d2 > d1/β∗,
+(1.5)
+where ˆC is independent of ε (but depends on r, a, b, α, β∗, α0, β0, d1 and d2). Explicit forms for ˆC
+in (1.5) are given in Section 3.
+3
+
+i
+j
+k
+Unknown α and β
+0
+1
+2
+3
+4
+5
+0
+10
+20
+30
+40
+50
+j
+k
+Known α
+-5
+0
+5
+0
+10
+20
+30
+40
+50
+i
+k
+Known β
+Figure 1: Level curves of h = 50 for the schedule criterion functions h in Corollary 3.3 (left panel), Corol-
+lary 3.4 (middle panel) and Corollary 3.5 (right panel) with c1 = c2 = 2. The level curves describe the search
+order. The red dots show the corresponding indices (i, j, k) in the set defined in (3.4). The index i indicates
+the parameter search value aiα0 for α. The index j indicates the parameter search value bjβ0 for β. The
+height (i.e., k) indicates the total number of inner iterations for a fixed (i, j).
+A few comments are in order. First, note that ε is not a parameter of the algorithm: it is only
+used to describe the algorithm’s behavior as the number of iterations increases. Second, it is possible
+for a problem (1.1) to satisfy the approximate sharpness condition (1.2) for different parameters
+α, β and η, which may give different convergence rates and constant ˆC in (1.5). If so, Theorem 1.1
+says that for a given accuracy threshold ε ≥ η, we can take the best rate of convergence/iteration
+bound over different approximate sharpness constants. Third, Theorem 1.1 does not guarantee a
+decrease of the objective function error below η as ε → 0+. This is quite reasonable in practice.
+For example, in the case of sparse recovery from noisy measurements, η is the magnitude of the
+noise level. Therefore there is little benefit in decreasing the objective function error below η, since
+the error in the recovered vector will generally be O (η). Fourth, the assumption in (1.4) is generic
+for convergence rates of first-order methods. We present some examples in Section 4. The +1 term
+is included in (1.4) since we often have a bound of the form
+CΓ(δ, ϵ) ≤
+�
+Cδd1/ϵd2�
+.
+Finally, the parameters α0 > 0 and β0 ≥ 1 in Algorithm 2 are estimates for the true α, β. If no
+estimates are known, we can set α0 = β0 = 1. We also include the case that either or both of α
+and β are known in our analysis (see the Corollaries in Section 3.2). The parameter r ∈ (0, 1) is a
+scale factor that adjusts the parameters of the first-order method at each restart. As we discuss in
+Section 2, a good choice is r = e−1/d2.
+Our scheme performs a grid search over parameters α, β using the bases a, b > 1. The order of
+the search is based on a so-called schedule criterion (see Definition 3.1 and Fig. 1). This new idea
+allows flexibility depending on which parameters are known and which are unknown, and leads to
+a unified framework for proving convergence results (e.g., using Theorem 3.2). We postpone the
+details until Section 3, but, in particular, this new framework allows us to search over a nonuniform
+grid (Corollary 3.3) that searches more in iteration space as opposed to parameter index space (see
+left panel of Fig. 1). This is key to developing a search method for unknown parameters that does
+not suffer from reduced convergence rates.
+Suppose now that η ≲ ε. When Algorithm 2 is applied with a suitable first-order method, it leads
+to optimal1 complexity bounds for a wide range of different convex optimization problems, without
+1By optimal, we mean optimal in the number of oracle calls to f, its gradient (where appropriate) or suitable
+proximal maps. For the first-order methods we discuss, this number will always be bounded by a small multiple of
+the number of iterations.
+4
+
+50
+40.
+30
+20、
+10、
+0
+0
+-5
+5
+0
+5Objective function class/structure
+Asymptotic bound for K(ε)
+Example method
+L−smooth
+See Definition 4.2
+(NB: must have β ≥ 2)
+β = 2:
+�
+L/α · log(1/ε)
+Nesterov’s method
+d1 = 1, d2 = 1/2
+See Section 4.1
+β > 2:
+√
+L
+α1/β∗ ·
+1
+ε1/2−1/β∗
+(u, v)−smoothable
+See Definition 4.5
+β = 1:
+√
+ab
+α · log(1/ε)
+Nesterov’s method
+with smoothing
+d1 = 1, d2 = 1
+See Section 4.2
+β > 1:
+√
+ab
+α1/β∗ ·
+1
+ε1−1/β∗
+H¨older smooth, parameter ν ∈ [0, 1]
+See Definition 4.8
+(NB: must have β ≥ 1 + ν)
+β = 1+ν:
+M
+2
+1+3ν
+ν
+α
+2
+(1+3ν) · log(1/ε)
+Universal fast
+gradient method
+d1 = (2 + 2ν)/(1 + 3ν)
+d2 = 2/(1 + 3ν)
+See Section 4.3
+β > 1+ν:
+M
+2
+1+3ν
+ν
+α
+2+2ν
+β∗(1+3ν) ·
+1
+ε
+2(β∗−1−ν)
+β∗(1+3ν)
+f(x)=q(x)+g(x)+h(Bx), q is Lq−smooth,
+supz∈dom(h) infy∈∂h(z) ∥y∥ ≤ Lh,
+∥B∥ ≤ LB
+β = 1:
+LBLh+Lq
+α
+· log(1/ε)
+Primal-dual algorithm
+d1 = 1, d2 = 1
+See Section 4.4
+β > 1:
+LBLh+Lq
+α1/β∗
+·
+1
+ε1−1/β∗
+f(x)=q(x)+g(x)+h(Bx), q is Lq−smooth,
+supz∈dom(h) infy∈∂h(z) ∥y∥ ≤ Lh,
+∥A∥ ≤ LA, ∥B∥ ≤ LB,
+Q={x : Ax ∈ C}, gQ(x)=κ infz∈C∥Ax − z∥
+β = 1: κLA+LBLh+Lq
+α
+· log(1/ε)
+Primal-dual algorithm
+with constraints
+d1 = 1, d2 = 1
+See Section 4.5
+β > 1:
+κLA+LBLh+Lq
+α1/β∗
+·
+1
+ε1−1/β∗
+Table 1: Asymptotic cost bounds (as ε ↓ 0 for η ≲ ε) and suitable first-order methods for Algorithm 2 when
+applied to different classes of objective functions. Note that whenever the bound is a polynomial in log(1/ε),
+we have β∗ = β.
+knowledge of α and β. Table 1 summarizes some of these bounds and the following correspond to
+an example for each row:
+• For L-smooth functions (Definition 4.2) with β = 2, a well-known lower bound for the subclass
+of strongly convex smooth functions is O(
+�
+L/α log(1/ε)) [54]. If β > 2 then the optimal
+lower bound is O(
+√
+Lα−1/β/ε1/2−1/β) [53, page 26]. In both cases, we achieve these optimal
+bounds with our algorithm using, for example, Nesterov’s method. This is an improvement
+(by at least a factor of log(1/ε)) over the restart scheme presented in [66].
+• Suppose that the objective function f is Lf-Lipschitz and has linear growth. Such functions
+are (1, L2
+f/2)-smoothable (Definition 4.5). When β = 1, the combination of our algorithm and
+Nesterov’s method with smoothing has complexity O (log(1/ε)). This is an improvement over
+the restart scheme presented in [66], which has complexity O
+�
+log2(1/ε)
+�
+for such functions.
+Similarly, for general (u, v)-smoothable objective functions, we improve (by at least a factor
+of log(1/ε)) on the results over the restart scheme presented in [66].
+• For H¨older smooth functions (see Definition 4.8), the bound in Table 1 matches (with β
+replaced by β∗) the optimal bound from [53, page 26]. This is an improvement (by at least a
+factor of log(1/ε)) over the restart scheme presented in [66].
+5
+
+• There is little work on optimal rates for saddle point problems, a challenge being that there
+are different measures of error (see [70]). Hence we cannot claim that the final two rows of
+Table 1 yield optimal rates. Nevertheless, they yield significantly faster convergence rates
+than unrestarted first-order methods for saddle point problems.
+Finally, it is worth pointing out two straightforward generalization of the assumptions in Sec-
+tion 1.1 where our algorithms and results also hold.
+First, the approximate sharpness condition (1.2) can be further generalized to consider any fixed
+set Y ⊆ D as opposed to �
+X. This is expressed as
+d(x, y) ≤
+�f(x) − f(y) + gQ(x) + η
+α
+�1/β
+,
+∀x ∈ D, y ∈ Y,
+where α, β, η, and gQ are defined the same way as in the (1.2). With a suitable generalization
+of (1.3), much of the work presented here can be extended to this general setting. Note that this
+is of particular interest whenever the exact minimizer of the associated optimization problem is
+not desired. In sparse recovery, the ground truth vector being recovered from noisy measurements
+is often not the minimizer of the associated optimization problem (e.g., see Section 5 or [27]).
+It is sufficient when the recovered vector’s measurements match the original measurements up to
+a noise level. Similarly, when training overparameterized models in machine learning, e.g., deep
+neural networks, a balance between training error and generalization error is preferred as opposed
+to solely minimizing the training error.
+Second, our restart procedure for unknown constants always decreases the sum of the objective
+and feasibility gap functions after each restart. Moreover, we only make use of (1.2) in our analysis
+each time we restart, so it suffices that we only need (1.2) to hold in the sublevel set
+{x ∈ D : f(x) + gQ(x) ≤ f(x0) + gQ(x0)}
+for a starting vector x0 ∈ D.
+1.4
+Connections with previous work
+Recently, there has been a large amount of work on adaptive first-order methods [33, 34, 37, 39,
+52,62,71]. Adaptive methods seek to learn when to restart a first-order method by trying various
+values for the method’s parameters and observing consequences over a number of iterations. A
+catalyst for this body of work was provided by Nesterov [57], where he designed an accelerated (line
+search) method for L-smooth objective functions f (see Section 4.1) with an optimal convergence
+rate O(
+�
+L/ε) without needing L as an input. In the same paper, Nesterov considered strongly
+convex objective functions with a grid search for approximating the strong convexity parameter.
+By narrowing the class of objective functions, this led to an adaptive method with a dramatically
+improved convergence rate (O(log(1/ε)) vs. O(1/√ε)), even without having to know the Lipschitz
+constant or strong convexity parameter.
+The complexity of first-order methods is usually controlled by smoothness assumptions on the
+objective function, such as Lipschitz continuity of its gradient.
+Additional assumptions on the
+objective function such as strong and uniform convexity provide, respectively, linear and faster
+polynomial rates of convergence [55]. Restart schemes for strongly convex or uniformly convex
+functions have been studied in [44, 49, 53, 57]. However, strong or uniform convexity is often too
+restrictive an assumption in many applications.
+6
+
+An assumption more general than strong or uniform convexity is sharpness:
+d(x, �
+X) ≤
+�
+f(x) − ˆf
+α
+�1/β
+,
+∀x ∈ Q,
+(1.6)
+also known as a H¨olderian growth/error bound or a �Lojasiewicz-type inequality.
+For example,
+Nemirovskii and Nesterov [53] linked a “strict minimum” condition similar to (1.6) (with known
+constants) with faster convergence rates using restart schemes for smooth objective functions. For
+further use of �Lojasiewicz-type inequalities for first-order methods, see [7,18,19,36,45]. H¨olderian
+error bounds were first introduced by Hoffman [43] to study systems of linear inequalities, and
+extended to convex optimization in [8,20,21,51,67]. �Lojasiewicz showed that (1.6) holds generically
+for real analytic and subanalytic functions [50], and Bolte, Daniilidis, and Lewis extended this
+result to nonsmooth subanalytic convex functions [17]. However, the proofs of these results use
+topological arguments that are far from constructive. Hence, without further case-by-case analysis
+of problems and outside of some particular cases (e.g., strong convexity), we cannot assume that
+suitable constants in (1.6) are known.
+An example of (1.6) for β = 1 was considered in [68] (see also [16]), where the authors use a
+restarted NESTA algorithm [12] for the exact recovery of sparse vectors from noiseless measure-
+ments. The approximate sharpness condition (1.2) was first considered in [27] for the case of β = 1,
+and known α and η, to allow the recovery of approximately sparse vectors from noisy measurements
+and further related examples. Here the parameter η > 0 is crucial, both in practice and to allow
+analysis. See also [1, 61]. Though similar to the sharpness condition in (1.6), our more general
+assumption in (1.2) differs in two important ways, discussed above. First, we do not assume that
+the sharpness condition is exact (η > 0), and, second, we do not require iterates of our algorithm
+to be feasible (the function gQ). It is also important to re-emphasize that, in this paper, we do not
+assume that the approximate sharpness constants are known.
+The η term in (1.2) is expected and natural. For example, in [28] it was shown that there
+are well-conditioned recovery problems for which stable and accurate neural networks exist, but
+no training algorithm can obtain them.
+The existence of a training algorithm depends on the
+amount/type of training data and the accuracy required. However, under certain conditions, one
+can train an appropriate neural network: [28] links trainability to a special case of (1.2), and links
+the accuracy possible via training to the corresponding η term. In the setting of inexact input,
+the noise parameter appears as a limitation on the ability of an algorithm [9]. These phenomena
+occur even if the algorithm is only expected to work on a restricted class of inputs that are ‘nice’
+or ‘natural’ for the problem under consideration. The results of [9,28] lead to the phenomenon of
+generalized hardness of approximation (see also [38]), where it is possible to obtain solutions up
+to some threshold, but beyond that threshold it becomes impossible. This threshold is strongly
+related to η in the standard cases.
+Most restart schemes are designed for a narrow family of first-order methods, and typically
+rely on learning approximations of the parameter values characterizing functions in a particular
+class (e.g., learning the Lipschitz constant L when f is assumed to be L-smooth, or the constants
+α and β in (1.6)). There are two notable exceptions related to the present paper. First, Roulet
+and d’Aspremont [69] consider all f possessing sharpness, and having H¨older continuous gradient
+with exponent 0 < ν ≤ 1.
+The restart schemes of [69] result in optimal complexity bounds
+when particular algorithms are employed in the schemes, assuming scheme parameters are set
+to appropriate values that, however, are generally unknown in practice. However, for smooth f
+(i.e., ν = 1), [69] develops an adaptive grid search procedure within the scheme to accurately
+approximate the required values, leading to an overall complexity that is optimal up to logarithmic
+7
+
+Notation
+Meaning
+f
+Proper convex function
+D
+Effective domain of f
+Q
+Closed, convex subset of Rn or Cn
+gQ
+Sharpness feasibility gap function, identically zero on Q
+ˆf
+Minimum value of objective function over Q
+�
+X
+Set of minimizers of f
+d
+Metric on Rn or Cn
+η
+Sharpness gap constant
+α
+Sharpness scaling constant
+β
+Sharpness exponentiation constant
+δ
+Distance bound between initial point to optimum points
+ε
+Bound on sum of objective function error and feasibility gap
+ϵj
+Sum of objective function error and feasibility gap at jth restart initial point
+Γ
+Optimization algorithm
+CΓ
+Cost function that outputs the number of iterates
+φ
+Mapping of current algorithm step to parameter subscripts (i, j, k)
+h
+Function defining classes of maps φ as abstract execution order of restart scheme
+χC
+Indicator function of a set C (χC(x) = 0 if x ∈ C, χC(x) = ∞ otherwise)
+∥·∥
+Unless otherwise stated, the Euclidean norm on Cn or the induced 2-norm on Cm×n
+⟨·, ·⟩
+Unless otherwise stated, the Euclidean inner product on Cn
+⟨·, ·⟩R
+Unless otherwise stated, ⟨x, y⟩R = Re (⟨x, y⟩) for x, y ∈ Cn
+R+
+Non-negative real numbers
+R++
+Positive real numbers
+N0
+Non-negative integers {0} ∪ N
+Table 2: Notation used throughout the paper.
+factors. Second, Renegar and Grimmer [66] provide a simple scheme for restarting (generic) first-
+order methods. Multiple instances are run that communicate their improvements in objective value
+to one another, possibly triggering restarts. Their restart scheme only depends on how much the
+objective value has been decreased and does not attempt to learn parameter values. The scheme
+in [66] leads to nearly optimal complexity bounds for quite general classes of functions. This method
+differs quite significantly from ours in that it does not assume an underlying sharpness condition
+(1.6) (although such a condition is used in the analysis to obtain explicit complexity bounds).
+However, as observed previously, by assuming (1.2) we are able to obtain better and essentially
+optimal rates that avoid additional factors of log(1/ε). Moreover, in contrast to [66], our method
+is independent of the total number of iterations, and we do not need to specify the total number
+of iterations in advance. Further, we also address the practical case of approximate sharpness and
+allow the case of infeasible iterates (the convergence analysis of [66] relies on η = 0 and that iterates
+are feasible).
+1.5
+Notation and outline
+For ease of reference, Section 1.5 outlines the notation used throughout the paper. The remainder
+of this paper is organized as follows. In Section 2, we introduce a restart scheme in the case where
+η is unknown, but α and β are known. This transpires to be significantly simpler than the general
+8
+
+Algorithm 1: Restart scheme for unknown η.
+Input
+: Optimization algorithm Γ for (1.1), initial vector x0 ∈ D, upper bound ϵ0 such
+that f(x0) − ˆf + gQ(x0) ≤ ϵ0, constants α > 0 and β ≥ 1 such that (1.2) holds
+(for possibly unknown η ≥ 0), r ∈ (0, 1), and number of restart iterations t ∈ N.
+Output: Final iterate xt approximating a solution to (1.1)
+1 for k = 0, 1, . . . , t − 1 do
+2
+ϵk+1 ← rϵk ;
+3
+δk+1 ←
+� 2ϵk
+α
+�1/β ;
+4
+z ← Γ (δk+1, ϵk+1, xk);
+5
+xk+1 ← argmin {f(x) + gQ(x) : x = xk or x = z};
+6 end
+case. Next, in Section 3 we introduce and analyze the full restart scheme when all three constants
+are potentially unknown. In Section 4, we apply this restart scheme to different problems with
+various first-order methods, leading, in particular, to the results described in Table 1. Next, in
+Section 5 we present a series of numerical experiments illustrating the restart schemes in different
+applications. Finally, we end in Section 6 with conclusions and open problems.
+2
+Restart scheme for unknown η but known α and β
+To formulate a restart scheme within the setup of Section 1.1, observe that the approximate sharp-
+ness condition (1.2) relates d(x, �
+X) to the objective function error f(x)− ˆf and feasibility gap gQ(x).
+The upper bound in the approximate sharpness condition can be used as δ for the algorithm Γ, and
+ϵ set as a rescaling of the previous sum of objective error and feasibility gap f(x)− ˆf +gQ(x). How-
+ever, in practice, we may not know the exact values of the objective error f(x) − ˆf and feasibility
+gap gQ(x). It is, instead, enough to know upper bounds for these quantities.
+We first consider the case where α, β are known, but η is unknown. This simpler case provides
+insight into the solution of the full problem considered in Section 3. We define a restart scheme
+under this assumption in Algorithm 1. Using (1.2) and Γ, it is easy to see inductively that for any
+t with ϵt ≥ η, Algorithm 1 produces iterates x0, x1, . . . , xt ∈ D that satisfy
+f(xk) − ˆf + gQ(xk) ≤ ϵk,
+d(xk, �
+X) ≤
+�
+f(xk) − ˆf + gQ(xk) + η
+α
+�1/β
+≤
+�ϵk + η
+α
+�1/β
+≤
+�2ϵk
+α
+�1/β
+,
+0 ≤ k ≤ t.
+(2.1)
+In addition, the total number of inner iterations used in Algorithm 1 is at most
+t−1
+�
+k=0
+CΓ
+��2ϵk
+α
+�1/β
+, ϵk+1
+�
+.
+Under further assumptions about the function CΓ, we can show that the iterates produced by the
+restart scheme yield linear (if d2 = d1β) or fast algebraic (if d2 > d1β) decay of f(xk) − ˆf + gQ(xk)
+in k down to a finite tolerance proportional to η. Hence, this property holds for both the objective
+error f(xk) − ˆf and feasibility gap gQ(xk). We state and prove this in the following theorem. Note
+that these additional assumptions are not arbitrary and will appear in our examples later.
+9
+
+Theorem 2.1. Consider Algorithm 1 and its corresponding inputs. For any ε ∈ (0, ϵ0), if we run
+Algorithm 1 with t ≥ ⌈log(ϵ0/ε)/ log(1/r)⌉, then
+f(xt) − ˆf + gQ(xt) ≤ max{η, ε}.
+(2.2)
+Suppose, in addition, that for all δ, ϵ > 0, CΓ satisfies
+CΓ(δ, ϵ) ≤ Cδd1/ϵd2 + 1,
+C, d1, d2 > 0.
+Then the total number of iterations of Γ needed to compute an xt with (2.2) is at most
+�log(ϵ0/ε)
+log(1/r)
+�
++ C2d1/β
+αd1/βrd2 ·
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1−r⌈log(ϵ0/ε)/log(1/r)⌉|d2−d1/β|
+1−r|d2−d1/β|
+·
+1
+ϵd2−d1/β
+0
+,
+if d2 < d1/β,
+�
+log(ϵ0/ε)
+log(1/r)
+�
+,
+if d2 = d1/β,
+1−r⌈log(ϵ0/ε)/log(1/r)⌉|d2−d1/β|
+1−r|d2−d1/β|
+·
+1
+εd2−d1/β ,
+if d2 > d1/β.
+(2.3)
+Note that the cases in (2.3) match in the limit d2 − d1/β → 0.
+Proof of Theorem 2.1. The statement of the theorem is unchanged if we assume that ε ≥ η. Hence,
+we may assume without loss of generality that ε ≥ η. Let s = ⌈log(ϵ0/ε)/ log(1/r)⌉, then ϵs−1 =
+rs−1ϵ0 ≥ ε ≥ η. It follows that we are in the regime where (2.1) holds and hence
+f(xs−1) − ˆf + gQ(xs−1) ≤ ϵs−1,
+d(xs−1, �
+X) ≤
+�2ϵs−1
+α
+�1/β
+.
+Then by line 4 of Algorithm 1 and the choice of s, we have
+f(z) − ˆf + gQ(z) ≤ ϵs ≤ ε,
+z = Γ(δs, ϵs, xs−1).
+Due to the argmin taken in Algorithm 1, (2.2) follows. The total number of iterations, T, needed
+to reach such an xs is bounded by
+T ≤
+s−1
+�
+k=0
+CΓ
+��2ϵk
+α
+�1/β
+, ϵk+1
+�
+≤ s + C
+s−1
+�
+k=0
+(2ϵk)d1/β
+αd1/βϵd2
+k+1
+= s + C2d1/β
+αd1/βrd2
+s−1
+�
+k=0
+1
+ϵd2−d1/β
+k
+.
+In the case that d2 = d1/β, then ϵd2−d1/β
+k
+= 1 and we obtain
+T ≤ s + C2d1/β
+αd1/βrd2 s =
+�
+1 + C2d1/β
+αd1/βrd2
+� �log(ϵ0/ε)
+log(1/r)
+�
+.
+If d2 ̸= d1/β, we use that ϵk = rkϵ0 and sum the geometric series to obtain
+T ≤
+�log(ϵ0/ε)
+log(1/r)
+�
++ C2d1/β
+αd1/βrd2
+1 − r⌈log(ϵ0/ε)/log(1/r)⌉(d1/β−d2)
+1 − rd1/β−d2
+1
+ϵd2−d1/β
+0
+.
+(2.4)
+If d2 > d1/β, then since ϵ0 ≥ ε/rs−1, we have ϵd2−d1/β
+0
+≥ εd2−d1/βrd2−d1/β/rs(d2−d1/β). Substituting
+this into (2.4) and rearranging yields
+T ≤
+�log(ϵ0/ε)
+log(1/r)
+�
++ C2d1/β
+αd1/βrd2
+1 − r⌈log(ϵ0/ε)/log(1/r)⌉(d2−d1/β)
+1 − rd2−d1/β
+1
+εd2−d1/β .
+The result follows by considering the three separate cases in (2.3).
+10
+
+Remark 2.2 (How to choose r). Suppose that d2 = d1/β and that
+�log(ϵ0/ε)
+log(1/r)
+�
+≤ 2log(ϵ0/ε)
+log(1/r) .
+Using this new bound instead, the total number of iterations T performed by Γ is bounded by
+T ≤
+�log(ϵ0/ε)
+log(1/r)
+�
++ C2d1/β+1
+αd1/β
+log(ϵ0/ε)
+r−d2
+log(1/r).
+Hence T is bounded by an ε-dependent constant times r−d2/ log(1/r), which can be minimized
+analytically by choosing r = e−1/d2. Note that the optimal r here does not depend on the approximate
+sharpness constants. Therefore, one has
+T ≤ ⌈d2 log(ϵ0/ε)⌉ + Ced22d1/β+1
+αd1/β
+log(ϵ0/ε)
+This is meaningful in terms of choosing one less parameter, namely r for Algorithm 1.
+An optimal value of r can also be found for the case d2 > d1/β. However, this optimal value
+depends on ε in a complicated manner. In the limit ε ↓ 0, the optimal choice is
+r =
+�
+d2
+2d2 − d1/β
+�
+1
+d2−d1/β
+,
+which does depend on the sharpness constant β. As d2 − d1/β ↓ 0, this choice converges to the
+choice r = e−1/d2, that is obtained when d2 = d1/β. Similarly, if d2 < d1/β, then the optimal choice
+depends on ε in a complicated manner but converges to the choice r = e−1/d2 as d2 − d1/β ↑ 0.
+In any of these cases, the same argument for optimal r applies to the algorithms in Section 3.
+In the case that β is unknown, we recommend the choice r = e−1/d2.
+♦
+3
+Restart scheme for unknown α, β and η
+In the event that the constants α, β of (1.2) are unknown, we introduce a logarithmic grid search on
+each of α and β, running multiple instances of Algorithm 1, and aggregating results that minimize
+the objective error and feasibility gap. Even if suitable global α and β are known, the following
+algorithm is useful since it also takes advantage of sharper versions of (1.2) that only hold locally
+around optimal points.
+3.1
+The algorithm
+To introduce the algorithm, we need some additional notation and definitions. This will allow us
+to define a new general scheme for logarithmic grid searches, with examples given in Section 3.2.
+Definition 3.1. Consider an infinite subset S ⊆ Z × N0 × N. Let h : R+ × R+ × R++ → R++ be a
+function that is nondecreasing in its first and second arguments, and strictly increasing in its third
+argument. We call such an h a schedule criterion function, or simply a schedule criterion. Given a
+schedule criterion h, an h-assignment over S is a bijection φ : N → S satisfying
+h(|i′|, j′, k′) ≤ h(|i|, j, k)
+⇐⇒
+φ−1(i′, j′, k′) ≤ φ−1(i, j, k),
+(3.1)
+for all (i, j, k), (i′, j′, k′) ∈ S.
+▲
+11
+
+Algorithm 2: Restart scheme for unknown α, β and η in (1.2) via grid search.
+Input
+: Optimization algorithm Γ for (1.1), bijection φ as in Definition 3.1, initial vector
+x(0) ∈ D, upper bound ϵ0 such that f(x(0)) − ˆf + gQ(x(0)) ≤ ϵ0, constants
+a, b > 1, r ∈ (0, 1), α0 > 0, β0 ≥ 1 and total number of inner iterations t ∈ N.
+Output: Final iterate x(t) approximating a solution to (1.1).
+1 Initialize x(0) = x0, Ui,j = 0, Vi,j = 0, ϵi,j,0 = ϵ0 for all i ∈ Z, j ∈ N0;
+2 for m = 0, 1, . . . , t − 1 do
+3
+(i, j, k) ← φ(m + 1) ;
+4
+αi ← aiα0, βj ← bjβ0, U ← Ui,j, V ← Vi,j;
+5
+ϵi,j,U+1 ← rϵi,j,U;
+6
+if 2ϵi,j,U > αi then
+7
+δi,j,U+1 ←
+� 2ϵi,j,U
+αi
+�min{b/βj,1/β0}
+;
+8
+else
+9
+δi,j,U+1 ←
+� 2ϵi,j,U
+αi
+�1/βj;
+10
+end
+11
+if V + CΓ (δi,j,U+1, ϵi,j,U+1) ≤ k then
+12
+z(m) ← Γ
+�
+δi,j,U+1, ϵi,j,U+1, x(m)�
+;
+13
+x(m+1) ← argmin
+�
+f(x) + gQ(x) : x = z(m) or x = x(m)�
+;
+14
+Vi,j ← V + CΓ (δi,j,U+1, ϵi,j,U+1);
+15
+Ui,j ← U + 1;
+16
+else
+17
+x(m+1) = x(m) ;
+18
+end
+19 end
+Let a, b > 1 be constants. Our algorithm employs logarithmic search grids for the unknown
+parameters α and β. Specifically, we consider the values αi = aiα0 for i ∈ Z and βj = bjβ0 for
+j ∈ N0, where we assume that α0, β0 are additional inputs with α0 > 0 and β ≥ β0 ≥ 1. In essence,
+our algorithm applies the restart scheme described in Algorithm 1 with the values αi and βj for
+each i and j. However, it does so according to a particular schedule, specified by the functions h
+and φ. The schedule criterion and assignment together control the execution order of Algorithm 1
+instances for each i and j. Note that the lower bound β0 in the definition of the βj is to capture
+additional knowledge that may be available (see, e.g., the examples in Section 4), and may be set
+to 1 if no such knowledge is available. Similarly, the constant α0 centers the search grid for α and
+can be set to 1.
+Our algorithm is presented in Algorithm 2. It proceeds as follows. At step m ∈ {0, . . . , t − 1}
+it first applies the bijection φ to obtain the tuple (i, j, k) = φ(m + 1). The first two entries give
+the approximate sharpness parameter values αi = aiα0 and βj = bjβ0. The final entry k is a
+counter, which is an upper bound for the total number of iterations used by the algorithm for
+these parameter values. We also have two further counters associated with each double (i, j). The
+counter Vi,j counts the total number of inner iterations of Γ used by the restart scheme with these
+parameters. The second counter Ui,j counts the number of completed restarts (outer iterations)
+corresponding to these parameters.
+Having obtained a tuple (i, j, k) = φ(m + 1), the algorithm proceeds as follows. First, much as
+12
+
+in line 2 of Algorithm 1, it updates the first scaling parameter in line 5. Then, reminiscent of line
+3 of Algorithm 1, it updates the other scaling parameter in lines 6-10. This step is more involved,
+a complication that arises because the true parameter β is unknown.
+The next lines, lines 11-16, are similar to lines 4-5 of Algorithm 1. The main difference is the
+inclusion of the if statement, which is done to control the computational cost. It stipulates that a
+restart be performed (line 12) if the total cost (including the proposed restart) does not exceed the
+counter k (line 11). If this is not the case, then no restart is performed, and the algorithm moves
+on to the next step.
+We now present a general result on this algorithm. It relates the total number of inner iterations
+of Γ used by Algorithm 2 to produce a solution within a desired error to intrinsic properties of the
+schedule criterion function h. With this in hand, we derive explicit bounds for specific choices of h
+in Section 3.2.
+Theorem 3.2. Let S ⊆ Z × N0 × N be an infinite subset, h be a schedule criterion, and φ an
+h-assignment over S. Let α, β and η be approximate sharpness constants of f in (1.2). Consider
+Algorithm 2 for fixed a, b > 1. Define the (unknown) indices
+I = ⌊loga(α/α0)⌋,
+J = ⌈logb(β/β0)⌉
+and the corresponding constants
+α∗ = aIα0 ≤ α,
+β∗ = bJβ0 ≥ β.
+Then for q ∈ N we have
+δI,J,q =
+�
+max
+�
+1, 2rq−1ϵ0
+α∗
+��min{b/β∗,1/β0} �
+min
+�
+1, 2rq−1ϵ0
+α∗
+��1/β∗
+(3.2)
+Now, for any ε ∈ (0, ϵ0), let
+K(ε) := K(ε, α, β, η) =
+⌈log(ϵ0/ε)/ log(1/r)⌉
+�
+q=1
+CΓ (δI,J,q, rqϵ0)
+(3.3)
+and suppose that (I, J, K(ε)) ∈ S.
+Then the total number of inner iterations of Γ needed by
+Algorithm 2 to compute x(t) with
+f(x(t)) − ˆf + gQ(x(t)) ≤ max{η, ε},
+is bounded by the cardinality of the set
+�
+(i′, j′, k′) ∈ S : h(|i′|, j′, l′) ≤ h (|I|, J, K(ε))
+�
+.
+(3.4)
+In addition, if CΓ satisfies
+CΓ(δ, ϵ) ≤ Cδd1/ϵd2 + 1,
+C, d1, d2 > 0,
+(3.5)
+for all δ, ϵ > 0, then we have
+K(ε) ≤
+�log(ϵ0/ε)
+log(1/r)
+�
++ max
+�
+�
+�
+�2ϵ0
+α∗
+�d1 min
+�
+b−1
+β∗ , 1
+β0 − 1
+β∗
+�
+, 1
+�
+�
+� ×
+C2d1/β∗
+αd1/β∗
+∗
+rd2 ·
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1−r⌈log(ϵ0/ε)/log(1/r)⌉|d2−d1/β∗|
+1−r|d2−d1/β∗|
+·
+1
+ϵd2−d1/β∗
+0
+,
+if d2 < d1/β∗,
+�
+log(ϵ0/ε)
+log(1/r)
+�
+,
+if d2 = d1/β∗,
+1−r⌈log(ϵ0/ε)/log(1/r)⌉|d2−d1/β∗|
+1−r|d2−d1/β∗|
+·
+1
+εd2−d1/β∗ ,
+if d2 > d1/β∗.
+(3.6)
+13
+
+Proof. Since ϵi,j,q−1 = rq−1ϵ0 for all q ∈ N, (3.2) must hold by considering the two separate cases
+defining δI,J,q. Similar to the proof of Theorem 2.1, we may assume without loss of generality that
+ε ≥ η. Note that, due to (1.2),
+d(x, �
+X) ≤
+�
+f(x) − ˆf + gQ(x) + η
+α∗
+�1/β
+,
+∀x ∈ D.
+(3.7)
+Now consider the following adapted version of the iterates in Algorithm 1:
+1 for p = 0, 1, . . . do
+2
+ϵp+1 ← rϵp ;
+3
+if 2ϵp > α∗ then
+4
+δp+1 ←
+�
+2ϵp
+α∗
+�min{b/β∗,1/β0}
+;
+5
+else
+6
+δp+1 ←
+�
+2ϵp
+α∗
+�1/β∗;
+7
+end
+8
+z ← Γ (δp+1, ϵp+1, xp);
+9
+xp+1 ← argmin {f(x) + gQ(x) : x = xp or x = z};
+10 end
+It is easy to see inductively that for any l with ϵl ≥ η the above produces iterates {x0, x1, . . . , xl} ⊂
+D satisfying
+f(xp) − ˆf + gQ(xp) ≤ ϵp,
+d(xp, �
+X) ≤ δp+1,
+0 ≤ p ≤ l.
+The only difference to the previous argument for Algorithm 1 is the use of (3.7), and the fact that
+�
+f(xp) − ˆf + gQ(xp) + η
+α∗
+�1/β
+≤
+�2ϵp
+α∗
+�1/β
+≤
+�
+�
+�
+�
+�
+�
+2ϵp
+α∗
+�min{b/β∗,1/β0}
+,
+if 2ϵp > α∗
+�
+2ϵp
+α∗
+�1/β∗ ,
+otherwise.
+Here, we use the fact that β ≥ β0 in the first case.
+In Algorithm 2, each Ui,j plays the role of the index p in the above iterates (i.e., counting the
+number of restarts for a fixed (i, j)) and Vi,j counts the total number of inner iterations that have
+been executed by the algorithm Γ for the approximate sharpness constants given by the double
+index (i, j). The fact that we take minimizers of f + gQ across different indices does not alter the
+above inductive argument, since the argument only depends on bounding the value of f − ˆf + gQ.
+Moreover, since h is strictly increasing in its final argument and satisfies (3.1), the counter index k
+counts successively through N for any fixed (i, j) as the for loop in Algorithm 2 proceeds. It follows
+that if φ(m + 1) = (I, J, k), VI,J + CΓ
+�
+δI,J,UI,J+1, ϵI,J,UI,J+1, x(m)�
+≤ k and ϵI,J,UI,J ≥ η, then
+f(x(m+1)) − ˆf + gQ(x(m+1)) ≤ ϵI,J,UI,J+1 = rUI,J+1ϵ0.
+(3.8)
+Hence, for Algorithm 2 to produce an iterate with
+f(x(t)) − ˆf + gQ(x(t)) ≤ max{η, ε},
+(3.9)
+14
+
+it is sufficient to reach an m with φ(m + 1) = (I, J, k) such that
+k ≥
+⌈log(ϵ0/ε)/ log(1/r)⌉
+�
+q=1
+CΓ (δI,J,q, ϵI,J,q) = K(ε)
+(3.10)
+and execute the resulting restart. To see why this is the case, notice that if k satisfies this inequality,
+then the number of restart iterations performed by the algorithm for the parameter values (I, J)
+is at least ⌈log(ϵ0/ε)/ log(1/r)⌉. Plugging this into (3.8) gives the desired bound (3.9).
+Now consider the set in (3.4). Due to (3.1), we notice that this set is equivalent to
+{(i′, j′, k′) ∈ S : φ−1(i′, j′, k′) ≤ m + 1},
+where φ(m + 1) = (I, J, K(ε)). Notice that if a tuple (i′, j′, k′) belongs to this set, then (i′, j′, k′′)
+belongs to the set for every 1 ≤ k′′ ≤ k′. Thus, the number of terms in this set corresponding to
+the pair (i′, j′) is precisely the total number of inner iterations performed by the algorithm at the
+corresponding parameter values up to step m. We immediately deduce that the cardinality of the
+set (3.4) is a bound for the total number of inner iterations performed by the algorithm across all
+parameter values up to step m, as required.
+To finish the proof, we must show that (3.6) holds under the additional assumption (3.5) on CΓ.
+Suppose first that δI,J,q > 1, then
+CΓ (δI,J,q, rqϵ0) ≤ C
+�2rq−1ϵ0
+α∗
+�d1 min{b/β∗,1/β0}
+(rqϵ0)−d2 + 1
+≤ C
+�2ϵ0
+α∗
+�d1[min{b/β∗,1/β0}−1/β∗] �2rq−1ϵ0
+α∗
+�d1/β∗
+(rqϵ0)−d2 + 1
+= C
+rd2
+�2ϵ0
+α∗
+�d1[min{b/β∗,1/β0}−1/β∗] � 2
+α∗
+�d1/β∗ �
+rq−1ϵ0
+�−d2+d1/β∗ + 1.
+Similarly, if δI,J,q ≤ 1, then
+CΓ (δI,J,q, rqϵ0) ≤ C
+rd2
+� 2
+α∗
+�d1/β∗ �
+rq−1ϵ0
+�−d2+d1/β∗ + 1.
+From (3.10), it follows that
+K(ε) ≤
+�log(ϵ0/ε)
+log(1/r)
+�
++max
+��2ϵ0
+α∗
+�d1[min{b/β∗,1/β0}−1/β∗]
+, 1
+�
+· C2d1/β∗
+αd1/β∗
+∗
+rd2 ·
+⌈ log(ϵ0/ε)
+log(1/r) ⌉−1
+�
+k=0
+1
+(rkϵ0)d2−d1/β∗ .
+We now note that the only difference between this bound for K(ε) and the bound for T in the proof
+of Theorem 2.1 is the factor that maximizes over the terms in curly brackets and the replacement
+of α and β by α∗ and β∗, respectively. The result now follows by using the same arguments as in
+the proof of Theorem 2.1.
+3.2
+Choices of schedule criterion functions and assignments
+The total number of inner iterations of Γ needed for Algorithm 2 depends on the choice of h and
+φ. We examine some choices and state them as corollaries. Examples are shown in Fig. 1.
+15
+
+Corollary 3.3 (Unknown α and β). Suppose that S = Z × N0 × N and let
+h(x1, x2, x3) = (x1 + 1)c1(x2 + 1)c2x3,
+c1, c2 > 1
+be a schedule criterion with h-assignment φ. Then for any ε ∈ (0, ϵ0), running Algorithm 2 with
+t ≥ 2c1c2τ/[(c1 − 1)(c2 − 1)],
+τ = (|⌊loga(α/α0)⌋| + 1)c1(|⌈logb(β/β0)⌉| + 1)c2K(ε),
+where K(ε) is as in (3.3), implies that
+f(x(t)) − ˆf + gQ(x(t)) ≤ max{η, ε}.
+Proof. It suffices to prove that the stated lower bound on t is an upper bound for the cardinality of
+the set (3.4) from Theorem 3.2. We do this by finding an upper bound on the number of solutions
+to nc1
+1 nc2
+2 n3 ≤ τ where n1, n2, n3 ∈ N. By directly counting, the number of solutions is bounded by
+τ 1/c1
+�
+n1=1
+�
+τ
+nc1
+1
+� 1
+c2
+�
+n2=1
+τ
+nc1
+1 nc2
+2
+≤ τ
+∞
+�
+n1=1
+1
+nc1
+1
+∞
+�
+n2=1
+1
+nc2
+2
+.
+We have that
+∞
+�
+n1=1
+1
+nc1
+1
+≤ 1 +
+� ∞
+1
+dx
+xc1 =
+c1
+c1 − 1.
+It follows that the number of solutions is bounded by τc1c2/((c1−1)(c2−1)). Each counted solution
+(n1, n2, n3) defines at most two tuples (i′, j′, k′) in the set (3.4), namely i′ = ±(n1 − 1), j′ = n2 − 1,
+k′ = n3. In reverse, each tuple (i′, j′, k′) of the set (3.4) is always associated with a single solution
+(n1, n2, n3), namely n1 = |i′| + 1, n2 = j′ + 1, n3 = k′. It then follows that that the set (3.4) is
+bounded by 2τc1c2/((c1 − 1)(c2 − 1)).
+We compare the cost in Corollary 3.3 to that of Theorem 2.1 under the assumption (3.5). Let
+ˆK(ε) be the cost in (2.3).
+K(ε) ≲ ˆK(ε)
+�
+1,
+if β = β∗ or d2 ≤ d1/β∗,
+1
+εd1(1/β−1/β∗) ,
+otherwise.
+(3.11)
+It follows that if β = β∗ or d2 ≤ d1/β∗, the cost of Algorithm 2 is of the same order as ˆK(ε). If
+neither of these hold, then the cost of Algorithm 2 is of the order of ε−d1(1/β−1/β∗) times the cost of
+Algorithm 1. Note that the order of this extra algebraic dependence can be made arbitrarily small
+by taking b close to 1, at the expense of a factor in the term τ that grows as logb(β/β0)c2.
+We now consider the cases where either α or β is known.
+Corollary 3.4 (Known α). Suppose that α = aiα0. Let S = {i} × N0 × N and h(x1, x2, x3) =
+(x2 + 1)c2x3, c2 > 1, be a schedule criterion. Then given any h-assignment φ and any ε ∈ (0, ϵ0),
+running Algorithm 2 with
+t ≥ c2τ/(c2 − 1),
+τ = (|⌈logb(β/β0)⌉| + 1)c2K(ε),
+where K(ε) is as in (3.3), implies that
+f(x(t)) − ˆf + gQ(x(t)) ≤ max{η, ε}.
+16
+
+Proof. The result follows after modifying the proof of Corollary 3.3 as follows. First, find an upper
+bound to the number of solutions to nc2
+2 n3 ≤ τ for n2, n3 ∈ N.
+Now find the correspondence
+between the solutions and the tuples (i′, j′, k′) of (3.4), where i′ is now fixed.
+For the case of known β, we alter Algorithm 2 by removing the if statement in line 6 and always
+using the update rule in line 9.
+Corollary 3.5 (Known β). Suppose that β = β0 is known, S = Z × {0} × N and h(x1, x2, x3) =
+(x1 + 1)c1x3, c1 > 1, is a schedule criterion. Then given any h-assignment φ and any ε ∈ (0, ϵ0),
+running Algorithm 2 and
+t ≥ 2c1τ/(c1 − 1),
+τ = (|⌊loga(α/α0)⌋| + 1)c1K(ε),
+where K(ε) is as in (3.3), implies that
+f(x(t)) − ˆf + gQ(x(t)) ≤ max{η, ε}.
+Proof. Similar to the previous proof, the result follows after modifying the proof of Corollary 3.3.
+First, find an upper bound to the number of solutions to nc1
+1 n3 ≤ τ for n1, n3 ∈ N. Now find the
+correspondence between the solutions and the tuples (i′, j′, k′) of (3.4), where j′ is now fixed.
+Remark 3.6 (How to choose a, b). In the case of Corollary 3.5 and assuming (3.5), we can select
+an optimal value of a.
+From Corollary 3.5 and α∗ ≥ α/a, the part of τ that depends on a is
+bounded by O((|⌊loga(α/α0)⌋| + 1)c1ad1/β). We can upper bound this further by both dropping the
+floor function and, then dropping the +1 in brackets. We are then led to minimizing
+| loga(α/α0)|c1ad1/β = | log(α/α0)|c1ad1/β/ log(a)c1.
+Under these assumptions, the optimal value of a is ec1β/d1. Note that in the case of Corollary 3.4,
+there is no clear optimal choice for b since the optimal choice is ε-dependent.
+♦
+Remark 3.7 (How to choose c1, c2). For Corollaries 3.3 and 3.5, an optimal choice of c1 > 1
+exists but it depends on the unknown parameter α. To see this, minimize the lower bound of t in
+the aforementioned corollaries with respect to c1, noting that the only term in τ that depends on c1
+is (|⌊loga(α/α0)⌋| + 1)c1. Assuming α0 ̸= α, this gives
+c1 =
+1 +
+�
+1 +
+4
+log(|⌊loga(α/α0)⌋|+1)
+2
+.
+By the same reasoning, for Corollaries 3.3 and 3.4 and β0 ̸= β, the optimal choice of c2 > 1 depends
+on the unknown parameter β and is given by
+c2 =
+1 +
+�
+1 +
+4
+log(|⌈logb(β/β0)⌉|+1)
+2
+.
+Intuitively, if α0 is far from α then c1 should be closer to 1, and similarly for β0 and β regarding
+c2. In the absence of prior knowledge, we recommend a sensible default such as c1 = c2 = 2.
+♦
+Finally, to emphasize the generality of our algorithm, we consider the case where α and β are
+known to lie within explicit ranges. In this case, we modify set S based on these ranges and choose
+a schedule criterion function h(x1, x2, x3) depending on x3 only. The following result is immediate.
+17
+
+Corollary 3.8 (Known ranges for α, β). Suppose we have integers
+imin ≤ imax,
+0 ≤ jmin ≤ jmax,
+for which
+α ∈ [aiminα0, aimaxα0],
+β ∈ [bjminβ0, bjmaxβ0].
+Let
+S = {imin, imin + 1, . . . , imax} × {jmin, jmin + 1, . . . , jmax} × N,
+and h(x1, x2, x3) = x3 be a schedule criterion. Then given any h-assignment φ and any ε ∈ (0, ϵ0),
+running Algorithm 2 with
+t ≥ (imax − imin + 1)(jmax − jmin + 1)K(ε),
+where K(ε) is as in (3.3), implies f(x(t)) − ˆf + gQ(x(t)) ≤ max{η, ε}.
+Note that Algorithm 2 is sequential. However, one can readily devise a parallel implementation
+that runs Algorithm 1 in parallel over each pair (i, j) and then minimizes f + gQ over all instances
+at the end of the process.
+4
+Examples
+In this section, we present various examples of first-order methods that can be used in our restart
+scheme for different problem settings. In particular, we describe the methods that lead to the
+various results in Table 1. We do this by explicitly deriving a method Γ : R++ ×R++ ×D → D that
+satisfies (1.3) and give an explicit bound for the cost function CΓ(δ, ϵ, x0) of the form Cδd1/ϵd2 + 1
+for suitable d1 and d2.
+Remark 4.1 (Optimization over C). In convex analysis and continuous optimization, it is standard
+to consider function inputs lying in a finite-dimensional vector space over R. The results described
+below are extended to C, but this treatment does not always arise in the original papers for the
+first-order methods. We are interested in the domain of f being a subset of Cn. Hence, we consider
+the natural isomorphism between Cn and R2n given by: if z = x + iy ∈ Cn with x, y ∈ Rn, then
+z �→ (x, y). We refer to z as the complex representation and (x, y) as the real representation. Now,
+one proceeds to do convex analysis and continuous optimization in the real representation, then
+express the results in the equivalent complex representation. Fortunately, not much needs to change
+(at least symbolically) when switching between real and complex representations.
+For example, the Euclidean inner products ⟨·, ·⟩ have to be substituted with their real part, i.e.,
+⟨·, ·⟩R := Re ⟨·, ·⟩. Another example pertains to the differentiability of f. Specifically, for x, y ∈ Rn,
+we say that f is differentiable at z = x+iy ∈ D ⊆ Cn if and only if Re (f) is (real) differentiable at
+(x, y). To define the gradient, denote ∇x and ∇y as the vector of partial derivatives corresponding
+to variables x and y, respectively. Then ∇f := ∇xRe (f) + i∇yRe (f), noting that because f is
+real-valued, we have Im (f) ≡ 0.
+Other parts of convex analysis, such as convexity, functions,
+proximal mappings, subgradients, and so on, also extend to a complex vector domain by applying
+the definitions to the real representation of complex vectors.
+♦
+18
+
+Algorithm 3: Nesterov’s method
+Input
+: An L-smooth function f and closed, convex set Q ⊆ Cn as in (1.1), prox-function
+p(·; x0) with strong convexity constant σp and unique minimizer x0 ∈ Q,
+sequences {γj}∞
+j=0 and {τj}∞
+j=0, and number of iterations N.
+Output: The vector xN, which estimates a minimizer of (1.1).
+1 z0 ← x0
+2 for j = 0, 1, . . . , N − 1 do
+3
+xj+1 ← argmin
+x∈Q
+L
+2 ∥x − zj∥2
+ℓ2 + ⟨∇f(zj), x − zj⟩R
+4
+vj ← argmin
+x∈Q
+L
+σp p(x; x0) + �j
+i=0 γi⟨∇f(zi), x − zi⟩R
+5
+zj+1 ← τjvj + (1 − τj)xj+1
+6 end
+4.1
+Nesterov’s method for L-smooth functions
+For our first example, we consider Nesterov’s method [56], an accelerated projected gradient descent
+algorithm for general constrained convex optimization problems. Specifically, the algorithm aims
+to solve (1.1) in the special case when f is convex and L-smooth:
+Definition 4.2. A function f : Cn → R is L-smooth over Q ⊆ Cn if it is Fr´echet differentiable in
+an open set containing Q, and for all x, y in this set, its gradient ∇f has the Lipschitz property
+∥∇f(x) − ∇f(y)∥ℓ2 ≤ L∥x − y∥ℓ2.
+▲
+Nesterov’s method is given in Algorithm 3. The algorithm uses the notion of a prox-function
+p. Here p : Q → R is a proper, closed and strongly convex function with strong convexity constant
+σp > 0, that, in addition, satisfies minx∈Q p(x) = 0.
+Let x0 = argminx∈Qp(x) be the unique
+minimizer of p. To make this dependence explicit, we write p(·) = p(·; x0). A common and simple
+choice of prox-function is p(x; x0) = 1
+2∥x − x0∥2
+ℓ2 with σp = 1. This will be useful when we express
+Nesterov’s method with smoothing, in terms of Γ. We now state Nesterov’s main result that gives
+a bound for f(xk) − f(x), for any x ∈ Q.
+Lemma 4.3 (Nesterov’s theorem). Let Q ⊆ Cn be nonempty, closed and convex, f a convex L-
+smooth function over Q. In addition, let p : Q → R be a proper, closed and strongly convex function
+over Q with strong convexity constant σp > 0 with minx∈Q p(x) = 0. Then Algorithm 3 with
+γj = j + 1
+2
+,
+τj =
+2
+j + 3,
+x0 = argmin
+x∈Q
+p(x),
+generates a sequence {xk}∞
+k=1 ⊂ Q satisfying
+f(xk) − f(x) ≤ 4Lp(x; x0)
+k(k + 1)σp
+,
+∀x ∈ Q.
+(4.1)
+Lemma 4.3 consists of two modifications of [56, Theorem 2]. First, we do not assume Q is
+bounded, as the results in the original work do not use this. Second, we allow x ∈ Q instead of
+x ∈ �
+X. The proof in the original work does not use the optimality of x, and only requires x to
+be feasible. We utilize this property when considering Nesterov’s method with smoothing. The
+following is now immediate.
+19
+
+Proposition 4.4. Let Q ⊆ Cn be nonempty, closed and convex, f a convex L-smooth function over
+Q (Definition 4.2). Given input (δ, ϵ, x0) ∈ R+ ×R+ ×Q, let Γ(δ, ϵ, x0) be the output of Algorithm 3
+with
+p(x; x0) = 1
+2∥x − x0∥2
+ℓ2,
+γj = j + 1
+2
+,
+τj =
+2
+j + 3,
+N =
+�
+δ
+√
+2L
+√ϵ
+�
+.
+Then (1.3) holds with gQ ≡ 0. Specifically,
+f(Γ(δ, ϵ, x0)) − ˆf ≤ ϵ,
+∀x0 ∈ Q with d(x0, �
+X) ≤ δ,
+(4.2)
+where d is the metric induced by the ℓ2-norm. It follows that we can take
+CΓ(δ, ϵ) =
+�
+δ
+√
+2L
+√ϵ
+�
+.
+(4.3)
+Proposition 4.4 shows that we can take d1 = 1 and d2 = 1/2 in the cost bound (3.5) for
+Nesterov’s method (without smoothing). If f is L−smooth and satisfies (1.2) with η = 0, then
+β ≥ 2. It follows that we can take β0 = 2. Theorem 3.2 now implies the rates in the first row of
+Table 1.
+Several other remarks are in order. First, in Nesterov’s method, the iterates xj are always
+feasible since the corresponding update step returns a point in Q. Thus in Proposition 4.4 we do
+not have to define gQ since Γ trivially satisfies (1.3) with gQ ≡ 0. Finally, in Nesterov’s method,
+the requirement x0 ∈ Q can be relaxed. For instance, we only require f is L-smooth over the union
+of Q and an open neighborhood of x0 for some L > 0 to start with x0 /∈ Q.
+4.2
+Nesterov’s method for (u, v)-smoothable functions
+We can extend Nesterov’s method to solve (1.1) without assuming that f is differentiable. This is
+done via smoothing. For this, we need the following definition from [10, Definition 10.43] (extended
+to functions with complex-vector domains).
+Definition 4.5. Let u, v > 0. A convex function f : Cn → R is called (u, v)-smoothable if for any
+µ > 0 there exists a convex differentiable function fµ : Cn → R such that
+1. fµ(x) ≤ f(x) ≤ fµ(x) + vµ for all x ∈ Cn
+2. fµ is u
+µ-smooth over Cn
+The function fµ is referred to as a 1
+µ-smooth approximation of f with parameters (u, v), and µ is
+referred to as the smoothing parameter.
+▲
+Smoothing is a framework that approximates f arbitrarily closely by a family of smooth func-
+tions, i.e., functions with Lipschitz gradients. This means that we can apply Nesterov’s method to
+a smooth approximation of f, and also analyze the objective error in terms of f. The following
+provides a modified version of Lemma 4.3 for (a, b)-smoothable f, and is proven in Appendix A.1.
+Lemma 4.6. Let f : Cn → R be a convex (u, v)-smoothable function. Given any µ > 0, let fµ be a
+1
+µ-smooth approximation of f with parameters (u, v). Then taking Q, p, γj, τj, x0 as in Lemma 4.3
+and applying Algorithm 3 to the function fµ produces a sequence {xk}∞
+k=1 satisfying
+f(xk) − f(x) ≤
+4up(x; x0)
+µk(k + 1)σp
++ vµ,
+x ∈ Q.
+(4.4)
+20
+
+The following proposition shows that Nesterov’s method with smoothing can be formulated as
+an algorithm Γ in our framework, and is proven in Appendix A.1.
+Proposition 4.7. Let Q ⊆ Cn be nonempty, closed and convex, and f : Cn → R a convex (u, v)-
+smoothable function (Definition 4.5). Given input (δ, ϵ, x0) ∈ R+ × R+ × Q, let Γ(δ, ϵ, x0) be the
+output of Algorithm 3 applied to function fµ with
+µ = ϵ
+2v,
+p(x; x0) = 1
+2∥x − x0∥2
+ℓ2,
+γj = j + 1
+2
+,
+τj =
+2
+j + 3,
+N =
+�
+2
+√
+2uv · δ
+ϵ
+�
+.
+Then
+f(Γ(δ, ϵ, x0)) − ˆf ≤ ϵ,
+∀x0 ∈ Q satisfying d(x0, �
+X) ≤ δ,
+where d is the metric induced by the ℓ2-norm. It follows that we can set
+CΓ(δ, ϵ, x0) =
+�
+2
+√
+2uv · δ
+ϵ
+�
+.
+This result shows that we can take d1 = 1 and d2 = 1 in (3.5) in the case of Nesterov’s method
+with smoothing. Theorem 3.2 now implies the rates in the second row of Table 1.
+The following discussion considers a standard example of smoothing that is closely related to
+proximal maps, from [10, Theorem 10.51].
+If f : Cn → R is convex and Lipschitz continuous
+with Lipschitz constant Lf, then it is (1, L2
+f/2)-smoothable. In particular, the Moreau envelope
+with parameter µ > 0 is a 1
+µ-smooth approximation of f with parameters (1, L2
+f). Given a convex
+function f : Cn → R and µ > 0, the Moreau envelope of f is the function
+Mµ
+f (x) = min
+y∈Cn
+�
+f(y) + 1
+2µ∥x − y∥2
+ℓ2
+�
+.
+(4.5)
+The number µ is referred to as the smoothing parameter. The Moreau envelope Mµ
+f is well-defined,
+and the minimization problem defined in (4.5) has a unique solution corresponding to proxµf(x),
+i.e., the proximal map of µf at x [10, Theorem 6.3]. The Moreau envelope of f is also 1
+µ-smooth
+over its domain, where for any x we have
+∇Mµ
+f (x) = 1
+µ(x − proxµf(x)).
+Examples of Moreau envelopes of functions can be found in [10, Section 6.7].
+4.3
+The universal fast gradient method
+We next consider H¨older smooth functions, which are a natural way of interpolating between
+nonsmooth and smooth objective functions.
+Definition 4.8. A function q : Cn → R is H¨older smooth over Q ⊆ Cn with parameter ν ∈ [0, 1] if
+∥∇q(x) − ∇q(y)∥ℓ2 ≤ Mν∥x − y∥ν
+ℓ2,
+∀ x, y ∈ Q, ∇q(x) ∈ ∂q(x), ∇q(y) ∈ ∂q(y).
+▲
+We consider the universal fast gradient method [58] for the problem
+min
+x∈Q f(x),
+f(x) := q(x) + g(x),
+(4.6)
+21
+
+Algorithm 4: Universal fast gradient method
+Input
+: ϵ > 0, L0 > 0, φ0(x) = 0, y0 = x0, A0 = 0.
+Output: The vector xN, which estimates a minimizer of (4.6).
+1 for k = 0, 1, . . . , N do
+2
+vk ← proxφk,Q(x0)
+3
+ik ← −1
+4
+do
+5
+ik ← ik + 1
+6
+Compute ak+1,ik from the equation a2
+k+1,ik =
+1
+2ikLk (Ak + ak+1,ik).
+7
+Ak+1,ik ← Ak + ak+1,ik
+8
+τk,ik ← ak+1,ik/Ak+1,ik
+9
+xk+1,ik ← τk,ikvk + (1 − τk,ik)yk
+10
+Choose a subgradient ∇q(xk+1,ik) ∈ ∂q(xk+1,ik).
+11
+ˆφk+1,ik(x) ← ak+1,ik[⟨∇q(xk+1,ik), x⟩R + g(x)]
+12
+ˆxk+1,ik ← proxˆφk+1,ik,Q(vk)
+13
+yk+1,ik ← τk,ik ˆxk+1,ik + (1 − τk,ik)yk
+14
+while q(yk+1,ik)>q(xk+1,ik)+⟨∇q(xk+1,ik), yk+1,ik −xk+1,ik⟩R+2ik−1Lk∥yk+1,ik −xk+1,ik∥2
+ℓ2+ ϵ
+2τk,ik
+15
+xk+1 ← xk+1,ik, yk+1 ← yk+1,ik, ak+1 ← ak+1,ik, τk ← τk,ik
+16
+Ak+1 ← Ak + ak+1, Lk+1 ← 2ik−1Lk
+17
+φk+1(x) ← φk(x) + ak+1[q(xk+1) + ⟨∇q(xk+1), x − xk+1⟩R + g(x)].
+18 end
+where q is a proper convex function that is H¨older smooth for some ν ∈ [0, 1], and g is a closed
+convex function whose proximal map,
+proxcg,Q(x) = argmin
+y∈Q
+�
+c · g(y) + 1
+2∥x − y∥2
+ℓ2
+�
+,
+is straightforward to compute. The iterates of the universal fast gradient method are summarized
+in Algorithm 4.
+Lemma 4.9 (Theorem 3 of [58]). Let Q ⊆ Cn be nonempty, closed and convex, q a proper convex
+function that is H¨older smooth for some ν ∈ [0, 1] and Mν < ∞ (Definition 4.8), and g a closed
+convex function. Then Algorithm 4 generates a sequence {xk}∞
+k=1 ⊂ Q satisfying
+f(xk) − ˆf ≤
+� 22+4νM2
+ν
+ϵ1−νk1+3ν
+�
+1
+1+ν d(x0, �
+X)2
+2
++ ϵ
+2,
+∀x ∈ Q,
+(4.7)
+where d is the metric induced by the ℓ2-norm.
+By choosing k to match the two terms on the right-hand side of (4.7), the following proposition
+is immediate.
+Proposition 4.10. Let Q ⊆ Cn be nonempty, closed and convex, q a proper convex function is
+H¨older smooth for some ν ∈ [0, 1] and Mν ≥ 0 (Definition 4.8), and g a closed convex function.
+Given input (δ, ϵ, x0) ∈ R+ × R+ × Q, let Γ(δ, ϵ, x0) be the output of Algorithm 4 with
+N =
+�
+���
+2
+2+4ν
+1+3ν M
+2
+1+3ν
+ν
+δ
+2+2ν
+1+3ν
+ϵ
+2
+1+3ν
+�
+���
+.
+22
+
+Algorithm 5: Primal-dual algorithm for the problem (4.8).
+Input
+: Initial vectors x0 ∈ Cn and y0 ∈ Cm, proximal step sizes τ, σ > 0, number of
+iterations N, matrix B ∈ Cm×n, and routines for appropriate proximal maps.
+Output: Final ergodic average XN approximating a solution to (4.8).
+1 Initiate with x(0) = x0, y(0)
+1
+= y0, X0 = 0, and Y0 = 0.
+2 for j = 0, . . . , N − 1 do
+3
+x(j+1) ← proxτg
+�
+x(j) − τB∗y(j) − τ∇q(x(j))
+�
+;
+4
+y(j+1) ← proxσh∗
+�
+y(j) + σB(2x(j+1) − x(j))
+�
+;
+5
+Xj+1 ←
+1
+j+1
+�
+jXj + x(j+1)�
+;
+6
+Yj+1 ←
+1
+j+1
+�
+jYj + y(j+1)�
+;
+7 end
+Then
+f(Γ(δ, ϵ, x0)) − ˆf ≤ ϵ,
+∀x0 ∈ Q satisfying d(x0, �
+X) ≤ δ,
+where d is the metric induced by the ℓ2-norm. It follows that we can set
+CΓ(δ, ϵ, x0) =
+�
+���
+2
+2+4ν
+1+3ν M
+2
+1+3ν
+ν
+δ
+2+2ν
+1+3ν
+ϵ
+2
+1+3ν
+�
+���
+.
+Proposition 4.10 shows that we can take d1 = (2 + 2ν)/(1 + 3ν) and d2 = 2/(1 + 3ν) for the
+universal fast gradient method. Note that if q satisfies both (1.2) for η = 0 and Definition 4.8, then
+β ≥ 1 + ν [69]. Therefore, we take β0 = 1 + ν. Theorem 3.2 now implies the rates in the third row
+of Table 1.
+4.4
+The primal-dual iteration for unconstrained problems
+We now consider Chambolle and Pock’s primal-dual algorithm [23, 25]. The primal-dual hybrid
+gradient (PDHG) algorithm is a popular method to solve saddle point problems [22,31,63]. Consider
+the problem
+min
+x∈Cn f(x),
+f(x) := q(x) + g(x) + h(Bx),
+(4.8)
+where: B ∈ Cm×n with ∥B∥ ≤ LB; q is a proper, lower semicontinuous, convex function, and is
+Lq-smooth; and g, h are proper, lower semicontinuous, convex functions whose proximal maps are
+straightforward to compute. We also use the standard Euclidean metric for d in (1.2) and write
+the primal-dual iterates in their simplified form accordingly.
+The saddle-point problem associated with (4.8) is
+min
+x∈Cn max
+y∈Cm L(x, y) := ⟨Bx, y⟩R + q(x) + g(x) − h∗(y).
+(4.9)
+The primal-dual iterates are summarized in Algorithm 5, where the output is the ergodic average
+of the primal-dual iterates. Note that the primal-dual algorithm allows us to easily deal with the
+matrix B, which can be difficult with other first-order methods. If τ(σL2
+B + Lq) ≤ 1, then [25,
+Theorem 1] shows that for any x ∈ Cn and y ∈ Cm,
+L (Xk, y) − L (x, Yk) ≤ 1
+k
+�
+∥x − x(0)∥
+2
+τ
++ ∥y − y(0)∥
+2
+σ
+�
+.
+(4.10)
+The following lemma is a simple consequence of this bound and is proven in Appendix A.2.
+23
+
+Lemma 4.11. Consider the primal-dual iterates in Algorithm 5. If τ(σL2
+B + Lq) ≤ 1, then
+f(Xk) − f(x) ≤ 1
+k
+�
+∥x − x(0)∥
+2
+τ
++ ∥y − y(0)∥
+2
+σ
+�
+,
+∀x ∈ Cn, y ∈ ∂h(BXk).
+(4.11)
+We can take the infimum over y ∈ ∂h(BXk) on the right-hand side of (4.11) to obtain
+f(Xk) − f(x) ≤ 1
+k
+�
+∥x − x(0)∥
+2
+τ
++
+supz∈dom(h) infy∈∂h(z) ∥y − y(0)∥
+2
+σ
+�
+,
+∀x ∈ Cn.
+(4.12)
+To bound the right-hand side, we take y(0) = 0 and consider the case where h is such that there
+always exist points y in the subdifferential of h for which ∥y∥ is not too large. Note that this
+always holds if, for example, h is Lipschitz continuous and its domain is open [10, Theorem 3.61].
+The following proposition now shows how this falls into the framework of our restart scheme and
+is proven in Appendix A.2.
+Proposition 4.12. Suppose that
+sup
+z∈dom(h)
+inf
+y∈∂h(z) ∥y∥ ≤ Lh < ∞.
+(4.13)
+Given input (δ, ϵ, x0) ∈ R+ × R+ × Cn, let Γ(δ, ϵ, x0) be the output of Algorithm 5 with
+y0 = 0,
+τ =
+δ
+LBLh + δLq
+,
+σ = Lh
+δLB
+,
+N =
+�δ
+ϵ (2LBLh + δLq)
+�
+.
+Then
+f(Γ(δ, ϵ, x0)) − ˆf ≤ ϵ,
+∀x0 with d(x0, �
+X) ≤ δ.
+(4.14)
+It follows that we can take
+CΓ(δ, ϵ, x0) =
+�δ
+ϵ (2LBLh + δLq)
+�
+.
+(4.15)
+Assuming that δ is bounded, Proposition 4.12 shows that we can take d1 = 1 and d2 = 1 for
+the primal-dual algorithm. Theorem 3.2 now implies the rates in the fourth row of Table 1.
+4.5
+The primal-dual iterations for constrained problems
+We now consider primal-dual iterations, but for the constrained problem
+min
+x∈Cn f(x) + χC(Ax),
+f(x) := q(x) + g(x) + h(Bx),
+(4.16)
+with the same assumptions on q, g, h and B as in Section 4.4, but now with the additional term
+χC(Ax). Here, C is a closed and non-empty convex set, χC is its indicator function of C and
+A ∈ Cm′×n with ∥A∥ ≤ LA. This fits into our framework with the choice
+Q = {x ∈ Cn : Ax ∈ C},
+gQ(x) = gQ(κ; x) = κ · inf
+z∈C ∥Ax − z∥,
+for κ > 0.
+Note that κ is an additional parameter that can be chosen to balance the rate of
+reduction in the feasibility gap versus the objective function error. It is possible to formulate a
+projected version of the primal-dual iteration. However, like with Nesterov’s method, this is only
+24
+
+Algorithm 6: Primal-dual algorithm for the constrained problem (4.16).
+Input
+: Initial vectors x0 ∈ Cn, [y0]1 ∈ Cm and [y0]2 ∈ Cm′, proximal step sizes
+τ, σ1, σ2 > 0, number of iterations N, matrices B ∈ Cm×n and A ∈ Cm′×n, and
+routines for appropriate proximal maps.
+Output: Final ergodic average XN approximating a solution to (4.16).
+1 Initiate with x(0) = x0, y(0)
+1
+= [y0]1, y(0)
+2
+= [y0]2, X0 = 0, [Y0]1 = 0, and [Y0]2 = 0.
+2 for j = 0, . . . , N − 1 do
+3
+x(j+1) ← proxτg
+�
+x(j) − τB∗y(j)
+1
+− τA∗y(j)
+2
+− τ∇q(x(j))
+�
+;
+4
+y(j+1)
+1
+← proxσ1h∗
+�
+y(j)
+1
++ σ1B(2x(j+1) − x(j))
+�
+;
+5
+y(j+1)
+2
+← y(j)
+2
++ σ2A(2x(j+1) − x(j)) − σ2PC
+�
+y(j)
+2 /σ2 + A(2x(j+1) − x(j))
+�
+;
+6
+Xj+1 ←
+1
+j+1
+�
+jXj + x(j+1)�
+;
+7
+[Yj+1]1 ←
+1
+j+1
+�
+j[Yj]1 + y(j+1)
+1
+�
+;
+8
+[Yj+1]2 ←
+1
+j+1
+�
+j[Yj]2 + y(j+1)
+2
+�
+;
+9 end
+possible when the projection onto Q can be easily computed. In this section, we consider a primal-
+dual iteration for (4.16) that only involves computing the projection onto the set C, at the price of
+producing non-feasible iterates.
+The saddle-point problem associated with (4.16) is
+min
+x∈Cn max
+y1∈Cm max
+y2∈Cm′ LC(x, y1, y2) := ⟨Bx, y1⟩R +q(x)+g(x)−h∗(y1)+⟨Ax, y2⟩R −sup
+z∈C
+⟨z, y2⟩R. (4.17)
+The primal-dual iterates are summarized in Algorithm 6, where, again, the output is the ergodic
+average of the primal-dual iterates. We have included three proximal step sizes τ, σ1 and σ2, which
+correspond to the primal variable and the two dual variables, respectively. To compute the proximal
+map associated with the second dual variable, we use Moreau’s identity to write
+proxσ2χ∗
+C(y) = y − σ2PC(y/σ2),
+where PC denotes the projection onto C (with respect to the standard Euclidean norm).
+If τ(σ1L2
+B + σ2L2
+A + Lq) ≤ 1, then a straightforward adaption of [25, Theorem 1] shows that for
+any x ∈ Cn, y1 ∈ Cm and y2 ∈ Cm′,
+LC (Xk, y1, y2) − LC (x, [Yk]1, [Yk]2) ≤ 1
+k
+�
+�∥x − x(0)∥
+2
+τ
++ ∥y1 − y(0)
+1 ∥
+2
+σ1
++ ∥y2 − y(0)
+2 ∥
+2
+σ2
+�
+� .
+(4.18)
+We now have the following lemma and resulting proposition, both of which are proven in Ap-
+pendix A.3.
+Lemma 4.13. Consider the primal-dual algorithm in Algorithm 6 with y(0)
+2
+= 0. If τ(σ1L2
+B +
+σ2L2
+A + Lq) ≤ 1, then for any κ > 0
+f(Xk) − f(x) + gQ(κ; Xk) ≤ 1
+k
+�
+�∥x − x(0)∥
+2
+τ
++ ∥y1 − y(0)
+1 ∥
+2
+σ1
++ κ2
+σ2
+�
+� ,
+∀x ∈ Q, y1 ∈ ∂h(BXk).
+(4.19)
+25
+
+Proposition 4.14. Suppose that
+sup
+z∈dom(h)
+inf
+y∈∂h(z) ∥y∥ ≤ Lh < ∞.
+(4.20)
+Given input (δ, ϵ, x0) ∈ R+ × R+ × Cn, let Γ(δ, ϵ, x0) be the output of Algorithm 6 with
+[y0]1 = 0, [y0]2 = 0, τ =
+δ
+κLA + LhLB + δLq
+, σ1 = Lh
+δLB
+, σ2 =
+κ
+δLA
+, N =
+�δ (2κLA + 2LhLB + δLq)
+ϵ
+�
+.
+Then
+f(Γ(δ, ϵ, x0)) − ˆf + gQ(κ; ˆx) ≤ ϵ,
+∀x0 with d(x0, �
+X) ≤ δ.
+(4.21)
+It follows that we can take
+CΓ(δ, ϵ, x0) =
+�δ (2κLA + 2LhLB + δLq)
+ϵ
+�
+.
+(4.22)
+Assuming that δ is bounded, Proposition 4.14 shows that we can take d1 = 1 and d2 = 1 for
+the primal-dual algorithm. Theorem 3.2 now implies the rates in the final row of Table 1.
+5
+Numerical experiments
+We implement several numerical experiments for the general restart scheme (Algorithm 2) applied
+to three different problems. The first is a simple sparse recovery problem modeled as QCBP, which is
+solved using the primal-dual iteration for constrained problems (Algorithm 6). Second, we consider
+image reconstruction from Fourier measurements via TV minimization.
+The reconstruction is
+computed using NESTA [12], where NESTA is an accelerated projected gradient descent algorithm
+derived from Nesterov’s method (Algorithm 3) with smoothing. Third, we perform feature selection
+on three real-world datasets. This selection is done by solving a SR-LASSO problem on the data
+with unconstrained primal-dual iterations (Algorithm 5).
+Before discussing the examples in turn, we make some general remarks about the implemen-
+tation.
+First, we use the schedule criteria from Section 3.2, and for parameters we always set
+c1 = c2 = 2, b = e, r = e−1, and a = ec1β/d1 for unknown α but known β (Corollary 3.5), other-
+wise a = ec1/d1 if both are unknown (Corollary 3.3). Assignments from the schedule criteria are
+obtained by enumerating and sorting solutions of the respective Diophantine equations found in
+the proofs of Corollaries 3.3 to 3.5. The choice of r is motivated by Remark 2.2 and the choice of a
+by Remark 3.6. The choice of c1 and c2 were arbitrary, with the intent of being sane defaults, and
+otherwise can be tuned to improve performance.
+Second, when using the restart scheme for primal-dual iterations, we store and perform restarts
+on the dual variables for each instance indexed by (i, j).
+Third, we use a simple workaround to handle finite precision arithmetic. In the grid search
+for the restart scheme, the sharpness parameter αi can be arbitrarily large or small, and βj can
+be arbitrarily large. Also, the adaptive restart parameters δ = δi,j,U and ϵ = ϵi,j,U can become
+arbitrarily small. Regarding the grid indices, we limit i and j so that
+|i| ≤ ⌊loga(1/ϵmach)⌋,
+j ≤ ⌊logb(1/ϵmach)⌋,
+where ϵmach is machine epsilon.
+Regarding the adaptive parameters, after the assignments of
+δi,j,U+1 and ϵi,j,U+1 in Algorithm 2, we insert the updates δi,j,U+1 := max(δi,j,U+1, 10ϵmach) and
+ϵi,j,U+1 := max(ϵi,j,U+1, 10ϵmach) to avoid setting them to zero.
+26
+
+Fourth, we slightly modify the primal-dual algorithm to improve overall performance. For each
+j ≥ 1, we track a separate iterate �
+Xj defined by
+�
+Xj = argmini=1,...,jf(Xi) + κgQ(Xi),
+j ≥ 1.
+The iterates { �
+Xj}j≥1 are not used in the primal-dual algorithm, but are instead used to evaluate
+the reconstruction or objective error in our experiments. In addition, the algorithm returns �
+XN as
+its final iterate. We similarly track a separate iterate for the dual variables, selecting them based
+on an evaluation of the Lagrangian (4.17) with �
+Xj. Note that choosing to output �
+XN instead of
+XN is theoretically justified, since if (1.3) holds, then our modification would still satisfy (1.3) for
+the same parameters (δ, ϵ, x0).
+5.1
+Sparse recovery via QCBP
+We consider reconstructing a vector x ∈ Rn from noisy measurements y = Ax + e ∈ Rm, where
+A ∈ Rm×n is a matrix whose entries are i.i.d. Gaussian random variables with mean zero and
+variance 1/m, and e ∈ Rm is a noise vector satisfying ∥e∥ℓ2 ≤ ς for some noise level ς > 0. For a
+positive integer n, we write [n] = {1, 2, . . . , M}. Given a vector z = (zi)n
+i=1 ∈ Cn and S ⊆ [n], the
+vector zS has ith entry zi if i ∈ S, and is zero otherwise. The best s-term approximation error of
+z is defined as
+σs(z)ℓ1 = min{∥uS − z∥ℓ1 : u ∈ Cn, S ⊆ [n], |S| ≤ s}.
+We assume that x is approximately s-sparse, in the sense that its best s-term approximation error
+σs(x)ℓ1 is small. The recovery of x is formulated as solving the QCBP problem
+min
+z∈Rn ∥z∥ℓ1 subject to ∥Az − y∥ℓ2 ≤ ς.
+(5.1)
+We use the following condition on the matrix A to ensure that approximate sharpness holds.
+Definition 5.1 (Robust null space property, e.g., Definition 5.14 of [4]). The matrix A ∈ Cm×n
+satisfies the robust Null Space Property (rNSP) with constants 0 < ρ < 1 and γ > 0 if
+∥vS∥ℓ2 ≤ ρ
+√s∥vS∁∥ℓ1 + γ∥Av∥ℓ2,
+for all v ∈ Cn and S ⊆ [M] with |S| ≤ s.
+▲
+In [27, Theorem 3.3], it was shown that the robust null space property (rNSP) implies approx-
+imate sharpness. We restate the result in the notation of this paper for completeness.
+Proposition 5.2 (Approximate sharpness of ℓ1-norm for QCBP sparse recovery). Let ς > 0.
+Suppose A ∈ Cm×n has the rNSP of order s with constants 0 < ρ < 1, γ > 0. Let y ∈ Cm, D = Cn,
+Q = {x ∈ Cn : ∥Ax − y∥ℓ2 ≤ ς} and f(x) = ∥x∥ℓ1. Then the approximate sharpness condition (1.2)
+holds with
+gQ(z; √s) = √s max{∥Az − y∥ℓ2 − ς, 0},
+α = ˆc1
+√s,
+β = 1,
+η = ˆc2σs(x)ℓ1 + ˆc3ς√s,
+for constants ˆc1, ˆc2, ˆc3 > 0 are constants depending only on ρ and γ.
+The theory of compressed sensing [4, 35] aims to construct (random) matrices satisfying the
+rNSP, which is itself implied by the better-known Restricted Isometry Property (RIP). For example,
+if A is a Gaussian random matrix, then it satisfies the rNSP with probability at least 1−ε, provided
+m ≥ C · (s · log(eN/s) + log(2/ε)) (see, e.g., [4, Theorem 5.22]). However, a sharp value of the
+constant C, and therefore also the rNSP constants ρ and γ, is unknown. This implies that the
+approximate sharpness constants α and η are also unknown.
+This motivates using the restart
+scheme (Algorithm 2), which does not require knowledge of α or η, to solve (5.1).
+27
+
+0
+500
+1000
+1500
+2000
+10 -6
+10 -4
+10 -2
+10 0
+Total inner iterations t
+∥xt − x∥ℓ2
+0
+1000
+2000
+3000
+4000
+5000
+10 -6
+10 -4
+10 -2
+10 0
+Total inner iterations t
+∥xt − x∥ℓ2
+Figure 2: Reconstruction error of restarted primal-dual iteration for QCBP with ς = 10−6. Left: The restart
+scheme with fixed sharpness constants β = 1 and various α. Right: Various different schemes (including
+restarted and unrestarted schemes).
+5.1.1
+Experimental setup
+We use the primal-dual iteration for constrained problems (Algorithm 6) to solve the sparse recovery
+problem. This can be done by expressing QCBP in (5.1) as (4.16) with
+q ≡ 0,
+h ≡ 0,
+B = 0,
+g(x) = ∥x∥ℓ1,
+C = {z ∈ CN : ∥z − y∥ℓ2 ≤ ς}.
+Given these choices, the proximal map of τg is the shrinkage-thresholding operator, and the projec-
+tion map is straightforward to compute since C is a shifted ℓ2-ball. Moreover, we have h∗(z) = +∞
+whenever z ̸= 0, and is zero otherwise. Therefore the proximal map proxσ1h∗(x) = ∥x∥2
+ℓ2/2, and
+thus y(j)
+1
+= 0 for all j > 0 if the initial data y(0)
+1
+= 0. In essence, we can ignore the parameter σ1
+and updating the iterates y(j)
+1
+in the primal-dual iterations (Algorithm 6). The error bound derived
+in Lemma 4.13 holds with the σ1 term omitted.
+Unless stated otherwise, the parameters used are ambient dimension n = 128, sparsity level
+s = 10, measurements m = 60, noise level ς = 10−6. The ground truth vector x is exactly sparse
+with s of its entries (randomly selected) corresponding to i.i.d. standard normal entries. The noise
+vector e is selected uniformly random on the ℓ2-ball of radius ς and thus ∥e∥ℓ2 = ς. The objective
+function is f(x) = ∥x∥ℓ1 and the feasibility gap is given by gQ(x; κ) = κ · max{∥Ax − y∥ℓ2 − ς, 0},
+which is derived from Section 4.5. The feasibility gap weight is set to κ = √m from Proposition 5.2,
+noting that s ≤ m in general. In addition, α0 = √m, β0 = 1. The choice of α0 is also motivated
+by Proposition 5.2.
+5.1.2
+Results
+Fig. 2 shows the performance of the restart scheme in Algorithm 1 for various fixed values of α and
+β = 1. For smaller α, the error decreases linearly down to the noise level ς = 10−6. This agrees
+with Theorem 2.1. Increasing α leads to fast linear convergence, up to a threshold (between 101
+and 101.2). After this point, the performance of the restart scheme abruptly breaks down since
+large α violates the approximate sharpness condition (1.2).
+To overcome such parameter sensitivity, we use Algorithm 2. Fig. 2 also compares the perfor-
+mance of the restart scheme with fixed (α, β) = (√m, 1) with restart schemes that (i) perform a
+28
+
+0
+500
+1000
+1500
+2000
+2500
+3000
+10 -6
+10 -4
+10 -2
+10 0
+Total inner iterations t
+∥xt − x∥ℓ2
+Grid search over α
+0
+1000
+2000
+3000
+4000
+5000
+10 -6
+10 -4
+10 -2
+10 0
+Total inner iterations t
+∥xt − x∥ℓ2
+Grid search over β
+Figure 3: Reconstruction error of restarted primal-dual iteration for QCBP with ς = 10−6. Left: The restart
+scheme with grid search over α and various fixed β. Right: The restart scheme with grid search over β and
+various fixed α.
+grid search over α, for fixed β = 1, and (ii) perform a grid search over both α and β. Both grid
+search schemes exhibit linear convergence, in agreement with Theorem 1.1. They converge less
+rapidly than the scheme with fixed (α, β), but require no empirical parameter tuning. Note that
+all restart schemes significantly outperform the unrestarted primal-dual iteration (“no restarts”).
+Next, we consider two cases of grid searching over exactly one sharpness constant and leaving the
+other fixed. Fig. 3 shows the results for fixed α with β grid search and fixed β with α grid search.
+Both yield linear decay, although at a slightly worse rate. A key point to note is the potential
+benefit of grid searching. Compare the reconstruction error with those for the fixed restart schemes
+in Fig. 2 with log10(α) ≥ 1.2 and β = 1. In the fixed constant scheme, these parameter choices stall
+the error. However, β grid search overcomes this and manages to reconstruct x within a tolerance
+proportional to ς after sufficiently many restarts.
+Finally, Fig. 4 considers the effect on the restart schemes when changing the noise level ς. In
+all cases, the restart schemes linearly decay to a tolerance proportional to ς, and outperform the
+unrestarted primal-dual iterations.
+5.2
+Image reconstruction via TV minimization
+In this experiment, we consider image reconstruction with Fourier measurements – a sensing modal-
+ity with applications notably in Magnetic Resonance Imaging (MRI) [4]. Specifically, we consider
+the recovery of a vector x ∈ Rn from noisy Fourier measurements y = Ax+e ∈ Cm, where A ∈ Cm×n
+corresponds to a subsampled Fourier matrix and e ∈ Cm models noise or perturbations. The vector
+x is a vectorized complex 2-D image X ∈ CR×R, where n = R2 for some positive power-of-two
+integer R. The matrix A has the form A = m−1/2PΩF, where F ∈ Cn×n is the 2-D discrete Fourier
+transform and Ω ⊆ n is a sampling mask with |Ω| = m. Here, Ω defines the matrix PΩ ∈ Cm×n,
+which selects the rows of F by index according to the indices in Ω. Lastly, ∥e∥ℓ2 ≤ ς for some noise
+level ς > 0. A widely used tool for reconstructing x from y is the total variation (TV) minimization
+problem
+min
+z∈Cn ∥V z∥ℓ1 subject to ∥Az − y∥ℓ2 ≤ ς,
+where V is the 2-D (anisotropic) discrete gradient transform with periodic boundary conditions [3].
+29
+
+0
+1000
+2000
+3000
+4000
+5000
+10 -10
+10 -5
+10 0
+Total inner iterations t
+ς = 10−2
+∥xt − x∥ℓ2
+0
+1000
+2000
+3000
+4000
+5000
+10 -10
+10 -5
+10 0
+Total inner iterations t
+ς = 10−4
+∥xt − x∥ℓ2
+0
+1000
+2000
+3000
+4000
+5000
+10 -10
+10 -5
+10 0
+Total inner iterations t
+ς = 10−6
+∥xt − x∥ℓ2
+0
+1000
+2000
+3000
+4000
+5000
+10 -10
+10 -5
+10 0
+Total inner iterations t
+ς = 10−8
+∥xt − x∥ℓ2
+0
+1000
+2000
+3000
+4000
+5000
+10 -10
+10 -5
+10 0
+Total inner iterations t
+ς = 10−10
+∥xt − x∥ℓ2
+0
+1000
+2000
+3000
+4000
+5000
+10 -10
+10 -5
+10 0
+Total inner iterations t
+ς = 10−12
+∥xt − x∥ℓ2
+Figure 4: Reconstruction error of restarted primal-dual iteration for QCBP with ς = 10−2k for k = 1, 2, . . . , 6.
+Each plot includes the various (restarted and unrestarted) schemes.
+Similar to the sparse recovery problem described in the previous section, the TV-Fourier image
+reconstruction problem can be shown to have the approximate sharpness condition (1.2) with high
+probability under a suitable random sampling pattern Ω. Stating and proving this is somewhat
+more involved, but can be done with a careful adaptation of the analysis within [3, Sec. 7.4].
+5.2.1
+Experimental setup
+The first-order solver we use is NESTA (NESTerov’s Algorithm), an accelerated projected gradient
+descent algorithm used to solve problems of the form
+min
+z∈Cn ∥W ∗z∥ℓ1 subject to ∥Az − y∥ℓ2 ≤ ς,
+W ∈ Cn×m′,
+where TV minimization is a special case with W = V ⊤. NESTA is derived from Nesterov’s method
+with smoothing, where the objective function f(z) = ∥W ∗z∥ℓ1 is smoothed by replacing the ℓ1-norm
+with its Moreau envelope. This yields a 1/µ-smooth approximation fµ(z) = ∥W ∗z∥ℓ1,µ of f with
+parameters (∥W ∗∥2
+ℓ2, m′/2). Here ∥w∥ℓ1,µ = �m′
+i=1 |wi|µ for w = (wi)m′
+i=1 and | · |µ is the complex
+Huber function (see, e.g., [61]). In particular, we have ∥V ∥ℓ2 = 2
+√
+2 for TV minimization in 2-D.
+The second part of the derivation of NESTA is finding closed-form expressions for the update
+steps. In general, this is not possible to do except in special cases. However, NESTA considers A
+with orthonormal rows up to a constant factor, i.e., AA∗ = νI for some ν > 0. Such an assumption
+yields a closed form for the update formulas and is not unreasonable since many forward operators
+in compressive imaging have orthonormal rows. For example, with the subsampled Fourier matrix
+we have AA∗ = (N/m)I, and hence the desired property holds with ν = N/m.
+We reconstruct an R×R GPLU phantom image [42] with R = 512 so that the ambient dimension
+is n = 5122. The noise e is uniformly sampled from an ℓ2-ball of radius ς = 10−5, and so ∥e∥ℓ2 = ς.
+Two sampling masks are considered for the subsampled Fourier matrix A and are shown in Fig. 5.
+30
+
+Near-optimal sampling mask
+Radial sampling mask
+Figure 5: Sampling patterns for the Fourier measurements used in the image reconstruction experiments.
+0
+500
+1000
+1500
+2000
+10 -4
+10 -2
+10 0
+Total inner iterations t
+∥xt − x∥ℓ2
+Near-optimal sampling mask
+0
+500
+1000
+1500
+2000
+10 -4
+10 -2
+10 0
+Total inner iterations t
+∥xt − x∥ℓ2
+Radial sampling mask
+Figure 6: Reconstruction error of restarted NESTA for TV minimization with ς = 10−5, and with the near-
+optimal and radial sampling masks, respectively. The restart scheme uses fixed sharpness constants β = 1
+and various α.
+The first is a near-optimal sampling scheme [3, Sec. 4.2] and the second is a radial sampling scheme,
+where the latter is common in practice. Each mask yields approximately a 12.5% sampling rate.
+For the restart scheme, the objective function is f(z) = ∥V x∥ℓ1 and the feasibility gap gQ ≡ 0
+since NESTA always produces feasible iterates. The smoothing parameters µ are handled directly
+by the restarting procedure and explicitly depend on ϵi,j,U (see Proposition 4.7). The main two
+experiments are done for each of the two sampling masks. Lastly, we choose α0 = √m, β0 = 1.
+The choice of α0 is motivated by [27, Theorem 6.3] which generalizes Proposition 5.2.
+5.2.2
+Results
+First, we run the restart scheme with fixed sharpness constants (no grid search) corresponding to
+pairs (α, β) with β = 1 and various α values. The reconstruction error versus total inner iterations
+is plotted in Fig. 6 with near-optimal sampling (left) and radial sampling (right). The results are
+very similar to the first sparse recovery via QCBP experiment. Again, the rate of decay corresponds
+31
+
+:.
+=
+::
+..
+:=
+:
+:?
+:
+.
+: 5
+P:
+L
+:
+....
+..2.
+:
+.
+::
+..
+"..
+.".'
+.
+..
+:
+.
+:0
+0.5
+1
+1.5
+2
+2.5
+3
+10 4
+10 -4
+10 -2
+10 0
+Total inner iterations t
+∥xt − x∥ℓ2
+Near-optimal sampling mask
+0
+0.5
+1
+1.5
+2
+2.5
+3
+10 4
+10 -4
+10 -2
+10 0
+Total inner iterations t
+∥xt − x∥ℓ2
+Radial sampling mask
+Figure 7: Reconstruction error of restarted NESTA for TV minimization with ς = 10−5, and with the near-
+optimal and radial sampling masks, respectively. Various different (restarted and unrestarted) schemes are
+used.
+to linear decay as anticipated from Theorem 3.2. The convergence rate improves as α increases, up
+to a threshold (about α = 630 for near-optimal sampling, and about α = 446 for radial sampling),
+where afterwards the limiting tolerance increases steadily, yielding poor reconstruction results. This
+phenomenon is discussed in the first experiment of sparse recovery via QCBP. A key observation is
+how changing the sampling mask changes the threshold α value. This motivates using a grid search
+to avoid having to tune α as a parameter for different sampling masks.
+In the second experiment, we compare the reconstruction errors of several restart schemes,
+together with standalone NESTA (i.e., no restarts) with various smoothing parameters. This is
+shown in Fig. 7 with near-optimal sampling (left) and radial sampling (right). The smoothing
+parameters used are µ = 10iς, i ∈ {−2, 1, 0, 1}. The results are analogous to the fourth experiment
+with sparse recovery via QCBP. We note that the radial sampling mask produces slightly lower
+convergence rates than the near-optimal scheme.
+Moreover, we observe that converging to the
+limiting tolerance of NESTA is sensitive to the choice of smoothing parameter µ. By making µ
+smaller, we better approximate the original problem and thus the reconstruction, but require more
+iterations to achieve a better approximation. In contrast, restarting NESTA via Algorithm 2 does
+not require any tuning of the smoothing parameter and outperforms the non-restarted algorithm.
+5.3
+Feature selection via SR-LASSO
+Our final experiment considers feature selection via the Square Root LASSO (SR-LASSO) problem
+[2, 14, 15, 72]. Let X ∈ Rm×n be a data matrix, where each row corresponds to a data point and
+each column corresponds to a feature, and y ∈ Rm the label vector for the data points. Since we
+wish to learn an affine mapping from data points to labels, we augment X by appending a new
+column consisting of ones, with the augmentation denoted by A ∈ Rm×(n+1). Now fix λ > 0. Then
+we seek a vector x ∈ Rn+1 that solves the SR-LASSO problem
+min
+z∈Rn+1 ∥Az − y∥ℓ2 + λ∥z∥ℓ1.
+An advantage of this problem over the classical LASSO is that it requires less tuning of the parame-
+ter λ as the problem instance or noise level changes. See [72] for discussion and recovery conditions
+for this problem. Feature selection is done by identifying the indices of close-to-zero entries of x,
+32
+
+0
+2000
+4000
+6000
+8000
+10000
+10 -10
+10 -5
+10 0
+Total inner iterations t
+f(xt) − ˆf
+wine
+0
+1
+2
+3
+4
+5
+10 4
+10 -10
+10 -5
+10 0
+Total inner iterations t
+f(xt) − ˆf
+cc
+0
+1
+2
+3
+4
+5
+10 4
+10 -15
+10 -10
+10 -5
+10 0
+Total inner iterations t
+f(xt) − ˆf
+leu
+Figure 8: Objective error versus the total inner iteration of various (restarted and unrestarted) schemes of
+primal-dual iteration for SR-LASSO. The plots correspond to three different datasets.
+which are the features to discard. This reduces the number of columns of X for future processing
+or analysis.
+The SR-LASSO is a well-known tool in high-dimensional statistics. It can also be used for
+sparse recovery problems, in which case approximate sharpness follows (like it did with QCBP)
+from the rNSP (Definition 5.1) [27]. However, in the feature selection problem, properties such as
+the rNSP are unlikely to hold. In this case, more general recovery conditions for SR-LASSO (and
+LASSO), such as the compatibility condition [72], are more useful. Under these conditions, one also
+has approximate sharpness with unknown constants.
+5.3.1
+Setup
+We use the unconstrained primal-dual iterations (Algorithm 5) to solve SR-LASSO. We can express
+SR-LASSO as (4.16) by
+q ≡ 0,
+g(x) = λ∥x∥ℓ1,
+h(Bx) = ∥Bx − y∥ℓ2,
+B = A.
+From this, the primal-dual updates can be computed explicitly.
+The proximal map τg is the
+shrinkage-thresholding operator and the proximal map of σh∗ is a projection map onto the ℓ2-ball.
+In either case, the proximal maps are straightforward to compute. For three different datasets,
+we compare the SR-LASSO objective error of various unrestarted and restarted schemes.
+The
+minimum of SR-LASSO for each dataset is computed using CVX [40,41] with high precision and
+the SDPT3 solver and is used to compute the objective errors in Figs. 8 and 9.
+We use three datasets: wine quality (wine) [29] with m = 6497 points and n = 11 features, colon
+cancer (cc) [26] with m = 62 points and n = 2000 features, and leukemia (leu) [26] with m = 38
+points and n = 7129 features. The wine data corresponds to a regression task of predicting wine
+quality, cc and leu are two-class classification tasks of diagnosing illness based on data features.
+We use λ = 3, 2, and 4 for the wine, cc, and leu datasets, respectively. We measure sparsity s of
+ˆx by interpreting an entry to be non-zero if its absolute value is greater than 10−5. The values α0
+and β0 are chosen empirically as estimates of the true sharpness constants α and β, respectively.
+5.3.2
+Results
+Fig. 8 shows the performance of various restart schemes for this problem on the three different
+datasets. In all cases, the restarted schemes outperform the unrestarted scheme. The suitable values
+of α and β differ significantly across the datasets, indicating that the optimal sharpness parameters
+are problem-dependent. This further demonstrated in Fig. 9, where we show the restart scheme
+for various fixed β and grid search over α - the restart schemes with choices of β > 1 outperform
+33
+
+0
+1000
+2000
+3000
+4000
+5000
+10 -10
+10 -5
+10 0
+f(xt) − ˆf
+wine
+0
+2000
+4000
+6000
+8000
+10000
+10 -10
+10 -5
+10 0
+Total inner iterations t
+f(xt) − ˆf
+cc
+Total inner iterations t
+f(xt) − ˆf
+leu
+Figure 9: Objective error versus the total inner iteration of restarted primal-dual iteration for SR-LASSO.
+The plots correspond to grid search over α with various fixed β for three different datasets.
+the schemes that use β = 1. This is in contrast to the sparse recovery example, where theory and
+experiment suggest β = 1 as a good choice. This phenomenon is unsurprising since the approximate
+sharpness condition (see (1.2)) for this problem is expected to be highly dependent on the data.
+Nonetheless, using our grid search scheme, we obviate the need for estimating or tuning these
+parameters.
+6
+Conclusion
+We provided a framework for the optimal acceleration of first-order methods under approximate
+sharpness conditions. These conditions generalize sharpness by incorporating an unknown constant
+perturbation to the objective error, offering greater robustness to noise or model classes.
+Our
+scheme can achieve optimal convergence rates for a wide variety of problems, despite not assuming
+knowledge of the constants appearing in (1.2). Moreover, we do not require the first-order method
+to produce feasible iterates, a flexibility that is useful when employing methods such as primal-dual
+iterations. As illustrated by our numerical experiments, our schemes are also practical, and often
+lead to significant improvements over unrestarted schemes or restart schemes with poor parameter
+choices.
+There are numerous possible avenues for future research and extensions of our framework. One
+avenue involves replacing the metric in (1.2) by a Bregman distance, and acceleration for convex
+optimization problems in Banach spaces. Another involves applications to (non-convex) bilevel
+optimization schemes. For saddle-point problems such as (4.9) and (4.17), it may be possible to
+develop similar restart schemes based on primal-dual gaps replacing f(x)− ˆf in (1.2), see [5] and [32]
+for primal-dual gap sharpness and restart schemes in the cases of β = 1 and β = 2, respectively. See
+also [46,47] for recent work on restarts based on gap functions for Frank-Wolfe algorithms. Finally,
+there is the extension of our restart schemes to handle stochastic first-order methods, including
+larger-scale machine learning problems.
+A
+Miscellaneous proofs
+In this section, we prove several results that were stated in Section 4.
+34
+
+3 = 1.0
+B = 2.0
+β = 3.0
+β = 4.0
+5.010-10
+10-15
+0
+5000
+10β = 6.0
+β = 7.0
+β = 8.0
+000
+1500010-5A.1
+Nesterov’s method with smoothing
+Proof of Lemma 4.6. Applying Lemma 4.3 with the function fµ and using the second part of Def-
+inition 4.5 gives
+fµ(xk) − fµ(x) ≤
+4up(x; x0)
+µk(k + 1)σp
+.
+Now using both inequalities in the first part of Definition 4.5 gives the result.
+Proof of Proposition 4.7. Suppose that x0 ∈ Q with d(x0, �
+X) ≤ δ.
+Then by Lemma 4.6 with
+ˆx ∈ �
+X ⊆ Q, we have
+f(xN) − ˆf ≤
+4up(ˆx; x0)
+µN(N + 1)σp
++ vµ.
+Using
+1
+N(N+1) ≤
+1
+N2 , σp = 1 and p(ˆx) ≤ 1
+2δ2 by choice of p, we get
+f(xN) − ˆf ≤ 2uδ2
+µN2 + vµ.
+Substituting µ =
+ϵ
+2v and using that N ≥ 2
+√
+2uv · δ
+ϵ gives the result.
+A.2
+Primal-dual iterations for unconstrained problems
+Proof of Lemma 4.11. We use (4.10) and prove bounds on each of the terms on the left-hand side.
+First, we have
+L (Xk, y) = ⟨BXk, y⟩R + q(Xk) + g(Xk) − h∗(y).
+Since h is convex and lower semicontinuous, h∗∗ = h. It follows that
+h(BXk) = max
+y∈Cm⟨BXk, y⟩R − h∗(y) = − min
+y∈Cm(h∗(y) − ⟨BXk, y⟩R).
+The objective function is convex and lower semicontinuous, and the set of minimizers is y such that
+0 ∈ ∂ (h∗(·) − ⟨·, BXk⟩) (y) = ∂h∗(y) − BXk.
+Rearranging and using the Legendre–Fenchel identity, we deduce that this set of minimizers is
+precisely ∂h(BXk). It follows that
+L (Xk, y) = f(Xk),
+∀y ∈ ∂h(BXk).
+(A.1)
+Second, we have
+L (x, Yk) = ⟨Bx, Yk⟩R + q(x) + g(x) − h∗(Yk).
+The above argument shows that
+h(Bx) = max
+y∈Cm⟨Bx, y⟩R − h∗(y) ≥ ⟨Bx, Yk⟩R − h∗(Yk).
+It follows that
+L (x, Yk) ≤ f(x).
+(A.2)
+The bound (4.11) now follows by combining (A.1) and (A.2).
+35
+
+Proof of Proposition 4.12. First, consider general τ, σ > 0 with τ(σL2
+B + Lq) = 1. For input x0
+with d(x0, �
+X) ≤ δ, (4.13) and (4.12) imply that for x ∈ �
+X,
+f(XN) − ˆf ≤ 1
+N
+�δ2
+τ + L2
+h
+σ
+�
+= 1
+N
+�
+σδ2L2
+B + L2
+h
+σ + δ2Lq
+�
+.
+Choosing the step size σ > 0 to minimize the right-hand side leads to
+σ = Lh
+δLB
+,
+τ =
+δ
+LBLh + δLq
+,
+f(XN) − ˆf ≤ δ
+N (2LBLh + δLq) .
+Equations (4.14) and (4.15) now follow by taking N =
+� δ
+ϵ (2LBLh + δLq)
+�
+.
+A.3
+Primal-dual iterations for constrained problems
+Proof of Lemma 4.13. Using the same arguments as the proof of Lemma 4.11, (4.18) implies that
+for y(0)
+2
+= 0,
+f(Xk) − f(x) + ⟨AXk, y2⟩R − sup
+z∈C
+⟨z, y2⟩R − ⟨Ax, [Yk]2⟩R + sup
+z∈C
+⟨z, [Yk]2⟩R
+≤ 1
+k
+�
+�∥x − x(0)∥
+2
+τ
++ ∥y1 − y(0)
+1 ∥
+2
+σ1
++ ∥y2∥2
+σ2
+�
+� ,
+∀x ∈ Cn, y1 ∈ ∂h(BXk), y2 ∈ Cm′.
+If x ∈ Q, then
+−⟨Ax, [Yk]2⟩R + sup
+z∈C
+⟨z, [Yk]2⟩R ≥ 0.
+Let ˆz ∈ C be of minimal distance to AXk and let y2 be a multiple of AXk − ˆz such that y2 has
+norm κ. Since C is convex, the following holds [10, Theorem 6.41]
+⟨z, y2⟩R ≤ ⟨ˆz, y2⟩R,
+∀z ∈ C.
+It follows that
+⟨AXk, y2⟩R − sup
+z∈C
+⟨z, y2⟩R ≥ ⟨AXk − ˆz, y2⟩R = κ · inf
+z∈C ∥AXk − z∥ = gQ(κ; Xk).
+Combining the inequalities yields (4.19).
+Proof of Proposition 4.14. First, consider general τ, σ1, σ2 > 0 with τ(σ1L2
+B + σ2L2
+A + Lq) = 1.
+For input x0 with d(x0, �
+X) ≤ δ, we argue as in the proof of Proposition 4.12 (but now using
+Lemma 4.13) to obtain
+f(XN) − ˆf + gQ(κ; XN) ≤ 1
+N
+�δ2
+τ + L2
+h
+σ1
++ κ2
+σ2
+�
+= 1
+N
+�
+σ1δ2L2
+B + L2
+h
+σ1
++ σ2δ2L2
+A + κ2
+σ2
++ δ2Lq
+�
+.
+(A.3)
+Optimizing the proximal step sizes leads to
+τ =
+δ
+κLA + LhLB + δLq
+,
+σ1 = Lh
+δLB
+,
+σ2 =
+κ
+δLA
+.
+Substituting these values into (A.3) leads to
+f(XN) − ˆf + gQ(XN) ≤ δ
+N (2κLA + 2LhLB + δLq) .
+The rest of the proof follows the same argument as the proof of Proposition 4.12.
+36
+
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diff --git a/WdE0T4oBgHgl3EQfVwBf/content/tmp_files/load_file.txt b/WdE0T4oBgHgl3EQfVwBf/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..bcdf213f1329ccf795ce27a3b932d8abd1c0325e
--- /dev/null
+++ b/WdE0T4oBgHgl3EQfVwBf/content/tmp_files/load_file.txt
@@ -0,0 +1,1910 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf,len=1909
+page_content='Restarts subject to approximate sharpness: A parameter-free and optimal scheme for first-order methods Ben Adcock, Matthew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Colbrook, Maksym Neyra-Nesterenko January 9, 2023 Abstract Sharpness is an almost generic assumption in continuous optimization that bounds the dis- tance from minima by objective function suboptimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' It leads to the acceleration of first-order methods via restarts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' However, sharpness involves problem-specific constants that are typically unknown, and previous restart schemes reduce convergence rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Moreover, such schemes are challenging to apply in the presence of noise or approximate model classes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=', in compressive imaging or learning problems), and typically assume that the first-order method used produces feasible iterates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We consider the assumption of approximate sharpness, a generalization of sharpness that incorporates an unknown constant perturbation to the objective function er- ror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This constant offers greater robustness (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=', with respect to noise or relaxation of model classes) for finding approximate minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' By employing a new type of search over the un- known constants, we design a restart scheme that applies to general first-order methods and does not require the first-order method to produce feasible iterates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Our scheme maintains the same convergence rate as when assuming knowledge of the constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The rates of convergence we obtain for various first-order methods either match the optimal rates or improve on previ- ously established rates for a wide range of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We showcase our restart scheme on several examples and point to future applications and developments of our framework and theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Keywords: First-order methods, Restarting and acceleration, Approximate sharpness, Convex optimization, Convergence rates, Inverse problems Mathematics Subject Classification: 65K0, 65B99, 68Q25, 90C25, 90C60 1 Introduction First-order methods are the workhorse of much of modern continuous optimization [6, 10, 24, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' They are widely used to solve large-scale problems because of their excellent scalability and easiness of implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' However, standard first-order methods often converge slowly, for instance, when applied to nonsmooth objective functions or functions lacking strong convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This has motivated a large amount of work on speeding up such methods [11,30,48,55,60,64,65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Recently there has been significant interest in using restarts to accelerate the convergence of first-order methods [1,13,27,33,34,37,39,44,46,47,49,52,57,61,62,66,68,69,71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' A restart scheme repeatedly takes the output of an optimization algorithm instance as the initial point of a new instance or “restart”, and additionally may reselect the algorithm parameters before executing the 2Corresponding author: m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='colbrook@damtp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='uk DAMTP, Centre for Mathematical Sciences, University of Cambridge, UK 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='02268v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='OC] 5 Jan 2023 new instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Under the right conditions, the objective error and feasibility gap decay faster for the restarted scheme than for the underlying (unrestarted) first-order method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' However, as discussed below, existing restart schemes either require somewhat restrictive as- sumptions in which various constants are known, or attain suboptimal convergence rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This paper overcomes these limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We introduce a general restart scheme that applies to a broad class of convex optimization problems, generalizes and improves upon various existing schemes, and leads to optimal complexity bounds for a wide range of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1 The problem We consider the general convex optimization problem min x∈Q f(x), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1) where f : D → R is a proper, closed convex function with non-empty effective domain D ⊆ Cn, and Q ⊆ Cn is a closed, convex set with Q ⊂ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Let ˆf denote the optimal value of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1) and � X ⊂ Q denote its set of minimizers, where we assume that � X is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Our key assumption is that f satisfies the following approximate sharpness condition d(x, � X) ≤ � f(x) − ˆf + gQ(x) + η α �1/β , ∀x ∈ D, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) for a metric d on Cn and some constants α > 0, β ≥ 1, η ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We slightly abuse notation by defining d(x, S) := infz∈S d(x, z) for a set S ⊆ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Here, gQ : D → R+ is a function satisfying gQ(x) = 0 ⇐⇒ x ∈ Q and for any sequence {xm} ⊂ D, d(xm, Q) → 0 implies g(xm) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In this paper, we assume that the function gQ is known, but that the constants η, α and β (or a subset thereof) are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We refer to gQ as the feasibility gap function and f − ˆf as the objective (function) error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' To formulate a restart scheme that accelerates an optimization algorithm solving (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1), we assume that f satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2), and that we have access to an optimization algorithm Γ : R++ × R++ × D → D that defines a map (δ, ϵ, x0) �→ x, with the property that d(x0, � X) ≤ δ =⇒ f(x) − ˆf + gQ(x) ≤ ϵ, where x = Γ(δ, ϵ, x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3) In essence, for an initial value x0 within distance δ of an optimal solution, the algorithm produces an output x that is ϵ-suboptimal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=', f(x) − ˆf ≤ ϵ, and ϵ-feasible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=', gQ(x) ≤ ϵ, for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3) is a generic condition that appear in typical convergence analysis of first-order methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In Section 4, we describe various examples of first-order optimization methods that yield algorithms satisfying this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' See also [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The algorithms Γ we consider in this paper are iterative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We define the cost function CΓ : R++ × R++ → N, where CΓ(δ, ϵ) represents an upper bound on the number of iterations Γ needs to compute x = Γ(δ, ϵ, x0) for any starting value x0 satisfying d(x0, � X) ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' One can generalize this framework to also consider cost in terms of floating point operations or other measures of time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' It is assumed that CΓ is nondecreasing in its first argument and nonincreasing in its second argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Examples are given in Section 4 for various first-order methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2 Motivations The assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) is much weaker than typical assumptions for acceleration, such as strong convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' It can be considered an approximate version of the sharpness condition considered in [69] (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We discuss its links to other error bounds in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' There are two key differences between (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) and sharpness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' First, we do not assume that the sharpness condition is exact, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=', we have an additional η ≥ 0 term that controls the approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This is very important in many applications and for noisy data, and provides greater robustness of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' For example, when considering sparse recovery, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) covers both noisy measurements and approximately sparse vectors [27], which is more realistic than exact sparse recovery from noiseless measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Second, we do not require iterates of our algorithm to be feasible, and this is captured by the additional feasibility gap function gQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This adds further flexibility and efficiency when selecting the first-order method for the restart scheme (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=', the primal-dual algorithm considered in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The other key motivation for this work is that we do not assume knowledge of the constants α, β, and η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' When these parameters are known, it is relatively straightforward to derive a restart scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' However, the constants are rarely known in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' For example, sharpness holds for general subanalytic convex functions [17], but the proof of this result uses topological arguments that are far from constructive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' As another example, in a sparse recovery problem, η depends on the noise level and the sparsity level of the unknown vector, neither of which are typically known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In some applications, one may have bounds for one or more of these constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Nevertheless, if such bounds are loose – for instance, global bounds may be highly pessimistic near minimizers – this can lead to inefficient schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Our method obviates the need for such bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' However, it also allows the user to input such prior information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=', exact values of or ranges for the constants) if these are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3 Contributions The following theorem, which follows directly from the results presented in Section 3, summarizes our main convergence rates result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Let α, β and η be (unknown) approximate sharpness constants of f in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Consider Algorithm 2 for fixed a, b > 1, r < 1, α0 > 0, β0 ≥ 1 and the choices of schedule criterion and assignment functions described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Then running Algorithm 2 with t ≳ K(ε), ε → 0+, (total inner) iterations, where K(ε) is given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3), implies that f(x(t)) − ˆf + gQ(x(t)) ≤ max{η, ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Let β∗ = b⌈logb(β/β0)⌉β0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' If, in addition, CΓ satisfies CΓ(δ, ϵ) ≤ Cδd1/ϵd2 + 1, C, d1, d2 > 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4) for all δ, ϵ > 0, then we have K(ε) ≤ ˆCεd1/β∗−d2 · � ⌈log(1/ε)⌉ , if d2 ≤ d1/β∗, 1, if d2 > d1/β∗, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5) where ˆC is independent of ε (but depends on r, a, b, α, β∗, α0, β0, d1 and d2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Explicit forms for ˆC in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5) are given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 3 i j k Unknown α and β 0 1 2 3 4 5 0 10 20 30 40 50 j k Known α 5 0 5 0 10 20 30 40 50 i k Known β Figure 1: Level curves of h = 50 for the schedule criterion functions h in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3 (left panel), Corol- lary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4 (middle panel) and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5 (right panel) with c1 = c2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The level curves describe the search order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The red dots show the corresponding indices (i, j, k) in the set defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The index i indicates the parameter search value aiα0 for α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The index j indicates the parameter search value bjβ0 for β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The height (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=', k) indicates the total number of inner iterations for a fixed (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' A few comments are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' First, note that ε is not a parameter of the algorithm: it is only used to describe the algorithm’s behavior as the number of iterations increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Second, it is possible for a problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1) to satisfy the approximate sharpness condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) for different parameters α, β and η, which may give different convergence rates and constant ˆC in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' If so, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1 says that for a given accuracy threshold ε ≥ η, we can take the best rate of convergence/iteration bound over different approximate sharpness constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Third, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1 does not guarantee a decrease of the objective function error below η as ε → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This is quite reasonable in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' For example, in the case of sparse recovery from noisy measurements, η is the magnitude of the noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Therefore there is little benefit in decreasing the objective function error below η, since the error in the recovered vector will generally be O (η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Fourth, the assumption in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4) is generic for convergence rates of first-order methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We present some examples in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The +1 term is included in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4) since we often have a bound of the form CΓ(δ, ϵ) ≤ � Cδd1/ϵd2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Finally, the parameters α0 > 0 and β0 ≥ 1 in Algorithm 2 are estimates for the true α, β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' If no estimates are known, we can set α0 = β0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We also include the case that either or both of α and β are known in our analysis (see the Corollaries in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The parameter r ∈ (0, 1) is a scale factor that adjusts the parameters of the first-order method at each restart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' As we discuss in Section 2, a good choice is r = e−1/d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Our scheme performs a grid search over parameters α, β using the bases a, b > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The order of the search is based on a so-called schedule criterion (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This new idea allows flexibility depending on which parameters are known and which are unknown, and leads to a unified framework for proving convergence results (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=', using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We postpone the details until Section 3, but, in particular, this new framework allows us to search over a nonuniform grid (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3) that searches more in iteration space as opposed to parameter index space (see left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This is key to developing a search method for unknown parameters that does not suffer from reduced convergence rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Suppose now that η ≲ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' When Algorithm 2 is applied with a suitable first-order method, it leads to optimal1 complexity bounds for a wide range of different convex optimization problems, without 1By optimal, we mean optimal in the number of oracle calls to f, its gradient (where appropriate) or suitable proximal maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' For the first-order methods we discuss, this number will always be bounded by a small multiple of the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 4 50 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 30 20、 10、 0 0 5 5 0 5Objective function class/structure Asymptotic bound for K(ε) Example method L−smooth See Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2 (NB: must have β ≥ 2) β = 2: � L/α · log(1/ε) Nesterov’s method d1 = 1, d2 = 1/2 See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1 β > 2: √ L α1/β∗ · 1 ε1/2−1/β∗ (u, v)−smoothable See Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5 β = 1: √ ab α · log(1/ε) Nesterov’s method with smoothing d1 = 1, d2 = 1 See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2 β > 1: √ ab α1/β∗ · 1 ε1−1/β∗ H¨older smooth, parameter ν ∈ [0, 1] See Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='8 (NB: must have β ≥ 1 + ν) β = 1+ν: M 2 1+3ν ν α 2 (1+3ν) · log(1/ε) Universal fast gradient method d1 = (2 + 2ν)/(1 + 3ν) d2 = 2/(1 + 3ν) See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3 β > 1+ν: M 2 1+3ν ν α 2+2ν β∗(1+3ν) · 1 ε 2(β∗−1−ν) β∗(1+3ν) f(x)=q(x)+g(x)+h(Bx), q is Lq−smooth, supz∈dom(h) infy∈∂h(z) ∥y∥ ≤ Lh, ∥B∥ ≤ LB β = 1: LBLh+Lq α log(1/ε) Primal-dual algorithm d1 = 1, d2 = 1 See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4 β > 1: LBLh+Lq α1/β∗ 1 ε1−1/β∗ f(x)=q(x)+g(x)+h(Bx), q is Lq−smooth, supz∈dom(h) infy∈∂h(z) ∥y∥ ≤ Lh, ∥A∥ ≤ LA, ∥B∥ ≤ LB, Q={x : Ax ∈ C}, gQ(x)=κ infz∈C∥Ax − z∥ β = 1: κLA+LBLh+Lq α log(1/ε) Primal-dual algorithm with constraints d1 = 1, d2 = 1 See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5 β > 1: κLA+LBLh+Lq α1/β∗ 1 ε1−1/β∗ Table 1: Asymptotic cost bounds (as ε ↓ 0 for η ≲ ε) and suitable first-order methods for Algorithm 2 when applied to different classes of objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Note that whenever the bound is a polynomial in log(1/ε), we have β∗ = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' knowledge of α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Table 1 summarizes some of these bounds and the following correspond to an example for each row: For L-smooth functions (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) with β = 2, a well-known lower bound for the subclass of strongly convex smooth functions is O( � L/α log(1/ε)) [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' If β > 2 then the optimal lower bound is O( √ Lα−1/β/ε1/2−1/β) [53, page 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In both cases, we achieve these optimal bounds with our algorithm using, for example, Nesterov’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This is an improvement (by at least a factor of log(1/ε)) over the restart scheme presented in [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Suppose that the objective function f is Lf-Lipschitz and has linear growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Such functions are (1, L2 f/2)-smoothable (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' When β = 1, the combination of our algorithm and Nesterov’s method with smoothing has complexity O (log(1/ε)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This is an improvement over the restart scheme presented in [66], which has complexity O � log2(1/ε) � for such functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Similarly, for general (u, v)-smoothable objective functions, we improve (by at least a factor of log(1/ε)) on the results over the restart scheme presented in [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' For H¨older smooth functions (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='8), the bound in Table 1 matches (with β replaced by β∗) the optimal bound from [53, page 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This is an improvement (by at least a factor of log(1/ε)) over the restart scheme presented in [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 5 There is little work on optimal rates for saddle point problems, a challenge being that there are different measures of error (see [70]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Hence we cannot claim that the final two rows of Table 1 yield optimal rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Nevertheless, they yield significantly faster convergence rates than unrestarted first-order methods for saddle point problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Finally, it is worth pointing out two straightforward generalization of the assumptions in Sec- tion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1 where our algorithms and results also hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' First, the approximate sharpness condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) can be further generalized to consider any fixed set Y ⊆ D as opposed to � X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This is expressed as d(x, y) ≤ �f(x) − f(y) + gQ(x) + η α �1/β , ∀x ∈ D, y ∈ Y, where α, β, η, and gQ are defined the same way as in the (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' With a suitable generalization of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3), much of the work presented here can be extended to this general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Note that this is of particular interest whenever the exact minimizer of the associated optimization problem is not desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In sparse recovery, the ground truth vector being recovered from noisy measurements is often not the minimizer of the associated optimization problem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=', see Section 5 or [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' It is sufficient when the recovered vector’s measurements match the original measurements up to a noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Similarly, when training overparameterized models in machine learning, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=', deep neural networks, a balance between training error and generalization error is preferred as opposed to solely minimizing the training error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Second, our restart procedure for unknown constants always decreases the sum of the objective and feasibility gap functions after each restart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Moreover, we only make use of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) in our analysis each time we restart, so it suffices that we only need (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) to hold in the sublevel set {x ∈ D : f(x) + gQ(x) ≤ f(x0) + gQ(x0)} for a starting vector x0 ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4 Connections with previous work Recently, there has been a large amount of work on adaptive first-order methods [33, 34, 37, 39, 52,62,71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Adaptive methods seek to learn when to restart a first-order method by trying various values for the method’s parameters and observing consequences over a number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' A catalyst for this body of work was provided by Nesterov [57], where he designed an accelerated (line search) method for L-smooth objective functions f (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1) with an optimal convergence rate O( � L/ε) without needing L as an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In the same paper, Nesterov considered strongly convex objective functions with a grid search for approximating the strong convexity parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' By narrowing the class of objective functions, this led to an adaptive method with a dramatically improved convergence rate (O(log(1/ε)) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' O(1/√ε)), even without having to know the Lipschitz constant or strong convexity parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The complexity of first-order methods is usually controlled by smoothness assumptions on the objective function, such as Lipschitz continuity of its gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Additional assumptions on the objective function such as strong and uniform convexity provide, respectively, linear and faster polynomial rates of convergence [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Restart schemes for strongly convex or uniformly convex functions have been studied in [44, 49, 53, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' However, strong or uniform convexity is often too restrictive an assumption in many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 6 An assumption more general than strong or uniform convexity is sharpness: d(x, � X) ≤ � f(x) − ˆf α �1/β , ∀x ∈ Q, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='6) also known as a H¨olderian growth/error bound or a �Lojasiewicz-type inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' For example, Nemirovskii and Nesterov [53] linked a “strict minimum” condition similar to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='6) (with known constants) with faster convergence rates using restart schemes for smooth objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' For further use of �Lojasiewicz-type inequalities for first-order methods, see [7,18,19,36,45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' H¨olderian error bounds were first introduced by Hoffman [43] to study systems of linear inequalities, and extended to convex optimization in [8,20,21,51,67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' �Lojasiewicz showed that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='6) holds generically for real analytic and subanalytic functions [50], and Bolte, Daniilidis, and Lewis extended this result to nonsmooth subanalytic convex functions [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' However, the proofs of these results use topological arguments that are far from constructive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Hence, without further case-by-case analysis of problems and outside of some particular cases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=', strong convexity), we cannot assume that suitable constants in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='6) are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' An example of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='6) for β = 1 was considered in [68] (see also [16]), where the authors use a restarted NESTA algorithm [12] for the exact recovery of sparse vectors from noiseless measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The approximate sharpness condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) was first considered in [27] for the case of β = 1, and known α and η, to allow the recovery of approximately sparse vectors from noisy measurements and further related examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Here the parameter η > 0 is crucial, both in practice and to allow analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' See also [1, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Though similar to the sharpness condition in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='6), our more general assumption in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) differs in two important ways, discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' First, we do not assume that the sharpness condition is exact (η > 0), and, second, we do not require iterates of our algorithm to be feasible (the function gQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' It is also important to re-emphasize that, in this paper, we do not assume that the approximate sharpness constants are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The η term in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) is expected and natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' For example, in [28] it was shown that there are well-conditioned recovery problems for which stable and accurate neural networks exist, but no training algorithm can obtain them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The existence of a training algorithm depends on the amount/type of training data and the accuracy required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' However, under certain conditions, one can train an appropriate neural network: [28] links trainability to a special case of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2), and links the accuracy possible via training to the corresponding η term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In the setting of inexact input, the noise parameter appears as a limitation on the ability of an algorithm [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' These phenomena occur even if the algorithm is only expected to work on a restricted class of inputs that are ‘nice’ or ‘natural’ for the problem under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The results of [9,28] lead to the phenomenon of generalized hardness of approximation (see also [38]), where it is possible to obtain solutions up to some threshold, but beyond that threshold it becomes impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This threshold is strongly related to η in the standard cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Most restart schemes are designed for a narrow family of first-order methods, and typically rely on learning approximations of the parameter values characterizing functions in a particular class (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=', learning the Lipschitz constant L when f is assumed to be L-smooth, or the constants α and β in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' There are two notable exceptions related to the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' First, Roulet and d’Aspremont [69] consider all f possessing sharpness, and having H¨older continuous gradient with exponent 0 < ν ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The restart schemes of [69] result in optimal complexity bounds when particular algorithms are employed in the schemes, assuming scheme parameters are set to appropriate values that, however, are generally unknown in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' However, for smooth f (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' ν = 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' [69] develops an adaptive grid search procedure within the scheme to accurately approximate the required values,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' leading to an overall complexity that is optimal up to logarithmic 7 Notation Meaning f Proper convex function D Effective domain of f Q Closed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' convex subset of Rn or Cn gQ Sharpness feasibility gap function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' identically zero on Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='ˆf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='Minimum value of objective function over Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='Set of minimizers of f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='Metric on Rn or Cn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='Sharpness gap constant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='Sharpness scaling constant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='Sharpness exponentiation constant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='Distance bound between initial point to optimum points ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='ε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='Bound on sum of objective function error and feasibility gap ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='ϵj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='Sum of objective function error and feasibility gap at jth restart initial point ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='Optimization algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='CΓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='Cost function that outputs the number of iterates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='Mapping of current algorithm step to parameter subscripts (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' k) h Function defining classes of maps φ as abstract execution order of restart scheme χC Indicator function of a set C (χC(x) = 0 if x ∈ C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' χC(x) = ∞ otherwise) ∥·∥ Unless otherwise stated,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' the Euclidean norm on Cn or the induced 2-norm on Cm×n ⟨·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' ·⟩ Unless otherwise stated,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' the Euclidean inner product on Cn ⟨·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' ·⟩R Unless otherwise stated,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' ⟨x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' y⟩R = Re (⟨x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' y⟩) for x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' y ∈ Cn R+ Non-negative real numbers R++ Positive real numbers N0 Non-negative integers {0} ∪ N Table 2: Notation used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Second, Renegar and Grimmer [66] provide a simple scheme for restarting (generic) first- order methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Multiple instances are run that communicate their improvements in objective value to one another, possibly triggering restarts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Their restart scheme only depends on how much the objective value has been decreased and does not attempt to learn parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The scheme in [66] leads to nearly optimal complexity bounds for quite general classes of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This method differs quite significantly from ours in that it does not assume an underlying sharpness condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='6) (although such a condition is used in the analysis to obtain explicit complexity bounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' However, as observed previously, by assuming (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) we are able to obtain better and essentially optimal rates that avoid additional factors of log(1/ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Moreover, in contrast to [66], our method is independent of the total number of iterations, and we do not need to specify the total number of iterations in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Further, we also address the practical case of approximate sharpness and allow the case of infeasible iterates (the convergence analysis of [66] relies on η = 0 and that iterates are feasible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5 Notation and outline For ease of reference, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5 outlines the notation used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In Section 2, we introduce a restart scheme in the case where η is unknown, but α and β are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This transpires to be significantly simpler than the general 8 Algorithm 1: Restart scheme for unknown η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Input : Optimization algorithm Γ for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1), initial vector x0 ∈ D, upper bound ϵ0 such that f(x0) − ˆf + gQ(x0) ≤ ϵ0, constants α > 0 and β ≥ 1 such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) holds (for possibly unknown η ≥ 0), r ∈ (0, 1), and number of restart iterations t ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Output: Final iterate xt approximating a solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1) 1 for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' , t − 1 do 2 ϵk+1 ← rϵk ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 3 δk+1 ← � 2ϵk α �1/β ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 4 z ← Γ (δk+1, ϵk+1, xk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 5 xk+1 ← argmin {f(x) + gQ(x) : x = xk or x = z};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 6 end case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Next, in Section 3 we introduce and analyze the full restart scheme when all three constants are potentially unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In Section 4, we apply this restart scheme to different problems with various first-order methods, leading, in particular, to the results described in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Next, in Section 5 we present a series of numerical experiments illustrating the restart schemes in different applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Finally, we end in Section 6 with conclusions and open problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 2 Restart scheme for unknown η but known α and β To formulate a restart scheme within the setup of Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1, observe that the approximate sharp- ness condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) relates d(x, � X) to the objective function error f(x)− ˆf and feasibility gap gQ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The upper bound in the approximate sharpness condition can be used as δ for the algorithm Γ, and ϵ set as a rescaling of the previous sum of objective error and feasibility gap f(x)− ˆf +gQ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' How- ever, in practice, we may not know the exact values of the objective error f(x) − ˆf and feasibility gap gQ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' It is, instead, enough to know upper bounds for these quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We first consider the case where α, β are known, but η is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This simpler case provides insight into the solution of the full problem considered in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We define a restart scheme under this assumption in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) and Γ, it is easy to see inductively that for any t with ϵt ≥ η, Algorithm 1 produces iterates x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' , xt ∈ D that satisfy f(xk) − ˆf + gQ(xk) ≤ ϵk, d(xk, � X) ≤ � f(xk) − ˆf + gQ(xk) + η α �1/β ≤ �ϵk + η α �1/β ≤ �2ϵk α �1/β , 0 ≤ k ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1) In addition, the total number of inner iterations used in Algorithm 1 is at most t−1 � k=0 CΓ ��2ϵk α �1/β , ϵk+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Under further assumptions about the function CΓ, we can show that the iterates produced by the restart scheme yield linear (if d2 = d1β) or fast algebraic (if d2 > d1β) decay of f(xk) − ˆf + gQ(xk) in k down to a finite tolerance proportional to η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Hence, this property holds for both the objective error f(xk) − ˆf and feasibility gap gQ(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We state and prove this in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Note that these additional assumptions are not arbitrary and will appear in our examples later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 9 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Consider Algorithm 1 and its corresponding inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' For any ε ∈ (0, ϵ0), if we run Algorithm 1 with t ≥ ⌈log(ϵ0/ε)/ log(1/r)⌉, then f(xt) − ˆf + gQ(xt) ≤ max{η, ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) Suppose, in addition, that for all δ, ϵ > 0, CΓ satisfies CΓ(δ, ϵ) ≤ Cδd1/ϵd2 + 1, C, d1, d2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Then the total number of iterations of Γ needed to compute an xt with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) is at most �log(ϵ0/ε) log(1/r) � + C2d1/β αd1/βrd2 · � � � � � � � � � 1−r⌈log(ϵ0/ε)/log(1/r)⌉|d2−d1/β| 1−r|d2−d1/β| 1 ϵd2−d1/β 0 , if d2 < d1/β, � log(ϵ0/ε) log(1/r) � , if d2 = d1/β, 1−r⌈log(ϵ0/ε)/log(1/r)⌉|d2−d1/β| 1−r|d2−d1/β| 1 εd2−d1/β , if d2 > d1/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3) Note that the cases in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3) match in the limit d2 − d1/β → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The statement of the theorem is unchanged if we assume that ε ≥ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Hence, we may assume without loss of generality that ε ≥ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Let s = ⌈log(ϵ0/ε)/ log(1/r)⌉, then ϵs−1 = rs−1ϵ0 ≥ ε ≥ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' It follows that we are in the regime where (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1) holds and hence f(xs−1) − ˆf + gQ(xs−1) ≤ ϵs−1, d(xs−1, � X) ≤ �2ϵs−1 α �1/β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Then by line 4 of Algorithm 1 and the choice of s, we have f(z) − ˆf + gQ(z) ≤ ϵs ≤ ε, z = Γ(δs, ϵs, xs−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Due to the argmin taken in Algorithm 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The total number of iterations, T, needed to reach such an xs is bounded by T ≤ s−1 � k=0 CΓ ��2ϵk α �1/β , ϵk+1 � ≤ s + C s−1 � k=0 (2ϵk)d1/β αd1/βϵd2 k+1 = s + C2d1/β αd1/βrd2 s−1 � k=0 1 ϵd2−d1/β k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In the case that d2 = d1/β, then ϵd2−d1/β k = 1 and we obtain T ≤ s + C2d1/β αd1/βrd2 s = � 1 + C2d1/β αd1/βrd2 � �log(ϵ0/ε) log(1/r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' If d2 ̸= d1/β, we use that ϵk = rkϵ0 and sum the geometric series to obtain T ≤ �log(ϵ0/ε) log(1/r) � + C2d1/β αd1/βrd2 1 − r⌈log(ϵ0/ε)/log(1/r)⌉(d1/β−d2) 1 − rd1/β−d2 1 ϵd2−d1/β 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4) If d2 > d1/β, then since ϵ0 ≥ ε/rs−1, we have ϵd2−d1/β 0 ≥ εd2−d1/βrd2−d1/β/rs(d2−d1/β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Substituting this into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4) and rearranging yields T ≤ �log(ϵ0/ε) log(1/r) � + C2d1/β αd1/βrd2 1 − r⌈log(ϵ0/ε)/log(1/r)⌉(d2−d1/β) 1 − rd2−d1/β 1 εd2−d1/β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The result follows by considering the three separate cases in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 10 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2 (How to choose r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Suppose that d2 = d1/β and that �log(ϵ0/ε) log(1/r) � ≤ 2log(ϵ0/ε) log(1/r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Using this new bound instead, the total number of iterations T performed by Γ is bounded by T ≤ �log(ϵ0/ε) log(1/r) � + C2d1/β+1 αd1/β log(ϵ0/ε) r−d2 log(1/r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Hence T is bounded by an ε-dependent constant times r−d2/ log(1/r), which can be minimized analytically by choosing r = e−1/d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Note that the optimal r here does not depend on the approximate sharpness constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Therefore, one has T ≤ ⌈d2 log(ϵ0/ε)⌉ + Ced22d1/β+1 αd1/β log(ϵ0/ε) This is meaningful in terms of choosing one less parameter, namely r for Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' An optimal value of r can also be found for the case d2 > d1/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' However, this optimal value depends on ε in a complicated manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In the limit ε ↓ 0, the optimal choice is r = � d2 2d2 − d1/β � 1 d2−d1/β , which does depend on the sharpness constant β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' As d2 − d1/β ↓ 0, this choice converges to the choice r = e−1/d2, that is obtained when d2 = d1/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Similarly, if d2 < d1/β, then the optimal choice depends on ε in a complicated manner but converges to the choice r = e−1/d2 as d2 − d1/β ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In any of these cases, the same argument for optimal r applies to the algorithms in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In the case that β is unknown, we recommend the choice r = e−1/d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' ♦ 3 Restart scheme for unknown α, β and η In the event that the constants α, β of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) are unknown, we introduce a logarithmic grid search on each of α and β, running multiple instances of Algorithm 1, and aggregating results that minimize the objective error and feasibility gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Even if suitable global α and β are known, the following algorithm is useful since it also takes advantage of sharper versions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) that only hold locally around optimal points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1 The algorithm To introduce the algorithm, we need some additional notation and definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This will allow us to define a new general scheme for logarithmic grid searches, with examples given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Consider an infinite subset S ⊆ Z × N0 × N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Let h : R+ × R+ × R++ → R++ be a function that is nondecreasing in its first and second arguments, and strictly increasing in its third argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We call such an h a schedule criterion function, or simply a schedule criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Given a schedule criterion h, an h-assignment over S is a bijection φ : N → S satisfying h(|i′|, j′, k′) ≤ h(|i|, j, k) ⇐⇒ φ−1(i′, j′, k′) ≤ φ−1(i, j, k), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1) for all (i, j, k), (i′, j′, k′) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' ▲ 11 Algorithm 2: Restart scheme for unknown α, β and η in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) via grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Input : Optimization algorithm Γ for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1), bijection φ as in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1, initial vector x(0) ∈ D, upper bound ϵ0 such that f(x(0)) − ˆf + gQ(x(0)) ≤ ϵ0, constants a, b > 1, r ∈ (0, 1), α0 > 0, β0 ≥ 1 and total number of inner iterations t ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Output: Final iterate x(t) approximating a solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 1 Initialize x(0) = x0, Ui,j = 0, Vi,j = 0, ϵi,j,0 = ϵ0 for all i ∈ Z, j ∈ N0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 2 for m = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' , t − 1 do 3 (i, j, k) ← φ(m + 1) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 4 αi ← aiα0, βj ← bjβ0, U ← Ui,j, V ← Vi,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 5 ϵi,j,U+1 ← rϵi,j,U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 6 if 2ϵi,j,U > αi then 7 δi,j,U+1 ← � 2ϵi,j,U αi �min{b/βj,1/β0} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 8 else 9 δi,j,U+1 ← � 2ϵi,j,U αi �1/βj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 10 end 11 if V + CΓ (δi,j,U+1, ϵi,j,U+1) ≤ k then 12 z(m) ← Γ � δi,j,U+1, ϵi,j,U+1, x(m)� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 13 x(m+1) ← argmin � f(x) + gQ(x) : x = z(m) or x = x(m)� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 14 Vi,j ← V + CΓ (δi,j,U+1, ϵi,j,U+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 15 Ui,j ← U + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 16 else 17 x(m+1) = x(m) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 18 end 19 end Let a, b > 1 be constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Our algorithm employs logarithmic search grids for the unknown parameters α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Specifically, we consider the values αi = aiα0 for i ∈ Z and βj = bjβ0 for j ∈ N0, where we assume that α0, β0 are additional inputs with α0 > 0 and β ≥ β0 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In essence, our algorithm applies the restart scheme described in Algorithm 1 with the values αi and βj for each i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' However, it does so according to a particular schedule, specified by the functions h and φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The schedule criterion and assignment together control the execution order of Algorithm 1 instances for each i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Note that the lower bound β0 in the definition of the βj is to capture additional knowledge that may be available (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=', the examples in Section 4), and may be set to 1 if no such knowledge is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Similarly, the constant α0 centers the search grid for α and can be set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Our algorithm is presented in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' It proceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' At step m ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' , t − 1} it first applies the bijection φ to obtain the tuple (i, j, k) = φ(m + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The first two entries give the approximate sharpness parameter values αi = aiα0 and βj = bjβ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The final entry k is a counter, which is an upper bound for the total number of iterations used by the algorithm for these parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We also have two further counters associated with each double (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The counter Vi,j counts the total number of inner iterations of Γ used by the restart scheme with these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The second counter Ui,j counts the number of completed restarts (outer iterations) corresponding to these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Having obtained a tuple (i, j, k) = φ(m + 1), the algorithm proceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' First, much as 12 in line 2 of Algorithm 1, it updates the first scaling parameter in line 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Then, reminiscent of line 3 of Algorithm 1, it updates the other scaling parameter in lines 6-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This step is more involved, a complication that arises because the true parameter β is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The next lines, lines 11-16, are similar to lines 4-5 of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The main difference is the inclusion of the if statement, which is done to control the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' It stipulates that a restart be performed (line 12) if the total cost (including the proposed restart) does not exceed the counter k (line 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' If this is not the case, then no restart is performed, and the algorithm moves on to the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We now present a general result on this algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' It relates the total number of inner iterations of Γ used by Algorithm 2 to produce a solution within a desired error to intrinsic properties of the schedule criterion function h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' With this in hand, we derive explicit bounds for specific choices of h in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Let S ⊆ Z × N0 × N be an infinite subset, h be a schedule criterion, and φ an h-assignment over S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Let α, β and η be approximate sharpness constants of f in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Consider Algorithm 2 for fixed a, b > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Define the (unknown) indices I = ⌊loga(α/α0)⌋, J = ⌈logb(β/β0)⌉ and the corresponding constants α∗ = aIα0 ≤ α, β∗ = bJβ0 ≥ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Then for q ∈ N we have δI,J,q = � max � 1, 2rq−1ϵ0 α∗ ��min{b/β∗,1/β0} � min � 1, 2rq−1ϵ0 α∗ ��1/β∗ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) Now, for any ε ∈ (0, ϵ0), let K(ε) := K(ε, α, β, η) = ⌈log(ϵ0/ε)/ log(1/r)⌉ � q=1 CΓ (δI,J,q, rqϵ0) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3) and suppose that (I, J, K(ε)) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Then the total number of inner iterations of Γ needed by Algorithm 2 to compute x(t) with f(x(t)) − ˆf + gQ(x(t)) ≤ max{η, ε}, is bounded by the cardinality of the set � (i′, j′, k′) ∈ S : h(|i′|, j′, l′) ≤ h (|I|, J, K(ε)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4) In addition, if CΓ satisfies CΓ(δ, ϵ) ≤ Cδd1/ϵd2 + 1, C, d1, d2 > 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5) for all δ, ϵ > 0, then we have K(ε) ≤ �log(ϵ0/ε) log(1/r) � + max � � � �2ϵ0 α∗ �d1 min � b−1 β∗ , 1 β0 − 1 β∗ � , 1 � � � × C2d1/β∗ αd1/β∗ ∗ rd2 · � � � � � � � � � 1−r⌈log(ϵ0/ε)/log(1/r)⌉|d2−d1/β∗| 1−r|d2−d1/β∗| 1 ϵd2−d1/β∗ 0 , if d2 < d1/β∗, � log(ϵ0/ε) log(1/r) � , if d2 = d1/β∗, 1−r⌈log(ϵ0/ε)/log(1/r)⌉|d2−d1/β∗| 1−r|d2−d1/β∗| 1 εd2−d1/β∗ , if d2 > d1/β∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='6) 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Since ϵi,j,q−1 = rq−1ϵ0 for all q ∈ N, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) must hold by considering the two separate cases defining δI,J,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Similar to the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1, we may assume without loss of generality that ε ≥ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Note that, due to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2), d(x, � X) ≤ � f(x) − ˆf + gQ(x) + η α∗ �1/β , ∀x ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='7) Now consider the following adapted version of the iterates in Algorithm 1: 1 for p = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' do 2 ϵp+1 ← rϵp ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 3 if 2ϵp > α∗ then 4 δp+1 ← � 2ϵp α∗ �min{b/β∗,1/β0} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 5 else 6 δp+1 ← � 2ϵp α∗ �1/β∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 7 end 8 z ← Γ (δp+1, ϵp+1, xp);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 9 xp+1 ← argmin {f(x) + gQ(x) : x = xp or x = z};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 10 end It is easy to see inductively that for any l with ϵl ≥ η the above produces iterates {x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' , xl} ⊂ D satisfying f(xp) − ˆf + gQ(xp) ≤ ϵp, d(xp, � X) ≤ δp+1, 0 ≤ p ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The only difference to the previous argument for Algorithm 1 is the use of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='7), and the fact that � f(xp) − ˆf + gQ(xp) + η α∗ �1/β ≤ �2ϵp α∗ �1/β ≤ � � � � � � 2ϵp α∗ �min{b/β∗,1/β0} , if 2ϵp > α∗ � 2ϵp α∗ �1/β∗ , otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Here, we use the fact that β ≥ β0 in the first case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In Algorithm 2, each Ui,j plays the role of the index p in the above iterates (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=', counting the number of restarts for a fixed (i, j)) and Vi,j counts the total number of inner iterations that have been executed by the algorithm Γ for the approximate sharpness constants given by the double index (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The fact that we take minimizers of f + gQ across different indices does not alter the above inductive argument, since the argument only depends on bounding the value of f − ˆf + gQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Moreover, since h is strictly increasing in its final argument and satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1), the counter index k counts successively through N for any fixed (i, j) as the for loop in Algorithm 2 proceeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' It follows that if φ(m + 1) = (I, J, k), VI,J + CΓ � δI,J,UI,J+1, ϵI,J,UI,J+1, x(m)� ≤ k and ϵI,J,UI,J ≥ η, then f(x(m+1)) − ˆf + gQ(x(m+1)) ≤ ϵI,J,UI,J+1 = rUI,J+1ϵ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='8) Hence, for Algorithm 2 to produce an iterate with f(x(t)) − ˆf + gQ(x(t)) ≤ max{η, ε}, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='9) 14 it is sufficient to reach an m with φ(m + 1) = (I, J, k) such that k ≥ ⌈log(ϵ0/ε)/ log(1/r)⌉ � q=1 CΓ (δI,J,q, ϵI,J,q) = K(ε) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='10) and execute the resulting restart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' To see why this is the case, notice that if k satisfies this inequality, then the number of restart iterations performed by the algorithm for the parameter values (I, J) is at least ⌈log(ϵ0/ε)/ log(1/r)⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Plugging this into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='8) gives the desired bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Now consider the set in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Due to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1), we notice that this set is equivalent to {(i′, j′, k′) ∈ S : φ−1(i′, j′, k′) ≤ m + 1}, where φ(m + 1) = (I, J, K(ε)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Notice that if a tuple (i′, j′, k′) belongs to this set, then (i′, j′, k′′) belongs to the set for every 1 ≤ k′′ ≤ k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Thus, the number of terms in this set corresponding to the pair (i′, j′) is precisely the total number of inner iterations performed by the algorithm at the corresponding parameter values up to step m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We immediately deduce that the cardinality of the set (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4) is a bound for the total number of inner iterations performed by the algorithm across all parameter values up to step m, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' To finish the proof, we must show that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='6) holds under the additional assumption (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5) on CΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Suppose first that δI,J,q > 1, then CΓ (δI,J,q, rqϵ0) ≤ C �2rq−1ϵ0 α∗ �d1 min{b/β∗,1/β0} (rqϵ0)−d2 + 1 ≤ C �2ϵ0 α∗ �d1[min{b/β∗,1/β0}−1/β∗] �2rq−1ϵ0 α∗ �d1/β∗ (rqϵ0)−d2 + 1 = C rd2 �2ϵ0 α∗ �d1[min{b/β∗,1/β0}−1/β∗] � 2 α∗ �d1/β∗ � rq−1ϵ0 �−d2+d1/β∗ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Similarly, if δI,J,q ≤ 1, then CΓ (δI,J,q, rqϵ0) ≤ C rd2 � 2 α∗ �d1/β∗ � rq−1ϵ0 �−d2+d1/β∗ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='10), it follows that K(ε) ≤ �log(ϵ0/ε) log(1/r) � +max ��2ϵ0 α∗ �d1[min{b/β∗,1/β0}−1/β∗] , 1 � C2d1/β∗ αd1/β∗ ∗ rd2 · ⌈ log(ϵ0/ε) log(1/r) ⌉−1 � k=0 1 (rkϵ0)d2−d1/β∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We now note that the only difference between this bound for K(ε) and the bound for T in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1 is the factor that maximizes over the terms in curly brackets and the replacement of α and β by α∗ and β∗, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The result now follows by using the same arguments as in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2 Choices of schedule criterion functions and assignments The total number of inner iterations of Γ needed for Algorithm 2 depends on the choice of h and φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We examine some choices and state them as corollaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Examples are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 15 Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3 (Unknown α and β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Suppose that S = Z × N0 × N and let h(x1, x2, x3) = (x1 + 1)c1(x2 + 1)c2x3, c1, c2 > 1 be a schedule criterion with h-assignment φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Then for any ε ∈ (0, ϵ0), running Algorithm 2 with t ≥ 2c1c2τ/[(c1 − 1)(c2 − 1)], τ = (|⌊loga(α/α0)⌋| + 1)c1(|⌈logb(β/β0)⌉| + 1)c2K(ε), where K(ε) is as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3), implies that f(x(t)) − ˆf + gQ(x(t)) ≤ max{η, ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' It suffices to prove that the stated lower bound on t is an upper bound for the cardinality of the set (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4) from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We do this by finding an upper bound on the number of solutions to nc1 1 nc2 2 n3 ≤ τ where n1, n2, n3 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' By directly counting, the number of solutions is bounded by τ 1/c1 � n1=1 � τ nc1 1 � 1 c2 � n2=1 τ nc1 1 nc2 2 ≤ τ ∞ � n1=1 1 nc1 1 ∞ � n2=1 1 nc2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We have that ∞ � n1=1 1 nc1 1 ≤ 1 + � ∞ 1 dx xc1 = c1 c1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' It follows that the number of solutions is bounded by τc1c2/((c1−1)(c2−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Each counted solution (n1, n2, n3) defines at most two tuples (i′, j′, k′) in the set (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4), namely i′ = ±(n1 − 1), j′ = n2 − 1, k′ = n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In reverse, each tuple (i′, j′, k′) of the set (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4) is always associated with a single solution (n1, n2, n3), namely n1 = |i′| + 1, n2 = j′ + 1, n3 = k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' It then follows that that the set (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4) is bounded by 2τc1c2/((c1 − 1)(c2 − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We compare the cost in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3 to that of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1 under the assumption (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Let ˆK(ε) be the cost in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' K(ε) ≲ ˆK(ε) � 1, if β = β∗ or d2 ≤ d1/β∗, 1 εd1(1/β−1/β∗) , otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='11) It follows that if β = β∗ or d2 ≤ d1/β∗, the cost of Algorithm 2 is of the same order as ˆK(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' If neither of these hold, then the cost of Algorithm 2 is of the order of ε−d1(1/β−1/β∗) times the cost of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Note that the order of this extra algebraic dependence can be made arbitrarily small by taking b close to 1, at the expense of a factor in the term τ that grows as logb(β/β0)c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We now consider the cases where either α or β is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4 (Known α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Suppose that α = aiα0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Let S = {i} × N0 × N and h(x1, x2, x3) = (x2 + 1)c2x3, c2 > 1, be a schedule criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Then given any h-assignment φ and any ε ∈ (0, ϵ0), running Algorithm 2 with t ≥ c2τ/(c2 − 1), τ = (|⌈logb(β/β0)⌉| + 1)c2K(ε), where K(ε) is as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3), implies that f(x(t)) − ˆf + gQ(x(t)) ≤ max{η, ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 16 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The result follows after modifying the proof of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' First, find an upper bound to the number of solutions to nc2 2 n3 ≤ τ for n2, n3 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Now find the correspondence between the solutions and the tuples (i′, j′, k′) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4), where i′ is now fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' For the case of known β, we alter Algorithm 2 by removing the if statement in line 6 and always using the update rule in line 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5 (Known β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Suppose that β = β0 is known, S = Z × {0} × N and h(x1, x2, x3) = (x1 + 1)c1x3, c1 > 1, is a schedule criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Then given any h-assignment φ and any ε ∈ (0, ϵ0), running Algorithm 2 and t ≥ 2c1τ/(c1 − 1), τ = (|⌊loga(α/α0)⌋| + 1)c1K(ε), where K(ε) is as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3), implies that f(x(t)) − ˆf + gQ(x(t)) ≤ max{η, ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Similar to the previous proof, the result follows after modifying the proof of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' First, find an upper bound to the number of solutions to nc1 1 n3 ≤ τ for n1, n3 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Now find the correspondence between the solutions and the tuples (i′, j′, k′) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4), where j′ is now fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='6 (How to choose a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In the case of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5 and assuming (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5), we can select an optimal value of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' From Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5 and α∗ ≥ α/a, the part of τ that depends on a is bounded by O((|⌊loga(α/α0)⌋| + 1)c1ad1/β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We can upper bound this further by both dropping the floor function and, then dropping the +1 in brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We are then led to minimizing | loga(α/α0)|c1ad1/β = | log(α/α0)|c1ad1/β/ log(a)c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Under these assumptions, the optimal value of a is ec1β/d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Note that in the case of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4, there is no clear optimal choice for b since the optimal choice is ε-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' ♦ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='7 (How to choose c1, c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' For Corollaries 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5, an optimal choice of c1 > 1 exists but it depends on the unknown parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' To see this, minimize the lower bound of t in the aforementioned corollaries with respect to c1, noting that the only term in τ that depends on c1 is (|⌊loga(α/α0)⌋| + 1)c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Assuming α0 ̸= α, this gives c1 = 1 + � 1 + 4 log(|⌊loga(α/α0)⌋|+1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' By the same reasoning, for Corollaries 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4 and β0 ̸= β, the optimal choice of c2 > 1 depends on the unknown parameter β and is given by c2 = 1 + � 1 + 4 log(|⌈logb(β/β0)⌉|+1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Intuitively, if α0 is far from α then c1 should be closer to 1, and similarly for β0 and β regarding c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In the absence of prior knowledge, we recommend a sensible default such as c1 = c2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' ♦ Finally, to emphasize the generality of our algorithm, we consider the case where α and β are known to lie within explicit ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In this case, we modify set S based on these ranges and choose a schedule criterion function h(x1, x2, x3) depending on x3 only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The following result is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 17 Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='8 (Known ranges for α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Suppose we have integers imin ≤ imax, 0 ≤ jmin ≤ jmax, for which α ∈ [aiminα0, aimaxα0], β ∈ [bjminβ0, bjmaxβ0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Let S = {imin, imin + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' , imax} × {jmin, jmin + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' , jmax} × N, and h(x1, x2, x3) = x3 be a schedule criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Then given any h-assignment φ and any ε ∈ (0, ϵ0), running Algorithm 2 with t ≥ (imax − imin + 1)(jmax − jmin + 1)K(ε), where K(ε) is as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3), implies f(x(t)) − ˆf + gQ(x(t)) ≤ max{η, ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Note that Algorithm 2 is sequential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' However, one can readily devise a parallel implementation that runs Algorithm 1 in parallel over each pair (i, j) and then minimizes f + gQ over all instances at the end of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 4 Examples In this section, we present various examples of first-order methods that can be used in our restart scheme for different problem settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In particular, we describe the methods that lead to the various results in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We do this by explicitly deriving a method Γ : R++ ×R++ ×D → D that satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3) and give an explicit bound for the cost function CΓ(δ, ϵ, x0) of the form Cδd1/ϵd2 + 1 for suitable d1 and d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1 (Optimization over C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In convex analysis and continuous optimization, it is standard to consider function inputs lying in a finite-dimensional vector space over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The results described below are extended to C, but this treatment does not always arise in the original papers for the first-order methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We are interested in the domain of f being a subset of Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Hence, we consider the natural isomorphism between Cn and R2n given by: if z = x + iy ∈ Cn with x, y ∈ Rn, then z �→ (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We refer to z as the complex representation and (x, y) as the real representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Now, one proceeds to do convex analysis and continuous optimization in the real representation, then express the results in the equivalent complex representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Fortunately, not much needs to change (at least symbolically) when switching between real and complex representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' For example, the Euclidean inner products ⟨·, ·⟩ have to be substituted with their real part, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=', ⟨·, ·⟩R := Re ⟨·, ·⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Another example pertains to the differentiability of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Specifically, for x, y ∈ Rn, we say that f is differentiable at z = x+iy ∈ D ⊆ Cn if and only if Re (f) is (real) differentiable at (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' To define the gradient, denote ∇x and ∇y as the vector of partial derivatives corresponding to variables x and y, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Then ∇f := ∇xRe (f) + i∇yRe (f), noting that because f is real-valued, we have Im (f) ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Other parts of convex analysis, such as convexity, functions, proximal mappings, subgradients, and so on, also extend to a complex vector domain by applying the definitions to the real representation of complex vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' ♦ 18 Algorithm 3: Nesterov’s method Input : An L-smooth function f and closed, convex set Q ⊆ Cn as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1), prox-function p(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' x0) with strong convexity constant σp and unique minimizer x0 ∈ Q, sequences {γj}∞ j=0 and {τj}∞ j=0, and number of iterations N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Output: The vector xN, which estimates a minimizer of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 1 z0 ← x0 2 for j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' , N − 1 do 3 xj+1 ← argmin x∈Q L 2 ∥x − zj∥2 ℓ2 + ⟨∇f(zj), x − zj⟩R 4 vj ← argmin x∈Q L σp p(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' x0) + �j i=0 γi⟨∇f(zi), x − zi⟩R 5 zj+1 ← τjvj + (1 − τj)xj+1 6 end 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1 Nesterov’s method for L-smooth functions For our first example, we consider Nesterov’s method [56], an accelerated projected gradient descent algorithm for general constrained convex optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Specifically, the algorithm aims to solve (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1) in the special case when f is convex and L-smooth: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' A function f : Cn → R is L-smooth over Q ⊆ Cn if it is Fr´echet differentiable in an open set containing Q, and for all x, y in this set, its gradient ∇f has the Lipschitz property ∥∇f(x) − ∇f(y)∥ℓ2 ≤ L∥x − y∥ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' ▲ Nesterov’s method is given in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The algorithm uses the notion of a prox-function p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Here p : Q → R is a proper, closed and strongly convex function with strong convexity constant σp > 0, that, in addition, satisfies minx∈Q p(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Let x0 = argminx∈Qp(x) be the unique minimizer of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' To make this dependence explicit, we write p(·) = p(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' A common and simple choice of prox-function is p(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' x0) = 1 2∥x − x0∥2 ℓ2 with σp = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This will be useful when we express Nesterov’s method with smoothing, in terms of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We now state Nesterov’s main result that gives a bound for f(xk) − f(x), for any x ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3 (Nesterov’s theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Let Q ⊆ Cn be nonempty, closed and convex, f a convex L- smooth function over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In addition, let p : Q → R be a proper, closed and strongly convex function over Q with strong convexity constant σp > 0 with minx∈Q p(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Then Algorithm 3 with γj = j + 1 2 , τj = 2 j + 3, x0 = argmin x∈Q p(x), generates a sequence {xk}∞ k=1 ⊂ Q satisfying f(xk) − f(x) ≤ 4Lp(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' x0) k(k + 1)σp , ∀x ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3 consists of two modifications of [56, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' First, we do not assume Q is bounded, as the results in the original work do not use this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Second, we allow x ∈ Q instead of x ∈ � X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The proof in the original work does not use the optimality of x, and only requires x to be feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We utilize this property when considering Nesterov’s method with smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The following is now immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 19 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Let Q ⊆ Cn be nonempty, closed and convex, f a convex L-smooth function over Q (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Given input (δ, ϵ, x0) ∈ R+ ×R+ ×Q, let Γ(δ, ϵ, x0) be the output of Algorithm 3 with p(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' x0) = 1 2∥x − x0∥2 ℓ2, γj = j + 1 2 , τj = 2 j + 3, N = � δ √ 2L √ϵ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3) holds with gQ ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Specifically, f(Γ(δ, ϵ, x0)) − ˆf ≤ ϵ, ∀x0 ∈ Q with d(x0, � X) ≤ δ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) where d is the metric induced by the ℓ2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' It follows that we can take CΓ(δ, ϵ) = � δ √ 2L √ϵ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3) Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4 shows that we can take d1 = 1 and d2 = 1/2 in the cost bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5) for Nesterov’s method (without smoothing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' If f is L−smooth and satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) with η = 0, then β ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' It follows that we can take β0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2 now implies the rates in the first row of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Several other remarks are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' First, in Nesterov’s method, the iterates xj are always feasible since the corresponding update step returns a point in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Thus in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4 we do not have to define gQ since Γ trivially satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3) with gQ ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Finally, in Nesterov’s method, the requirement x0 ∈ Q can be relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' For instance, we only require f is L-smooth over the union of Q and an open neighborhood of x0 for some L > 0 to start with x0 /∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2 Nesterov’s method for (u, v)-smoothable functions We can extend Nesterov’s method to solve (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1) without assuming that f is differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This is done via smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' For this, we need the following definition from [10, Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='43] (extended to functions with complex-vector domains).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Let u, v > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' A convex function f : Cn → R is called (u, v)-smoothable if for any µ > 0 there exists a convex differentiable function fµ : Cn → R such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' fµ(x) ≤ f(x) ≤ fµ(x) + vµ for all x ∈ Cn 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' fµ is u µ-smooth over Cn The function fµ is referred to as a 1 µ-smooth approximation of f with parameters (u, v), and µ is referred to as the smoothing parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' ▲ Smoothing is a framework that approximates f arbitrarily closely by a family of smooth func- tions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=', functions with Lipschitz gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This means that we can apply Nesterov’s method to a smooth approximation of f, and also analyze the objective error in terms of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The following provides a modified version of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3 for (a, b)-smoothable f, and is proven in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Let f : Cn → R be a convex (u, v)-smoothable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Given any µ > 0, let fµ be a 1 µ-smooth approximation of f with parameters (u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Then taking Q, p, γj, τj, x0 as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3 and applying Algorithm 3 to the function fµ produces a sequence {xk}∞ k=1 satisfying f(xk) − f(x) ≤ 4up(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' x0) µk(k + 1)σp + vµ, x ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4) 20 The following proposition shows that Nesterov’s method with smoothing can be formulated as an algorithm Γ in our framework, and is proven in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Let Q ⊆ Cn be nonempty, closed and convex, and f : Cn → R a convex (u, v)- smoothable function (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Given input (δ, ϵ, x0) ∈ R+ × R+ × Q, let Γ(δ, ϵ, x0) be the output of Algorithm 3 applied to function fµ with µ = ϵ 2v, p(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' x0) = 1 2∥x − x0∥2 ℓ2, γj = j + 1 2 , τj = 2 j + 3, N = � 2 √ 2uv · δ ϵ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Then f(Γ(δ, ϵ, x0)) − ˆf ≤ ϵ, ∀x0 ∈ Q satisfying d(x0, � X) ≤ δ, where d is the metric induced by the ℓ2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' It follows that we can set CΓ(δ, ϵ, x0) = � 2 √ 2uv · δ ϵ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This result shows that we can take d1 = 1 and d2 = 1 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5) in the case of Nesterov’s method with smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2 now implies the rates in the second row of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The following discussion considers a standard example of smoothing that is closely related to proximal maps, from [10, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' If f : Cn → R is convex and Lipschitz continuous with Lipschitz constant Lf, then it is (1, L2 f/2)-smoothable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In particular, the Moreau envelope with parameter µ > 0 is a 1 µ-smooth approximation of f with parameters (1, L2 f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Given a convex function f : Cn → R and µ > 0, the Moreau envelope of f is the function Mµ f (x) = min y∈Cn � f(y) + 1 2µ∥x − y∥2 ℓ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5) The number µ is referred to as the smoothing parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The Moreau envelope Mµ f is well-defined, and the minimization problem defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5) has a unique solution corresponding to proxµf(x), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=', the proximal map of µf at x [10, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The Moreau envelope of f is also 1 µ-smooth over its domain, where for any x we have ∇Mµ f (x) = 1 µ(x − proxµf(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Examples of Moreau envelopes of functions can be found in [10, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3 The universal fast gradient method We next consider H¨older smooth functions, which are a natural way of interpolating between nonsmooth and smooth objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' A function q : Cn → R is H¨older smooth over Q ⊆ Cn with parameter ν ∈ [0, 1] if ∥∇q(x) − ∇q(y)∥ℓ2 ≤ Mν∥x − y∥ν ℓ2, ∀ x, y ∈ Q, ∇q(x) ∈ ∂q(x), ∇q(y) ∈ ∂q(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' ▲ We consider the universal fast gradient method [58] for the problem min x∈Q f(x), f(x) := q(x) + g(x), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='6) 21 Algorithm 4: Universal fast gradient method Input : ϵ > 0, L0 > 0, φ0(x) = 0, y0 = x0, A0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Output: The vector xN, which estimates a minimizer of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 1 for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' , N do 2 vk ← proxφk,Q(x0) 3 ik ← −1 4 do 5 ik ← ik + 1 6 Compute ak+1,ik from the equation a2 k+1,ik = 1 2ikLk (Ak + ak+1,ik).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 7 Ak+1,ik ← Ak + ak+1,ik 8 τk,ik ← ak+1,ik/Ak+1,ik 9 xk+1,ik ← τk,ikvk + (1 − τk,ik)yk 10 Choose a subgradient ∇q(xk+1,ik) ∈ ∂q(xk+1,ik).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 11 ˆφk+1,ik(x) ← ak+1,ik[⟨∇q(xk+1,ik), x⟩R + g(x)] 12 ˆxk+1,ik ← proxˆφk+1,ik,Q(vk) 13 yk+1,ik ← τk,ik ˆxk+1,ik + (1 − τk,ik)yk 14 while q(yk+1,ik)>q(xk+1,ik)+⟨∇q(xk+1,ik), yk+1,ik −xk+1,ik⟩R+2ik−1Lk∥yk+1,ik −xk+1,ik∥2 ℓ2+ ϵ 2τk,ik 15 xk+1 ← xk+1,ik, yk+1 ← yk+1,ik, ak+1 ← ak+1,ik, τk ← τk,ik 16 Ak+1 ← Ak + ak+1, Lk+1 ← 2ik−1Lk 17 φk+1(x) ← φk(x) + ak+1[q(xk+1) + ⟨∇q(xk+1), x − xk+1⟩R + g(x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 18 end where q is a proper convex function that is H¨older smooth for some ν ∈ [0, 1], and g is a closed convex function whose proximal map, proxcg,Q(x) = argmin y∈Q � c · g(y) + 1 2∥x − y∥2 ℓ2 � , is straightforward to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The iterates of the universal fast gradient method are summarized in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='9 (Theorem 3 of [58]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Let Q ⊆ Cn be nonempty, closed and convex, q a proper convex function that is H¨older smooth for some ν ∈ [0, 1] and Mν < ∞ (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='8), and g a closed convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Then Algorithm 4 generates a sequence {xk}∞ k=1 ⊂ Q satisfying f(xk) − ˆf ≤ � 22+4νM2 ν ϵ1−νk1+3ν � 1 1+ν d(x0, � X)2 2 + ϵ 2, ∀x ∈ Q, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='7) where d is the metric induced by the ℓ2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' By choosing k to match the two terms on the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='7), the following proposition is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Let Q ⊆ Cn be nonempty, closed and convex, q a proper convex function is H¨older smooth for some ν ∈ [0, 1] and Mν ≥ 0 (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='8), and g a closed convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Given input (δ, ϵ, x0) ∈ R+ × R+ × Q, let Γ(δ, ϵ, x0) be the output of Algorithm 4 with N = � ��� 2 2+4ν 1+3ν M 2 1+3ν ν δ 2+2ν 1+3ν ϵ 2 1+3ν � ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 22 Algorithm 5: Primal-dual algorithm for the problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Input : Initial vectors x0 ∈ Cn and y0 ∈ Cm, proximal step sizes τ, σ > 0, number of iterations N, matrix B ∈ Cm×n, and routines for appropriate proximal maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Output: Final ergodic average XN approximating a solution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 1 Initiate with x(0) = x0, y(0) 1 = y0, X0 = 0, and Y0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 2 for j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' , N − 1 do 3 x(j+1) ← proxτg � x(j) − τB∗y(j) − τ∇q(x(j)) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 4 y(j+1) ← proxσh∗ � y(j) + σB(2x(j+1) − x(j)) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 5 Xj+1 ← 1 j+1 � jXj + x(j+1)� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 6 Yj+1 ← 1 j+1 � jYj + y(j+1)� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 7 end Then f(Γ(δ, ϵ, x0)) − ˆf ≤ ϵ, ∀x0 ∈ Q satisfying d(x0, � X) ≤ δ, where d is the metric induced by the ℓ2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' It follows that we can set CΓ(δ, ϵ, x0) = � ��� 2 2+4ν 1+3ν M 2 1+3ν ν δ 2+2ν 1+3ν ϵ 2 1+3ν � ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='10 shows that we can take d1 = (2 + 2ν)/(1 + 3ν) and d2 = 2/(1 + 3ν) for the universal fast gradient method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Note that if q satisfies both (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) for η = 0 and Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='8, then β ≥ 1 + ν [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Therefore, we take β0 = 1 + ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2 now implies the rates in the third row of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4 The primal-dual iteration for unconstrained problems We now consider Chambolle and Pock’s primal-dual algorithm [23, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The primal-dual hybrid gradient (PDHG) algorithm is a popular method to solve saddle point problems [22,31,63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Consider the problem min x∈Cn f(x), f(x) := q(x) + g(x) + h(Bx), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='8) where: B ∈ Cm×n with ∥B∥ ≤ LB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' q is a proper, lower semicontinuous, convex function, and is Lq-smooth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' and g, h are proper, lower semicontinuous, convex functions whose proximal maps are straightforward to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We also use the standard Euclidean metric for d in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) and write the primal-dual iterates in their simplified form accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The saddle-point problem associated with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='8) is min x∈Cn max y∈Cm L(x, y) := ⟨Bx, y⟩R + q(x) + g(x) − h∗(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='9) The primal-dual iterates are summarized in Algorithm 5, where the output is the ergodic average of the primal-dual iterates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Note that the primal-dual algorithm allows us to easily deal with the matrix B, which can be difficult with other first-order methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' If τ(σL2 B + Lq) ≤ 1, then [25, Theorem 1] shows that for any x ∈ Cn and y ∈ Cm, L (Xk, y) − L (x, Yk) ≤ 1 k � ∥x − x(0)∥ 2 τ + ∥y − y(0)∥ 2 σ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='10) The following lemma is a simple consequence of this bound and is proven in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 23 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Consider the primal-dual iterates in Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' If τ(σL2 B + Lq) ≤ 1, then f(Xk) − f(x) ≤ 1 k � ∥x − x(0)∥ 2 τ + ∥y − y(0)∥ 2 σ � , ∀x ∈ Cn, y ∈ ∂h(BXk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='11) We can take the infimum over y ∈ ∂h(BXk) on the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='11) to obtain f(Xk) − f(x) ≤ 1 k � ∥x − x(0)∥ 2 τ + supz∈dom(h) infy∈∂h(z) ∥y − y(0)∥ 2 σ � , ∀x ∈ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='12) To bound the right-hand side, we take y(0) = 0 and consider the case where h is such that there always exist points y in the subdifferential of h for which ∥y∥ is not too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Note that this always holds if, for example, h is Lipschitz continuous and its domain is open [10, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The following proposition now shows how this falls into the framework of our restart scheme and is proven in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Suppose that sup z∈dom(h) inf y∈∂h(z) ∥y∥ ≤ Lh < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='13) Given input (δ, ϵ, x0) ∈ R+ × R+ × Cn, let Γ(δ, ϵ, x0) be the output of Algorithm 5 with y0 = 0, τ = δ LBLh + δLq , σ = Lh δLB , N = �δ ϵ (2LBLh + δLq) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Then f(Γ(δ, ϵ, x0)) − ˆf ≤ ϵ, ∀x0 with d(x0, � X) ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='14) It follows that we can take CΓ(δ, ϵ, x0) = �δ ϵ (2LBLh + δLq) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='15) Assuming that δ is bounded, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='12 shows that we can take d1 = 1 and d2 = 1 for the primal-dual algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2 now implies the rates in the fourth row of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5 The primal-dual iterations for constrained problems We now consider primal-dual iterations, but for the constrained problem min x∈Cn f(x) + χC(Ax), f(x) := q(x) + g(x) + h(Bx), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='16) with the same assumptions on q, g, h and B as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4, but now with the additional term χC(Ax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Here, C is a closed and non-empty convex set, χC is its indicator function of C and A ∈ Cm′×n with ∥A∥ ≤ LA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This fits into our framework with the choice Q = {x ∈ Cn : Ax ∈ C}, gQ(x) = gQ(κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' x) = κ · inf z∈C ∥Ax − z∥, for κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Note that κ is an additional parameter that can be chosen to balance the rate of reduction in the feasibility gap versus the objective function error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' It is possible to formulate a projected version of the primal-dual iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' However, like with Nesterov’s method, this is only 24 Algorithm 6: Primal-dual algorithm for the constrained problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Input : Initial vectors x0 ∈ Cn, [y0]1 ∈ Cm and [y0]2 ∈ Cm′, proximal step sizes τ, σ1, σ2 > 0, number of iterations N, matrices B ∈ Cm×n and A ∈ Cm′×n, and routines for appropriate proximal maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Output: Final ergodic average XN approximating a solution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 1 Initiate with x(0) = x0, y(0) 1 = [y0]1, y(0) 2 = [y0]2, X0 = 0, [Y0]1 = 0, and [Y0]2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 2 for j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' , N − 1 do 3 x(j+1) ← proxτg � x(j) − τB∗y(j) 1 − τA∗y(j) 2 − τ∇q(x(j)) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 4 y(j+1) 1 ← proxσ1h∗ � y(j) 1 + σ1B(2x(j+1) − x(j)) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 5 y(j+1) 2 ← y(j) 2 + σ2A(2x(j+1) − x(j)) − σ2PC � y(j) 2 /σ2 + A(2x(j+1) − x(j)) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 6 Xj+1 ← 1 j+1 � jXj + x(j+1)� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 7 [Yj+1]1 ← 1 j+1 � j[Yj]1 + y(j+1) 1 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 8 [Yj+1]2 ← 1 j+1 � j[Yj]2 + y(j+1) 2 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 9 end possible when the projection onto Q can be easily computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In this section, we consider a primal- dual iteration for (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='16) that only involves computing the projection onto the set C, at the price of producing non-feasible iterates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The saddle-point problem associated with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='16) is min x∈Cn max y1∈Cm max y2∈Cm′ LC(x, y1, y2) := ⟨Bx, y1⟩R +q(x)+g(x)−h∗(y1)+⟨Ax, y2⟩R −sup z∈C ⟨z, y2⟩R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='17) The primal-dual iterates are summarized in Algorithm 6, where, again, the output is the ergodic average of the primal-dual iterates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We have included three proximal step sizes τ, σ1 and σ2, which correspond to the primal variable and the two dual variables, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' To compute the proximal map associated with the second dual variable, we use Moreau’s identity to write proxσ2χ∗ C(y) = y − σ2PC(y/σ2), where PC denotes the projection onto C (with respect to the standard Euclidean norm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' If τ(σ1L2 B + σ2L2 A + Lq) ≤ 1, then a straightforward adaption of [25, Theorem 1] shows that for any x ∈ Cn, y1 ∈ Cm and y2 ∈ Cm′, LC (Xk, y1, y2) − LC (x, [Yk]1, [Yk]2) ≤ 1 k � �∥x − x(0)∥ 2 τ + ∥y1 − y(0) 1 ∥ 2 σ1 + ∥y2 − y(0) 2 ∥ 2 σ2 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='18) We now have the following lemma and resulting proposition, both of which are proven in Ap- pendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Consider the primal-dual algorithm in Algorithm 6 with y(0) 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' If τ(σ1L2 B + σ2L2 A + Lq) ≤ 1, then for any κ > 0 f(Xk) − f(x) + gQ(κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Xk) ≤ 1 k � �∥x − x(0)∥ 2 τ + ∥y1 − y(0) 1 ∥ 2 σ1 + κ2 σ2 � � , ∀x ∈ Q, y1 ∈ ∂h(BXk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='19) 25 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Suppose that sup z∈dom(h) inf y∈∂h(z) ∥y∥ ≤ Lh < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='20) Given input (δ, ϵ, x0) ∈ R+ × R+ × Cn, let Γ(δ, ϵ, x0) be the output of Algorithm 6 with [y0]1 = 0, [y0]2 = 0, τ = δ κLA + LhLB + δLq , σ1 = Lh δLB , σ2 = κ δLA , N = �δ (2κLA + 2LhLB + δLq) ϵ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Then f(Γ(δ, ϵ, x0)) − ˆf + gQ(κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' ˆx) ≤ ϵ, ∀x0 with d(x0, � X) ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='21) It follows that we can take CΓ(δ, ϵ, x0) = �δ (2κLA + 2LhLB + δLq) ϵ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='22) Assuming that δ is bounded, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='14 shows that we can take d1 = 1 and d2 = 1 for the primal-dual algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2 now implies the rates in the final row of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 5 Numerical experiments We implement several numerical experiments for the general restart scheme (Algorithm 2) applied to three different problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The first is a simple sparse recovery problem modeled as QCBP, which is solved using the primal-dual iteration for constrained problems (Algorithm 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Second, we consider image reconstruction from Fourier measurements via TV minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The reconstruction is computed using NESTA [12], where NESTA is an accelerated projected gradient descent algorithm derived from Nesterov’s method (Algorithm 3) with smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Third, we perform feature selection on three real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This selection is done by solving a SR-LASSO problem on the data with unconstrained primal-dual iterations (Algorithm 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Before discussing the examples in turn, we make some general remarks about the implemen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' First, we use the schedule criteria from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2, and for parameters we always set c1 = c2 = 2, b = e, r = e−1, and a = ec1β/d1 for unknown α but known β (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5), other- wise a = ec1/d1 if both are unknown (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Assignments from the schedule criteria are obtained by enumerating and sorting solutions of the respective Diophantine equations found in the proofs of Corollaries 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The choice of r is motivated by Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2 and the choice of a by Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The choice of c1 and c2 were arbitrary, with the intent of being sane defaults, and otherwise can be tuned to improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Second, when using the restart scheme for primal-dual iterations, we store and perform restarts on the dual variables for each instance indexed by (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Third, we use a simple workaround to handle finite precision arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In the grid search for the restart scheme, the sharpness parameter αi can be arbitrarily large or small, and βj can be arbitrarily large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Also, the adaptive restart parameters δ = δi,j,U and ϵ = ϵi,j,U can become arbitrarily small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Regarding the grid indices, we limit i and j so that |i| ≤ ⌊loga(1/ϵmach)⌋, j ≤ ⌊logb(1/ϵmach)⌋, where ϵmach is machine epsilon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Regarding the adaptive parameters, after the assignments of δi,j,U+1 and ϵi,j,U+1 in Algorithm 2, we insert the updates δi,j,U+1 := max(δi,j,U+1, 10ϵmach) and ϵi,j,U+1 := max(ϵi,j,U+1, 10ϵmach) to avoid setting them to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 26 Fourth, we slightly modify the primal-dual algorithm to improve overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' For each j ≥ 1, we track a separate iterate � Xj defined by � Xj = argmini=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=',jf(Xi) + κgQ(Xi), j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The iterates { � Xj}j≥1 are not used in the primal-dual algorithm, but are instead used to evaluate the reconstruction or objective error in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In addition, the algorithm returns � XN as its final iterate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We similarly track a separate iterate for the dual variables, selecting them based on an evaluation of the Lagrangian (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='17) with � Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Note that choosing to output � XN instead of XN is theoretically justified, since if (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3) holds, then our modification would still satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3) for the same parameters (δ, ϵ, x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1 Sparse recovery via QCBP We consider reconstructing a vector x ∈ Rn from noisy measurements y = Ax + e ∈ Rm, where A ∈ Rm×n is a matrix whose entries are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Gaussian random variables with mean zero and variance 1/m, and e ∈ Rm is a noise vector satisfying ∥e∥ℓ2 ≤ ς for some noise level ς > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' For a positive integer n, we write [n] = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' , M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Given a vector z = (zi)n i=1 ∈ Cn and S ⊆ [n], the vector zS has ith entry zi if i ∈ S, and is zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The best s-term approximation error of z is defined as σs(z)ℓ1 = min{∥uS − z∥ℓ1 : u ∈ Cn, S ⊆ [n], |S| ≤ s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We assume that x is approximately s-sparse, in the sense that its best s-term approximation error σs(x)ℓ1 is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The recovery of x is formulated as solving the QCBP problem min z∈Rn ∥z∥ℓ1 subject to ∥Az − y∥ℓ2 ≤ ς.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1) We use the following condition on the matrix A to ensure that approximate sharpness holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1 (Robust null space property, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=', Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='14 of [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The matrix A ∈ Cm×n satisfies the robust Null Space Property (rNSP) with constants 0 < ρ < 1 and γ > 0 if ∥vS∥ℓ2 ≤ ρ √s∥vS∁∥ℓ1 + γ∥Av∥ℓ2, for all v ∈ Cn and S ⊆ [M] with |S| ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' ▲ In [27, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3], it was shown that the robust null space property (rNSP) implies approx- imate sharpness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We restate the result in the notation of this paper for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2 (Approximate sharpness of ℓ1-norm for QCBP sparse recovery).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Let ς > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Suppose A ∈ Cm×n has the rNSP of order s with constants 0 < ρ < 1, γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Let y ∈ Cm, D = Cn, Q = {x ∈ Cn : ∥Ax − y∥ℓ2 ≤ ς} and f(x) = ∥x∥ℓ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Then the approximate sharpness condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) holds with gQ(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' √s) = √s max{∥Az − y∥ℓ2 − ς, 0}, α = ˆc1 √s, β = 1, η = ˆc2σs(x)ℓ1 + ˆc3ς√s, for constants ˆc1, ˆc2, ˆc3 > 0 are constants depending only on ρ and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The theory of compressed sensing [4, 35] aims to construct (random) matrices satisfying the rNSP, which is itself implied by the better-known Restricted Isometry Property (RIP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' For example, if A is a Gaussian random matrix, then it satisfies the rNSP with probability at least 1−ε, provided m ≥ C · (s · log(eN/s) + log(2/ε)) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=', [4, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' However, a sharp value of the constant C, and therefore also the rNSP constants ρ and γ, is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This implies that the approximate sharpness constants α and η are also unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This motivates using the restart scheme (Algorithm 2), which does not require knowledge of α or η, to solve (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 27 0 500 1000 1500 2000 10 -6 10 -4 10 -2 10 0 Total inner iterations t ∥xt − x∥ℓ2 0 1000 2000 3000 4000 5000 10 -6 10 -4 10 -2 10 0 Total inner iterations t ∥xt − x∥ℓ2 Figure 2: Reconstruction error of restarted primal-dual iteration for QCBP with ς = 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Left: The restart scheme with fixed sharpness constants β = 1 and various α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Right: Various different schemes (including restarted and unrestarted schemes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1 Experimental setup We use the primal-dual iteration for constrained problems (Algorithm 6) to solve the sparse recovery problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This can be done by expressing QCBP in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1) as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='16) with q ≡ 0, h ≡ 0, B = 0, g(x) = ∥x∥ℓ1, C = {z ∈ CN : ∥z − y∥ℓ2 ≤ ς}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Given these choices, the proximal map of τg is the shrinkage-thresholding operator, and the projec- tion map is straightforward to compute since C is a shifted ℓ2-ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Moreover, we have h∗(z) = +∞ whenever z ̸= 0, and is zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Therefore the proximal map proxσ1h∗(x) = ∥x∥2 ℓ2/2, and thus y(j) 1 = 0 for all j > 0 if the initial data y(0) 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In essence, we can ignore the parameter σ1 and updating the iterates y(j) 1 in the primal-dual iterations (Algorithm 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The error bound derived in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='13 holds with the σ1 term omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Unless stated otherwise, the parameters used are ambient dimension n = 128, sparsity level s = 10, measurements m = 60, noise level ς = 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The ground truth vector x is exactly sparse with s of its entries (randomly selected) corresponding to i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' standard normal entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The noise vector e is selected uniformly random on the ℓ2-ball of radius ς and thus ∥e∥ℓ2 = ς.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The objective function is f(x) = ∥x∥ℓ1 and the feasibility gap is given by gQ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' κ) = κ · max{∥Ax − y∥ℓ2 − ς, 0}, which is derived from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The feasibility gap weight is set to κ = √m from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2, noting that s ≤ m in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In addition, α0 = √m, β0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The choice of α0 is also motivated by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2 Results Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 2 shows the performance of the restart scheme in Algorithm 1 for various fixed values of α and β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' For smaller α, the error decreases linearly down to the noise level ς = 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This agrees with Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Increasing α leads to fast linear convergence, up to a threshold (between 101 and 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' After this point, the performance of the restart scheme abruptly breaks down since large α violates the approximate sharpness condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' To overcome such parameter sensitivity, we use Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 2 also compares the perfor- mance of the restart scheme with fixed (α, β) = (√m, 1) with restart schemes that (i) perform a 28 0 500 1000 1500 2000 2500 3000 10 -6 10 -4 10 -2 10 0 Total inner iterations t ∥xt − x∥ℓ2 Grid search over α 0 1000 2000 3000 4000 5000 10 -6 10 -4 10 -2 10 0 Total inner iterations t ∥xt − x∥ℓ2 Grid search over β Figure 3: Reconstruction error of restarted primal-dual iteration for QCBP with ς = 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Left: The restart scheme with grid search over α and various fixed β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Right: The restart scheme with grid search over β and various fixed α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' grid search over α, for fixed β = 1, and (ii) perform a grid search over both α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Both grid search schemes exhibit linear convergence, in agreement with Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' They converge less rapidly than the scheme with fixed (α, β), but require no empirical parameter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Note that all restart schemes significantly outperform the unrestarted primal-dual iteration (“no restarts”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Next, we consider two cases of grid searching over exactly one sharpness constant and leaving the other fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 3 shows the results for fixed α with β grid search and fixed β with α grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Both yield linear decay, although at a slightly worse rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' A key point to note is the potential benefit of grid searching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Compare the reconstruction error with those for the fixed restart schemes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 2 with log10(α) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2 and β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In the fixed constant scheme, these parameter choices stall the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' However, β grid search overcomes this and manages to reconstruct x within a tolerance proportional to ς after sufficiently many restarts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Finally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 4 considers the effect on the restart schemes when changing the noise level ς.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In all cases, the restart schemes linearly decay to a tolerance proportional to ς, and outperform the unrestarted primal-dual iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2 Image reconstruction via TV minimization In this experiment, we consider image reconstruction with Fourier measurements – a sensing modal- ity with applications notably in Magnetic Resonance Imaging (MRI) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Specifically, we consider the recovery of a vector x ∈ Rn from noisy Fourier measurements y = Ax+e ∈ Cm, where A ∈ Cm×n corresponds to a subsampled Fourier matrix and e ∈ Cm models noise or perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The vector x is a vectorized complex 2-D image X ∈ CR×R, where n = R2 for some positive power-of-two integer R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The matrix A has the form A = m−1/2PΩF, where F ∈ Cn×n is the 2-D discrete Fourier transform and Ω ⊆ n is a sampling mask with |Ω| = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Here, Ω defines the matrix PΩ ∈ Cm×n, which selects the rows of F by index according to the indices in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Lastly, ∥e∥ℓ2 ≤ ς for some noise level ς > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' A widely used tool for reconstructing x from y is the total variation (TV) minimization problem min z∈Cn ∥V z∥ℓ1 subject to ∥Az − y∥ℓ2 ≤ ς, where V is the 2-D (anisotropic) discrete gradient transform with periodic boundary conditions [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='10 -10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='10 -5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='10 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='Total inner iterations t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='ς = 10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='∥xt − x∥ℓ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='10 -10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='10 -5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='10 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='Total inner iterations t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='ς = 10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='∥xt − x∥ℓ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='10 -10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='10 -5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='10 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='Total inner iterations t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='ς = 10−6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='∥xt − x∥ℓ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='10 -10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='10 -5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='10 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='Total inner iterations t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='ς = 10−8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='∥xt − x∥ℓ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='10 -10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='10 -5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='10 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='Total inner iterations t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='ς = 10−10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='∥xt − x∥ℓ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='10 -10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='10 -5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='10 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='Total inner iterations t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='ς = 10−12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='∥xt − x∥ℓ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='Figure 4: Reconstruction error of restarted primal-dual iteration for QCBP with ς = 10−2k for k = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' , 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Each plot includes the various (restarted and unrestarted) schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Similar to the sparse recovery problem described in the previous section, the TV-Fourier image reconstruction problem can be shown to have the approximate sharpness condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) with high probability under a suitable random sampling pattern Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Stating and proving this is somewhat more involved, but can be done with a careful adaptation of the analysis within [3, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1 Experimental setup The first-order solver we use is NESTA (NESTerov’s Algorithm), an accelerated projected gradient descent algorithm used to solve problems of the form min z∈Cn ∥W ∗z∥ℓ1 subject to ∥Az − y∥ℓ2 ≤ ς, W ∈ Cn×m′, where TV minimization is a special case with W = V ⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' NESTA is derived from Nesterov’s method with smoothing, where the objective function f(z) = ∥W ∗z∥ℓ1 is smoothed by replacing the ℓ1-norm with its Moreau envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This yields a 1/µ-smooth approximation fµ(z) = ∥W ∗z∥ℓ1,µ of f with parameters (∥W ∗∥2 ℓ2, m′/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Here ∥w∥ℓ1,µ = �m′ i=1 |wi|µ for w = (wi)m′ i=1 and | · |µ is the complex Huber function (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=', [61]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In particular, we have ∥V ∥ℓ2 = 2 √ 2 for TV minimization in 2-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The second part of the derivation of NESTA is finding closed-form expressions for the update steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In general, this is not possible to do except in special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' However, NESTA considers A with orthonormal rows up to a constant factor, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=', AA∗ = νI for some ν > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Such an assumption yields a closed form for the update formulas and is not unreasonable since many forward operators in compressive imaging have orthonormal rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' For example, with the subsampled Fourier matrix we have AA∗ = (N/m)I, and hence the desired property holds with ν = N/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We reconstruct an R×R GPLU phantom image [42] with R = 512 so that the ambient dimension is n = 5122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The noise e is uniformly sampled from an ℓ2-ball of radius ς = 10−5, and so ∥e∥ℓ2 = ς.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Two sampling masks are considered for the subsampled Fourier matrix A and are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 30 Near-optimal sampling mask Radial sampling mask Figure 5: Sampling patterns for the Fourier measurements used in the image reconstruction experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 0 500 1000 1500 2000 10 -4 10 -2 10 0 Total inner iterations t ∥xt − x∥ℓ2 Near-optimal sampling mask 0 500 1000 1500 2000 10 -4 10 -2 10 0 Total inner iterations t ∥xt − x∥ℓ2 Radial sampling mask Figure 6: Reconstruction error of restarted NESTA for TV minimization with ς = 10−5, and with the near- optimal and radial sampling masks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The restart scheme uses fixed sharpness constants β = 1 and various α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The first is a near-optimal sampling scheme [3, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2] and the second is a radial sampling scheme, where the latter is common in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Each mask yields approximately a 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5% sampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' For the restart scheme, the objective function is f(z) = ∥V x∥ℓ1 and the feasibility gap gQ ≡ 0 since NESTA always produces feasible iterates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The smoothing parameters µ are handled directly by the restarting procedure and explicitly depend on ϵi,j,U (see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The main two experiments are done for each of the two sampling masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Lastly, we choose α0 = √m, β0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The choice of α0 is motivated by [27, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3] which generalizes Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2 Results First, we run the restart scheme with fixed sharpness constants (no grid search) corresponding to pairs (α, β) with β = 1 and various α values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The reconstruction error versus total inner iterations is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 6 with near-optimal sampling (left) and radial sampling (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The results are very similar to the first sparse recovery via QCBP experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Again, the rate of decay corresponds 31 :.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' = :: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='. := : :?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' : 5 P: L : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='.2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' :: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='. ".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' ".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content="' ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='. : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' :0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5 3 10 4 10 -4 10 -2 10 0 Total inner iterations t ∥xt − x∥ℓ2 Near-optimal sampling mask 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5 3 10 4 10 -4 10 -2 10 0 Total inner iterations t ∥xt − x∥ℓ2 Radial sampling mask Figure 7: Reconstruction error of restarted NESTA for TV minimization with ς = 10−5, and with the near- optimal and radial sampling masks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Various different (restarted and unrestarted) schemes are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' to linear decay as anticipated from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The convergence rate improves as α increases, up to a threshold (about α = 630 for near-optimal sampling, and about α = 446 for radial sampling), where afterwards the limiting tolerance increases steadily, yielding poor reconstruction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This phenomenon is discussed in the first experiment of sparse recovery via QCBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' A key observation is how changing the sampling mask changes the threshold α value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This motivates using a grid search to avoid having to tune α as a parameter for different sampling masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In the second experiment, we compare the reconstruction errors of several restart schemes, together with standalone NESTA (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=', no restarts) with various smoothing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 7 with near-optimal sampling (left) and radial sampling (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The smoothing parameters used are µ = 10iς, i ∈ {−2, 1, 0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The results are analogous to the fourth experiment with sparse recovery via QCBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We note that the radial sampling mask produces slightly lower convergence rates than the near-optimal scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Moreover, we observe that converging to the limiting tolerance of NESTA is sensitive to the choice of smoothing parameter µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' By making µ smaller, we better approximate the original problem and thus the reconstruction, but require more iterations to achieve a better approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In contrast, restarting NESTA via Algorithm 2 does not require any tuning of the smoothing parameter and outperforms the non-restarted algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3 Feature selection via SR-LASSO Our final experiment considers feature selection via the Square Root LASSO (SR-LASSO) problem [2, 14, 15, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Let X ∈ Rm×n be a data matrix, where each row corresponds to a data point and each column corresponds to a feature, and y ∈ Rm the label vector for the data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Since we wish to learn an affine mapping from data points to labels, we augment X by appending a new column consisting of ones, with the augmentation denoted by A ∈ Rm×(n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Now fix λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Then we seek a vector x ∈ Rn+1 that solves the SR-LASSO problem min z∈Rn+1 ∥Az − y∥ℓ2 + λ∥z∥ℓ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' An advantage of this problem over the classical LASSO is that it requires less tuning of the parame- ter λ as the problem instance or noise level changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' See [72] for discussion and recovery conditions for this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Feature selection is done by identifying the indices of close-to-zero entries of x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 32 0 2000 4000 6000 8000 10000 10 -10 10 -5 10 0 Total inner iterations t f(xt) − ˆf wine 0 1 2 3 4 5 10 4 10 -10 10 -5 10 0 Total inner iterations t f(xt) − ˆf cc 0 1 2 3 4 5 10 4 10 -15 10 -10 10 -5 10 0 Total inner iterations t f(xt) − ˆf leu Figure 8: Objective error versus the total inner iteration of various (restarted and unrestarted) schemes of primal-dual iteration for SR-LASSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The plots correspond to three different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' which are the features to discard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This reduces the number of columns of X for future processing or analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The SR-LASSO is a well-known tool in high-dimensional statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' It can also be used for sparse recovery problems, in which case approximate sharpness follows (like it did with QCBP) from the rNSP (Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1) [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' However, in the feature selection problem, properties such as the rNSP are unlikely to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In this case, more general recovery conditions for SR-LASSO (and LASSO), such as the compatibility condition [72], are more useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Under these conditions, one also has approximate sharpness with unknown constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1 Setup We use the unconstrained primal-dual iterations (Algorithm 5) to solve SR-LASSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We can express SR-LASSO as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='16) by q ≡ 0, g(x) = λ∥x∥ℓ1, h(Bx) = ∥Bx − y∥ℓ2, B = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' From this, the primal-dual updates can be computed explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The proximal map τg is the shrinkage-thresholding operator and the proximal map of σh∗ is a projection map onto the ℓ2-ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In either case, the proximal maps are straightforward to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' For three different datasets, we compare the SR-LASSO objective error of various unrestarted and restarted schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The minimum of SR-LASSO for each dataset is computed using CVX [40,41] with high precision and the SDPT3 solver and is used to compute the objective errors in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 8 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We use three datasets: wine quality (wine) [29] with m = 6497 points and n = 11 features, colon cancer (cc) [26] with m = 62 points and n = 2000 features, and leukemia (leu) [26] with m = 38 points and n = 7129 features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The wine data corresponds to a regression task of predicting wine quality, cc and leu are two-class classification tasks of diagnosing illness based on data features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We use λ = 3, 2, and 4 for the wine, cc, and leu datasets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We measure sparsity s of ˆx by interpreting an entry to be non-zero if its absolute value is greater than 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The values α0 and β0 are chosen empirically as estimates of the true sharpness constants α and β, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2 Results Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 8 shows the performance of various restart schemes for this problem on the three different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' In all cases, the restarted schemes outperform the unrestarted scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The suitable values of α and β differ significantly across the datasets, indicating that the optimal sharpness parameters are problem-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This further demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 9, where we show the restart scheme for various fixed β and grid search over α - the restart schemes with choices of β > 1 outperform 33 0 1000 2000 3000 4000 5000 10 -10 10 -5 10 0 f(xt) − ˆf wine 0 2000 4000 6000 8000 10000 10 -10 10 -5 10 0 Total inner iterations t f(xt) − ˆf cc Total inner iterations t f(xt) − ˆf leu Figure 9: Objective error versus the total inner iteration of restarted primal-dual iteration for SR-LASSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The plots correspond to grid search over α with various fixed β for three different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' the schemes that use β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This is in contrast to the sparse recovery example, where theory and experiment suggest β = 1 as a good choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' This phenomenon is unsurprising since the approximate sharpness condition (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2)) for this problem is expected to be highly dependent on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Nonetheless, using our grid search scheme, we obviate the need for estimating or tuning these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 6 Conclusion We provided a framework for the optimal acceleration of first-order methods under approximate sharpness conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' These conditions generalize sharpness by incorporating an unknown constant perturbation to the objective error, offering greater robustness to noise or model classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Our scheme can achieve optimal convergence rates for a wide variety of problems, despite not assuming knowledge of the constants appearing in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Moreover, we do not require the first-order method to produce feasible iterates, a flexibility that is useful when employing methods such as primal-dual iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' As illustrated by our numerical experiments, our schemes are also practical, and often lead to significant improvements over unrestarted schemes or restart schemes with poor parameter choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' There are numerous possible avenues for future research and extensions of our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' One avenue involves replacing the metric in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) by a Bregman distance, and acceleration for convex optimization problems in Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Another involves applications to (non-convex) bilevel optimization schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' For saddle-point problems such as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='9) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='17), it may be possible to develop similar restart schemes based on primal-dual gaps replacing f(x)− ˆf in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2), see [5] and [32] for primal-dual gap sharpness and restart schemes in the cases of β = 1 and β = 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' See also [46,47] for recent work on restarts based on gap functions for Frank-Wolfe algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Finally, there is the extension of our restart schemes to handle stochastic first-order methods, including larger-scale machine learning problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' A Miscellaneous proofs In this section, we prove several results that were stated in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 34 3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='0 B = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='0 β = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='0 β = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='010-10 10-15 0 5000 10β = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='0 β = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='0 β = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='0 000 1500010-5A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1 Nesterov’s method with smoothing Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3 with the function fµ and using the second part of Def- inition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5 gives fµ(xk) − fµ(x) ≤ 4up(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' x0) µk(k + 1)σp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Now using both inequalities in the first part of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='5 gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Suppose that x0 ∈ Q with d(x0, � X) ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Then by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='6 with ˆx ∈ � X ⊆ Q, we have f(xN) − ˆf ≤ 4up(ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' x0) µN(N + 1)σp + vµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Using 1 N(N+1) ≤ 1 N2 , σp = 1 and p(ˆx) ≤ 1 2δ2 by choice of p, we get f(xN) − ˆf ≤ 2uδ2 µN2 + vµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Substituting µ = ϵ 2v and using that N ≥ 2 √ 2uv · δ ϵ gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2 Primal-dual iterations for unconstrained problems Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' We use (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='10) and prove bounds on each of the terms on the left-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' First, we have L (Xk, y) = ⟨BXk, y⟩R + q(Xk) + g(Xk) − h∗(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Since h is convex and lower semicontinuous, h∗∗ = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' It follows that h(BXk) = max y∈Cm⟨BXk, y⟩R − h∗(y) = − min y∈Cm(h∗(y) − ⟨BXk, y⟩R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The objective function is convex and lower semicontinuous, and the set of minimizers is y such that 0 ∈ ∂ (h∗(·) − ⟨·, BXk⟩) (y) = ∂h∗(y) − BXk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Rearranging and using the Legendre–Fenchel identity, we deduce that this set of minimizers is precisely ∂h(BXk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' It follows that L (Xk, y) = f(Xk), ∀y ∈ ∂h(BXk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1) Second, we have L (x, Yk) = ⟨Bx, Yk⟩R + q(x) + g(x) − h∗(Yk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The above argument shows that h(Bx) = max y∈Cm⟨Bx, y⟩R − h∗(y) ≥ ⟨Bx, Yk⟩R − h∗(Yk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' It follows that L (x, Yk) ≤ f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2) The bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='11) now follows by combining (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='1) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 35 Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' First, consider general τ, σ > 0 with τ(σL2 B + Lq) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' For input x0 with d(x0, � X) ≤ δ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='13) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='12) imply that for x ∈ � X, f(XN) − ˆf ≤ 1 N �δ2 τ + L2 h σ � = 1 N � σδ2L2 B + L2 h σ + δ2Lq � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Choosing the step size σ > 0 to minimize the right-hand side leads to σ = Lh δLB , τ = δ LBLh + δLq , f(XN) − ˆf ≤ δ N (2LBLh + δLq) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='14) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='15) now follow by taking N = � δ ϵ (2LBLh + δLq) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3 Primal-dual iterations for constrained problems Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Using the same arguments as the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='11, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='18) implies that for y(0) 2 = 0, f(Xk) − f(x) + ⟨AXk, y2⟩R − sup z∈C ⟨z, y2⟩R − ⟨Ax, [Yk]2⟩R + sup z∈C ⟨z, [Yk]2⟩R ≤ 1 k � �∥x − x(0)∥ 2 τ + ∥y1 − y(0) 1 ∥ 2 σ1 + ∥y2∥2 σ2 � � , ∀x ∈ Cn, y1 ∈ ∂h(BXk), y2 ∈ Cm′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' If x ∈ Q, then −⟨Ax, [Yk]2⟩R + sup z∈C ⟨z, [Yk]2⟩R ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Let ˆz ∈ C be of minimal distance to AXk and let y2 be a multiple of AXk − ˆz such that y2 has norm κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Since C is convex, the following holds [10, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='41] ⟨z, y2⟩R ≤ ⟨ˆz, y2⟩R, ∀z ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' It follows that ⟨AXk, y2⟩R − sup z∈C ⟨z, y2⟩R ≥ ⟨AXk − ˆz, y2⟩R = κ · inf z∈C ∥AXk − z∥ = gQ(κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Combining the inequalities yields (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' First, consider general τ, σ1, σ2 > 0 with τ(σ1L2 B + σ2L2 A + Lq) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' For input x0 with d(x0, � X) ≤ δ, we argue as in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='12 (but now using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='13) to obtain f(XN) − ˆf + gQ(κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' XN) ≤ 1 N �δ2 τ + L2 h σ1 + κ2 σ2 � = 1 N � σ1δ2L2 B + L2 h σ1 + σ2δ2L2 A + κ2 σ2 + δ2Lq � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3) Optimizing the proximal step sizes leads to τ = δ κLA + LhLB + δLq , σ1 = Lh δLB , σ2 = κ δLA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Substituting these values into (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='3) leads to f(XN) − ˆf + gQ(XN) ≤ δ N (2κLA + 2LhLB + δLq) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' The rest of the proof follows the same argument as the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' 36 References [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Adcock, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Brugiapaglia, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Dexter, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Moraga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' On efficient algorithms for comput- ing near-best polynomial approximations to high-dimensional, Hilbert-valued functions from limited samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content='13908, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' [2] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Adcock, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Brugiapaglia, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Webster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Sparse Polynomial Approximation of High- Dimensional Functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Society for Industrial and Applied Mathematics, Philadelphia, PA, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' [3] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Adcock, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Dexter, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Improved recovery guarantees and sampling strategies for TV minimization in compressive imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Imaging Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=', 14(3):1149–1183, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' [4] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Adcock and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Hansen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Compressive Imaging: Structure, Sampling, Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' CUP, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' [5] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Applegate, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Hinder, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Lu, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Lubin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Faster first-order primal-dual methods for linear programming using restarts and sharpness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Mathematical Programming, pages 1–52, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' [6] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Aster, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Borchers, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Thurber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Parameter estimation and inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Elsevier, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
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+page_content=' Soubeyran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
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+page_content=' [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Auslender and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
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+page_content=' Crouzeix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Global regularity theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
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+page_content=' [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' Bastounis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfVwBf/content/2301.02268v1.pdf'}
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diff --git a/X9FLT4oBgHgl3EQfUi8y/content/tmp_files/2301.12049v1.pdf.txt b/X9FLT4oBgHgl3EQfUi8y/content/tmp_files/2301.12049v1.pdf.txt
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+Photoinduced pseudospin-wave emission from
+charge-density-wave domain wall with superconductivity
+Yukihiro Matsubayashi,1, ∗ Yusuke Masaki,1, 2, † and Hiroaki Matsueda1, 3
+1Department of Applied Physics, Tohoku University, Sendai 980-8579, Japan
+2Research and Education Center for Natural Sciences,
+Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan
+3Center for Science and Innovation in Spintronics, Tohoku University,
+2-1-1 Katahira, Aoba, Sendai, Miyagi 980-8577 Japan
+(Dated: January 31, 2023)
+We study photoinduced dynamics triggered by an inhomogeneity due to competition between charge density
+waves (CDWs) and superconductivity. As a simple example, we consider the superconducting (SC) interface
+between two CDW domains with opposite signs.
+The real-time dynamics are calculated within the time-
+dependent Hartree–Fock–Bogoliubov framework, where the order parameter dynamics and the nonequilibrium
+quasiparticle distribution functions are studied. We also calculate the various dynamical response functions
+within a generalized random phase approximation. Through comparisons between the real time dynamics and
+the analysis of the response functions, it is found that the photo-driven SC interface can emit collective modes
+of the SC order parameter. This is analogous to the spin wave emission from the magnetic domain wall in an
+antiferromagnet, particularly in the case of a low driving frequency, where the order parameters can be mapped
+onto the pseudospin picture. In the high-frequency case, we find a domain wall melting caused by changes in
+the quasiparticle distribution, which induces superconductivity in the whole system.
+I.
+INTRODUCTION
+Recent advances in terahertz laser technologies have en-
+abled us to access the low-energy scales important for eluci-
+dating the fundamental properties of solids, e.g., the energy
+scales of phonons, excitons, plasmons, magnons, supercon-
+ducting (SC) gaps and density waves [1–3]. Terahertz lasers
+can be used to investigate not only the linear response of mat-
+ter but also the nonlinear response and highly nonequilibrium
+phenomena thanks to its strong intensity. For example, obser-
+vation of the Higgs mode, which is the amplitude mode of the
+SC pair potential, has been achieved by non-adiabatic exci-
+tation of the Bardeen–Cooper–Schrieffer (BCS) ground state
+with an ultrafast intense terahertz laser [4–7]. Among the var-
+ious research topics, optical control of superconductivity has
+crucial importance for both fundamental physics and techno-
+logical applications. Intensive studies have been conducted to
+enhance the transition temperature 𝑇𝑐 and the SC order [8–16]
+and to realize SC states different from the equilibrium one,
+such as photoinduced topological superconductivity [17–20].
+Electronic phases competing with superconductivity present
+the possibility of indirect photo-control of the SC phase. Such
+orders can be inferred to exist in various materials such as
+transition metal dichalcogenides [21–25] and cuprate super-
+condoctors [26–37]. Numerous experiments and theoretical
+studies have also indicated the importance of competing mul-
+tiple orders with inhomogeneity [38–45]. In addition, it is
+considered that inhomogeneity acts as a trigger in the initial
+process of the photoinduced phase transition [42, 46, 47]. Fur-
+ther investigations into the nonequilibrium dynamics of such
+systems are required in order to pave a way for discovering
+nontrivial phenomena.
+∗ yukihiro.matsubayashi.s8@dc.tohoku.ac.jp
+† yusuke.masaki.c1@tohoku.ac.jp
+The attractive Hubbard model has been widely used to
+study the nonequilibrium dynamics of superconductors [48–
+52]. This model has received much attention not only as a
+simple toy model of superconductivity but also in terms of its
+realization with ultracold atom [53–57]. In this model, the
+superconductivity and charge density wave (CDW) are degen-
+erate at half-filling [58–64]. Laser control of superconduc-
+tivity has been proposed for both the attractive and repulsive
+Hubbard model by suppressing the CDW order and charge in-
+homogeneity in recent theoretical studies [14, 49, 65]. These
+studies highlight the importance of degenerate orders when
+using laser irradiation to control electronic phases.
+In this paper, we investigate the photoinduced nonequilib-
+rium dynamics of a non-uniform system composed of SC and
+CDW orders. To describe such a system, we use the extended
+attractive Hubbard model (EAHM) with an onsite attractive
+interaction 𝑈 < 0 and nearest-neighbor repulsive interaction
+𝑉 > 0 on a square lattice. The EAHM is known to host a va-
+riety of exotic SC phases such as 𝑠-wave, 𝑝-wave and 𝑑-wave
+symmetries [45, 66–68]. At half-filling, because 𝑉 > 0, the
+CDW order is stabilized as the ground state, and the SC orders
+are destabilized. In order to drive the dynamics of supercon-
+ductivity, we consider a non-uniform system with an 𝑠-wave
+SC interface sandwiched by two CDW domains with opposite
+signs. This is a simple setup where both superconductivity and
+CDW exist. The time evolution in the laser field is calculated
+within the mean-field approximation. We classified our results
+by the driving frequency of the laser 𝜔ext and the CDW gap
+𝜔𝑔: (i) 𝜔ext ≪ 𝜔𝑔 and (ii) 𝜔ext ∼ 𝜔𝑔. In case (i), we find
+a collective mode emission from the domain wall. The col-
+lective mode can be interpreted as a pseudospin wave, where
+the pseudospin represents the CDW and SC order parameters.
+In case (ii), the CDW domains and the SC domain wall melt
+over time through the quasiparticle excitation. On the other
+hand, the emission of the SC collective mode from the do-
+main wall induces superconductivity throughout the system.
+arXiv:2301.12049v1 [cond-mat.supr-con] 28 Jan 2023
+
+2
+Their dynamics are discussed by analyzing the time-dependent
+quasiparticle population and pair potential.
+This paper is organized as follows: Sec. II introduces the
+EAHM and the time-dependent calculation method within the
+Hartree–Fock–Bogoliubov approximation. It also introduces
+the charge and pair correlation functions. The correlation func-
+tions are derived by linear response theory and calculated in
+the random phase approximation (RPA). Section III A explains
+the non-uniform self-consistent solution and its description in
+terms of pseudospins. Section III B discusses the collective
+mode emission from the domain wall under an electric field
+oscillating at a low frequency. In Sec. III C, we discuss the
+nonequilibrium dynamics induced by the resonant excitation.
+Finally, Sec. IV presents a summary and discussion of the
+overall results of the paper. Note that this paper uses the unit
+ℏ = |𝑒| = 𝑐 = 𝑘B = 1.
+II.
+MODEL AND METHODS
+In this section, we introduce the model Hamiltonian and the
+mean field approximation with its self-consistent condition.
+Then, we derive a set of equations of motion for the time
+evolution driven by an applied external field. In subsect. C, we
+develop a self-consistent linear response theory, which tells us
+information about the collective mode of the system from the
+imaginary part of the dynamical susceptibility.
+A.
+Extended attractive Hubbard model
+As a minimal model of the system with the superconduc-
+tivity and CDW, we introduce the extended attractive Hubbard
+model on a square lattice:
+𝐻 =
+∑︁
+𝑖, 𝑗,𝜎
+J𝑖 𝑗𝑐†
+𝑖𝜎𝑐 𝑗 𝜎 + 𝑈
+∑︁
+𝑖
+𝑛𝑖↑𝑛𝑖↓
++ 1
+2
+∑︁
+𝑖, 𝑗
+𝑉𝑖 𝑗𝑛𝑖𝑛 𝑗 − 𝜇
+∑︁
+𝑖
+𝑛𝑖,
+(1)
+where 𝑐†
+𝑖𝜎 (𝑐𝑖𝜎) is the creation (annihilation) operator at site
+𝑖 with spin 𝜎, and 𝑛𝑖 = �
+𝜎 𝑛𝑖𝜎 = �
+𝜎 𝑐†
+𝑖𝜎𝑐𝑖𝜎 is the number
+operator at site 𝑖. Here, 𝑖 identifies the two-dimensional lattice
+site r𝑖 = (𝑖𝑥, 𝑖𝑦) and so does 𝑗. We focus on the half-filling
+case by adjusting the chemical potential 𝜇. The first term is the
+hopping Hamiltonian, where the hopping parameter without
+any external field is given by J𝑖 𝑗 = 𝐽(< 0) for nearest-neighbor
+sites𝑖 and 𝑗 and otherwise 0, i.e., J𝑖 𝑗 = 𝐽𝛿|r𝑖−r 𝑗 |,1. The second
+term describes the on-site attractive interaction for 𝑈 < 0.
+The third term describes the nearest-neighbor interaction for
+𝑉𝑖 𝑗 = 𝑉𝛿|r𝑖−r 𝑗 |,1, which lifts the degeneracy between the SC
+and CDW states [63]. When the nearest-neighbor interaction
+𝑉 > 0 (𝑉 < 0), the CDW (SC) phase has lower energy. For the
+total number of the unit cells 𝑁, the Fourier transform 𝑐𝑖𝜎 =
+1/
+√
+𝑁 �
+k 𝑒𝑖k·r𝑖𝑐k𝜎 leads to the following representation of
+the Hamiltonian:
+𝐻 =
+∑︁
+k𝜎
+𝜖k𝜎𝑛k𝜎 + 1
+2𝑁
+∑︁
+k𝜎𝜎′
+𝛿𝜎, ¯𝜎′(𝑈 + 𝑉k)𝜌−k𝜎𝜌k𝜎′, (2)
+where 𝑛k𝜎 = 𝑐†
+k𝜎𝑐k𝜎, 𝜌q𝜎 = �
+k 𝑐†
+k𝜎𝑐k+q𝜎, 𝜖k = 2𝐽[cos(𝑘𝑥)
++ cos(𝑘𝑦)], 𝑉k = 2𝑉 [cos(𝑘𝑥) + cos(𝑘𝑦)]. The mean-field ap-
+proximation of the Hamiltonian (1) is performed as follows:
+The 𝑈 term takes into account the Hartree–Fock terms as well
+as the anomalous average, that is, the so-called Bogoliubov
+term. The 𝑉 term takes into account only the Hartree term in
+order to exclude the bond order wave and the 𝑑-wave super-
+conductivity for simplicity. By using the mean fields ⟨𝑛𝑖𝜎⟩
+and Δ𝑖 ≡ ⟨𝑐𝑖↓𝑐𝑖↑⟩, the interaction terms reduce to
+∑︁
+𝑖
+(𝑈 ⟨𝑛𝑖 ¯𝜎⟩ − 𝜇)𝑛𝑖𝜎 + 𝑈
+∑︁
+𝑖
+(Δ𝑖𝑐†
+𝑖↑𝑐†
+𝑖↓ + H.c.) + 𝑉
+∑︁
+⟨𝑖, 𝑗⟩
+⟨𝑛𝑖⟩ 𝑛 𝑗.
+(3)
+Hence, the EAHM can be rewritten in a quadratic form:
+𝐻 ≃ �𝐶†𝐻BdG �𝐶,
+(4)
+where �𝐶† = (𝑐†
+1↑, . . . , 𝑐†
+𝑁 ↑, 𝑐1↓, . . . , 𝑐𝑁 ↓). The static struc-
+tures of the mean fields, ⟨𝑛𝑖⟩ and Δ𝑖, and the electron states
+are determined self consistently. From the mean-field Hamil-
+tonian with a set of the mean fields, we obtain the one particle
+eigenenergies and eigenstates, which determine a new set of
+mean fields. We repeat this iterative procedure until the largest
+error in the updates becomes less than 𝜀err = 10−8.
+B.
+Equation of motion
+In order to calculate the real-space dynamics, we introduce
+normal and anomalous density matrices 𝒢𝑖𝜎; 𝑗 𝜎′ = ⟨𝑐†
+𝑗 𝜎′𝑐𝑖𝜎⟩,
+ℱ𝑖𝜎; 𝑗 𝜎′ = ⟨𝑐 𝑗 𝜎′𝑐𝑖𝜎⟩. The time evolution is calculated on the
+basis of the equation of motion for the density matrices given
+by
+−𝑖 𝑑
+𝑑𝑡𝒢 =
+�
+J − 𝜌𝑈 − 𝜌𝑉 �
+𝒢 − 𝒢
+�
+J − 𝜌𝑈 − 𝜌𝑉 �
++ Δ𝑈ℱ∗ − ℱΔ𝑈
+(5)
+−𝑖 𝑑
+𝑑𝑡 ℱ =
+�
+J − 𝜌𝑈 − 𝜌𝑉 �
+ℱ + ℱ
+�
+J ∗ − 𝜌𝑈 − 𝜌𝑉 + 2𝜇𝐼
+�
++ (𝒢 − 𝐼) Δ𝑈 − 𝑡 �
+𝒢Δ𝑈�
+,
+(6)
+where the matrix elements are given by J𝑖𝜎; 𝑗 𝜎′ = 𝛿𝜎,𝜎′J𝑖 𝑗,
+𝜌𝑈
+𝑖𝜎; 𝑗 𝜎′ = 𝑈𝛿𝑖, 𝑗𝛿𝜎,𝜎′𝒢𝑖 ¯𝜎;𝑖 ¯𝜎, Δ𝑈
+𝑖𝜎; 𝑗 𝜎′ = 𝑈𝛿𝑖, 𝑗 (𝑖𝜎𝑦)𝜎,𝜎′ℱ𝑖↑,𝑖↓,
+and 𝜌𝑉
+𝑖𝜎;𝑗 𝜎′ = 𝛿𝑖, 𝑗𝛿𝜎,𝜎′ �
+𝑘 𝜎 𝒢𝑘 𝜎;𝑘 𝜎𝑉𝑘 𝑗. The dimension of
+each matrix is 2𝑁 × 2𝑁. A similar formalism is derived in
+Refs. 50 and 69.
+We also define a time-dependent distribution function to
+track the time evolution of the electronic structure.
+The
+
+3
+Hamiltonian at time 𝑡 can be diagonalized as �𝐶†𝐻BdG(𝑡) �𝐶 =
+�𝐵†(𝑡)𝐷(𝑡) �𝐵(𝑡), where �𝐵(𝑡) = 𝑈†(𝑡) �𝐶, and 𝐷(𝑡) = diag(𝐸1(𝑡),
+· · · , 𝐸2𝑁 (𝑡)). The 𝜇-th component of �𝐵(𝑡), given by �𝐵𝜇(𝑡) =
+�
+𝑗 [𝑈(𝑡)]∗
+𝑗𝜇 �𝐶 𝑗, stands for the annihilation operator of the
+quasiparticle with eigenenergy 𝐸𝜇(𝑡) The time-dependent dis-
+tribution function for each 𝐸𝜇(𝑡), denoted by N𝜇(𝑡), is calcu-
+lated as
+N𝜇(𝑡) =
+�
+�𝐵†
+𝜇(𝑡) �𝐵𝜇(𝑡)
+�
+=
+∑︁
+𝑖, 𝑗
+� ˆ𝑈(𝑡)
+�
+𝑖𝜇
+� ˆ𝑈(𝑡)
+�∗
+𝑗𝜇
+�
+�𝐶†
+𝑖 (𝑡) �𝐶 𝑗 (𝑡)
+�
+,
+(7)
+where ⟨ �𝐶†
+𝑖 (𝑡) �𝐶 𝑗 (𝑡)⟩ is calculated from 𝒢(𝑡) and ℱ(𝑡).
+In
+addition, the time-dependent pair-potential for each 𝐸𝜇(𝑡),
+denoted by Δ𝜇(𝑡), is calculated as
+Δ𝜇(r𝑖, 𝑡) =
+� ˆ𝑈(𝑡)
+�
+𝑖𝜇
+� ˆ𝑈(𝑡)
+�∗
+𝑖+𝑁 𝜇
+�
+𝐵†
+𝜇(𝑡)𝐵𝜇(𝑡)
+�
+(8)
+This quantity satisfies the following relation: �
+𝜇 Δ𝜇(r𝑖, 𝑡) =
+⟨𝑐𝑖↓(𝑡)𝑐𝑖↑(𝑡)⟩.
+C.
+Linear response theory
+The charge and pair correlation functions for an imaginary
+time 𝜏 are, respectively, defined as
+Πc(q, q′, 𝜏) =
+∑︁
+𝜎𝜎′
+Πc,𝜎𝜎′(q, q′, 𝜏),
+(9)
+Πc,𝜎𝜎′(q, q′, 𝜏) = − 1
+𝑁
+�
+𝑇𝜏𝜌q𝜎(𝜏)𝜌−q′𝜎′(0)
+�
+,
+(10)
+ΠSC(q, q′, 𝜏) = − 1
+𝑁
+�
+𝑇𝜏Δq(𝜏)Δ†
+q′(0)
+�
+,
+(11)
+where 𝜌q𝜎 = �
+k 𝑐†
+k𝜎𝑐k+q𝜎, Δq = �
+k 𝑐−(k+q)↓𝑐k↑, and 𝑇𝜏 is
+the imaginary-time ordered product. Their Fourier transform
+in the frequency domain is given with the bosonic Matsubara
+frequency 𝜀ℓ = 2𝜋ℓ𝑇 by
+Πc(SC)(q, q′, 𝑖𝜖ℓ) =
+∫ 1/𝑇
+0
+𝑑𝜏𝑒𝑖𝜖ℓ 𝜏Πc(SC)(q, q′, 𝜏).
+(12)
+The Fourier transform of Eq. (10) is defined similarly. We
+calculate these correlation functions within the RPA. The RPA
+takes into account the self-consistent dynamics of the mean
+fields due to the applied external fields.
+We formulate the
+correlation functions within the RPA, by following Refs. 70
+and 71.
+In the following, we construct the RPA formalism in the
+presence of the CDW order characterized by the order vec-
+tor Q = (𝜋, 𝜋).
+First we should remark on the momenta
+in the argument of Πc(SC) in Eq. (12).
+To analyze the
+excitation structure, we are interested in the diagonal ele-
+ments Πc(SC) (q, q, 𝑖𝜖ℓ).
+However, the (q, q + Q) compo-
+nent is also taken into account through the intermediate pro-
+cess of the RPA, as can be seen below.
+For this purpose,
+we introduce a 4 by 4 matrix ˆΠc,q and a 2 by 2 matrix
+ˇΠSC,q defined as [ ˆΠc,q]q1 𝜎1:q2 𝜎2 = Πc,𝜎1 𝜎2(q1, q2, 𝑖𝜖ℓ) and
+[ ˇΠSC,q]q1:q2 = ΠSC(q1, q2, 𝑖𝜖ℓ), respectively, where q1,2 takes
+either q or q + Q ≡ ¯q. The definitions given by Eq. (12) are
+convenient for the following matrix RPA form. In the pres-
+ence of the CDW order, the following Green’s functions take
+nonzero values: 𝐺k𝜎(𝜏) = − ⟨𝑇𝜏 𝑐k𝜎(𝜏)𝑐†
+k𝜎(0)⟩ =: 𝐺k(𝜏)
+and 𝐷k𝜎(𝜏) = − ⟨𝑇𝜏 𝑐k𝜎(𝜏)𝑐†
+k+Q𝜎(0)⟩ =: 𝐷k(𝜏), where
+they are independent of the spin index because the SDW or-
+der is neglected. Within the RPA, correlation functions in the
+matrix form are given by
+ˆΠRPA
+c,q (𝑖𝜖ℓ) =
+� ˆ𝐼 − ˆΠ0
+c,q(𝑖𝜖ℓ) · ˆ𝑈c,q
+�−1 · ˆΠ0
+c,q(𝑖𝜖ℓ),
+(13)
+ˇΠRPA
+SC,q(𝑖𝜖ℓ) =
+�
+ˇ𝐼 − ˇΠ0
+SC,q(𝑖𝜖ℓ) ˇ𝑈SC,q
+�−1 ˇΠ0
+SC,q(𝑖𝜖ℓ),
+(14)
+where ˆ𝐼( ˇ𝐼) denotes the 4 by 4 (2 by 2) identity matrix, and
+ˆ𝑈c,q and ˇ𝑈SC,q are the interaction matrices defined for the
+basis sets (q ↑, q ↓, ¯q ↑, ¯q ↓) and (q, ¯q), respectively. Their
+matrix elements are, respectively, defined as
+ˆ𝑈c,q =
+����
+�
+𝑉q
+𝑈 + 𝑉q
+0
+0
+𝑈 + 𝑉q
+𝑉q
+0
+0
+0
+0
+−𝑉q
+𝑈 − 𝑉q
+0
+0
+𝑈 − 𝑉q
+−𝑉q
+����
+�
+,
+(15)
+ˇ𝑈SC,q =
+�
+𝑈 + 𝑉q
+0
+0
+𝑈 − 𝑉q
+�
+.
+(16)
+In Eqs. (13) and (14),
+ˆΠ0
+c and ˇΠ0
+sc are the lowest-order
+correlation functions that include the Hartree–Fock con-
+tributions in the single particle Green’s functions.
+By
+noting that Π0
+𝜎𝜎′(q, q′, 𝑖𝜖ℓ)
+=
+𝛿𝜎,𝜎′Π0
+c(q, q′, 𝑖𝜖ℓ)
+and
+Π0
+c,SC(q1, q2, 𝑖𝜖ℓ) = Π0
+c,SC(q2, q1, 𝑖𝜖ℓ), the independent com-
+ponents can be explicitly written as
+Π0
+c(q, q, 𝑖𝜖ℓ) = 𝑇
+𝑁
+∑︁
+k,ℓ
+�
+𝐺k(𝑖𝜔𝑛)𝐺k+q(𝑖𝜖ℓ + 𝑖𝜔𝑛)
++𝐷k(𝑖𝜔𝑛)†𝐷k+q(𝑖𝜖ℓ + 𝑖𝜔𝑛)
+�
+,
+(17)
+Π0
+c(q, ¯q, 𝑖𝜖ℓ) = 𝑇
+𝑁
+∑︁
+k,ℓ
+�
+𝐺k(𝑖𝜔𝑛)𝐷k+q(𝑖𝜖ℓ + 𝑖𝜔𝑛)
++𝐷†
+k(𝑖𝜔𝑛)𝐺k+q(𝑖𝜖ℓ + 𝑖𝜔𝑛)
+�
+,
+(18)
+Π0
+SC(q, q, 𝑖𝜖ℓ) = − 𝑇
+𝑁
+∑︁
+k,ℓ
+�
+𝐺k(−𝑖𝜔𝑛)𝐺−(k+q) (𝑖𝜖ℓ + 𝑖𝜔𝑛)
++𝐷k(−𝑖𝜔𝑛)𝐷−(k+q) (𝑖𝜖ℓ + 𝑖𝜔𝑛)
+�
+,
+(19)
+Π0
+SC(q, ¯q, 𝑖𝜖ℓ) = − 𝑇
+𝑁
+∑︁
+k,ℓ
+�
+𝐺k(−𝑖𝜔𝑛)𝐷−(k+q) (𝑖𝜖ℓ + 𝑖𝜔𝑛)
++𝐷k(−𝑖𝜔𝑛)𝐺−(k+q) (𝑖𝜖ℓ + 𝑖𝜔𝑛)
+�
+.
+(20)
+To
+investigate
+the
+excitation
+structure,
+we
+perform
+an analytic continuation of �
+𝜎,𝜎′[ ˆΠRPA
+c,q (𝑖𝜖ℓ)]q𝜎:q𝜎′ and
+[ ˇΠRPA
+SC,q(𝑖𝜖ℓ)]q:q: 𝑖𝜖ℓ → 𝜔 +𝑖𝛿, and describe them as ΠRPA
+c,q (𝜔)
+and ΠRPA
+SC,q(𝜔), respectively.
+
+4
+280
+285
+290
+295
+300
+305
+310
+315
+320
+rx
+1
+5
+10
+ry
+(a)
+−0.25
+0.00
+0.25
+280
+285
+290
+295
+300
+305
+310
+315
+320
+rx
+−0.50
+−0.25
+0.00
+0.25
+0.50
+(b)
+∆(r) = ⟨ci↓ci↑⟩
+mz(r) = 1
+2(�
+σ⟨niσ⟩ − 1)e−iQ·ri
+280 285 290 295 300 305 310 315 320
+rx
+−4
+−2
+0
+2
+4
+ω
+(c)
+10−3
+10−2
+10−1
+100
+10−2
+100
+DOS
+(d)
+←
+←
+FIG. 1.
+(a) Charge density distribution in real space. An interface
+is located at 𝑟𝑥 = 300 in the system of size 401 × 40. (b) Order
+parameters of SC Δ(r) and CDW 𝑚𝑧(r) along the 𝑥 direction. The
+interface is dominated by Δ(r) rather than 𝑚𝑧(r), while in the region
+far from the interface Δ(r) goes to 0 and 𝑚𝑧(r) is dominant. (c) Local
+density of states (LDOS) around the domain wall. The horizontal axis
+denotes the spatial direction across the domain wall (𝑟𝑥), while the
+vertical axis denotes the energy (𝜔).
+along the 𝑥 direction.
+The
+intensities of the LDOS are indicated by the color map. (d) Density
+of states plotted on a logarithmic scale. Arrows indicate the DWBSs.
+The vertical axis is the same as that in (c).
+III.
+RESULTS
+A.
+Setup
+Before showing numerical results based on the above for-
+mulation, we summarize the parameters and numerical condi-
+tions. As a unit of energy, we set |𝐽| = 1. In the following
+numerical results, we use 𝑈 = −2, 𝑉 = 0.05. The total number
+of electrons 𝑁e is fixed to 𝑁, corresponding to the half-filling
+case. A non-zero 𝑉 lifts the degeneracy between the CDW
+state and the SC state, and the CDW state is realized as the
+uniform ground state in the half-filling case.
+As an initial state of the time evolution, we consider two
+CDW domains with opposite signs, which induce the super-
+conductivity along their interface [67]. The geometry of the
+system is an 𝑁𝑥 × 𝑁𝑦 = 401 × 40 site lattice with periodic
+boundary conditions (PBCs) in both directions.
+When 𝑁𝑥
+is odd, the PBC in the 𝑥 direction naturally introduces an
+interface along the 𝑦 direction as a self-consistent solution.
+Here, we use the following integer notations 𝑖 = (𝑖𝑥, 𝑖𝑦) = r𝑖
+and r = (𝑟𝑥, 𝑟𝑦) interchangeably to represent a site position.
+Figure 1(a) shows the spatial profile of the charge density
+�
+𝜎(⟨𝑐†
+𝑖𝜎𝑐𝑖𝜎⟩−1). Figure 1(b) shows the 𝑟𝑥 dependence of the
+SC order parameter Δ(r) = ⟨𝑐𝑖↓𝑐𝑖↑⟩ using the dashed line, and
+that of the staggered density 𝑚𝑧(r) = (�
+𝜎 ⟨𝑛𝑖𝜎⟩ −1)𝑒𝑖Q·r𝑖/2
+using the solid line.
+Note that Δ(r) and 𝑚𝑧(r) are uni-
+form along the 𝑦 direction. At the center of the domain wall
+𝑟𝑥 = 300, Δ𝑖 reaches a maximum value, and 𝑚𝑧 is zero.
+In Figs. 1(c) and 1(d), we plot the local density of
+states (LDOS) and density of states (DOS), defined by
+LDOS(𝑟𝑥, 𝜔) = −1/(𝜋𝑁𝑦) �
+𝑟𝑦 Im tr [(𝜔 + 𝑖𝜂 − 𝐻BdG)−1]r,r
+and DOS(𝜔) = 1/𝑁𝑥
+�
+𝑟𝑥 LDOS(𝑟𝑥, 𝜔), where tr is the trace
+in Nambu space. The figures show that the system has a CDW
+gap 𝜔𝑔 ≃ 1.1. In (c), there are fermionic states bound in the
+domain wall indicated by the arrows in (d). In this paper, we
+call them domain-wall bound states (DWBSs).
+Such a non-uniform structure is related to a magnetic do-
+main wall [67].
+The low-energy state of the EAHM are
+described by the order parameters of the CDW and the
+superconductivity, which can be regarded as an antiferro-
+magnetic order in a classical pseudospin system. Consider
+the following map [58, 59, 72]: 𝑆𝑥
+𝑗 + 𝑖𝑆𝑦
+𝑗 = 𝑐 𝑗↓𝑐 𝑗↑𝑒𝑖Q·r 𝑗,
+𝑆𝑧
+𝑗 = (𝑛 𝑗↑ + 𝑛 𝑗↓ − 1)/2, the interaction between nearest-
+neighbor peseudospins are antiferromagnetic. The Neél or-
+der along the 𝑧 direction in pseudospin space corresponds to
+the CDW, and the Neél order in the 𝑥-𝑦 plane corresponds
+to uniform superconductivity.
+In the low-energy region, a
+spatially local order-parameter manifold is constructed on the
+SO(3) sphere of �𝑆 𝑗𝑒𝑖Q·r 𝑗. In the absence of 𝑉, the ground
+state has SO(3) symmetry. A small 𝑉(> 0) lifts this degen-
+eracy, so that the Neél order along the 𝑧-axis has a lower
+energy than that in the 𝑥-𝑦 plane; the north (south) pole de-
+notes a positive (negative) value of the CDW order parameter
+𝑚𝑧(r 𝑗) = ⟨𝑆𝑧
+𝑗𝑒𝑖Q·r 𝑗⟩ > 0 (𝑚𝑧(r 𝑗) < 0). The aforementioned
+structure can be regarded as a pseudospin antiferromagnetic
+domain. In the domain wall region, antiferromagnetic spin
+structures with 𝑥 and 𝑦 components ⟨𝑆𝑥,𝑦
+𝑗
+𝑒𝑖Q·r 𝑗⟩ correspond
+to the uniform SC structure with U(1) phase degrees of free-
+dom, Re Δ(r), Im Δ(r) [67]. The domain-wall width depends
+on the off-site interaction 𝑉 and the width diverges as 𝑉 → 0,
+because 𝑉 plays a role of easy-axis anisotropy along the 𝑧-axis
+in the pseudospin picture. For the above parameter set, the
+width along the 𝑥 direction in real space is estimated to be 10
+sites Fig. 1(b).
+The equation of motion is numerically solved using the
+fourth-order Runge–Kutta method (RK4) for the time-step
+Δ𝑡 = 0.01. By taking advantage of the translational symme-
+try along the 𝑦 direction Fig. 1(a), we performed the Fourier
+transform along the 𝑦 direction. The details are explained in
+Appendix A.
+In the following subsections, we investigate the photoin-
+duced dynamics of the above interface structure driven by
+an oscillating electric field with a frequency 𝜔ext via the
+Peierls substitution of the vector potential A.
+The vec-
+tor potential modifies the hopping amplitude to J𝑖 𝑗
+→
+J𝑖 𝑗 exp[−𝑖A(𝑡) · (r𝑖 − r 𝑗)]. We choose the vector potential
+to be A(𝑡) = e𝑝 𝐴(𝑡) = e𝑝 𝐴0 sin(𝜔ext𝑡) with amplitude 𝐴0,
+
+5
+polarization e𝑝, and driving frequency 𝜔ext. The polarization
+direction is set as e𝑝 = (1, 0), where we have checked that
+the 𝑦 component of e𝑝 does not bring about any noticeable
+dynamics.
+B.
+Pseudospin dynamics: 𝜔ext ≪ 𝜔𝑔
+In this subsection, we examine the photoinduced dynamics
+with driving frequency 𝜔ext = 0.1, which is lower than the
+CDW gap 𝜔𝑔 ≃ 1.1. In this regime, it is expected that hardly
+any quasiparticles are excited and the pseudospin picture is
+valid. The amplitude 𝐴0 is set to 0.02.
+The top panels in Fig. 2 show the real-space and real-time
+evolutions of the mean field 𝑂(𝑟𝑥, 𝑡) ≡ 𝑁𝑦−1 �
+𝑟𝑦 𝑂(r =
+(𝑟𝑥, 𝑟𝑦), 𝑡), where 𝑂 in each panel is as follows: (a) den-
+sity deviation from the averaged value 𝑛(r) − 1, (b) (c) the
+real and imaginary parts of the uniform SC order parameter
+ΔSC(r) ≡ Δ(r), and (d) (e) the real and imaginary parts of the
+staggered SC order parameter Δ𝑄
+SC(r) ≡ Δ(r)𝑒𝑖Q·r. In panels
+(b), the SC interface, initially at 𝑟𝑥 = 300, hardly changes
+its position and the SC phase during the time evolution. The
+other panels, (a) and (c)–(e), show the polarization of the mean
+fields in the interface, which represents a deformation of the
+domain wall.
+It should be also noted that the uniform and staggered SC
+components, propagate from the domain wall, as can be seen
+in panels (b)–(f), i.e., the collective mode propagates. The
+laser irradiation drives the deformation of the domain wall.
+Such an internal deformation generates a pseudospin wave
+propagating outwards. In the region far from the domain wall,
+the pseudospin wave can be regarded as precessional motion
+of pseudospins around the 𝑧-axis corresponding to the CDW,
+i.e., excitation of the 𝑥, 𝑦 components corresponding to the
+SC components. The precession propagates from the interface
+to the outside, which can be interpreted as the pseudospin
+wave emission from the domain walls. This pseudospin wave
+contains the uniform and staggered components as in the case
+of the low-energy antiferromagnetic spin waves. It is known
+that spin waves can also be emitted from a domain wall in the
+ferromagnetic case and the antiferromagnetic case by using
+oscillating magnetic fields or spin orbit torques [73, 74]. The
+pseudospin-wave emission in the present study is an analogous
+to these magnetic systems, but the emission is triggered by
+the electric field. (As mentioned in Sect. III A, the domain
+wall can be interpreted as an antiferromagnetic domain wall
+with easy-axis anisotropy along the 𝑧-axis.) In terms of the
+collective excitation of the SC order parameter, the propagating
+wave is a phase rotation of ΔSC with k ∼ (0, 0) and (𝜋, 𝜋).
+Note that the staggered component in this case can also be
+regarded as the 𝜂 pairing excitation of the attractive Hubbard
+model [64, 75–77]. The mass of the phase mode is due to the
+off-site Coulomb interaction 𝑉, whose effects on the collective
+mode are discussed at the end of this subsection.
+Next, to investigate the emitted collective mode in the uni-
+form region, we perform a Fourier transform into momentum
+and frequency spaces:
+𝑂(𝑘𝑥, 𝜔) = 𝑁𝑦
+∑︁
+𝑟𝑥
+∫ 𝑇max
+0
+𝑑𝑡 𝑒𝑖(𝑘𝑥𝑟𝑥−𝜔𝑡)𝑂(𝑟𝑥, 𝑡),
+(21)
+where 𝑇max = 400. The absolute values of the Fourier compo-
+nents are shown in Figs. 2(f)–2(j). By applying a laser with
+zero momentum, the spectral intensities can be acquired not
+only at k = 0 but also in the finite k region owing to the inho-
+mogeneity of the domain wall. The flat peaks in 𝜔 = 𝜔ext = 0.1
+in each panel correspond to the forced oscillation by the exter-
+nal electric field. Figures 2(g) and 2(h) [2(i) and 2(j)] show the
+dispersive SC collective modes for 𝜔 ≲ 𝜔𝑔 and 0 ≤ 𝑘𝑥 ≤ 𝜋
+with a uniform (staggered) profile along the 𝑦 direction.
+Note that using Δ(𝑘𝑥, 𝑘𝑦, 𝜔) = �
+r
+∫ 𝑇max
+0
+𝑑𝑡𝑒𝑖(k·r−𝜔𝑡)Δ(r),
+ΔSC(𝑘𝑥) = Δ(𝑘𝑥, 𝑘𝑦 = 0) and Δ𝑄
+SC(𝑘𝑥 − 𝜋) = Δ(𝑘𝑥, 𝑘𝑦 = 𝜋).
+As a result of the folding of the first Brillouin zone by
+the CDW order, the induced spectral intensities concentrate
+around k = (0, 0) and k = (𝜋, 𝜋). In addition, Floquet side
+bands surround the SC collective mode with energy difference
+±𝜔ext. We have checked that the Floquet side band exists at
+the other driving frequencies 𝜔ext. Those dispersion can be
+interpreted as a photon-dressed collective mode. A similar dis-
+persive branch is observed for 𝜔 ≳ 𝜔𝑔 in Fig. 2(f). However,
+this is not a collective mode of the charge density, but rather
+part of the continuum spectra of the particle-hole excitation,
+which is shown more clearly in the following analysis.
+In order to analyze the collective modes emitted from the
+domain wall, we calculated the dynamical charge and pair
+correlation functions based on the RPA (Eqs. (13) and (14)).
+Figures 3 show the correlation functions as a function of 𝑘𝑥 for
+𝑘𝑦 = 0 and those for 𝑘𝑦 = 𝜋. The imaginary parts of the dy-
+namical correlation functions in Fig. 3 are in good agreement
+with the results obtained from the real-space and real-time cal-
+culations in Fig. 2 [the charge density: Fig. 3(a) and Fig. 2(f),
+the uniform SC pair: Fig. 3(b) and Figs. 2(g), 2(h), and the
+staggered SC pair: Fig. 3(c) and Figs. 2(g), 2(h)]. Figure 3(a)
+shows continuum spectra rather than a collective mode.
+By contrast, Figs. 3(b) and 3(c) show a collective mode with
+an energy below the continuum spectra. When the inter-site
+Coulomb interaction𝑉 is zero, because of the SO(3) symmetry
+of the pseudospin, gapless NG modes appear at q = (0, 0) and
+(𝜋, 𝜋) [78? , 79], just as in the antiferromagnetic case with
+the Neél order. In the presence of a repulsive 𝑉(> 0), such a
+collective mode requires a finite excitation energy. In Figs. 3(b)
+and 3(c), the collective modes below the continuum are the
+massive Nambu–Goldstone (NG) modes, where the term “NG
+mode” refers due to the propagation of phase rotation of the
+SC mean fields. The lower panels of Fig. 2 show very small
+spectral intensities in the region 𝜔 ≥ 1, which indicates that the
+external field does not excite quasiparticles in the continuum-
+spectral region. This is why the pseudospin description works
+quite well.
+We should note that this calculation based on
+linear response theory cannot reveal any side bands around the
+collective mode, whereas that Floquet theory could be used to
+reveal the side band around the SC collective mode.
+We should also note that the energies of the SC collec-
+tive mode obtained by the RPA are slightly higher than those
+
+6
+1
+100 200 300 400
+rx
+0
+50
+100
+150
+200
+250
+300
+350
+t
+(a)
+n(rx) − 1
+−5
+0
+5
+×10−3
+1
+100 200 300 400
+rx
+(b)
+Re ∆SC(rx)
+−1
+0
+1
+×10−1
+1
+100 200 300 400
+rx
+(c)
+Im ∆SC(rx)
+−1
+0
+1
+×10−2
+1
+100 200 300 400
+rx
+(d)
+Re ∆Q
+SC(rx)
+−5
+0
+5
+×10−3
+1
+100 200 300 400
+rx
+(e)
+Im ∆Q
+SC(rx)
+−2.5
+0.0
+2.5
+×10−4
+0
+π/4
+π/2 3π/4
+π
+kx
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+ω
+(f)
+|n(kx)|
+10−2
+10−1
+100
+101
+0
+π/4
+π/2 3π/4
+π
+kx
+(g)
+|Re ∆SC(kx)|
+10−2 10−1 100 101 102 103
+0
+π/4
+π/2 3π/4
+π
+kx
+(h)
+|Im ∆SC(kx)|
+10−2
+10−1
+100
+101
+102
+0
+π/4
+π/2 3π/4
+π
+kx
+(i)
+|Re ∆Q
+SC(kx − π)|
+10−2
+10−1
+100
+101
+0
+π/4
+π/2 3π/4
+π
+kx
+(j)
+|Im ∆Q
+SC(kx − π)|
+10−2
+10−1
+100
+101
+FIG. 2. (upper panels) Time evolution of order parameters for 𝐴0 = 0.02 and 𝜔ext = 0.1. (a) Deviation of the charge density from the averaged
+value. (b)(c) Uniform and (d)(e) staggered SC order parameters. To visualize the propagation of the collective modes, we multiply the actual
+values in the range 1 ≤ 𝑟𝑥 ≤ 250 by 500 in (a)(b) and 50 in (c)(d). (lower panels) Intensities of the Fourier transforms of the corresponding
+time evolution in real space in the upper row.
+observed in the real-space calculation. This may have been
+because we neglected vertex-type diagrams due to 𝑉 for sim-
+plicity.
+C.
+Quasiparticle excitation: 𝜔ext ≳ 𝜔𝑔
+In this subsection, we examine the nonequilibrium dynamics
+at a frequency 𝜔 = 1.2 near the CDW gap, for which the
+quasiparticle responses as well as the collective response of
+the pseudospins are important. The amplitude is 𝐴0 = 0.02,
+and the system size is 121 × 40.
+Figure 4(a) shows suppression of the CDW order and im-
+plies a melting of the SC domain wall. The dashed line shows
+the spatial average of the absolute value of the CDW order
+parameter, which decreases with time. In this subsection, we
+take Δ(𝑟𝑥) to be the 𝑟𝑦-averaged SC mean-field instead of
+ΔSC(𝑟𝑥) for simplicity. For 𝑡 ≲ 200, the SC amplitude on the
+SC interface, represented by |Δ(𝑟𝑥,max)|, decreases as well.
+Here, 𝑟𝑥,max denotes the position at which |Δ(𝑟𝑥)| is a maxi-
+mum, that is, the domain wall center. Furthermore, |Δ(𝑟𝑥,max)|
+shows a sudden reduction for 𝑡 ∼ 200. By contrast, a gradual
+increase can be seen in the spatial average of |Δ(𝑟𝑥)| where
+the domain wall region 85 ≤ 𝑟𝑥 ≤ 95 is excluded, as shown by
+the dashed-dotted line. In other words, the SC interface struc-
+ture, represented by the locally strong SC amplitude, melts at
+around 𝑡 ∼ 200, and the SC order parameter extends to the
+whole system through propagation of the pseudospin wave.
+Figure 4(b) shows the time evolution of the quasiparticle pop-
+ulation. The line at 𝑡 = 0 represents the initial state, that is, the
+fully occupied states below 𝜔 = 0. The lines for 𝑡 > 0 show
+that the laser excitation reduces the CDW gap and increases
+the quasiparticle population in the conduction band.
+The time evolutions of the energy and spatial-resolved pair-
+potential amplitude are plotted in Figs. 4(c) and 4(d), where
+we have defined |Δ(𝑟𝑥, 𝑡, 𝜔)| = �
+𝜇 |Δ𝜇(𝑟𝑥, 𝑡)|𝛿(𝜔 − 𝐸𝜇) for
+Δ𝜇(𝑟𝑥, 𝑡) Eq. (8).
+First, let us discuss the energy-resolved
+structure of the domain-wall superconductivity and its time
+
+7
+0
+π/2
+π
+kx
+0.0
+0.5
+1.0
+1.5
+2.0
+ω
+(a)
+−1/π Im Πc(kx, ky = 0)
+0
+π/2
+π
+kx
+(b)
+−1/π Im ΠSC(kx, ky = 0)
+0
+π/2
+π
+kx
+(c)
+−1/π Im ΠSC(kx, ky = π)
+10−2
+10−1
+10−2
+10−1
+100
+10−2
+10−1
+100
+FIG. 3.
+Intensity map of the dynamical correlation functions for
+(a) charge and (b)(c) SC pair potential along the 𝑘𝑥 direction. The
+equilibrium state was a uniform CDW order and the system was of
+size 400 × 40. The top panels show the imaginary parts. The value
+of 𝑘𝑦 is shown in each panel.
+evolution. The energy-resolved pair potential at the domain-
+wall center 𝑟𝑥 = 𝑟𝑥,max is shown in Fig. 4(c). From the data
+at 𝑡 = 0, the domain wall superconductivity mainly consists of
+the DWBS at 𝜔 ≃ −0.27, which has a sharp and strong peak
+structure. It also includes a broader, less intense contribution
+from the continuum states for 𝜔 < −𝜔𝑔/2 ∼ −0.55.
+The
+following changes occur under an external field at the resonant
+frequency: (i) the absolute value of the energy of the DWBS
+decreases, as does their intensities; (ii) the energy distribution
+of the superconductivity from the DWBSs shifts to the positive
+side at 𝑡 ∼ 200. This population inversion is attributed to the
+phase rotation of the superconductivity inside the domain wall,
+as discussed in Appendix B. sAt 𝑡 ≳ 300, the intensities of the
+DWBSs at the positive and negative energies are comparable.
+By focusing on the phase of Δ(𝑟𝑥,max, 𝑡, 𝜔), however, the two
+contributions have opposite phase (not shown), which results
+in Δ(𝑟𝑥,max, 𝑡) having a small amplitude; (iii) the intensity of
+the continuum contribution also decreases. As a whole, the
+superconductivity in the domain wall region is reduced by
+resonant driving.
+Next, let us examine what happens away from the domain
+wall center.
+In such a region, the SC order parameter in-
+creases, as indicated in Fig. 4(a). This can be seen also in
+Fig. 4(d), which shows the energy-resolved amplitude of the
+pair potential averaged over 30 ≤ 𝑟𝑥 ≤ 60. In contrast to what
+is shown in panel (c), the continuum states |𝜔| > 𝜔𝑔/2 mainly
+contribute to the superconductivity, because the DWBSs are
+well localized around the domain wall. The small contribution
+at 𝑡 = 0 accounts for the exponential decay of the SC interface,
+in the region away from the domain wall center.
+Remark-
+ably, for 𝑡 ≥ 100 the energy-resolved pair potential increases
+particularly near the gap edge of the valence band. This en-
+hancement is more noticeable for 𝑡 ≳ 300, which is after the
+drastic reduction of the SC order at the domain wall.
+In summary, in the case of the resonant excitation, the ex-
+citation of the quasiparticles into the conduction band reduces
+the CDW gap 𝜔𝑔 (or the gap edge 𝜔𝑔/2). The quasiparticles
+near the gap edge of the valence band contribute to the for-
+mation of uniform superconductivity. This behavior which
+0
+100
+200
+300
+400
+t
+0.0
+0.1
+0.2
+0.3 (a)
+|∆(rx,max)|
+¯�
+rx|mz(rx)|
+10 × ¯�′
+rx|∆(rx)|
+−1.0
+−0.5
+0.0
+0.5
+1.0
+ω
+0
+2
+4
+N(t, ω)
+(b)
+t =0
+t =100
+t =200
+t =300
+t =400
+−1.0
+−0.5
+0.0
+0.5
+1.0
+ω
+0
+1
+2
+3
+|∆(rx,max, t, ω)|
+×10−1
+(c)
+−1.0
+−0.5
+0.0
+0.5
+1.0
+ω
+0
+2
+4
+¯�60
+rx=30|∆(rx, t, ω)|
+×10−3
+(d)
+FIG. 4. (a) Time evolution of the maximum value of |Δ(𝑟𝑥)| (solid),
+the averaged CDW order parameter 𝑚𝑧 (dashed), and the averaged
+SC order parameter in the region far from the domain wall (dashed-
+dotted).
+|Δ(𝑟𝑥)| is maximized at the center of the domain wall,
+denoted by 𝑟𝑥,max. (b) Time evolution of the quasiparticle population
+defined by N (𝑡, 𝜔) = �
+𝜇 N𝜇(𝑡)𝛿(𝜔 − 𝐸𝜇). Time evolution of the
+energy resolved pair potential at the domain-wall center, 𝑟𝑥 = 𝑟𝑥,max
+(c) and far from the domain wall (d). The vertical dotted lines are
+guides for the eye indicating the energies of the largest peaks for the
+DWBSs and their counterpart with the opposite signs. We define
+|Δ(𝑟𝑥, 𝑡, 𝜔)| = �
+𝜇 |Δ𝜇(𝑟𝑥, 𝑡)|𝛿(𝜔 − 𝐸𝜇). The broadening factor of
+the 𝛿-function in (b)–(d) is set to 0.01. We have introduced ¯� to
+represent the average per site, i.e., the summation divided by the
+number of terms in �. We use ¯�′
+𝑟𝑥 to denote exclusion of the domain
+wall region 85 ≤ 𝑟𝑥 ≤ 95 when taking the average.
+inclues the melting of the domain wall and the appearance of
+superconductivity in the bulk region, is triggered by the fol-
+lowing two factors: quasiparticle excitations, which cannot be
+described by the pseudospin picture, and the existence of the
+domain wall.
+
+8
+IV.
+SUMMARY AND DISCUSSION
+We have considered the laser-induced nonequilibrium dy-
+namics of the non-uniform system containing an SC domain
+wall sandwiched between CDW domains. We have found two
+driving-frequency regimes: (i) when the frequency of the driv-
+ing laser is below the CDW gap, its dynamics conform to the
+pseudospin picture. We have found that pseudospin waves are
+emitted from the domain wall; (ii) when the frequency of the
+laser is approximately equal to the CDW gap, excitation of the
+quasiparticles causes the CDW gap and the SC domain wall to
+melt. As a result, we have found that uniform superconductiv-
+ity is induced in the whole system.
+Laser control of superconductivity by utilizing a kind of
+NG mode was suggested in Ref. 49. This mode corresponds
+to a rotation of a vector in a plane composed of the SC and
+CDW order parameters, and can be controlled by tuning the
+frequency of the laser for 𝑉 = 0 where the SC and CDW orders
+are degenerate.
+However, the time scale of this collective
+rotation is very slow because the zero mode in equilibrium
+is utilized and the time scale is determined by an amount
+of 𝜂 pairing which is weakly excited to hold the pseudospin
+picture. In addition, we have not found such a collective mode
+in the presence of 𝑉 > 0 even in the uniform case; that is, the
+conditions under which it appears are restricted.
+Even in the presence of 𝑉 > 0, superconductivity may
+appear as an interface of two opposite CDW domains as pro-
+posed in Ref. 67. In this case, the oscillatory dynamics may
+have an analogy with that of the domain wall and the spin
+wave in the antiferromagnetic systems. We have clarified that
+the emission of the collective mode is possible in a uniformly
+oscillating electric field owing to the non-uniformity of the
+domain wall. We have also proposed a possibility of a kind
+of photoinduced uniform superconductivity via melting of the
+domain wall and the CDW order that is caused by the reso-
+nant excitation of the quasiparticles. These results suggest that
+the SC and CDW orders can be controlled by resonant laser
+excitation in a non-uniform CDW system.
+ACKNOWLEDGMENTS
+This work was supported by JST, the establishment of uni-
+versity fellowships towards the creation of science technol-
+ogy innovation, Grant Number JPMJFS2102 and JSPS KAK-
+ENHI, Grants Nos. JP19K14662 and JP22H01221. H.M. is
+supported by KAKENHI grant Nos. 21H04446, 21H03455,
+21K03380, and 20K03769, and by CSIS, Tohoku University.
+Appendix A: Fourier transform in the 𝑦 direction
+In this appendix, we explicitly show the equation of motion
+after performing a Fourier transform in the 𝑦 direction, which
+is possible thanks to the translational symmetry along the 𝑦
+direction (Fig. 1(a)). The Fourier transform of the electron an-
+nihilation operator at site 𝑖 with spin 𝜎 for the whole Brillouin
+zone −𝜋 < 𝑘𝑦 ≤ 𝜋 is given as
+𝑐𝑖𝜎 = 𝑐𝑖𝑥,𝑖𝑦 𝜎 =
+1
+√︁𝑁𝑦
+∑︁
+𝑘𝑦
+𝑒𝑖𝑘𝑦𝑟𝑦𝑐𝑖𝑥,𝑘𝑦 𝜎.
+(A1)
+The normal and anomalous density matrices introduced in
+Eq. (5) and (6) are transformed as
+𝒢𝑖𝑥,𝑘𝑦,𝜎;𝑖′𝑥,𝑘′𝑦,𝜎′ = 1
+𝑁𝑦
+∑︁
+𝑟𝑦,𝑟′𝑦
+𝑒𝑖𝑘′
+𝑦𝑟′
+𝑦−𝑖𝑘𝑦𝑟𝑦𝒢𝑖𝑥,𝑖𝑦,𝜎;𝑖′𝑥,𝑖′𝑦,𝜎′,
+(A2)
+ℱ𝑖𝑥,𝑘𝑦,𝜎;𝑖′𝑥,𝑘′𝑦,𝜎′ = 1
+𝑁𝑦
+∑︁
+𝑟𝑦,𝑟′𝑦
+𝑒−𝑖𝑘′
+𝑦𝑟′
+𝑦−𝑖𝑘𝑦𝑟𝑦ℱ𝑖𝑥,𝑖𝑦,𝜎;𝑖′𝑥,𝑖′𝑦,𝜎′.(A3)
+We introduce the charge density and SC order parameters di-
+agonalized for 𝑘𝑦:
+𝑛𝑖𝑥,𝑞,𝜎 = 1
+𝑁𝑦
+∑︁
+𝑘𝑦
+�
+𝑐†
+𝑖𝑥,𝑘𝑦,𝜎𝑐𝑖𝑥,𝑘𝑦+𝑞,𝜎
+�
+= 1
+𝑁𝑦
+∑︁
+𝑘𝑦
+𝒢𝑖𝑥,𝑘𝑦+𝑞,𝜎:𝑖𝑥,𝑘𝑦,𝜎,
+(A4)
+ΔSC
+𝑖𝑥,𝑞 = 1
+𝑁𝑦
+∑︁
+𝑘𝑦
+�
+𝑐𝑖𝑥,−(𝑘𝑦+𝑞),↓𝑐𝑖𝑥,𝑘𝑦,↑
+�
+= 1
+𝑁𝑦
+∑︁
+𝑘𝑦
+ℱ𝑖𝑥,𝑘𝑦,↑:𝑖𝑥,−(𝑘𝑦+𝑞)↓.
+(A5)
+Even in the nonequilibrium dynamics considered in this paper,
+𝑞 takes 0 or 𝜋 owing to the presence of the CDW order Q =
+(𝜋, 𝜋) and the translational symmetry along the 𝑦 direction.
+As a result, we have 2 × 2 × 𝑁𝑥 mean-fields. Note again that
+we set the polarization of the vector potential as e𝑝 = (1, 0) in
+order to restrict the photoinduced dynamics to the 𝑥 direction.
+The explicit forms of the equations of motion are
+
+9
+60
+90
+120
+rx
+0
+100
+200
+300
+t
+(a)
+Re ∆SC(rx)
+−0.1
+0.0
+0.1
+60
+90
+120
+rx
+0
+100
+200
+300
+t
+(b)
+Im ∆SC(rx)
+−0.1
+0.0
+0.1
+−1.0 −0.5 0.0
+0.5
+1.0
+ω
+0
+100
+200
+300
+t
+(c)
+�
+µ Nµ(t)δ(ω − Eµ)
+−2
+0
+0.0
+0.5
+1.0
+1.5
+2.0
+A0
+×10−2
+0.00
+0.05
+0.10
+0.15
+ω
+(d)
+10
+20
+30
+FIG. 5. Time evolution of the SC order parameter for (a) the real part and (b) the imaginary part. (c) Time evolution of the distribution function
+represented by �
+𝜇 N𝜇(𝑡)𝛿(𝜔 − 𝐸𝜇) plotted on a logarithmic scale. The broadening factor of the 𝛿-function is set to 0.01. The peaks around
+𝜔 = ±0.27 correspond to the DWBSs. (d) 𝐴0 dependence of the low-frequency mode in the domain wall. The gray dashed-line 𝜔 = 𝑎𝐴0 + 𝑏
+is calculated from the low-frequency modes by using the least-squares method, where 𝑎 = 4.26, 𝑏 = 6 × 10−4.
+− 𝑖 𝑑
+𝑑𝑡𝒢𝑖𝑥,𝑘𝑦,𝜎; 𝑗𝑥,𝑘′𝑦,𝜎′
+= 𝐽
+∑︁
+𝛿=±1
+�
+(𝑒𝑖𝑘′
+𝑦 𝛿 − 𝑒𝑖𝑘𝑦 𝛿)𝒢𝑖𝑥,𝑘𝑦,𝜎; 𝑗𝑥,𝑘′𝑦,𝜎′ + 𝑒−𝑖𝐴(𝑡) 𝛿𝒢𝑖𝑥,𝑘𝑦,𝜎; 𝑗𝑥+𝛿,𝑘′𝑦,𝜎′ − 𝑒𝑖𝐴(𝑡) 𝛿𝒢𝑖𝑥+𝛿,𝑘𝑦,𝜎; 𝑗𝑥,𝑘′𝑦,𝜎′
+�
++
+∑︁
+𝑞=0,𝜋
+�
+𝑈
+�
+(𝑛 𝑗𝑥,𝑞, ¯𝜎′𝒢𝑖𝑥,𝑘𝑦,𝜎; 𝑗𝑥,𝑘′𝑦+𝑞,𝜎′ − 𝑛𝑖𝑥,𝑞, ¯𝜎′𝒢𝑖𝑥,𝑘𝑦−𝑞,𝜎; 𝑗𝑥,𝑘′𝑦,𝜎′)
++ (𝛿𝜎,↑ − 𝛿𝜎,↓)ΔSC
+𝑖𝑥,𝑞ℱ∗
+𝑖𝑥,−(𝑘𝑦+𝑞), ¯𝜎: 𝑗𝑥,𝑘′𝑦,𝜎′ + (𝛿𝜎′,↑ − 𝛿𝜎′,↓)ΔSC∗
+𝑗𝑥,𝑞ℱ𝑖𝑥,𝑘𝑦,𝜎: 𝑗𝑥,−(𝑘′𝑦+𝑞), ¯𝜎′
+�
++ 𝑉
+∑︁
+𝑠
+∑︁
+ℓ= 𝑗𝑥,𝑖𝑥
+(2𝑒𝑖𝑞𝑛ℓ𝑥,𝑞,𝑠 + 𝑛ℓ𝑥+1,𝑞,𝑠 + 𝑛ℓ𝑥−1,𝑞,𝑠)(𝛿ℓ, 𝑗𝑥𝒢𝑖𝑥,𝑘𝑦,𝜎; 𝑗𝑥,𝑘′𝑦−𝑞,𝜎′ − 𝛿ℓ,𝑖𝑥𝒢𝑖𝑥,𝑘𝑦−𝑞,𝜎; 𝑗𝑥,𝑘′𝑦,𝜎′)
+�
+(A6)
+and
+− 𝑖 𝑑
+𝑑𝑡 ℱ𝑖𝑥,𝑘𝑦,𝜎;𝑗𝑥,𝑘′𝑦,𝜎′
+= −𝐽
+∑︁
+𝛿=±1
+�
+(𝑒𝑖𝑘′
+𝑦 𝛿 + 𝑒𝑖𝑘𝑦 𝛿 − 𝜇/𝐽)ℱ𝑖𝑥,𝑘𝑦,𝜎; 𝑗𝑥,𝑘′𝑦,𝜎′ + 𝑒𝑖𝐴(𝑡) 𝛿(ℱ𝑖𝑥+𝛿,𝑘𝑦,𝜎;𝑗𝑥,𝑘′𝑦,𝜎′ + ℱ𝑖𝑥,𝑘𝑦,𝜎;𝑗𝑥+𝛿,𝑘′𝑦,𝜎′)
+�
++
+∑︁
+𝑞=0,𝜋
+�
+𝑈
+�
+−𝑛𝑖𝑥,𝑞, ¯𝜎ℱ𝑖𝑥,𝑘𝑦−𝑞,𝜎: 𝑗𝑥,𝑘′𝑦,𝜎′ − 𝑛 𝑗𝑥,𝑞, ¯𝜎′ℱ𝑖𝑥,𝑘𝑦,𝜎:𝑗𝑥,𝑘′𝑦−𝑞,𝜎′ − 𝛿𝑖𝑥, 𝑗𝑥𝛿𝑘𝑦+𝑘′𝑦,−𝑞ΔSC
+𝑗𝑥,𝑞
++ΔSC
+𝑗𝑥,𝑞(𝛿𝜎′,↓ − 𝛿𝜎′,↑)𝒢𝑖𝑥,𝑘𝑦,𝜎:𝑗𝑥,−(𝑘′𝑦+𝑞), ¯𝜎′ + ΔSC
+𝑖𝑥,𝑞(𝛿𝜎,↑ − 𝛿𝜎,↓)𝒢𝑗𝑥,𝑘′𝑦,𝜎′:𝑖𝑥,−(𝑘𝑦+𝑞), ¯𝜎
+�
+− 𝑉
+∑︁
+𝑠
+∑︁
+ℓ=𝑗𝑥,𝑖𝑥
+(2𝑒𝑖𝑞𝑛ℓ𝑥,𝑞,𝑠 + 𝑛ℓ𝑥+1,𝑞,𝑠 + 𝑛ℓ𝑥−1,𝑞,𝑠)(𝛿ℓ, 𝑗𝑥ℱ𝑖𝑥,𝑘𝑦,𝜎; 𝑗𝑥,𝑘′𝑦−𝑞,𝜎′ − 𝛿ℓ,𝑖𝑥ℱ𝑖𝑥,𝑘𝑦−𝑞,𝜎; 𝑗𝑥,𝑘′𝑦,𝜎′)
+�
+.
+(A7)
+Note that 𝑘𝑦 − 𝑘′
+𝑦 = 0, 𝜋 (mod 2𝜋).
+Appendix B: Phase mode bound within the domain wall
+In this appendix, we report some results for 𝜔ext = 1.5 which
+is above the CDW gap 𝜔𝑔 = 1.1. Under this condition, we find
+a kind of phase mode within the domain wall. However, the
+incubation time for the first phase rotation depends on the
+system size with a single domain wall, and the phase mode
+does not appear in cases with two or more domain walls.
+These size effects suggest that the phase mode bound in the
+domain wall would not appear in a large system with multiple
+domain walls.
+Figures 5(a) and 5(b) show the case for a single domain wall.
+
+10
+The real and imaginary parts of the ΔSC around the domain
+wall start to oscillate alternately at around 𝑡 ∼ 100 with a
+finite frequency. This mode can be regarded as a phase mode
+bound within the domain wall. The time-dependence of the
+quasiparticle population is shown in Fig. 5(c). The excitation
+to the conduction band is due to the resonant pumping by the
+external field. Interestingly, in addition to the transition from
+the valence continuum to the conduction one, oscillation of
+the population of the DWBSs at 𝜔 ≃ ±0.27 is also observed.
+The frequency of the population dynamics is the same as that
+of the phase mode in the domain wall.
+We investigated the frequency of the above-mentioned phase
+mode as a function of 𝐴0 (Fig. 5(d)). The way of extracting the
+frequency of the mode is briefly explained as follows: There
+are one positive energy and one negative energy bound states
+for each 𝑘𝑦. For each 𝑘𝑦, we extract the time dependence of
+either positive or negative energy of the DWBSs and its popula-
+tion from 𝑡 = 0 to 𝑡 = 500. Next, after averaging over each 𝑘𝑦,
+we plot the 𝐴0 dependence of the low-frequency mode. The
+diamond markers indicate the positions of maximum intensity.
+Figure 5(d) shows a linear relation between the frequency of
+the phase mode and the amplitude 𝐴0, as in the case of the
+Rabi oscillation.
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diff --git a/X9FLT4oBgHgl3EQfUi8y/content/tmp_files/load_file.txt b/X9FLT4oBgHgl3EQfUi8y/content/tmp_files/load_file.txt
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@@ -0,0 +1,1295 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf,len=1294
+page_content='Photoinduced pseudospin-wave emission from charge-density-wave domain wall with superconductivity Yukihiro Matsubayashi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' ∗ Yusuke Masaki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' † and Hiroaki Matsueda1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 3 1Department of Applied Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Tohoku University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Sendai 980-8579,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Japan 2Research and Education Center for Natural Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Keio University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Hiyoshi 4-1-1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Yokohama,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Kanagawa 223-8521,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Japan 3Center for Science and Innovation in Spintronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Tohoku University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 2-1-1 Katahira,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Aoba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Sendai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Miyagi 980-8577 Japan (Dated: January 31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 2023) We study photoinduced dynamics triggered by an inhomogeneity due to competition between charge density waves (CDWs) and superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' As a simple example, we consider the superconducting (SC) interface between two CDW domains with opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The real-time dynamics are calculated within the time- dependent Hartree–Fock–Bogoliubov framework, where the order parameter dynamics and the nonequilibrium quasiparticle distribution functions are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' We also calculate the various dynamical response functions within a generalized random phase approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Through comparisons between the real time dynamics and the analysis of the response functions, it is found that the photo-driven SC interface can emit collective modes of the SC order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' This is analogous to the spin wave emission from the magnetic domain wall in an antiferromagnet, particularly in the case of a low driving frequency, where the order parameters can be mapped onto the pseudospin picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In the high-frequency case, we find a domain wall melting caused by changes in the quasiparticle distribution, which induces superconductivity in the whole system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' INTRODUCTION Recent advances in terahertz laser technologies have en- abled us to access the low-energy scales important for eluci- dating the fundamental properties of solids, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=', the energy scales of phonons, excitons, plasmons, magnons, supercon- ducting (SC) gaps and density waves [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Terahertz lasers can be used to investigate not only the linear response of mat- ter but also the nonlinear response and highly nonequilibrium phenomena thanks to its strong intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' For example, obser- vation of the Higgs mode, which is the amplitude mode of the SC pair potential, has been achieved by non-adiabatic exci- tation of the Bardeen–Cooper–Schrieffer (BCS) ground state with an ultrafast intense terahertz laser [4–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Among the var- ious research topics, optical control of superconductivity has crucial importance for both fundamental physics and techno- logical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Intensive studies have been conducted to enhance the transition temperature 𝑇𝑐 and the SC order [8–16] and to realize SC states different from the equilibrium one, such as photoinduced topological superconductivity [17–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Electronic phases competing with superconductivity present the possibility of indirect photo-control of the SC phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Such orders can be inferred to exist in various materials such as transition metal dichalcogenides [21–25] and cuprate super- condoctors [26–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Numerous experiments and theoretical studies have also indicated the importance of competing mul- tiple orders with inhomogeneity [38–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In addition, it is considered that inhomogeneity acts as a trigger in the initial process of the photoinduced phase transition [42, 46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Fur- ther investigations into the nonequilibrium dynamics of such systems are required in order to pave a way for discovering nontrivial phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' ∗ yukihiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='matsubayashi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='s8@dc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='tohoku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='jp † yusuke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='masaki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='c1@tohoku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='jp The attractive Hubbard model has been widely used to study the nonequilibrium dynamics of superconductors [48– 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' This model has received much attention not only as a simple toy model of superconductivity but also in terms of its realization with ultracold atom [53–57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In this model, the superconductivity and charge density wave (CDW) are degen- erate at half-filling [58–64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Laser control of superconduc- tivity has been proposed for both the attractive and repulsive Hubbard model by suppressing the CDW order and charge in- homogeneity in recent theoretical studies [14, 49, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' These studies highlight the importance of degenerate orders when using laser irradiation to control electronic phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In this paper, we investigate the photoinduced nonequilib- rium dynamics of a non-uniform system composed of SC and CDW orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' To describe such a system, we use the extended attractive Hubbard model (EAHM) with an onsite attractive interaction 𝑈 < 0 and nearest-neighbor repulsive interaction 𝑉 > 0 on a square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The EAHM is known to host a va- riety of exotic SC phases such as 𝑠-wave, 𝑝-wave and 𝑑-wave symmetries [45, 66–68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' At half-filling, because 𝑉 > 0, the CDW order is stabilized as the ground state, and the SC orders are destabilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In order to drive the dynamics of supercon- ductivity, we consider a non-uniform system with an 𝑠-wave SC interface sandwiched by two CDW domains with opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' This is a simple setup where both superconductivity and CDW exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The time evolution in the laser field is calculated within the mean-field approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' We classified our results by the driving frequency of the laser 𝜔ext and the CDW gap 𝜔𝑔: (i) 𝜔ext ≪ 𝜔𝑔 and (ii) 𝜔ext ∼ 𝜔𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In case (i), we find a collective mode emission from the domain wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The col- lective mode can be interpreted as a pseudospin wave, where the pseudospin represents the CDW and SC order parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In case (ii), the CDW domains and the SC domain wall melt over time through the quasiparticle excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' On the other hand, the emission of the SC collective mode from the do- main wall induces superconductivity throughout the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='12049v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='supr-con] 28 Jan 2023 2 Their dynamics are discussed by analyzing the time-dependent quasiparticle population and pair potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' This paper is organized as follows: Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' II introduces the EAHM and the time-dependent calculation method within the Hartree–Fock–Bogoliubov approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' It also introduces the charge and pair correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The correlation func- tions are derived by linear response theory and calculated in the random phase approximation (RPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Section III A explains the non-uniform self-consistent solution and its description in terms of pseudospins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Section III B discusses the collective mode emission from the domain wall under an electric field oscillating at a low frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' III C, we discuss the nonequilibrium dynamics induced by the resonant excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Finally, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' IV presents a summary and discussion of the overall results of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Note that this paper uses the unit ℏ = |𝑒| = 𝑐 = 𝑘B = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' MODEL AND METHODS In this section, we introduce the model Hamiltonian and the mean field approximation with its self-consistent condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Then, we derive a set of equations of motion for the time evolution driven by an applied external field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In subsect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' C, we develop a self-consistent linear response theory, which tells us information about the collective mode of the system from the imaginary part of the dynamical susceptibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Extended attractive Hubbard model As a minimal model of the system with the superconduc- tivity and CDW, we introduce the extended attractive Hubbard model on a square lattice: 𝐻 = ∑︁ 𝑖, 𝑗,𝜎 J𝑖 𝑗𝑐† 𝑖𝜎𝑐 𝑗 𝜎 + 𝑈 ∑︁ 𝑖 𝑛𝑖↑𝑛𝑖↓ + 1 2 ∑︁ 𝑖, 𝑗 𝑉𝑖 𝑗𝑛𝑖𝑛 𝑗 − 𝜇 ∑︁ 𝑖 𝑛𝑖, (1) where 𝑐† 𝑖𝜎 (𝑐𝑖𝜎) is the creation (annihilation) operator at site 𝑖 with spin 𝜎, and 𝑛𝑖 = � 𝜎 𝑛𝑖𝜎 = � 𝜎 𝑐† 𝑖𝜎𝑐𝑖𝜎 is the number operator at site 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Here, 𝑖 identifies the two-dimensional lattice site r𝑖 = (𝑖𝑥, 𝑖𝑦) and so does 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' We focus on the half-filling case by adjusting the chemical potential 𝜇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The first term is the hopping Hamiltonian, where the hopping parameter without any external field is given by J𝑖 𝑗 = 𝐽(< 0) for nearest-neighbor sites𝑖 and 𝑗 and otherwise 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=', J𝑖 𝑗 = 𝐽𝛿|r𝑖−r 𝑗 |,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The second term describes the on-site attractive interaction for 𝑈 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The third term describes the nearest-neighbor interaction for 𝑉𝑖 𝑗 = 𝑉𝛿|r𝑖−r 𝑗 |,1, which lifts the degeneracy between the SC and CDW states [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' When the nearest-neighbor interaction 𝑉 > 0 (𝑉 < 0), the CDW (SC) phase has lower energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' For the total number of the unit cells 𝑁, the Fourier transform 𝑐𝑖𝜎 = 1/ √ 𝑁 � k 𝑒𝑖k·r𝑖𝑐k𝜎 leads to the following representation of the Hamiltonian: 𝐻 = ∑︁ k𝜎 𝜖k𝜎𝑛k𝜎 + 1 2𝑁 ∑︁ k𝜎𝜎′ 𝛿𝜎, ¯𝜎′(𝑈 + 𝑉k)𝜌−k𝜎𝜌k𝜎′, (2) where 𝑛k𝜎 = 𝑐† k𝜎𝑐k𝜎, 𝜌q𝜎 = � k 𝑐† k𝜎𝑐k+q𝜎, 𝜖k = 2𝐽[cos(𝑘𝑥) + cos(𝑘𝑦)], 𝑉k = 2𝑉 [cos(𝑘𝑥) + cos(𝑘𝑦)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The mean-field ap- proximation of the Hamiltonian (1) is performed as follows: The 𝑈 term takes into account the Hartree–Fock terms as well as the anomalous average, that is, the so-called Bogoliubov term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The 𝑉 term takes into account only the Hartree term in order to exclude the bond order wave and the 𝑑-wave super- conductivity for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' By using the mean fields ⟨𝑛𝑖𝜎⟩ and Δ𝑖 ≡ ⟨𝑐𝑖↓𝑐𝑖↑⟩, the interaction terms reduce to ∑︁ 𝑖 (𝑈 ⟨𝑛𝑖 ¯𝜎⟩ − 𝜇)𝑛𝑖𝜎 + 𝑈 ∑︁ 𝑖 (Δ𝑖𝑐† 𝑖↑𝑐† 𝑖↓ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=') + 𝑉 ∑︁ ⟨𝑖, 𝑗⟩ ⟨𝑛𝑖⟩ 𝑛 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (3) Hence, the EAHM can be rewritten in a quadratic form: 𝐻 ≃ �𝐶†𝐻BdG �𝐶, (4) where �𝐶† = (𝑐† 1↑, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' , 𝑐† 𝑁 ↑, 𝑐1↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' , 𝑐𝑁 ↓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The static struc- tures of the mean fields, ⟨𝑛𝑖⟩ and Δ𝑖, and the electron states are determined self consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' From the mean-field Hamil- tonian with a set of the mean fields, we obtain the one particle eigenenergies and eigenstates, which determine a new set of mean fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' We repeat this iterative procedure until the largest error in the updates becomes less than 𝜀err = 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Equation of motion In order to calculate the real-space dynamics, we introduce normal and anomalous density matrices 𝒢𝑖𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 𝑗 𝜎′ = ⟨𝑐† 𝑗 𝜎′𝑐𝑖𝜎⟩, ℱ𝑖𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 𝑗 𝜎′ = ⟨𝑐 𝑗 𝜎′𝑐𝑖𝜎⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The time evolution is calculated on the basis of the equation of motion for the density matrices given by −𝑖 𝑑 𝑑𝑡𝒢 = � J − 𝜌𝑈 − 𝜌𝑉 � 𝒢 − 𝒢 � J − 𝜌𝑈 − 𝜌𝑉 � + Δ𝑈ℱ∗ − ℱΔ𝑈 (5) −𝑖 𝑑 𝑑𝑡 ℱ = � J − 𝜌𝑈 − 𝜌𝑉 � ℱ + ℱ � J ∗ − 𝜌𝑈 − 𝜌𝑉 + 2𝜇𝐼 � + (𝒢 − 𝐼) Δ𝑈 − 𝑡 � 𝒢Δ𝑈� , (6) where the matrix elements are given by J𝑖𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 𝑗 𝜎′ = 𝛿𝜎,𝜎′J𝑖 𝑗, 𝜌𝑈 𝑖𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 𝑗 𝜎′ = 𝑈𝛿𝑖, 𝑗𝛿𝜎,𝜎′𝒢𝑖 ¯𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='𝑖 ¯𝜎, Δ𝑈 𝑖𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 𝑗 𝜎′ = 𝑈𝛿𝑖, 𝑗 (𝑖𝜎𝑦)𝜎,𝜎′ℱ𝑖↑,𝑖↓, and 𝜌𝑉 𝑖𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='𝑗 𝜎′ = 𝛿𝑖, 𝑗𝛿𝜎,𝜎′ � 𝑘 𝜎 𝒢𝑘 𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='𝑘 𝜎𝑉𝑘 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The dimension of each matrix is 2𝑁 × 2𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' A similar formalism is derived in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 50 and 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' We also define a time-dependent distribution function to track the time evolution of the electronic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The 3 Hamiltonian at time 𝑡 can be diagonalized as �𝐶†𝐻BdG(𝑡) �𝐶 = �𝐵†(𝑡)𝐷(𝑡) �𝐵(𝑡), where �𝐵(𝑡) = 𝑈†(𝑡) �𝐶, and 𝐷(𝑡) = diag(𝐸1(𝑡), · · , 𝐸2𝑁 (𝑡)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The 𝜇-th component of �𝐵(𝑡), given by �𝐵𝜇(𝑡) = � 𝑗 [𝑈(𝑡)]∗ 𝑗𝜇 �𝐶 𝑗, stands for the annihilation operator of the quasiparticle with eigenenergy 𝐸𝜇(𝑡) The time-dependent dis- tribution function for each 𝐸𝜇(𝑡), denoted by N𝜇(𝑡), is calcu- lated as N𝜇(𝑡) = � �𝐵† 𝜇(𝑡) �𝐵𝜇(𝑡) � = ∑︁ 𝑖, 𝑗 � ˆ𝑈(𝑡) � 𝑖𝜇 � ˆ𝑈(𝑡) �∗ 𝑗𝜇 � �𝐶† 𝑖 (𝑡) �𝐶 𝑗 (𝑡) � , (7) where ⟨ �𝐶† 𝑖 (𝑡) �𝐶 𝑗 (𝑡)⟩ is calculated from 𝒢(𝑡) and ℱ(𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In addition, the time-dependent pair-potential for each 𝐸𝜇(𝑡), denoted by Δ𝜇(𝑡), is calculated as Δ𝜇(r𝑖, 𝑡) = � ˆ𝑈(𝑡) � 𝑖𝜇 � ˆ𝑈(𝑡) �∗ 𝑖+𝑁 𝜇 � 𝐵† 𝜇(𝑡)𝐵𝜇(𝑡) � (8) This quantity satisfies the following relation: � 𝜇 Δ𝜇(r𝑖, 𝑡) = ⟨𝑐𝑖↓(𝑡)𝑐𝑖↑(𝑡)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Linear response theory The charge and pair correlation functions for an imaginary time 𝜏 are, respectively, defined as Πc(q, q′, 𝜏) = ∑︁ 𝜎𝜎′ Πc,𝜎𝜎′(q, q′, 𝜏), (9) Πc,𝜎𝜎′(q, q′, 𝜏) = − 1 𝑁 � 𝑇𝜏𝜌q𝜎(𝜏)𝜌−q′𝜎′(0) � , (10) ΠSC(q, q′, 𝜏) = − 1 𝑁 � 𝑇𝜏Δq(𝜏)Δ† q′(0) � , (11) where 𝜌q𝜎 = � k 𝑐† k𝜎𝑐k+q𝜎, Δq = � k 𝑐−(k+q)↓𝑐k↑, and 𝑇𝜏 is the imaginary-time ordered product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Their Fourier transform in the frequency domain is given with the bosonic Matsubara frequency 𝜀ℓ = 2𝜋ℓ𝑇 by Πc(SC)(q, q′, 𝑖𝜖ℓ) = ∫ 1/𝑇 0 𝑑𝜏𝑒𝑖𝜖ℓ 𝜏Πc(SC)(q, q′, 𝜏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (12) The Fourier transform of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (10) is defined similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' We calculate these correlation functions within the RPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The RPA takes into account the self-consistent dynamics of the mean fields due to the applied external fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' We formulate the correlation functions within the RPA, by following Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 70 and 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In the following, we construct the RPA formalism in the presence of the CDW order characterized by the order vec- tor Q = (𝜋, 𝜋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' First we should remark on the momenta in the argument of Πc(SC) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' To analyze the excitation structure, we are interested in the diagonal ele- ments Πc(SC) (q, q, 𝑖𝜖ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' However, the (q, q + Q) compo- nent is also taken into account through the intermediate pro- cess of the RPA, as can be seen below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' For this purpose, we introduce a 4 by 4 matrix ˆΠc,q and a 2 by 2 matrix ˇΠSC,q defined as [ ˆΠc,q]q1 𝜎1:q2 𝜎2 = Πc,𝜎1 𝜎2(q1, q2, 𝑖𝜖ℓ) and [ ˇΠSC,q]q1:q2 = ΠSC(q1, q2, 𝑖𝜖ℓ), respectively, where q1,2 takes either q or q + Q ≡ ¯q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The definitions given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (12) are convenient for the following matrix RPA form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In the pres- ence of the CDW order, the following Green’s functions take nonzero values: 𝐺k𝜎(𝜏) = − ⟨𝑇𝜏 𝑐k𝜎(𝜏)𝑐† k𝜎(0)⟩ =: 𝐺k(𝜏) and 𝐷k𝜎(𝜏) = − ⟨𝑇𝜏 𝑐k𝜎(𝜏)𝑐† k+Q𝜎(0)⟩ =: 𝐷k(𝜏), where they are independent of the spin index because the SDW or- der is neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Within the RPA, correlation functions in the matrix form are given by ˆΠRPA c,q (𝑖𝜖ℓ) = � ˆ𝐼 − ˆΠ0 c,q(𝑖𝜖ℓ) · ˆ𝑈c,q �−1 · ˆΠ0 c,q(𝑖𝜖ℓ), (13) ˇΠRPA SC,q(𝑖𝜖ℓ) = � ˇ𝐼 − ˇΠ0 SC,q(𝑖𝜖ℓ) ˇ𝑈SC,q �−1 ˇΠ0 SC,q(𝑖𝜖ℓ), (14) where ˆ𝐼( ˇ𝐼) denotes the 4 by 4 (2 by 2) identity matrix, and ˆ𝑈c,q and ˇ𝑈SC,q are the interaction matrices defined for the basis sets (q ↑, q ↓, ¯q ↑, ¯q ↓) and (q, ¯q), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Their matrix elements are, respectively, defined as ˆ𝑈c,q = ���� � 𝑉q 𝑈 + 𝑉q 0 0 𝑈 + 𝑉q 𝑉q 0 0 0 0 −𝑉q 𝑈 − 𝑉q 0 0 𝑈 − 𝑉q −𝑉q ���� � , (15) ˇ𝑈SC,q = � 𝑈 + 𝑉q 0 0 𝑈 − 𝑉q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (16) In Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (13) and (14), ˆΠ0 c and ˇΠ0 sc are the lowest-order correlation functions that include the Hartree–Fock con- tributions in the single particle Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' By noting that Π0 𝜎𝜎′(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' q′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 𝑖𝜖ℓ) = 𝛿𝜎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='𝜎′Π0 c(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' q′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 𝑖𝜖ℓ) and Π0 c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='SC(q1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 𝑖𝜖ℓ) = Π0 c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='SC(q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' q1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 𝑖𝜖ℓ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' the independent com- ponents can be explicitly written as Π0 c(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 𝑖𝜖ℓ) = 𝑇 𝑁 ∑︁ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='ℓ � 𝐺k(𝑖𝜔𝑛)𝐺k+q(𝑖𝜖ℓ + 𝑖𝜔𝑛) +𝐷k(𝑖𝜔𝑛)†𝐷k+q(𝑖𝜖ℓ + 𝑖𝜔𝑛) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (17) Π0 c(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' ¯q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 𝑖𝜖ℓ) = 𝑇 𝑁 ∑︁ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='ℓ � 𝐺k(𝑖𝜔𝑛)𝐷k+q(𝑖𝜖ℓ + 𝑖𝜔𝑛) +𝐷† k(𝑖𝜔𝑛)𝐺k+q(𝑖𝜖ℓ + 𝑖𝜔𝑛) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (18) Π0 SC(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 𝑖𝜖ℓ) = − 𝑇 𝑁 ∑︁ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='ℓ � 𝐺k(−𝑖𝜔𝑛)𝐺−(k+q) (𝑖𝜖ℓ + 𝑖𝜔𝑛) +𝐷k(−𝑖𝜔𝑛)𝐷−(k+q) (𝑖𝜖ℓ + 𝑖𝜔𝑛) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (19) Π0 SC(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' ¯q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 𝑖𝜖ℓ) = − 𝑇 𝑁 ∑︁ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='ℓ � 𝐺k(−𝑖𝜔𝑛)𝐷−(k+q) (𝑖𝜖ℓ + 𝑖𝜔𝑛) +𝐷k(−𝑖𝜔𝑛)𝐺−(k+q) (𝑖𝜖ℓ + 𝑖𝜔𝑛) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (20) To investigate the excitation structure, we perform an analytic continuation of � 𝜎,𝜎′[ ˆΠRPA c,q (𝑖𝜖ℓ)]q𝜎:q𝜎′ and [ ˇΠRPA SC,q(𝑖𝜖ℓ)]q:q: 𝑖𝜖ℓ → 𝜔 +𝑖𝛿, and describe them as ΠRPA c,q (𝜔) and ΠRPA SC,q(𝜔), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 4 280 285 290 295 300 305 310 315 320 rx 1 5 10 ry (a) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='25 280 285 290 295 300 305 310 315 320 rx −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='50 (b) ∆(r) = ⟨ci↓ci↑⟩ mz(r) = 1 2(� σ⟨niσ⟩ − 1)e−iQ·ri 280 285 290 295 300 305 310 315 320 rx −4 −2 0 2 4 ω (c) 10−3 10−2 10−1 100 10−2 100 DOS (d) ← ← FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (a) Charge density distribution in real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' An interface is located at 𝑟𝑥 = 300 in the system of size 401 × 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (b) Order parameters of SC Δ(r) and CDW 𝑚𝑧(r) along the 𝑥 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The interface is dominated by Δ(r) rather than 𝑚𝑧(r), while in the region far from the interface Δ(r) goes to 0 and 𝑚𝑧(r) is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (c) Local density of states (LDOS) around the domain wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The horizontal axis denotes the spatial direction across the domain wall (𝑟𝑥), while the vertical axis denotes the energy (𝜔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' along the 𝑥 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The intensities of the LDOS are indicated by the color map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (d) Density of states plotted on a logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Arrows indicate the DWBSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The vertical axis is the same as that in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Setup Before showing numerical results based on the above for- mulation, we summarize the parameters and numerical condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' As a unit of energy, we set |𝐽| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In the following numerical results, we use 𝑈 = −2, 𝑉 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The total number of electrons 𝑁e is fixed to 𝑁, corresponding to the half-filling case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' A non-zero 𝑉 lifts the degeneracy between the CDW state and the SC state, and the CDW state is realized as the uniform ground state in the half-filling case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' As an initial state of the time evolution, we consider two CDW domains with opposite signs, which induce the super- conductivity along their interface [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The geometry of the system is an 𝑁𝑥 × 𝑁𝑦 = 401 × 40 site lattice with periodic boundary conditions (PBCs) in both directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' When 𝑁𝑥 is odd, the PBC in the 𝑥 direction naturally introduces an interface along the 𝑦 direction as a self-consistent solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Here, we use the following integer notations 𝑖 = (𝑖𝑥, 𝑖𝑦) = r𝑖 and r = (𝑟𝑥, 𝑟𝑦) interchangeably to represent a site position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Figure 1(a) shows the spatial profile of the charge density � 𝜎(⟨𝑐† 𝑖𝜎𝑐𝑖𝜎⟩−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Figure 1(b) shows the 𝑟𝑥 dependence of the SC order parameter Δ(r) = ⟨𝑐𝑖↓𝑐𝑖↑⟩ using the dashed line, and that of the staggered density 𝑚𝑧(r) = (� 𝜎 ⟨𝑛𝑖𝜎⟩ −1)𝑒𝑖Q·r𝑖/2 using the solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Note that Δ(r) and 𝑚𝑧(r) are uni- form along the 𝑦 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' At the center of the domain wall 𝑟𝑥 = 300, Δ𝑖 reaches a maximum value, and 𝑚𝑧 is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 1(c) and 1(d), we plot the local density of states (LDOS) and density of states (DOS), defined by LDOS(𝑟𝑥, 𝜔) = −1/(𝜋𝑁𝑦) � 𝑟𝑦 Im tr [(𝜔 + 𝑖𝜂 − 𝐻BdG)−1]r,r and DOS(𝜔) = 1/𝑁𝑥 � 𝑟𝑥 LDOS(𝑟𝑥, 𝜔), where tr is the trace in Nambu space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The figures show that the system has a CDW gap 𝜔𝑔 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In (c), there are fermionic states bound in the domain wall indicated by the arrows in (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In this paper, we call them domain-wall bound states (DWBSs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Such a non-uniform structure is related to a magnetic do- main wall [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The low-energy state of the EAHM are described by the order parameters of the CDW and the superconductivity, which can be regarded as an antiferro- magnetic order in a classical pseudospin system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Consider the following map [58, 59, 72]: 𝑆𝑥 𝑗 + 𝑖𝑆𝑦 𝑗 = 𝑐 𝑗↓𝑐 𝑗↑𝑒𝑖Q·r 𝑗, 𝑆𝑧 𝑗 = (𝑛 𝑗↑ + 𝑛 𝑗↓ − 1)/2, the interaction between nearest- neighbor peseudospins are antiferromagnetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The Neél or- der along the 𝑧 direction in pseudospin space corresponds to the CDW, and the Neél order in the 𝑥-𝑦 plane corresponds to uniform superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In the low-energy region, a spatially local order-parameter manifold is constructed on the SO(3) sphere of �𝑆 𝑗𝑒𝑖Q·r 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In the absence of 𝑉, the ground state has SO(3) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' A small 𝑉(> 0) lifts this degen- eracy, so that the Neél order along the 𝑧-axis has a lower energy than that in the 𝑥-𝑦 plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' the north (south) pole de- notes a positive (negative) value of the CDW order parameter 𝑚𝑧(r 𝑗) = ⟨𝑆𝑧 𝑗𝑒𝑖Q·r 𝑗⟩ > 0 (𝑚𝑧(r 𝑗) < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The aforementioned structure can be regarded as a pseudospin antiferromagnetic domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In the domain wall region, antiferromagnetic spin structures with 𝑥 and 𝑦 components ⟨𝑆𝑥,𝑦 𝑗 𝑒𝑖Q·r 𝑗⟩ correspond to the uniform SC structure with U(1) phase degrees of free- dom, Re Δ(r), Im Δ(r) [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The domain-wall width depends on the off-site interaction 𝑉 and the width diverges as 𝑉 → 0, because 𝑉 plays a role of easy-axis anisotropy along the 𝑧-axis in the pseudospin picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' For the above parameter set, the width along the 𝑥 direction in real space is estimated to be 10 sites Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The equation of motion is numerically solved using the fourth-order Runge–Kutta method (RK4) for the time-step Δ𝑡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' By taking advantage of the translational symme- try along the 𝑦 direction Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 1(a), we performed the Fourier transform along the 𝑦 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The details are explained in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In the following subsections, we investigate the photoin- duced dynamics of the above interface structure driven by an oscillating electric field with a frequency 𝜔ext via the Peierls substitution of the vector potential A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The vec- tor potential modifies the hopping amplitude to J𝑖 𝑗 → J𝑖 𝑗 exp[−𝑖A(𝑡) · (r𝑖 − r 𝑗)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' We choose the vector potential to be A(𝑡) = e𝑝 𝐴(𝑡) = e𝑝 𝐴0 sin(𝜔ext𝑡) with amplitude 𝐴0, 5 polarization e𝑝, and driving frequency 𝜔ext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The polarization direction is set as e𝑝 = (1, 0), where we have checked that the 𝑦 component of e𝑝 does not bring about any noticeable dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Pseudospin dynamics: 𝜔ext ≪ 𝜔𝑔 In this subsection, we examine the photoinduced dynamics with driving frequency 𝜔ext = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='1, which is lower than the CDW gap 𝜔𝑔 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In this regime, it is expected that hardly any quasiparticles are excited and the pseudospin picture is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The amplitude 𝐴0 is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The top panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 2 show the real-space and real-time evolutions of the mean field 𝑂(𝑟𝑥, 𝑡) ≡ 𝑁𝑦−1 � 𝑟𝑦 𝑂(r = (𝑟𝑥, 𝑟𝑦), 𝑡), where 𝑂 in each panel is as follows: (a) den- sity deviation from the averaged value 𝑛(r) − 1, (b) (c) the real and imaginary parts of the uniform SC order parameter ΔSC(r) ≡ Δ(r), and (d) (e) the real and imaginary parts of the staggered SC order parameter Δ𝑄 SC(r) ≡ Δ(r)𝑒𝑖Q·r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In panels (b), the SC interface, initially at 𝑟𝑥 = 300, hardly changes its position and the SC phase during the time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The other panels, (a) and (c)–(e), show the polarization of the mean fields in the interface, which represents a deformation of the domain wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' It should be also noted that the uniform and staggered SC components, propagate from the domain wall, as can be seen in panels (b)–(f), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=', the collective mode propagates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The laser irradiation drives the deformation of the domain wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Such an internal deformation generates a pseudospin wave propagating outwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In the region far from the domain wall, the pseudospin wave can be regarded as precessional motion of pseudospins around the 𝑧-axis corresponding to the CDW, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=', excitation of the 𝑥, 𝑦 components corresponding to the SC components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The precession propagates from the interface to the outside, which can be interpreted as the pseudospin wave emission from the domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' This pseudospin wave contains the uniform and staggered components as in the case of the low-energy antiferromagnetic spin waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' It is known that spin waves can also be emitted from a domain wall in the ferromagnetic case and the antiferromagnetic case by using oscillating magnetic fields or spin orbit torques [73, 74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The pseudospin-wave emission in the present study is an analogous to these magnetic systems, but the emission is triggered by the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (As mentioned in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' III A, the domain wall can be interpreted as an antiferromagnetic domain wall with easy-axis anisotropy along the 𝑧-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=') In terms of the collective excitation of the SC order parameter, the propagating wave is a phase rotation of ΔSC with k ∼ (0, 0) and (𝜋, 𝜋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Note that the staggered component in this case can also be regarded as the 𝜂 pairing excitation of the attractive Hubbard model [64, 75–77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The mass of the phase mode is due to the off-site Coulomb interaction 𝑉, whose effects on the collective mode are discussed at the end of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Next, to investigate the emitted collective mode in the uni- form region, we perform a Fourier transform into momentum and frequency spaces: 𝑂(𝑘𝑥, 𝜔) = 𝑁𝑦 ∑︁ 𝑟𝑥 ∫ 𝑇max 0 𝑑𝑡 𝑒𝑖(𝑘𝑥𝑟𝑥−𝜔𝑡)𝑂(𝑟𝑥, 𝑡), (21) where 𝑇max = 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The absolute values of the Fourier compo- nents are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 2(f)–2(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' By applying a laser with zero momentum, the spectral intensities can be acquired not only at k = 0 but also in the finite k region owing to the inho- mogeneity of the domain wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The flat peaks in 𝜔 = 𝜔ext = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='1 in each panel correspond to the forced oscillation by the exter- nal electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Figures 2(g) and 2(h) [2(i) and 2(j)] show the dispersive SC collective modes for 𝜔 ≲ 𝜔𝑔 and 0 ≤ 𝑘𝑥 ≤ 𝜋 with a uniform (staggered) profile along the 𝑦 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Note that using Δ(𝑘𝑥, 𝑘𝑦, 𝜔) = � r ∫ 𝑇max 0 𝑑𝑡𝑒𝑖(k·r−𝜔𝑡)Δ(r), ΔSC(𝑘𝑥) = Δ(𝑘𝑥, 𝑘𝑦 = 0) and Δ𝑄 SC(𝑘𝑥 − 𝜋) = Δ(𝑘𝑥, 𝑘𝑦 = 𝜋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' As a result of the folding of the first Brillouin zone by the CDW order, the induced spectral intensities concentrate around k = (0, 0) and k = (𝜋, 𝜋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In addition, Floquet side bands surround the SC collective mode with energy difference ±𝜔ext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' We have checked that the Floquet side band exists at the other driving frequencies 𝜔ext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Those dispersion can be interpreted as a photon-dressed collective mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' A similar dis- persive branch is observed for 𝜔 ≳ 𝜔𝑔 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 2(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' However, this is not a collective mode of the charge density, but rather part of the continuum spectra of the particle-hole excitation, which is shown more clearly in the following analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In order to analyze the collective modes emitted from the domain wall, we calculated the dynamical charge and pair correlation functions based on the RPA (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (13) and (14)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Figures 3 show the correlation functions as a function of 𝑘𝑥 for 𝑘𝑦 = 0 and those for 𝑘𝑦 = 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The imaginary parts of the dy- namical correlation functions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 3 are in good agreement with the results obtained from the real-space and real-time cal- culations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 2 [the charge density: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 3(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 2(f), the uniform SC pair: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 3(b) and Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 2(g), 2(h), and the staggered SC pair: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 3(c) and Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 2(g), 2(h)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Figure 3(a) shows continuum spectra rather than a collective mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' By contrast, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 3(b) and 3(c) show a collective mode with an energy below the continuum spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' When the inter-site Coulomb interaction𝑉 is zero, because of the SO(3) symmetry of the pseudospin, gapless NG modes appear at q = (0, 0) and (𝜋, 𝜋) [78?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' , 79], just as in the antiferromagnetic case with the Neél order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In the presence of a repulsive 𝑉(> 0), such a collective mode requires a finite excitation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 3(b) and 3(c), the collective modes below the continuum are the massive Nambu–Goldstone (NG) modes, where the term “NG mode” refers due to the propagation of phase rotation of the SC mean fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The lower panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 2 show very small spectral intensities in the region 𝜔 ≥ 1, which indicates that the external field does not excite quasiparticles in the continuum- spectral region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' This is why the pseudospin description works quite well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' We should note that this calculation based on linear response theory cannot reveal any side bands around the collective mode, whereas that Floquet theory could be used to reveal the side band around the SC collective mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' We should also note that the energies of the SC collec- tive mode obtained by the RPA are slightly higher than those 6 1 100 200 300 400 rx 0 50 100 150 200 250 300 350 t (a) n(rx) − 1 −5 0 5 ×10−3 1 100 200 300 400 rx (b) Re ∆SC(rx) −1 0 1 ×10−1 1 100 200 300 400 rx (c) Im ∆SC(rx) −1 0 1 ×10−2 1 100 200 300 400 rx (d) Re ∆Q SC(rx) −5 0 5 ×10−3 1 100 200 300 400 rx (e) Im ∆Q SC(rx) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='5 ×10−4 0 π/4 π/2 3π/4 π kx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='0 ω (f) |n(kx)| 10−2 10−1 100 101 0 π/4 π/2 3π/4 π kx (g) |Re ∆SC(kx)| 10−2 10−1 100 101 102 103 0 π/4 π/2 3π/4 π kx (h) |Im ∆SC(kx)| 10−2 10−1 100 101 102 0 π/4 π/2 3π/4 π kx (i) |Re ∆Q SC(kx − π)| 10−2 10−1 100 101 0 π/4 π/2 3π/4 π kx (j) |Im ∆Q SC(kx − π)| 10−2 10−1 100 101 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (upper panels) Time evolution of order parameters for 𝐴0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='02 and 𝜔ext = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (a) Deviation of the charge density from the averaged value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (b)(c) Uniform and (d)(e) staggered SC order parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' To visualize the propagation of the collective modes, we multiply the actual values in the range 1 ≤ 𝑟𝑥 ≤ 250 by 500 in (a)(b) and 50 in (c)(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (lower panels) Intensities of the Fourier transforms of the corresponding time evolution in real space in the upper row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' observed in the real-space calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' This may have been because we neglected vertex-type diagrams due to 𝑉 for sim- plicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Quasiparticle excitation: 𝜔ext ≳ 𝜔𝑔 In this subsection, we examine the nonequilibrium dynamics at a frequency 𝜔 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='2 near the CDW gap, for which the quasiparticle responses as well as the collective response of the pseudospins are important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The amplitude is 𝐴0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='02, and the system size is 121 × 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Figure 4(a) shows suppression of the CDW order and im- plies a melting of the SC domain wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The dashed line shows the spatial average of the absolute value of the CDW order parameter, which decreases with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In this subsection, we take Δ(𝑟𝑥) to be the 𝑟𝑦-averaged SC mean-field instead of ΔSC(𝑟𝑥) for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' For 𝑡 ≲ 200, the SC amplitude on the SC interface, represented by |Δ(𝑟𝑥,max)|, decreases as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Here, 𝑟𝑥,max denotes the position at which |Δ(𝑟𝑥)| is a maxi- mum, that is, the domain wall center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Furthermore, |Δ(𝑟𝑥,max)| shows a sudden reduction for 𝑡 ∼ 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' By contrast, a gradual increase can be seen in the spatial average of |Δ(𝑟𝑥)| where the domain wall region 85 ≤ 𝑟𝑥 ≤ 95 is excluded, as shown by the dashed-dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In other words, the SC interface struc- ture, represented by the locally strong SC amplitude, melts at around 𝑡 ∼ 200, and the SC order parameter extends to the whole system through propagation of the pseudospin wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Figure 4(b) shows the time evolution of the quasiparticle pop- ulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The line at 𝑡 = 0 represents the initial state, that is, the fully occupied states below 𝜔 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The lines for 𝑡 > 0 show that the laser excitation reduces the CDW gap and increases the quasiparticle population in the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The time evolutions of the energy and spatial-resolved pair- potential amplitude are plotted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 4(c) and 4(d), where we have defined |Δ(𝑟𝑥, 𝑡, 𝜔)| = � 𝜇 |Δ𝜇(𝑟𝑥, 𝑡)|𝛿(𝜔 − 𝐸𝜇) for Δ𝜇(𝑟𝑥, 𝑡) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' First, let us discuss the energy-resolved structure of the domain-wall superconductivity and its time 7 0 π/2 π kx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='0 ω (a) −1/π Im Πc(kx, ky = 0) 0 π/2 π kx (b) −1/π Im ΠSC(kx, ky = 0) 0 π/2 π kx (c) −1/π Im ΠSC(kx, ky = π) 10−2 10−1 10−2 10−1 100 10−2 10−1 100 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Intensity map of the dynamical correlation functions for (a) charge and (b)(c) SC pair potential along the 𝑘𝑥 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The equilibrium state was a uniform CDW order and the system was of size 400 × 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The top panels show the imaginary parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The value of 𝑘𝑦 is shown in each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The energy-resolved pair potential at the domain- wall center 𝑟𝑥 = 𝑟𝑥,max is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' From the data at 𝑡 = 0, the domain wall superconductivity mainly consists of the DWBS at 𝜔 ≃ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='27, which has a sharp and strong peak structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' It also includes a broader, less intense contribution from the continuum states for 𝜔 < −𝜔𝑔/2 ∼ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The following changes occur under an external field at the resonant frequency: (i) the absolute value of the energy of the DWBS decreases, as does their intensities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (ii) the energy distribution of the superconductivity from the DWBSs shifts to the positive side at 𝑡 ∼ 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' This population inversion is attributed to the phase rotation of the superconductivity inside the domain wall, as discussed in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' sAt 𝑡 ≳ 300, the intensities of the DWBSs at the positive and negative energies are comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' By focusing on the phase of Δ(𝑟𝑥,max, 𝑡, 𝜔), however, the two contributions have opposite phase (not shown), which results in Δ(𝑟𝑥,max, 𝑡) having a small amplitude;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (iii) the intensity of the continuum contribution also decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' As a whole, the superconductivity in the domain wall region is reduced by resonant driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Next, let us examine what happens away from the domain wall center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In such a region, the SC order parameter in- creases, as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' This can be seen also in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 4(d), which shows the energy-resolved amplitude of the pair potential averaged over 30 ≤ 𝑟𝑥 ≤ 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In contrast to what is shown in panel (c), the continuum states |𝜔| > 𝜔𝑔/2 mainly contribute to the superconductivity, because the DWBSs are well localized around the domain wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The small contribution at 𝑡 = 0 accounts for the exponential decay of the SC interface, in the region away from the domain wall center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Remark- ably, for 𝑡 ≥ 100 the energy-resolved pair potential increases particularly near the gap edge of the valence band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' This en- hancement is more noticeable for 𝑡 ≳ 300, which is after the drastic reduction of the SC order at the domain wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In summary, in the case of the resonant excitation, the ex- citation of the quasiparticles into the conduction band reduces the CDW gap 𝜔𝑔 (or the gap edge 𝜔𝑔/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The quasiparticles near the gap edge of the valence band contribute to the for- mation of uniform superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' This behavior which 0 100 200 300 400 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='3 (a) |∆(rx,max)| ¯� rx|mz(rx)| 10 × ¯�′ rx|∆(rx)| −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='0 ω 0 2 4 N(t, ω) (b) t =0 t =100 t =200 t =300 t =400 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='0 ω 0 1 2 3 |∆(rx,max, t, ω)| ×10−1 (c) −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='0 ω 0 2 4 ¯�60 rx=30|∆(rx, t, ω)| ×10−3 (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (a) Time evolution of the maximum value of |Δ(𝑟𝑥)| (solid), the averaged CDW order parameter 𝑚𝑧 (dashed), and the averaged SC order parameter in the region far from the domain wall (dashed- dotted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' |Δ(𝑟𝑥)| is maximized at the center of the domain wall, denoted by 𝑟𝑥,max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (b) Time evolution of the quasiparticle population defined by N (𝑡, 𝜔) = � 𝜇 N𝜇(𝑡)𝛿(𝜔 − 𝐸𝜇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Time evolution of the energy resolved pair potential at the domain-wall center, 𝑟𝑥 = 𝑟𝑥,max (c) and far from the domain wall (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The vertical dotted lines are guides for the eye indicating the energies of the largest peaks for the DWBSs and their counterpart with the opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' We define |Δ(𝑟𝑥, 𝑡, 𝜔)| = � 𝜇 |Δ𝜇(𝑟𝑥, 𝑡)|𝛿(𝜔 − 𝐸𝜇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The broadening factor of the 𝛿-function in (b)–(d) is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' We have introduced ¯� to represent the average per site, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=', the summation divided by the number of terms in �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' We use ¯�′ 𝑟𝑥 to denote exclusion of the domain wall region 85 ≤ 𝑟𝑥 ≤ 95 when taking the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' inclues the melting of the domain wall and the appearance of superconductivity in the bulk region, is triggered by the fol- lowing two factors: quasiparticle excitations, which cannot be described by the pseudospin picture, and the existence of the domain wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 8 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' SUMMARY AND DISCUSSION We have considered the laser-induced nonequilibrium dy- namics of the non-uniform system containing an SC domain wall sandwiched between CDW domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' We have found two driving-frequency regimes: (i) when the frequency of the driv- ing laser is below the CDW gap, its dynamics conform to the pseudospin picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' We have found that pseudospin waves are emitted from the domain wall;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (ii) when the frequency of the laser is approximately equal to the CDW gap, excitation of the quasiparticles causes the CDW gap and the SC domain wall to melt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' As a result, we have found that uniform superconductiv- ity is induced in the whole system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Laser control of superconductivity by utilizing a kind of NG mode was suggested in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' This mode corresponds to a rotation of a vector in a plane composed of the SC and CDW order parameters, and can be controlled by tuning the frequency of the laser for 𝑉 = 0 where the SC and CDW orders are degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' However, the time scale of this collective rotation is very slow because the zero mode in equilibrium is utilized and the time scale is determined by an amount of 𝜂 pairing which is weakly excited to hold the pseudospin picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In addition, we have not found such a collective mode in the presence of 𝑉 > 0 even in the uniform case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' that is, the conditions under which it appears are restricted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Even in the presence of 𝑉 > 0, superconductivity may appear as an interface of two opposite CDW domains as pro- posed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' In this case, the oscillatory dynamics may have an analogy with that of the domain wall and the spin wave in the antiferromagnetic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' We have clarified that the emission of the collective mode is possible in a uniformly oscillating electric field owing to the non-uniformity of the domain wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' We have also proposed a possibility of a kind of photoinduced uniform superconductivity via melting of the domain wall and the CDW order that is caused by the reso- nant excitation of the quasiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' These results suggest that the SC and CDW orders can be controlled by resonant laser excitation in a non-uniform CDW system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' ACKNOWLEDGMENTS This work was supported by JST, the establishment of uni- versity fellowships towards the creation of science technol- ogy innovation, Grant Number JPMJFS2102 and JSPS KAK- ENHI, Grants Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' JP19K14662 and JP22H01221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' is supported by KAKENHI grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 21H04446, 21H03455, 21K03380, and 20K03769, and by CSIS, Tohoku University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Appendix A: Fourier transform in the 𝑦 direction In this appendix, we explicitly show the equation of motion after performing a Fourier transform in the 𝑦 direction, which is possible thanks to the translational symmetry along the 𝑦 direction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The Fourier transform of the electron an- nihilation operator at site 𝑖 with spin 𝜎 for the whole Brillouin zone −𝜋 < 𝑘𝑦 ≤ 𝜋 is given as 𝑐𝑖𝜎 = 𝑐𝑖𝑥,𝑖𝑦 𝜎 = 1 √︁𝑁𝑦 ∑︁ 𝑘𝑦 𝑒𝑖𝑘𝑦𝑟𝑦𝑐𝑖𝑥,𝑘𝑦 𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (A1) The normal and anomalous density matrices introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (5) and (6) are transformed as 𝒢𝑖𝑥,𝑘𝑦,𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='𝑖′𝑥,𝑘′𝑦,𝜎′ = 1 𝑁𝑦 ∑︁ 𝑟𝑦,𝑟′𝑦 𝑒𝑖𝑘′ 𝑦𝑟′ 𝑦−𝑖𝑘𝑦𝑟𝑦𝒢𝑖𝑥,𝑖𝑦,𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='𝑖′𝑥,𝑖′𝑦,𝜎′, (A2) ℱ𝑖𝑥,𝑘𝑦,𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='𝑖′𝑥,𝑘′𝑦,𝜎′ = 1 𝑁𝑦 ∑︁ 𝑟𝑦,𝑟′𝑦 𝑒−𝑖𝑘′ 𝑦𝑟′ 𝑦−𝑖𝑘𝑦𝑟𝑦ℱ𝑖𝑥,𝑖𝑦,𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='𝑖′𝑥,𝑖′𝑦,𝜎′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (A3) We introduce the charge density and SC order parameters di- agonalized for 𝑘𝑦: 𝑛𝑖𝑥,𝑞,𝜎 = 1 𝑁𝑦 ∑︁ 𝑘𝑦 � 𝑐† 𝑖𝑥,𝑘𝑦,𝜎𝑐𝑖𝑥,𝑘𝑦+𝑞,𝜎 � = 1 𝑁𝑦 ∑︁ 𝑘𝑦 𝒢𝑖𝑥,𝑘𝑦+𝑞,𝜎:𝑖𝑥,𝑘𝑦,𝜎, (A4) ΔSC 𝑖𝑥,𝑞 = 1 𝑁𝑦 ∑︁ 𝑘𝑦 � 𝑐𝑖𝑥,−(𝑘𝑦+𝑞),↓𝑐𝑖𝑥,𝑘𝑦,↑ � = 1 𝑁𝑦 ∑︁ 𝑘𝑦 ℱ𝑖𝑥,𝑘𝑦,↑:𝑖𝑥,−(𝑘𝑦+𝑞)↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (A5) Even in the nonequilibrium dynamics considered in this paper, 𝑞 takes 0 or 𝜋 owing to the presence of the CDW order Q = (𝜋, 𝜋) and the translational symmetry along the 𝑦 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' As a result, we have 2 × 2 × 𝑁𝑥 mean-fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Note again that we set the polarization of the vector potential as e𝑝 = (1, 0) in order to restrict the photoinduced dynamics to the 𝑥 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The explicit forms of the equations of motion are 9 60 90 120 rx 0 100 200 300 t (a) Re ∆SC(rx) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='1 60 90 120 rx 0 100 200 300 t (b) Im ∆SC(rx) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='0 ω 0 100 200 300 t (c) � µ Nµ(t)δ(ω − Eµ) −2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='0 A0 ×10−2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='15 ω (d) 10 20 30 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Time evolution of the SC order parameter for (a) the real part and (b) the imaginary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (c) Time evolution of the distribution function represented by � 𝜇 N𝜇(𝑡)𝛿(𝜔 − 𝐸𝜇) plotted on a logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The broadening factor of the 𝛿-function is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The peaks around 𝜔 = ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='27 correspond to the DWBSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (d) 𝐴0 dependence of the low-frequency mode in the domain wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The gray dashed-line 𝜔 = 𝑎𝐴0 + 𝑏 is calculated from the low-frequency modes by using the least-squares method, where 𝑎 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='26, 𝑏 = 6 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' − 𝑖 𝑑 𝑑𝑡𝒢𝑖𝑥,𝑘𝑦,𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 𝑗𝑥,𝑘′𝑦,𝜎′ = 𝐽 ∑︁ 𝛿=±1 � (𝑒𝑖𝑘′ 𝑦 𝛿 − 𝑒𝑖𝑘𝑦 𝛿)𝒢𝑖𝑥,𝑘𝑦,𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 𝑗𝑥,𝑘′𝑦,𝜎′ + 𝑒−𝑖𝐴(𝑡) 𝛿𝒢𝑖𝑥,𝑘𝑦,𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 𝑗𝑥+𝛿,𝑘′𝑦,𝜎′ − 𝑒𝑖𝐴(𝑡) 𝛿𝒢𝑖𝑥+𝛿,𝑘𝑦,𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 𝑗𝑥,𝑘′𝑦,𝜎′ � + ∑︁ 𝑞=0,𝜋 � 𝑈 � (𝑛 𝑗𝑥,𝑞, ¯𝜎′𝒢𝑖𝑥,𝑘𝑦,𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 𝑗𝑥,𝑘′𝑦+𝑞,𝜎′ − 𝑛𝑖𝑥,𝑞, ¯𝜎′𝒢𝑖𝑥,𝑘𝑦−𝑞,𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 𝑗𝑥,𝑘′𝑦,𝜎′) + (𝛿𝜎,↑ − 𝛿𝜎,↓)ΔSC 𝑖𝑥,𝑞ℱ∗ 𝑖𝑥,−(𝑘𝑦+𝑞), ¯𝜎: 𝑗𝑥,𝑘′𝑦,𝜎′ + (𝛿𝜎′,↑ − 𝛿𝜎′,↓)ΔSC∗ 𝑗𝑥,𝑞ℱ𝑖𝑥,𝑘𝑦,𝜎: 𝑗𝑥,−(𝑘′𝑦+𝑞), ¯𝜎′ � + 𝑉 ∑︁ 𝑠 ∑︁ ℓ= 𝑗𝑥,𝑖𝑥 (2𝑒𝑖𝑞𝑛ℓ𝑥,𝑞,𝑠 + 𝑛ℓ𝑥+1,𝑞,𝑠 + 𝑛ℓ𝑥−1,𝑞,𝑠)(𝛿ℓ, 𝑗𝑥𝒢𝑖𝑥,𝑘𝑦,𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 𝑗𝑥,𝑘′𝑦−𝑞,𝜎′ − 𝛿ℓ,𝑖𝑥𝒢𝑖𝑥,𝑘𝑦−𝑞,𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 𝑗𝑥,𝑘′𝑦,𝜎′) � (A6) and − 𝑖 𝑑 𝑑𝑡 ℱ𝑖𝑥,𝑘𝑦,𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='𝑗𝑥,𝑘′𝑦,𝜎′ = −𝐽 ∑︁ 𝛿=±1 � (𝑒𝑖𝑘′ 𝑦 𝛿 + 𝑒𝑖𝑘𝑦 𝛿 − 𝜇/𝐽)ℱ𝑖𝑥,𝑘𝑦,𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 𝑗𝑥,𝑘′𝑦,𝜎′ + 𝑒𝑖𝐴(𝑡) 𝛿(ℱ𝑖𝑥+𝛿,𝑘𝑦,𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='𝑗𝑥,𝑘′𝑦,𝜎′ + ℱ𝑖𝑥,𝑘𝑦,𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='𝑗𝑥+𝛿,𝑘′𝑦,𝜎′) � + ∑︁ 𝑞=0,𝜋 � 𝑈 � −𝑛𝑖𝑥,𝑞, ¯𝜎ℱ𝑖𝑥,𝑘𝑦−𝑞,𝜎: 𝑗𝑥,𝑘′𝑦,𝜎′ − 𝑛 𝑗𝑥,𝑞, ¯𝜎′ℱ𝑖𝑥,𝑘𝑦,𝜎:𝑗𝑥,𝑘′𝑦−𝑞,𝜎′ − 𝛿𝑖𝑥, 𝑗𝑥𝛿𝑘𝑦+𝑘′𝑦,−𝑞ΔSC 𝑗𝑥,𝑞 +ΔSC 𝑗𝑥,𝑞(𝛿𝜎′,↓ − 𝛿𝜎′,↑)𝒢𝑖𝑥,𝑘𝑦,𝜎:𝑗𝑥,−(𝑘′𝑦+𝑞), ¯𝜎′ + ΔSC 𝑖𝑥,𝑞(𝛿𝜎,↑ − 𝛿𝜎,↓)𝒢𝑗𝑥,𝑘′𝑦,𝜎′:𝑖𝑥,−(𝑘𝑦+𝑞), ¯𝜎 � − 𝑉 ∑︁ 𝑠 ∑︁ ℓ=𝑗𝑥,𝑖𝑥 (2𝑒𝑖𝑞𝑛ℓ𝑥,𝑞,𝑠 + 𝑛ℓ𝑥+1,𝑞,𝑠 + 𝑛ℓ𝑥−1,𝑞,𝑠)(𝛿ℓ, 𝑗𝑥ℱ𝑖𝑥,𝑘𝑦,𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 𝑗𝑥,𝑘′𝑦−𝑞,𝜎′ − 𝛿ℓ,𝑖𝑥ℱ𝑖𝑥,𝑘𝑦−𝑞,𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 𝑗𝑥,𝑘′𝑦,𝜎′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' (A7) Note that 𝑘𝑦 − 𝑘′ 𝑦 = 0, 𝜋 (mod 2𝜋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Appendix B: Phase mode bound within the domain wall In this appendix, we report some results for 𝜔ext = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='5 which is above the CDW gap 𝜔𝑔 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Under this condition, we find a kind of phase mode within the domain wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' However, the incubation time for the first phase rotation depends on the system size with a single domain wall, and the phase mode does not appear in cases with two or more domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' These size effects suggest that the phase mode bound in the domain wall would not appear in a large system with multiple domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Figures 5(a) and 5(b) show the case for a single domain wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 10 The real and imaginary parts of the ΔSC around the domain wall start to oscillate alternately at around 𝑡 ∼ 100 with a finite frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' This mode can be regarded as a phase mode bound within the domain wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The time-dependence of the quasiparticle population is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 5(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The excitation to the conduction band is due to the resonant pumping by the external field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Interestingly, in addition to the transition from the valence continuum to the conduction one, oscillation of the population of the DWBSs at 𝜔 ≃ ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content='27 is also observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The frequency of the population dynamics is the same as that of the phase mode in the domain wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' We investigated the frequency of the above-mentioned phase mode as a function of 𝐴0 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 5(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The way of extracting the frequency of the mode is briefly explained as follows: There are one positive energy and one negative energy bound states for each 𝑘𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' For each 𝑘𝑦, we extract the time dependence of either positive or negative energy of the DWBSs and its popula- tion from 𝑡 = 0 to 𝑡 = 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Next, after averaging over each 𝑘𝑦, we plot the 𝐴0 dependence of the low-frequency mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' The diamond markers indicate the positions of maximum intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Figure 5(d) shows a linear relation between the frequency of the phase mode and the amplitude 𝐴0, as in the case of the Rabi oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Giannetti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Capone, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Fausti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Fabrizio, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Parmigiani, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Mihailovic, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 65, 58 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Ishihara, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Jpn 88, 072001 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' de la Torre, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Kennes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Claassen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Gerber, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' McIver, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Sentef, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 93, 041002 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
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+page_content=' Matsunaga, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Hamada, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Makise, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Uzawa, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Terai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Wang, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Shimano, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 111, 057002 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
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+page_content=' Tsuji, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Fujita, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Sugioka, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Makise, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Uzawa, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Terai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Aoki, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Shimano, Science 345, 1145 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' [6] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Katsumi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Tsuji, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
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+page_content=' Matsunaga, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Schnee- loch, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Zhong, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Gu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Aoki, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Gallais, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Shi- mano, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 120, 117001 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' [7] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Shimano and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Tsuji, Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Matter Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' 11, 103 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
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+page_content=' Flicker, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
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+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
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+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
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+page_content=' Chan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Gall, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Wurz, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Köhl, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
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+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
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+page_content=' Wurz, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Köhl, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
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+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
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+page_content=' Sugimoto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Yunoki, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Ohta, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Matter 30, 135601 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
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+page_content=' Vanacken, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
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+page_content=' Grilli, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
+page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FLT4oBgHgl3EQfUi8y/content/2301.12049v1.pdf'}
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+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+1
+Structure-Informed Shadow Removal Networks
+Yuhao Liu, Qing Guo, Lan Fu, Zhanghan Ke, Ke Xu, Wei Feng, Ivor W. Tsang, and Rynson W.H. Lau
+Abstract—Shadow removal is a fundamental task in computer vision. Despite the success, existing deep learning-based shadow
+removal methods still produce images with shadow remnants. These shadow remnants typically exist in homogeneous regions with low
+intensity values, making them untraceable in the existing image-to-image mapping paradigm. We observe from our experiments that
+shadows mainly degrade object colors at the image structure level (in which humans perceive object outlines filled with continuous
+colors). Hence, in this paper, we propose to remove shadows at the image structure level. Based on this idea, we propose a novel
+structure-informed shadow removal network (StructNet) to leverage the image structure information to address the shadow remnant
+problem. Specifically, StructNet first reconstructs the structure information of the input image without shadows, and then uses the
+restored shadow-free structure prior to guiding the image-level shadow removal. StructNet contains two main novel modules: (1) a
+mask-guided shadow-free extraction (MSFE) module to extract image structural features in a non-shadow to shadow directional manner,
+and (2) a multi-scale feature & residual aggregation (MFRA) module to leverage the shadow-free structure information to regularize feature
+consistency. In addition, we also propose to extend StructNet to exploit multi-level structure information (MStructNet), to further boost
+the shadow removal performance with minimum computational overheads. Extensive experiments on three shadow removal
+benchmarks demonstrate that our method outperforms existing shadow removal methods, and our StructNet can be integrated with
+existing methods to further boost their performances.
+Index Terms—Single-image shadow removal, Image structure, Structure-level shadow removal, Structure-informed shadow removal
+network.
+!
+1
+INTRODUCTION
+S
+HADOWS exist everywhere. They appear on surfaces
+where light cannot reach, due to occlusions. The pres-
+ence of shadows causes color and texture inconsistency,
+which in turn poses challenges for many downstream tasks,
+e.g., object tracking [1], detection [2], video segmentation [3],
+and face recognition [4]. Faithfully recovering the original
+color and textures of shadow regions helps facilitate the
+above tasks as well as other applications, e.g., game creation
+and relighting. Hence, shadow removal is a long-standing
+problem in computer vision and graphics with many meth-
+ods proposed.
+Conventional shadow removal methods are typically
+based on modeling intensity inconsistency [5] and illumi-
+nation variations [6], or involving user interaction [7]. These
+methods usually fail when the prior assumptions are not
+satisfied, or when the scene colors and textures are intricate.
+In recent years, deep learning-based shadow removal meth-
+•
+Yuhao Liu, Zhanghan Ke, Ke Xu and Rynson W.H. Lau are with the
+Department of Computer Science, City University of Hong Kong. E-
+mail: yuhliu9-c@my.cityu.edu.hk; zhanghake2-c@my.cityu.edu.hk; kkang-
+wing@gmail.com; Rynson.Lau@cityu.edu.hk.
+•
+Qing Guo and Ivor W. Tsang are with the Centre for Frontier AI Research,
+A*STAR, Singapore. E-mail: tsingqguo@ieee.org, ivor tsang@ihpc.a-
+star.edu.sg.
+•
+Lan
+Fu
+is
+with
+the
+InnoPeak
+Technology
+Inc..
+E-mail:
+lan.fu@innopeaktech.com.
+•
+Wei Feng is with the College of Intelligence and Computing, Tianjin
+University, China E-mail: wfeng@ieee.org.
+•
+Yuhao Liu and Qing Guo are the joint first authors.
+•
+Qing Guo and Rynson W.H. Lau are the joint corresponding authors.
+Manuscript received April 19, 2005; revised August 26, 2015.
+ods [8], [9], [10], [11], [12] achieve impressive performances,
+due to the high generalization capability of advanced neural
+networks as well as the availability of large-scale shadow
+removal datasets [9], [11]. These methods typically formu-
+late the shadow removal problem as an image-to-image
+mapping task. For example, Qu et al. [9] and Hu et al.
+[10] use CNNs to extract shadow-related information (i.e.,
+location, appearance, and semantic information) and then
+predict the shadow matte/mask for shadow removal. Fu
+et al. [8] use CNNs to predict exposure parameters and
+then remove shadows by fusing shadow images of different
+exposures. Hu et al. [13] propose a CycleGAN-based method
+to train a shadow removal network using unpaired train-
+ing data. However, these state-of-the-art methods may still
+produce unsatisfactory results with shadow remnants and
+color artifacts. In Fig. 1(a), we can see yellowish shadow
+remnants in the result from AEFNet [8]. These remnants
+are usually internally homogeneous and of low intensity
+values, making them hard to detect by the existing image-
+level shadow removal paradigm represented by [8].
+In this work, we propose to address the shadow remnant
+problem by incorporating the image structure information
+(which consists of object colors and outlines), as shown
+in Fig. 1(b)). The structure of an image is the primary
+information perceived by the human vision system [14],
+[15]. It separates objects into homogeneous regions with
+similar intensities [16]. Hence, we speculate that it should be
+much easier to locate and much cleaner to remove shadows
+in the image structure layer, due to the absence of high-
+frequency texture details. With the recovered shadow-free
+image structure layer as a guidance, it may then be possible
+to perform image-level shadow removal more effectively.
+To verify our idea, we first construct a naive UNet-
+like model that performs structure-level shadow removal
+arXiv:2301.03182v1 [cs.CV] 9 Jan 2023
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+2
+Structure-level
+ShadowRemoval
+Structure Extraction
+Image-level
+ShadowRemoval
+Image-level
+ShadowRemoval
+(a)
+(b)
+Shadow Images
+Shadow Removal
+w.o. Structure
+Shadow Removal
+w. Structure
+Ground Truth
+(e)
+RMSE=18.72
+RMSE=14.69
+RMSE=6.14
+RMSE=5.35
+RMSE=10.01
+RMSE=4.27
+(c) w.o. Structure
+(d) w. Structure
+Fig. 1: (a) State-of-the-art shadow removal methods (such as AEFNet [8] used in here) typically learn an image-level pixel-to-pixel mapping
+directly, and may often produce shadow remnants with color artifacts. (b) We propose to incorporate image structure information into the shadow
+removal process. We visualize the features of approaches (a) and (b) in (c) and (d), respectively, which show that features of our method are
+structured according to region homogeneity, helping remove shadow remnants and their resulting color artifacts (e). Two visual examples of
+original AEF and its structure-enhanced counterpart. Red arrows indicate the region with shadow remnants. The RMSE in the lower right corner
+indicates the error of this sample compared to the shadow-free ground truth.
+and uses the restored shadow-free structure to guide the
+image-level shadow removal process (Fig. 1(b)). With this
+model, we show that structure-level shadow removal can
+help boost the performances of a state-of-the-art shadow
+removal method [8] (Fig. 1(b) and Fig. 1(e)). We visualize
+the feature maps of the traditional approach Fig. 1(a) and
+our proposed approach Fig. 1(b) in Fig. 1(c) and Fig. 1(d),
+respectively. We can see that features in (d) are structured
+based on region homogeneity, which helps alleviate the
+color bleeding artifact of Fig. 1(a). However, we also note
+that the standard convolution used in our naive model (as
+well as in almost all existing methods) adopts spatially-
+shared weights to process both shadow and non-shadow
+regions, and neglects their distinct patterns, resulting in
+color shifts.
+Based on the above analysis, we propose the structure-
+informed shadow removal network (StructNet), which consists
+of the structure-level shadow removal step in stage-1 and
+the image-level shadow removal step in stage-2. We propose
+two novel modules to help remove shadow in the structure-
+level: mask-guided shadow-free extraction (MSFE) module and
+multi-scale feature & residual aggregation (MFRA) module. The
+MSFE module aims to model non-shadow to shadow struc-
+ture information conditioned on the non-shadow regions,
+while the MFRA module focuses on incorporating the ex-
+tracted shadow-free structure information into the shadow
+removal process with feature consistency regularization.
+They can dynamically extracts shadow-free structure in-
+formation and propagates them into shadow-regions for
+shadow removal. We conduct extensive experiments on
+three standard benchmarks to evaluate the performances of
+our method, and show that StructNet outperforms state-of-
+the-art shadow removal methods. Our results also show that
+StructNet can be incorporated into existing fully-supervised
+shadow removal methods to help enhance their perfor-
+mances. Finally, we propose to conduct the shadow removal
+task at multiple structure levels with a single architecture
+(named MStructNet), which is not only efficient but also
+further outperforms state-of-the-art methods. In summary,
+we make the following key contributions:
+• We construct a naive model (i.e., the vanilla UNet)
+for structure-level shadow removal and conduct exten-
+sive empirical studies on it. We show that removing
+shadows at structure-level is more effective than at the
+image-level, and the restored shadow-free structures
+can greatly improve the quality of the output images.
+• We propose the structure-informed shadow removal net-
+work (StructNet), which contains two novel modules for
+structure-level shadow removal: mask-guided shadow-
+free extraction (MSFE) module and multi-scale fea-
+ture & residual aggregation (MFRA) module. MSFE
+learns directional shadow-free structure information
+from non-shadow to shadow regions, while MFRA
+regularizes feature consistency by dynamically fusing
+the output from MSFE with whole image features.
+• We further propose a self-contained shadow removal
+method, multi-level StructNet (MStructNet), which uti-
+lizes multi-level shadow structures at the feature-level
+with low parameters for high-quality shadow removal.
+• Extensive evaluations and ablation studies on three
+public datasets show that the proposed StructNet can
+help enhance the performances of three SOTA methods
+(STCGAN [11], AEF [8], SADC [17]) and MStructNet
+achieves high-quality image restoration, outperforming
+all SOTA shadow removal methods.
+2
+RELATED WORK
+2.1
+Shadow Removal
+Traditional shadow removal methods [18], [19], [20], [21]
+mainly rely on image statistical priors (e.g., image gradients
+
+Information
+南京理三大学
+现众入口
+AudienceEntranc
+BENOA
+训妹馆入口
+Training Hall EntranceInformation
+南京理三大学
+观众入口
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+训练馆入口Information
+南京理三大学
+观众入口
+AudienceEntrance
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+7
+训练馆入口
+Training Halli EntranceJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+3
+and colors). Finlayson et al. [20], [22] solve shadow detection
+and removal via gradient consistency of illumination invari-
+ation. Shor and Lischinki [23] propose an illumination-based
+model in which pixel-wise relationship between shadow
+and shadow-free pixel intensities are modelled with shadow
+parameters. Guo et al. [6] propose a relative illumination
+model based on paired data modelling. However, conven-
+tional methods often do not work well when their hand-
+crafted features do not represent the real-world scenes.
+Deep learning-based approaches bring significant progress
+on the shadow removal task, with the help of large-scale
+datasets [9], [11]. DeShadowNet [9] is the first deep learning-
+based shadow removal method, which models the shadow
+removal as an image-to-image mapping process. It uses
+a multi-branch CNNs to extract multi-level contexts for
+shadow removal. Since then, many methods [10], [11], [24],
+[25], [26], [27], [28] have been proposed following this
+image-to-image mapping paradigm. They focus on design-
+ing intricate network architectures and exploiting distinctive
+properties (e.g., contexts, residuals and illuminations).
+Unsupervised methods [12], [13], [29], [30], [31], [32]
+have also been proposed to alleviate the labeling cost of
+training data. They also fall into the category of image-
+to-image translation by generating pseudo ground truth
+images or using unpaired shadow-free images. Le et al. [33],
+[34] establish a physical shadow formation model and use
+linear illumination transformations to remove shadows. Fu
+et al. [8] formulate shadow removal as an multi-exposure
+fusion problem, which lights the shadow regions by fus-
+ing images of multiple exposures. BEDSR [35] focuses on
+document shadow removal and utilizes document-specific
+priori to develop a background color parameter estimation
+module and a text supplementation module. EMD-Net [36]
+removes shadows by using a shadow illumination model
+and formulates the shadow removal as a variable opti-
+misation process. Although their formulations vary, these
+methods are still based on the image-to-image mapping
+paradigm. Due to the abundant of details in images, such
+a learning paradigm may not remove shadows accurately,
+resulting in shadow remnants and color artifacts.
+In this paper, we propose to exploit image structure
+to constrain the solution space. We demonstrate that our
+structure-to-image hierarchy can help improve the effective-
+ness of the shadow removal process significantly.
+2.2
+Structure-aware Vision Tasks
+The use of structure [14], [15] has been receiving attention
+for its ability to reflect the primary data of the human
+visual system in processing visual signals. Benefited from
+the evolution of two-layer separation [37], [38], structure
+has also been used for other tasks. For example, Ren et
+al. [39] approaches image inpainting from the perspective
+of frequency differences, first using structure to generate
+information such as edges at low frequencies, and then
+supplementing it with high frequency details. In video
+interpolation, Gui et al. [40] develop a structure-to-texture
+strategy by exploiting intermediate structure to maintain the
+smoothness of consecutive frames. Wang et al. [41] introduce
+structure into the cartoon representation to capture global
+structure information and sparse color blocks.
+However, these methods are fundamentally different
+from our approach in the utilisation of structure. Structure-
+aware inpainting [39] and video interpolation [40] rely
+on the structure to provide edge and contour information
+through the available background appearance and two con-
+secutive frames, respectively. In cartoonization [41], on the
+other hand, the structure provides a globally consistent view
+of the entire image that can be processed directly. So, it
+still belongs to image-to-image translation. In this paper,
+we focus on shadow images, where the patterns of shadow
+and non-shadow areas are highly variable and should be
+treated separately rather than uniformly. Meanwhile, the in-
+formation (e.g., color and illumination) in shadow and non-
+shadow regions of an image varies greatly, and restoring
+the shadow regions while ensuring consistency inside and
+outside of the shadow regions may be guided by shadow-
+free clues. Thus, this is a challenge that cannot be directly
+solved by existing structure-aware methods. To the best of
+our knowledge, we are the first to investigate how to utilize
+structure information in the shadow removal task.
+3
+STRUCTURE-LEVEL SHADOW REMOVAL
+In this section, we introduce our structure-level shadow
+removal approach.
+Specifically, we first formulate the
+structure-level shadow removal problem (Sec. 3.1). We
+then investigate the application of structure information in
+shadow removal (Sec. 3.2).
+3.1
+Formulation of Structure-Level Shadow Removal
+In structure-level shadow removal, we first use a struc-
+ture extraction method ϕ(·) to map the input image I ∈
+RH×W ×3 to a structure image, in which image inherent col-
+ors and main outlines are preserved while detailed textures
+are removed (see examples in Fig. 1(b) and Fig. 2), as
+Sl = ϕ(I, l),
+(1)
+where l > 0.0 is a hyper-parameter determining the struc-
+ture level, and Sl ∈ RH×W ×3 is the structure image at the
+lth structure level. A higher l will remove more detailed
+textures (see the first row of Fig. 2). We follow the setups in
+[8], [33] to formulate the lth structure-level shadow removal:
+ˆSl = φl(Sl, M),
+(2)
+where φl(·) is the shadow removal model corresponding to
+Sl, and M ∈ RH×W is a binary mask that indicates shadow
+and non-shadow pixels with 1 and 0, respectively. Note that
+the shadow mask is an input to the shadow removal task.
+The output ˆSl is a shadow-free structure at the lth structure
+level, i.e., the result of structure-level shadow removal.
+3.2
+Empirical Studies
+To study how the structure information affect shadow re-
+moval results, we employ the structure extraction model
+proposed by Xu et al. [42] as ϕ(·). We design a variant
+of vanilla UNet [43], which consists of an encoder with 5
+convolution layers and a decoder with 5 de-convolution
+layers, as φl(·). Each layer in our φl(·) is followed by
+an Instance Norm [44] function and a Leaky-ReLU [45]
+(for the encoder) or ReLU (for the decoder) function. We
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+4
+Structure level=0.0
+Structure level=0.005
+Structure level=0.015
+Structure level=0.045
+Structure level=0.1
+Input Structure
+First-stage
+Shaodw Removal
+Second-stage
+Shadow Removal
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+(g)
+(h)
+(i)
+(j)
+(k)
+(l)
+(m)
+(n)
+(o)
+Fig. 2: Shadow removal results on different image structure levels. The 1st row shows the original shadow image (a) and its structures (b)-(e)
+extracted by [42] at four different structure levels (i.e., l ∈ {0.005, 0.015, 0.045, 0.1}). The 2nd row shows the shadow removal results by feeding
+the shadow structures in the 1st row to respective vanilla UNets. Image (f) represents the result of the image-level shadow removal, while images
+(g)-(j) are the results of structure-level shadow removal with l > 0.0. The 3rd row shows restoration results of our naive two-stage shadow
+removal network by feeding the restored shadow-free structures (i.e., the images at 2nd row) into the second vanilla UNets.
+set the kernel size, padding, and stride of each layer to
+4, 2, and 1, respectively. We provide more details on the
+used models in the Supplemental. Based on the above net-
+work configurations, we aim to answer the following three
+questions: how does the capability of shadow removal
+vary at different structure levels? whether the structure-
+level shadow removal results could guide the image-level
+shadow removal? whether existing model architectures
+are suitable for structure-level shadow removal?
+3.2.1
+Shadow Removal at Different Structure Levels
+Since the shadow removal results may vary at different
+structure levels (lth), we train and test φl(·) at five structure
+levels l ∈ {0.0, 0.005, 0.015, 0.045, 0.1}. Note that l = 0.0 is
+equivalent to image-level shadow removal, i.e., Eq. 1 with
+l = 0.0 is an identity function. To avoid the possible
+influence of elaborately designed loss functions, we only
+optimize the prediction ˆSl = φl(ϕ(I, l), M) via the mean ab-
+solute error L1(ˆSl, S∗
+l ) = ∥ˆSl − S∗
+l ∥1, where S∗
+l = ϕ(I∗, l) is
+the ground truth structure generated from the shadow-free
+image I∗. On the validation set, we calculate the root mean
+square error (RMSE) between ˆSl and S∗
+l after converting
+them into the LAB color space. A smaller RMSE indicates a
+better shadow removal result.
+We conduct evaluations on two widely used datasets,
+including ISTD+ [34] and SRD [9]. Based on the results
+shown in Fig. 3, we observe that the RMSE on shadow
+regions decreases continuously as l increases. This suggests
+that it is easier to obtain high quality shadow removal
+results at the structure level (i.e., l > 0) than at the image
+level (i.e., l = 0). Such a phenomenon is also reflected in
+visual results shown in Fig. 2, in which there are obvious
+artifacts in the image-level shadow removal result (Fig. 2(f)),
+but such artifacts are greatly reduced at the structure-level
+shadow removal results (Fig. 2(g)-(j)). The RMSE curves
+ISTD+
+SRD
+Structure level (l)
+Structure level (l)
+RMSE
+RMSE
+Fig. 3: Comparison of the image-level (i.e., l = 0.0) and four structure-
+level shadow removal process with l ∈ {0.005, 0.015, 0.045, 0.1} on
+two public datasets (i.e., ISTD+ [34] and SRD [9]). We employ the root
+mean square error (RMSE) in the LAB color space as metric to evaluate
+the shadow-removal performances in the non-shadow regions, shadow
+regions, and the whole (i.e.All) image, respectively.
+in the non-shadow regions descend at the beginning then
+become flat when l increases to reach a certain level. The
+RMSE curves of the whole images have similar shapes to
+those of non-shadow regions. For non-shadow regions of
+the ISTD+ dataset, l = 0.1 has even worse RMSE than that
+of l = 0.045 (Fig. 3 left).
+These experiments show that a higher structure level l
+generally facilitates shadow removal by making the shadow
+removal network focus more on color and structure infor-
+mation instead of texture information. However, if l is too
+large, it may lead to shadow spreading, i.e., similar shadow
+visual patterns may appear in neighboring non-shadow
+regions (see Fig. 2(e)), which in turn causes a higher error in
+the non-shadow regions.
+3.2.2
+Shadow Removal with Structure-level Guidance
+We would like to investigate if structure-level shadow re-
+moval is beneficial to image-level shadow removal. We
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+5
+(c)
+(b)
+(d)
+(e)
+(f)
+(a)
+(a). Shadow Structure
+(b). Results of Vanilla UNet
+(c). Features in Vanilla UNet
+(d). Results of StructNet
+(f). Quantitative Comparision
+(e). Features in StructNet
+Fig. 4: Visualization and quantitative comparison of vanilla UNet and StructNet for structure-level shadow removal. (a) is the input shadow
+structure, which is fed to the vanilla UNet and StructNet to obtain (b) and (d), respectively. Images (c) and (e) show the randomly sampled three
+feature channels produced by the 2nd convolutional layer of the two networks. In addition, we also extract the features from the 2nd convolution
+layer of the vanilla UNet and StructNet of all images in the ISTD+ test set. For each image, we calculate the absolute difference between the
+shadow and non-shadow regions in each feature channel and obtain the average difference across all channels. Image (f) shows the average
+feature differences of all images using the vanilla UNet (green points) and StructNet (blue points).
+formulate a two-stage pipeline of which the first stage com-
+bines Eq. 1 and 2 for structure-level shadow removal. The
+second stage uses a new model ψl(·), which additionally
+takes the predicted shadow-free structures ˆSl as inputs for
+image-level shadow removal, as:
+ˆIl = ψl(I, ˆSl, M),
+(3)
+where ˆIl denotes the image-level shadow removal results
+guided by ˆSl. Theoretically, ψl can be an arbitrary image-
+level shadow removal method (e.g., ST-CGAN [11] or AEF
+[8]). For simplicity, we simply assume ψl to have the same
+architecture as φl. When training the pipeline corresponding
+to l ∈ {0.0, 0.005, 0.015, 0.045, 0.1}, we fix φl optimized in
+Sec. 3.2.1 and learn ψl. We apply the same L1 loss function
+and RMSE metric as in Sec. 3.2.1.
+Table 1 shows the results (on the ISTD+ dataset) of the
+two-stage shadow removal pipeline with different l and a
+single-stage image-level shadow removal model. Note that
+the pipeline with l = 0.0 can be regarded as a stack of
+two vanilla UNet models for image-level shadow removal.
+We observe that two-stage image-level shadow removal
+does not yield better performance, compared to single-stage
+image-level shadow removal, and shadow remnants cannot
+be eliminated by simply adding more CNN parameters as
+shown in Fig. 2(a,f,k); two-stage shadow removal with
+l > 0.0 achieves lower RMSE than that of image-level
+shadow removal (either two-stage shadow removal with
+l = 0.0 or single-stage shadow removal), which shows that
+the restored shadow-free structures can help image-level
+shadow removal. We also observe from the results in Fig. 2
+that the artifacts in Fig. 2(f) (i.e., the result of single-stage
+image-level shadow removal) are eliminated by the two-
+stage shadow removal with l > 0.0, as shown in Fig. 2(l-o)).
+3.2.3
+Limitations of using the Vanilla UNet
+In Sec. 3.2.2, we have demonstrated that the structure-level
+shadow removal results can benefit image-level shadow
+removal to some degree. Here, we would like to know if
+the vanilla UNet is good enough for this two-stage shadow
+removal (i.e., first at structure-level and then at image-
+level). We observe that the standard convolution operations
+TABLE 1: Comparison between direct single-stage image-level shadow
+removal and four variants of two-stage structure-level shadow removal.
+All experiments are conducted on the ISTD+ dataset with the vanilla
+UNet and L1 loss function.
+Structure level l
+for the first stage
+Shadow ↓
+Non-shadow ↓
+All ↓
+Two-stage
+shadow removal
+0.0
+6.28
+2.99
+3.53
+0.005
+5.98
+2.55
+3.11
+0.015
+5.89
+2.49
+3.05
+0.045
+6.17
+2.57
+3.16
+0.1
+6.15
+2.56
+3.15
+Single-stage Image-level
+shadow removal
+6.33
+2.78
+3.36
+used in the vanilla UNet process shadow and non-shadow
+regions uniformly, and ignore the distinctions between them
+(e.g., color-bias). In other words, the standard convolution
+used in the vanilla UNet attempts to map shadow and non-
+shadow regions that have very different appearances to the
+same pattern, which makes the learning of the convolution
+weights challenging.
+As a result, the vanilla UNet may
+produce obvious color shifts between shadow and non-
+shadow regions in the output image, as shown in Fig. 4(b).
+To support the above analysis, we visualize three ran-
+domly selected feature channels of the 2nd convolution
+layer in the vanilla UNet in Fig. 4(c). We can see that
+the features of the shadow and non-shadow regions show
+obvious divergences, although we expect them to be consis-
+tent in order to recover colors and textures of the shadow
+regions. We further conduct a quantitative experiment on
+the test set of ISTD+. For each sample, we first extract the
+features output by the 2nd convolution layer 1 of the vanilla
+UNet. We then compute the means of it feature maps in the
+shadow and non-shadow regions separately, and show the
+absolute difference between the two with a single point in
+Fig. 4(f). We can see that there are huge differences between
+shadow and non-shadow regions in the feature space. Such
+feature differences are caused by the uniform processing
+of standard convolutions used in the vanilla UNet. As a
+result, the vanilla UNet produces results with color shift.
+This motivates us to design a novel solution to overcome
+1. The difference between shadow and non-shadow regions in deeper
+layers is minimal and indistinguishable. Thus, we choose conv2.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+6
+(a)
+MSFE
+Conv.
+MFRA
+Dalition Conv.
+Constant Conv. for Mask
+MSFE( )
+(c)
+.
+(d)
+ MFRA( )
+...
+.
+...
+Image-level shadow removal
+(b)
+I
+I
+ Structure-level
+shadow removal
+B
+B
+B
+1
+B
+w1
+w2
+wS
+Fig. 5: Pipeline of the proposed StructNet. (a) shows the structure-level shadow removal. (b) shows the image-level shadow removal with the
+assistance of predicted shadow-free structure from (a). (c) shows the architecture of the mask-guided shadow-free extraction (MSFE) module. (d)
+shows the multi-scale feature & residual aggregation (MFRA) module for the fusion function.
+the problems of applying the vanilla UNet to structure-level
+shadow removal.
+4
+STRUCTNET
+In this section, we propose a novel two-stage model, named
+structure-informed shadow removal network (StructNet), to
+better utilize the structure-level shadow removal results to
+guide the image-level shadow removal step. StructNet con-
+tains two novel designs, including a mask-guided shadow-
+free extraction (MSFE) module as detailed in Sec. 4.1, and a
+multi-scale feature & residual aggregation (MFRA) module
+as detailed in Sec. 4.2. The configuration details of StructNet
+are described in Sec. 4.3.
+As discussed in Sec. 3.2.3, using a vanilla UNet in the
+first stage cannot differentiate between shadow and non-
+shadow regions properly, which in turn can exacerbate the
+differences between them and lead to artifacts, as shown in
+Fig. 4. To address such a spatial-invariant problem caused
+by the standard convolution operations in the vanilla UNet,
+we propose to make the convolution layer shadow-aware.
+Normally, pixels in the shadow regions are not only associ-
+ated with their neighboring pixels, but also with the distant
+non-shadow pixels of the same characteristic. Hence, we
+propose to add a directional bridge to the standard con-
+volution operations, which are guided by the non-shadow
+regions to make the elements of the shadow features similar
+to those of the non-shadow features. Specifically, given the
+input features Xj
+in ∈ RHj
+in×W j
+in×Cj
+in at the jth layer, we
+propose to process the features as:
+Xj
+out = Fusion(Xj
+in ∗ Wj, Bj),
+(4)
+where Xj
+out
+∈
+RHj
+out×W j
+out×Cj
+out are the output features,
+Wj are the learnable weights, and Bj ∈ RHj
+out×W j
+out×Cj
+out
+is a learned shifting tensor aiming to reduce the feature
+difference between the non-shadow and shadow regions.
+Fusion(·) is a function to fuse the shifting information in Bj
+and the features Xj
+in ∗ Wj effectively, thus regularizing the
+output features to be consistent between the shadow and
+non-shadow regions. Bj is computed by Bridge(·), as:
+Bj = Bridge(Xj
+in, Bj−1, Mj
+in),
+(5)
+where Mj
+in ∈ RHj
+in×W j
+in is a binary map that indicates the
+shadow regions with 1’s and non-shadow regions with 0’s.
+Bj−1 is the shifting tensor of the previous layer. Intuitively,
+Bridge(·) is trained to extract directional shifting from non-
+shadow to shadow regions. Note that such a solution has
+two benefits: The advantages of the standard convolution
+are preserved via Eq. 4, which can extract perception across
+the whole scene/image; The potential shifting between
+shadow and non-shadow regions is supplemented via Eq. 5.
+With the above formulation, we propose the structure-
+informed shadow removal network (StructNet), as shown
+in Fig. 5. StructNet consists of two stages. The first stage
+performs structure-level shadow removal, while the second
+stage conducts image-level shadow removal guided by the
+results from the first stage. In the first stage, we propose the
+two novel modules, i.e., MSFE and MFRA, to extensively
+exploit the structure information. As for the second stage, it
+can be one existing supervised shadow removal method.
+4.1
+The MSFE Module
+Inspired by the segmentation-aware convolution [46], we
+propose to embed the shadow mask in the convolution
+operation explicitly and formulate Bridge(·) as:
+Bj[p] = αp
+�
+q∈Np
+Bj−1[q](1 − Mj
+in[q])Wj
+R[q − p],
+(6)
+where Wj
+R ∈ RKj×Kj×Cj
+in×Cj
+out are the weights of a convo-
+lution layer, p and q are the coordinates of elements in Xj
+in,
+Mj, Bj, or Wj
+R. The set Np contains neighboring elements
+of p, and its size is equal to the kernel size of Wj
+B (i.e., K2).
+The normalization term αp is defined as
+1
+�
+q∈Np Mj
+in[q]. The
+mask (i.e., Mj
+in) is obtained by convoluting the mask from
+the previous layer (i.e., Mj−1
+in
+) with a constant weight (i.e.,
+W1 whose elements are one) through Mj
+in = Mj−1
+in
+∗ Wj
+1.
+Intuitively, with Eq. 6, the output tensor Bj is only depen-
+dent on the non-shadow regions due to the guidance of the
+mask Mj and is able to fill the gap across shadow and non-
+shadow regions.
+As the examples shown in Fig. 6, the features of the
+whole scene from the standard convolution (i.e., Xj
+in ∗ Wj)
+present obvious shadow regions while the shifting (i.e., Bj)
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+7
+B
+Fig. 6: Feature visualization of the global perception (Xj
+in ∗ Wj), the
+offset (Bj), and output features by fusing the former two (Xj
+out).
+contains the shifting information predicted from the non-
+shadow regions. As a result, compared with Xj
+in ∗ Wj, the
+final fusion result (i.e., Xj
+out) shows similar and consistent
+appearances across shadow and non-shadow regions. In
+addition, instead of training the weight Wj
+B for all examples,
+we propose to make it dynamically changed according to
+different input features, i.e., Wj
+R = η(Xj
+in), where η(·) is
+a sub-network having two convolution layers. Specifically,
+we set the stride of the first layer as 2, the kernel with the
+size of Kj, which halves the spatial resolution of Xj
+in and
+maintains low computation cost. The output size becomes
+Hj
+in
+2
+× W j
+in
+2
+× Cj
+in. For the second layer, we set the size of
+convolution kernel as 1∗1, which produces a tensor with
+size Hj
+in
+2 × W j
+in
+2 ×
+�
+Cj
+in × Kj × Kj�
+, and is then reshaped to
+produce the weight Wj
+R.
+Our method is similar to the partial convolution [47] but
+has the following differences: The convolution weights of
+the proposed bridge function are conditional on the input
+whole scene features, while those of the partial convolution
+are fixed after training; The operations with Eq. 4 and
+Eq. 5 are a combination of standard and dynamic partial
+convolution. The former aims to extract the perception of
+the whole image, while the latter is to bridge the shifting
+between shadow and non-shadow regions.
+4.2
+The MFRA Module
+Beyond the element-wise additive fusion of the convolu-
+tional perception (i.e., Xj
+in ∗ Wj) and the shifting Bj, we
+propose to conduct the fusion at multiple scales with a
+dynamic aggregation module. The main motivation stems
+from the fact that the desired output features should be
+consistent across shadow and non-shadow regions, and then
+each desired element in the output features is dependent on
+the context of the shifting bridge and the convolutional per-
+ception around its position. However, the effective context
+range for different elements and examples are different, and
+how to make the fusion adaptive to this change is critical.
+Specifically, given the convolutional perception (i.e., Xj
+in∗
+Wj) and the shifting Bj in Eq. 4, we first conduct multiple
+atrous convolutions with different dilation rates to obtain
+multi-scale features, i.e.,
+Xj
+s = σ([Xj
+in ∗ Wj, Bj] ∗ Dj
+s),
+(7)
+where σ(·) is the ReLU function, Dj
+s is the weight of an
+atrous convolution with dilation rate s. Here, we consider
+s ∈ S = {1, 24, 12, 6} and get the first three features
+{Xj
+s|s ∈ {1, 24, 12}} via Eq. 7. The size of the weights of
+all Dj
+s is 3 × 3 × Cj
+in × Cj
+out with strides {1, 1, 2, 2}. For the
+last and smallest scale features (i.e., s = 6), we do not extract
+from Xj
+in like Eq. 7, but feed Xj
+12 to a dilation convolution
+to obtain Xj
+6 (see Fig. 5(d)). Such an implementation can
+alleviate heavy information loss caused by directly down-
+sampling the input features two times, and reduce the
+computation cost. Then, the key problem is how to combine
+the four sets of features according to different inputs. To this
+end, we propose to estimate the combination parameters
+dynamically according to the inputs, i.e.,
+Xj
+out =
+S
+�
+s
+wj
+s ⊙ Xj
+s, with
+{wj
+s|s ∈ S} = Φ([Xj
+in ∗ Wj, wj]),
+(8)
+where wj
+s ∈ RHj
+in×W j
+in×Cj
+in assigns weights for each element
+in Xj
+s. Note that the elements at the same positions but dif-
+ferent channels share the same weights. Φ(·) is a subnetwork
+containing two convolution layers and a softmax layer. Each
+convolution is followed by a ReLU layer.
+4.3
+Configuration Details
+In this subsection, we detail the configuration of the first
+stage of StructNet, which contains three branches. The first
+branch takes the shadow structure (i.e., Sl) and the shadow
+mask (i.e., M) as inputs, and aims to estimate the shadow-
+free structure. Its encoder contains five convolutional layers.
+It receives shifting predicted by the second branch and con-
+ducts fusion. The second branch is to estimate the shifting
+(i.e., Bj) for the convolutional layer in the first branch. Each
+layer of the second branch (e.g., the jth layer) takes the
+shifting from the previous layer (e.g., Bj−1), the shadow
+mask from the third branch (i.e., Mj
+in), and the jth features
+from the first branch (i.e., Xj) as inputs. The third branch is
+to produce the binary masks for all layers along the encoder.
+The ten convolution layers (i.e., 5 for the encoder and 5
+for the decoder) in the first branch share the same settings
+with the vanila UNet in Sec. 3.2. In terms of the fusion
+function (i.e., MFRA), we set the kernel size of all convo-
+lutional layers to 3, and the number of kernels/filters (i.e.,
+Cj
+out) is {64, 128, 256, 512, 512} except for the second layer
+of weight generation (i.e., Φ(·)) where the number of kernels
+is equal to the number of parallel branches (i.e., 4). For
+the second branch, we have a total of five convolutional
+layers, each corresponding to one of the encoder layers in
+the first branch and having the same strides and number
+of kernels as the encoder layers. The kernel size of each
+convolutional layer (i.e., Kj) is set to {7, 5, 3, 3, 3}, and
+each layer is followed by a Batch-Norm (except for the first
+layer) and a ReLU function. For the third branch, we fix
+the constant convolutional kernels Wj
+1 of size Kj and with
+stride 2. As for the production of mask Mj, when the kernel
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+8
+MSFE
+Conv.
+MFRA
+Constant Conv. for Mask
+(a)
+(b)
+Block l
+Structure-aware
+encoder
+Fusion-oriented
+encoder
+Decoder
+B
+Fig. 7: Pipeline of the multi-level StructNet (MStructNet). (a) presents
+the whole pipeline, while (b) shows the detail of the blue blocks in (a).
+Wj
+1 is sliding on Mj−1
+in , the updated mask value of that
+window is set to 0 in Mj
+in if the convolved value is smaller
+than the sum of kernels Wj
+1; otherwise, it is set to 1. In
+addition, as shown in Table 1, the maximum performance
+gain is delivered when the structure level is 0.015, which
+also coincides with the observation of trade-off between the
+smoothness and performance of two-stage shadow removal.
+Thus, unless otherwise stated, we assume l = 0.015 in the
+proposed StructNet.
+5
+MULTI-LEVEL STRUCTNETS (MSTRUCTNET)
+Although our StructNet presented in Sec. 4 is able to re-
+store the shadow structure effectively and performs better
+than the vanilla UNet, benefiting the image-level shadow
+removal step significantly, such a two-stage solution leads
+to large computational overheads due to the naive com-
+bination of two networks. To address this problem, we
+further propose a self-contained shadow removal method
+that utilizes multi-level structures at the feature level with
+only a small increase in the parameter numbers. Specifically,
+we omits the step for predicting the shadow-free structure
+image through the first stage of StructNet, but use the
+non-shadow structure information directly. We refer to this
+method as MStructNet. Fig. 7 shows the pipeline.
+5.1
+Pipeline
+Given a shadow image I, we extract structures via Eq. 1 and
+consider four levels, i.e., l ∈ L = {0.005, 0.015, 0.045, 0.1}.
+Thus, we obtain four levels of structure, {Sl|l
+∈ L}.
+MStructNet takes the original shadow image, all the struc-
+tures, and the shadow mask as inputs to predict the shadow-
+free image directly. The whole pipeline contains three com-
+ponents, i.e., structure-aware encoder, fusion-oriented en-
+coder, and decoder. The structure-aware encoder contains
+|L| blocks to address |L| structures. Note that each block
+follows the design of StructNet and makes the shadow
+elements of the output features similar to the shadow-
+free elements under the guidance of the input shadow
+image and structures. The fusion-oriented encoder consists
+of standard convolutions and is to further extract deep
+feature embedding from the structure-aware features. The
+decoder is to map the feature embedding to the shadow-
+free image. We show the whole pipeline in Fig. 7(a). As the
+main difference between this pipeline and the vanilla UNet
+lies in the structure-aware encoder as shown in Fig. 7(b), we
+discuss the design of this encoder in detail below.
+In terms of the lth block in the structure-aware encoder,
+we have the original shadow image I, the lth structure Sl,
+and shadow mask M as inputs. We then feed them to the
+block having two convolutional layers equipped with the
+proposed MSFE and MFRA modules (see Fig. 7(b)), which
+produce two features denoted as X1
+l and X2
+l corresponding
+to the outputs of the first and second convolution layers, re-
+spectively. For the four levels of structures (i.e., {Sl|l ∈ L}),
+we obtain eight output features, i.e., {X1
+l }l∈L and {X2
+l }l∈L.
+We combine the four sets of features in {X1
+l }l∈L or {X2
+l }l∈L
+via an element-wise additive operation. The combined fea-
+tures are fed to the fusion-oriented encoder and decoder to
+estimate the shadow-free image ˆI.
+5.2
+Configuration Details
+Same as the first stage of StructNet in Sec. 4, each block l
+in the structure-aware encoder in MStructNet also has three
+branches. Take the lth block as an example, the inputs to
+the first branch include shadow image I, shadow structure
+Sl with level l and shadow mask M. The three inputs
+are concatenated along the channel axis and further fed
+to the standard convolution to perceive the global scene.
+The inputs to the second branch are shadow structure Sl
+and shadow mask M. Then, with the global perceptual
+features of the first branch as the guiding weights, the
+shifting features can be obtained by Eq. 6. The third branch
+updates the shadow mask Mj in the same way as in Sec. 4.3.
+Regarding the fusion-oriented encoder and the decoder,
+they contain only standard convolutional layers, instance-
+norm and activation function (e.g., Leaky-ReLU or ReLU),
+and all the settings are the same as those in the vanilla UNet
+in Sec. 3.2.
+6
+EXPERIMENT
+6.1
+Implementation Details
+6.1.1
+Loss Functions
+Instead of using just the L1 loss function in Sec. 3.2 as the re-
+construction loss, we add the perceptual loss to train Struct-
+Net and MStructNet for high restoration quality. Specifically,
+given a restored image ˆI and its ground truth I∗, we have
+L(ˆI, I∗) = λ1L1(ˆI, I∗) + λ2Lperc(ˆI, I∗),
+(9)
+where L1(ˆI, I∗) is the L1-norm distance to ensure pixel-
+level visual consistency. Lperc is the perceptual loss [48],
+which aims to ensure the restored image to have the same
+perception as the ground truth. It is formulated as:
+Lperc(ˆI, I∗) =
+3
+�
+i=1
+∥VGG16i(ˆI) − VGG16i(I∗)∥1,
+(10)
+where VGG16i(·) represents the activation map of the ith
+max-pooling layer in the VGG16 [49] pretrained on the
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+9
+ImageNet [50]. The two trade-off parameters in Eq. 9, i.e.,
+λ1 and λ2, ensure the numerical and gradient equivalence
+during training, and we empirically set λ1 = 1 and λ2 = 0.1
+for all the experiments on StructNet and MStructNet.
+We employ L(ˆI, I∗) to end-to-end train MStructNet
+directly. For StructNet, we use the same loss but with
+< ˆSl, S∗
+l > to train the first stage, i.e., L(ˆSl, S∗
+l ). After that,
+we fix the parameters of the first-stage network and use
+L(ˆI, I∗) to train the second-stage network.
+6.1.2
+Training and Testing Configurations
+We have implemented the proposed methods and all vari-
+ants in PyTorch on a single NVIDIA TITAN GPU with 12G
+memory. We optimize all networks by Adam [51] optimizer
+with the initial learning rate, β1 and β2 as 0.0002, 0.9, and
+0.999, respectively. We adopt the warmup for the first 2k it-
+erations and cosine decay strategy [52] to adjust the learning
+rate during training. For the training schedule, we train the
+first-stage network of StructNet for 100k iterations and the
+second-stage network for another 100k iterations to reach
+convergence on all datasets. For MStructNet, we train it for
+300k iterations to reach convergence for all datasets. Besides,
+we perform data augmentation to prevent over-fitting for
+both methods, which includes random left-right flipping,
+random rotation of angles ranging between -20o to 20o,
+random cropping, and resolution adjustment (i.e., all inputs
+are resized to 256×256). We then conduct a normalization to
+transform the inputs into a range of -1 to 1, and feed them
+to the networks. During inference, for a fair comparison and
+accommodating the hardware limitation (i.e., GPU memory
+usage), we follow [8], [11], [30], [53] and directly resize the
+inputs to 256×256, and obtain the prediction. The batch
+sizes for training and testing are 6 and 1, respectively.
+6.2
+Method Settings
+6.2.1
+Baseline Methods
+Baseline methods enhanced by StructNet. Our StructNet
+proposed in Sec. 4 is able to enhance existing shadow
+removal methods by first conducting structure-level shadow
+removal, and then regarding the restored shadow-free struc-
+ture as an auxiliary prior for the baseline method to predict
+the shadow-free image in the second stage. We use four
+baseline methods for comparison: the vanilla UNet in Sec. 3,
+and three state-of-the-art methods, i.e., STCGAN [11], AEF
+[8] and SADC [17]. We regard these baseline methods as the
+second-stage networks in StructNet, resulting in four vari-
+ants. Note that we select these three SOTA methods as they
+have very different frameworks, which can demonstrate the
+high extensibility and flexibility of StructNet.
+For the three SOTA baseline methods to receive the
+structure input from the first stage, we need to make small
+modifications to them. (The modification of the vanilla UNet
+is already discussed in Sec. 3.2.2.)
+• STCGAN [11]: It takes the shadow image as input,
+and performs shadow detection and removal with two
+successive sub-networks. It can be formulated as:
+ˆI = Γrem (Γdet (I) , I) ,
+(11)
+where Γdet(·) and Γrem(·) refer to the shadow detec-
+tion and removal sub-networks, respectively. We first
+use Γdet (I) to get the shadow mask M and compute
+the structure Sl of I. We then feed M and Sl to
+our structure-level shadow removal (i.e., first stage of
+StructNet) to obtain ˆSl. We insert ˆSl to Γrem(·) by:
+ˆI = Γrem
+�
+Γdet (I) , I, ˆSl
+�
+(12)
+• AEF [8]: It concatenates shadow image and shadow
+mask at inputs (i.e., I and M) and conducts image-level
+shadow removal first by automatically adjusting the
+illumination of shadow region via multiple exposure
+combinations. It then performs boundary refinement
+(Γedge). It can be formulated as:
+ˆI = Γedge
+�Γexp (I, M) , I, M
+� ,
+(13)
+where
+Γexp(·)
+denotes
+the
+exposure-fusion-based
+shadow removal function. To enhance AEF with our
+StructNet, we plug in the predicted shadow-free struc-
+ture ˆSl as an input to Γexp(·). The updated method is
+formulated as:
+ˆI = Γedge
+�
+Γexp
+�
+I, M, ˆSl
+�
+, I, M
+�
+,
+(14)
+• SADC [17]: It uses a two-branch network to extract fea-
+tures of shadow image I and mask M separately, and
+then decodes them to predict the shadow-free image.
+For simplicity, we formulate it as:
+ˆI = Γrem (Γfeat (I) , M) ,
+(15)
+where Γfeat(·) is the pre-trained VGG16 [49]. It is to
+extract the input image features. We implement SADC-
+based StructNet by embedding the restored structure ˆSl
+to Eq. 15, as:
+ˆI = Γrem
+�
+Γfeat (I) , M, ˆSl
+�
+.
+(16)
+Based on above modifications, we obtain three vari-
+ants of StructNet denoted as StructNet-STCGAN, StructNet-
+AEF, and StructNet-SADC, respectively. Note that the three
+versions share the same structure-level shadow removal
+network. We fix the first-stage network and retrain only the
+second-stage networks. Since the code and model weights of
+ST-CGAN are not publicly available, we re-implement this
+method and re-train it with the same training strategy and
+hyper-parameters. For AEF, we use the publicly released
+model weights for the ISTD and ISTD+ datasets. We strictly
+follow their original settings and parameters to train the en-
+hanced model that incorporates the predicted shadow-free
+structure. SADC only provides model weights for ISTD. For
+ISTD+, we strictly follow its code to train the corresponding
+models (i.e., original and enhanced ISTD+ models).
+Baseline methods for comparison. To further demon-
+strate the advantages of the structure-informed shadow
+removal networks, we compare the StructNet variants and
+MStructNet with state-of-the-art methods including Guo et
+al. [6], Gong et al. [7], DeshadwoNet [9], STCGAN [11],
+DSC [10], Mask-ShadowGAN [13], AR-GAN [53], SP+M-
+Net [33], CLA-GAN [25], RIS-GAN [26], Param+M+D-Net
+[12], DHAN [24], G2R [29], AEF [8], DC-ShadowGAN [30],
+SP+M+I-Net [34], BMNet [27], SADC [17], EMD-Net [36],
+and SG-ShadowNet [28]. Since the settings of different
+methods (e.g., test environment and inference resolution)
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+10
+TABLE 2: Validation results of StructNet-equipped shadow removal methods on ISTD and ISTD+ datasets. We embed four existing models, i.e.,
+vanilla UNet, STCGAN [11], AEF [8] and SADC [17], in our StructNet framework as four variants, and compare them with the original methods.
+Datasets
+Methods
+RMSE ↓
+PSNR ↑
+SSIM ↑
+Shad.
+No.Shad.
+All
+Shad.
+No.Shad.
+All
+Shad.
+No.Shad.
+All
+ISTD+ [33]
+vanilla UNet
+5.89
+2.49
+3.05
+37.86
+38.15
+34.34
+0.990
+0.985
+0.970
+StructNet-UNet
+5.31
+2.52
+2.97
+38.43
+37.33
+34.31
+0.990
+0.979
+0.963
+STCGAN [11]
+9.39
+4.25
+5.09
+35.09
+33.92
+30.36
+0.983
+0.961
+0.937
+StructNet-STCGAN
+6.25
+3.58
+4.02
+37.44
+34.52
+32.00
+0.988
+0.968
+0.949
+AEF [8]
+6.55
+3.77
+4.23
+36.04
+31.16
+29.44
+0.978
+0.892
+0.861
+StructNet-AEF
+6.35
+3.75
+4.17
+36.08
+31.18
+29.52
+0.978
+0.892
+0.861
+SADC [17]
+6.21
+3.05
+3.57
+37.18
+37.69
+33.88
+0.991
+0.982
+0.968
+StructNet-SADC
+5.82
+2.83
+3.32
+37.92
+37.72
+34.26
+0.991
+0.983
+0.969
+ISTD [11]
+vanilla UNet
+7.29
+4.73
+5.09
+35.69
+31.70
+29.74
+0.987
+0.970
+0.951
+StructNet-UNet
+6.33
+4.71
+4.98
+36.60
+31.57
+29.94
+0.988
+0.970
+0.952
+STCGAN [11]
+10.11
+5.76
+6.47
+33.93
+30.18
+27.90
+0.981
+0.959
+0.932
+StructNet-STCGAN
+7.52
+5.64
+5.95
+35.46
+30.52
+28.75
+0.985
+0.961
+0.939
+AEF [8]
+7.98
+5.54
+5.94
+34.39
+28.61
+27.11
+0.974
+0.880
+0.844
+StructNet-AEF
+7.49
+5.67
+5.97
+34.72
+28.09
+26.86
+0.975
+0.880
+0.844
+SADC [17]
+7.19
+5.06
+5.41
+35.52
+31.97
+29.85
+0.989
+0.976
+0.961
+StructNet-SADC
+6.83
+4.69
+5.04
+36.40
+32.27
+30.32
+0.989
+0.978
+0.963
+may not be the same, we list the ways that we obtain
+the quantitative results (a.k.a., metrics values) for a better
+evaluation, as follows:
+△ - We retrain their models strictly according to their
+hyper-parameters, but at a resolution of 256, and then
+compute the metrics.
+♦ - We down-sample their publicly released predictions to
+a resolution of 256, and then compute the metrics.
+⋆ - We replicate their reported data from their papers.
+6.2.2
+Datasets
+We conduct our experiments on three shadow removal
+benchmark datasets, i.e., SRD [9], ISTD [11] and ISTD+ [34],
+to evaluate the effectiveness of the proposed methods.
+SRD [9] is the first large-scale shadow removal dataset,
+consisting of 3,088 paired shadow and shadow-free images,
+of which 2,680 are for training and 408 for testing. Since
+shadow masks are not available in SRD, we follow AEF
+[8] to utilize the Otsu’s algorithm to extract the shadow
+masks from the difference between the shadow and shadow-
+free images. We adopt the extracted masks for network
+usage (i.e., training and testing), and use the available masks
+from DHAN [24] for metric evaluation. ISTD [11] is another
+benchmark dataset that comprises 1,870 triplets (i.e., shadow
+image, shadow mask and shadow-free image). This dataset
+is collected under 135 scenes and split into 1,330 for training
+and 540 triplets for testing. As the images in ISTD suffers
+from the color consistency problem in the non-shadow re-
+gions, Le et al., [33] corrected this problem to form the ISTD+
+dataset, which has the same data setting as the original ISTD
+dataset. For ISTD and ISTD+ datasets, we follow AEF [8]
+to use the ground-truth shadow masks for training, and
+extracted masks from Otsu’s algorithm for testing.
+6.2.3
+Evaluation Metrics
+We follow methods [8], [10], [12], [34] to compute the root
+mean square error (RMSE) between shadow-removed im-
+age and ground-truth shadow-free image in the LAB color
+space, which is also named as image-level RMSE. When
+evaluating structure-level shadow removal, we compute
+RMSE between the predicted shadow-free and ground truth
+structures as described in Sec. 3.2, which is denoted as
+structure-level RMSE. Following [27], [29], we also report
+the peak signal-to-noise ratio (PSNR) and structural similar-
+ity index (SSIM) to measure the visual quality scores. Note
+that all metrics are computed in the shadow region (Shad.),
+non-shadow regions (No. Shad.), and the whole image (All),
+respectively.
+6.3
+Evaluation of StructNet
+In this section, we validate the effectiveness of StructNet
+in Sec. 6.3.1 by using it to enhance four existing baseline
+methods and comparing with their original versions. We
+then conduct extensive ablation studies in Sec. 6.3.2 and
+Sec. 6.3.3 to analyze and validate the effectiveness of the
+proposed MSFE and MFRA modules.
+6.3.1
+Validation Results
+Quantitative comparison. With the implementations dis-
+cussed in Sec. 6.2.1, we evaluate four StructNet variants
+(i.e., StructNet-UNet/-STCGAN/-AEF/-SADC) on ISTD+
+and ISTD, and compare them with their original versions.
+We show the results in Table 2. We can see that the proposed
+StructNet improves all four baselines with a significant mar-
+gin on RSME in the shadow region over the two datasets.
+For example, the RMSE of STCGAN decreases from 9.39 to
+6.25 (an improvement of 33.4%) on ISTD+, and from 10.11
+to 7.52 (an improvement of 25.6%) on ISTD, which leads to
+obvious RMSE reduction in the whole image. As StructNet-
+UNet obtains the best results across all counterparts, for
+convenience, we refer to StructNet-UNet as StructNet in all
+subsequent experiments.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+11
+TABLE 3: Quantitative comparison with the SOTA methods on the ISTD dataset. The symbol to the left of each method indicates the source of
+results (refer to end of Sec. 6.2.1). STCGAN is re-implemented and trained by us, and we mark it as ‘△’. ‘-’ indicates values that are not available.
+The best and second best results are highlighted in red and green respectively.
+Methods
+RMSE ↓
+PSNR ↑
+SSIM ↑
+# Params
+(M:106)
+# FLOPs
+(G: 109)
+Shad.
+No.Shad.
+All
+Shad.
+No.Shad.
+All
+Shad.
+No.Shad.
+All
+⋆Guo et al. [6]
+18.65
+7.76
+9.26
+27.76
+26.44
+23.08
+0.964
+0.975
+0.919
+-
+-
+△STCGAN [11]
+10.11
+5.76
+6.47
+33.93
+30.18
+27.90
+0.981
+0.959
+0.932
+29.24
+17.88
+♦Mask-ShadowGAN [13]
+10.57
+5.91
+6.67
+31.73
+29.02
+26.36
+0.980
+0.959
+0.928
+11.38
+56.83
+♦DSC [10]
+8.45
+5.03
+5.59
+34.64
+31.26
+29.00
+0.984
+0.969
+0.944
+22.30
+123.47
+♦DHAN [24]
+7.49
+5.30
+5.66
+35.53
+31.05
+29.11
+0.988
+0.971
+0.954
+21.75
+262.87
+⋆AR-GAN [53]
+7.21
+5.83
+6.68
+-
+-
+-
+-
+-
+-
+-
+-
+⋆RIS-GAN [26]
+8.99
+6.33
+6.95
+-
+-
+-
+-
+-
+-
+-
+-
+⋆CLA-GAN [25]
+9.01
+6.25
+6.62
+-
+-
+-
+-
+-
+-
+-
+-
+⋆CANet [54]
+8.86
+6.07
+6.15
+-
+-
+-
+-
+-
+-
+-
+-
+△AEF [8]
+7.98
+5.54
+5.94
+34.39
+28.61
+27.11
+0.974
+0.880
+0.844
+143.01
+160.32
+♦DC-ShadowGAN [30]
+10.55
+5.79
+6.57
+31.69
+28.99
+26.38
+0.976
+0.958
+0.922
+21.16
+105.00
+⋆BMNet [27]
+7.60
+4.59
+5.02
+35.61
+32.80
+30.28
+0.988
+0.976
+0.959
+0.37
+10.99
+△SADC [17]
+7.19
+5.06
+5.41
+35.52
+31.97
+29.85
+0.989
+0.976
+0.961
+16.7
+81.92
+⋆SADC [17]
+6.64
+4.97
+5.22
+36.68
+30.81
+29.21
+0.991
+0.972
+0.960
+⋆EMD-Net [36]
+8.29
+4.55
+5.09
+36.95
+31.54
+29.85
+0.987
+0.978
+0.960
+-
+-
+StructNet
+6.33
+4.71
+4.98
+36.60
+31.57
+29.94
+0.988
+0.970
+0.952
+71.64
+48.23
+MStructNet
+6.34
+4.35
+4.68
+36.85
+32.49
+30.65
+0.989
+0.972
+0.955
+21.47
+32.72
+We also compare StructNet (i.e., StructNet-UNet) with
+the state-of-the-art methods in Table 3, Table 4, and Table 5.
+We can see from Table 3 and Table 4 that the proposed
+StructNet outperforms all baseline methods on both ISTD+
+and ISTD, achieving the lowest RMSE, demonstrating the
+advantages of our structure-informed approach. In partic-
+ular, on ISTD as shown in Table 3, StructNet obtains 6.33
+in the shadow regions, an improvement of 12.0% over the
+second best method (i.e., SADC△). Although StructNet has
+larger model size with more parameters (i.e., 71.64 M), its
+FLOPs (i.e., 48.23 G) is much lower than the baseline meth-
+ods (i.e., SADC and AEF). Although Mash-ShadowGAN has
+a smaller model size, it requires more computations for the
+cycle process and thus has a higher FLOPS. AEF involves
+multiple sub-networks, and hence more parameters and
+higher FLOPS. SADC, which processes features at high
+resolution by reducing the number of downsampling steps.
+Hence, the FLOPS is increased.
+6.3.2
+Effectiveness of the MSFE Module
+Here, we study the effectiveness of the mask-guided shadow-
+free extraction (MSFE) module on the ISTD+ dataset. Based
+on StructNet (i.e., StructNet-UNet), we construct different
+StructNet variants by using different structure-level shadow
+removal networks. We then evaluate the quality of the
+restored structures (i.e., structure-level RMSEs) from the first
+stage as well as the quality of the restored images (i.e.,
+image-level RMSEs) from the second stage.
+Adding MSFE to a single convolution layer in the
+vanilla UNet. We add the MSFE module to different convo-
+lution layers in the encoder of the vanilla UNet for structure-
+level shadow removal. To avoid the influence of the fusion
+function carried out by the MFRA module, we replace it
+with a naive element-wise additive operation instead. We
+TABLE 4: Quantitative comparison with the SOTA methods on the
+ISTD+ dataset. ‘-’ indicates values that are not available. The best and
+second best results are highlighted in red and green, respectively.
+Method \ RMSE ↓
+Shad.
+No.Shad.
+All
+Input Images
+40.2
+2.6
+8.5
+⋆Guo et al. [6]
+22.0
+3.1
+6.1
+⋆Gong et al. [7]
+13.3
+-
+-
+△STCGAN [11]
+9.4
+4.3
+5.1
+⋆SP+M-Net [33]
+7.9
+3.1
+3.9
+⋆Param+M+D-Net [12]
+9.7
+3.0
+4.0
+⋆G2R (w sup) [29]
+7.3
+2.9
+3.6
+⋆AEF [8]
+6.5
+3.8
+4.2
+⋆DC-ShadowGAN [30]
+10.3
+3.5
+4.6
+⋆SP+M+I-Net [34]
+6.0
+3.1
+3.6
+⋆BMNet [27]
+6.1
+2.9
+3.5
+△SADC [17]
+6.2
+3.1
+3.6
+⋆SG-ShadowNet [28]
+5.9
+2.9
+3.4
+StructNet
+5.3
+2.5
+3.0
+MStructNet
+5.3
+2.7
+3.1
+use StructNet(MSFE, j, Add) to denote the StructNet whose
+first-stage network uses the MSFE as Bridge() at the jth layer
+and the element-wise additive operation as Fusion(). We
+then obtain five variants, i.e., {StructNet(MSFE, j, Add)|j ∈
+{1, 2, 3, 4, 5}}. Table 6 shows the results. From these re-
+sults, we observe: Compared with the naive two-stage
+shadow removal method (i.e., vanilla UNet), StructNets with
+a single MSFE achieves lower structure-level and image-
+level RMSEs (i.e., StructNet(MSFE, 1/2/3/4/5, Add) in Ta-
+ble 6 vs. two-stage shadow removal in Table 1) in the
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+12
+TABLE 5: Quantitative comparison with the SOTA methods on the SRD
+dataset. ‘-’ indicates values that are not available. The best and second
+best results are highlighted in red and green, respectively.
+Method \ RMSE ↓
+Shad.
+No.Shad.
+All
+⋆Guo et al. [6]
+29.89
+6.47
+12.60
+⋆DeShadowNet [9]
+11.78
+4.84
+6.64
+⋆DSC [10]
+10.89
+4.99
+6.23
+⋆Mask-ShadowGAN [13]
+-
+-
+7.32
+⋆AR-GAN [53]
+7.24
+4.71
+5.74
+⋆DHAN [24]
+8.39
+4.67
+5.46
+⋆RIS-GAN [26]
+8.22
+6.05
+6.78
+⋆CLA-GAN [25]
+8.10
+6.01
+6.59
+⋆AEF [8]
+8.56
+5.75
+6.51
+⋆DC-ShadowGAN [30]
+7.70
+3.39
+4.66
+⋆CANet [54]
+7.82
+5.88
+5.98
+⋆BMNet [27]
+6.61
+3.61
+4.46
+△SADC [17] (256)
+9.68
+4.45
+5.89
+⋆EMD-Net [36]
+7.44
+3.74
+4.79
+⋆SG-ShadowNet [28]
+7.53
+2.97
+4.23
+StructNet
+6.93
+3.94
+4.81
+MStructNet
+6.69
+4.28
+4.97
+shadow regions, which demonstrates that the MSFE does
+benefit the structure-level shadow removal and enhance the
+image-level shadow removal. In general, if we embed
+MSFE in a deeper convolution layer, we get lower RMSEs
+in the shadow regions while higher RMSEs in the non-
+shadow regions at the structure level. For example, the
+structure-level RMSE of the shadow region decreases from
+5.10 to 4.73 if we add MSFE from the 1st to the 5th layers.
+We have similar observations on the image-level RMSEs.
+The above observations hint that bridging the difference
+between shadow and non-shadow regions at higher level
+helps enhance restoration quality of shadow regions but
+slightly harms the restoration quality of non-shadow re-
+gions.
+Adding MSFE to multiple convolutions in the vanilla
+UNet. We further add MSFE to more layers to study
+the change in RMSEs. As shown in Table 6, we de-
+note the variant as StructNet(MSFE, (1 · · · 5), Add), where
+we add MSFE to all convolutions between the 1st layer
+and 5th layer. Compared with StructNet(MSFE, 5, Add),
+StructNet(MSFE, (1 · · · 5), Add)
+has
+a
+lower
+structure-
+level RMSE (i.e., 1.87) in the non-shadow regions but
+a
+slightly
+higher
+structure-level
+RMSE
+(i.e.,
+4.82)
+in
+the shadow regions. The overall RMSE becomes 2.35,
+which is smaller than that of StructNet(MSFE, 5, Add).
+In
+contrast,
+compared
+with
+StructNet(MSFE, 1, Add),
+StructNet(MSFE, (1 · · · 5), Add) has a much lower structure-
+level RMSE in the shadow regions and the same RSME
+in the non-shadow regions. Such observations imply that
+equipping more convolutions with MSFE can balance the
+restoration quality in the non-shadow and shadow regions.
+Comparison with the convolutional skip function. To
+validate the necessity of MSFE, we implement a variant,
+i.e., StructNet(CONV, (1 · · · 5), Add), which formulates the
+fusion function as an additive operation and the bridge as a
+convolution layer like the convolutional skip connection in
+the residual network [55], i.e.,
+Xj
+out = Xj
+in ∗ Wj + Bj, where
+Bj = Conv([Xj
+in, Bj−1, Mj
+in]),
+(17)
+However,
+such
+a
+solution
+is
+not
+easy
+to
+address
+the
+limitations
+of
+the
+standard
+convolution.
+With
+skip
+connections,
+although
+the
+spatial
+information
+is
+embedded via the mask Mj
+in, skip connections with the
+convolution layer inherit the spatial-invariant property
+and cannot address the shadow and non-shadow shifting
+explicitly. Similarly, the element-wise addition neglects the
+relationship between different shadow and non-shadow
+regions and does not benefit the shifting prediction.
+As reported in Table 6, StructNet(CONV, (1 · · · 5), Add)
+obtains lower RMSEs in the shadow regions, which
+shows that it also benefits shadow removal. However,
+compared
+with
+StructNet(MSFE, (1 · · · 5), Add),
+StructNet(CONV, (1 · · · 5), Add) has much higher RMSEs
+in both shadow and non-shadow regions, even leading to
+a higher RMSE in “All” than the vanilla UNet (i.e., 2.48 vs
+2.46). This demonstrates the advantages of our MSFE.
+Model size comparison. Compared with the naive two-
+stage shadow removal, other variants have similar model
+size (See the “Params” column in Table 6) since the em-
+bedded module (e.g., MSFE or CONV) does not introduce a
+large amount of parameters. Hence, the improvement with
+MSFE mainly lies in the utilization of shadow-free structure
+information rather than the increase of model parameters.
+Note that the computation of Params and FLOPs covers
+two sub-networks, structure-level shadow removal at stage-
+1 and image-level shadow removal at stage-2. By default,
+we resort to THOP2 to conduct the calculation.
+6.3.3
+Effectiveness of the MFRA Module
+Adding MFRA to the convolution in the MSFE-based
+UNet. To validate the effectiveness of the MFRA module,
+we replace the element-wise additive fusion (i.e., “Add”)
+of the variants in Table 6 (i.e., StructNet(MSFE, ⋆, Add))
+with
+the
+proposed
+MFRA
+to
+obtain
+new
+variants,
+StructNet(MSFE, ⋆, MFRA), where ‘⋆’ denotes specific layer
+indexes used by StructNet. We show the results in Table 6.
+We have the following observations: All single-MSFE-
+based variants with MFRA (i.e., StructNet(MSFE, ⋆, MFRA))
+outperform the variants with the element-wise addition
+operation, which demonstrates that the proposed aggre-
+gation function does enhance shadow removal signifi-
+cantly. For example, StructNet(MSFE, 2, MFRA) achieves
+4.32 structure-level RMSE in the shadow regions, outper-
+forming StructNet(MSFE, 2, Add) by 10.0%. In addition, we
+also see improvements on image-level RMSEs. When
+we embed MSFE to multiple convolutions with MFRA,
+we find that StructNet(MSFE, (1 · · · 5), MFRA) achieves
+much better restoration quality in both shadow and non-
+shadow regions than StructNet(MSFE, (1 · · · 5), Add). As a
+result, in “All”, StructNet(MSFE, (1 · · · 5), MFRA) obtains
+2.12 on structure-level RMSE and 2.97 on image-level
+RMSE, which are 9.7% and 3.3% higher than those of
+StructNet(MSFE, (1 · · · 5), Add).
+2. https://github.com/Lyken17/pytorch-OpCounter
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+13
+TABLE 6: Comparison between StructNet variants. The comparisons are conducted on the ISTD+ dataset from two aspects, i.e., structure-level
+and image-level shadow removal. We denote all variants with StructNet(Factor1, Factor2, Factor3) where ‘Factor1’ represents the function for the
+Bridge(·), ‘Factor2’ means the positions to embed the ‘Factor1’, and ‘Factor3’ is the function for the Fusion(·) in Eq. 4.
+Methods for the first stage
+Structure-level RMSE ↓
+Image-level RMSE ↓
+# Params
+(M:106)
+# FLOPs
+(G:109)
+Shad.
+No.Shad.
+All
+Shad.
+No.Shad.
+All
+Two-stage shadow removal
+in Table 1 with l = 0.015
+5.54
+1.86
+2.46
+5.89
+2.49
+3.05
+39.59
+37.44
+StructNet(MSFE, 1, Add)
+5.10
+1.87
+2.41
+5.72
+2.59
+3.10
+39.60
+37.59
+StructNet(MSFE, 2, Add)
+4.80
+1.88
+2.36
+5.55
+2.59
+3.07
+39.80
+38.43
+StructNet(MSFE, 3, Add)
+4.76
+2.01
+2.46
+5.56
+2.58
+3.07
+40.62
+39.27
+StructNet(MSFE, 4, Add)
+4.73
+2.04
+2.49
+5.62
+2.61
+3.11
+41.80
+39.57
+StructNet(MSFE, 5, Add)
+4.74
+1.97
+2.43
+5.55
+2.57
+3.06
+44.16
+39.72
+StructNet(MSFE, 1, MFRA)
+4.82
+1.88
+2.36
+5.68
+2.59
+3.09
+39.78
+39.60
+StructNet(MSFE, 2, MFRA)
+4.32
+1.92
+2.31
+5.39
+2.59
+3.05
+40.54
+40.43
+StructNet(MSFE, 3, MFRA)
+4.55
+2.01
+2.43
+5.54
+2.58
+3.07
+43.57
+41.27
+StructNet(MSFE, 4, MFRA)
+4.65
+1.88
+2.33
+5.57
+2.52
+3.02
+53.60
+41.57
+StructNet(MSFE, 5, MFRA)
+4.58
+1.92
+2.35
+5.43
+2.52
+3.00
+55.97
+40.22
+StructNet(CONV, (1 · · · 5), Add)
+5.10
+1.97
+2.48
+5.82
+2.61
+3.14
+46.53
+39.38
+StructNet(MSFE, (1 · · · 5), Add)
+4.82
+1.87
+2.35
+5.50
+2.60
+3.07
+44.16
+39.72
+StructNet(MSFE, (1 · · · 5), MFRA)
+4.20
+1.71
+2.12
+5.31
+2.52
+2.97
+71.64
+48.23
+TABLE
+7:
+Ablation
+study
+on
+the
+proposed
+MFRA
+module.
+StructNet(MSFE, (1 · · · 5), MFRAv1) is the degraded MFRA by remov-
+ing the dynamic weights Bj
+s in Eq. 8 and adding different scale fea-
+tures directly. We include another degraded variant of MFRA (i.e.,
+StructNet(MSFE, (1 · · · 5), MFRAv2)) by computing all four scale fea-
+tures through Eq. 7 directly.
+Variants \ Structure-level RMSE ↓
+Shad.
+No.Shad.
+All
+StructNet(MSFE, (1 · · · 5), Add)
+4.82
+1.87
+2.35
+StructNet(MSFE, (1 · · · 5), ASPP)
+5.15
+1.81
+2.37
+StructNet(MSFE, (1 · · · 5), MFRAv1)
+4.65
+1.79
+2.26
+StructNet(MSFE, (1 · · · 5), MFRAv2)
+4.42
+1.82
+2.25
+StructNet(MSFE, (1 · · · 5), MFRA)
+4.20
+1.71
+2.12
+Comparison with alternative fusion solutions. We fur-
+ther compare the proposed MFRA with three potential
+fusion approaches to validate its advantages and effec-
+tiveness, by comparing the structure restoration quality
+(i.e., structure-level RMSE) on the ISTD+ dataset. First,
+we substitute MFRA with the well-known ASPP [56] that
+parallels atrous convolution layers with different rates to
+capture multi-scale information. We denote this variant
+as StructNet(MSFE, (1 · · · 5), ASPP). Second, we degrade
+MFRA by removing the dynamic weights Bj
+s in Eq. 8 and
+adding different scale features directly. We denote this vari-
+ant as StructNet(MSFE, (1 · · · 5), MFRAv1). Third, in Sec. 4.2,
+we extract four scale features where the first three-scale
+features are calculated by feeding the input features to Eq. 7
+and the fourth scale features (i.e., the smallest scale features)
+are extracted by feeding the third-scale features instead of
+the input features to a dilation convolution. To validate
+this specific design, we construct a degraded variant of
+MFRA, that is, we calculate all four scale features through
+Eq. 7 directly, and we name the corresponding StructNet as
+StructNet(MSFE, (1 · · · 5), MFRAv2).
+We report the comparison results in Table 7. We
+have the following observations: Compared with the
+baseline fusion strategy StructNet(MSFE, (1 · · · 5), Add),
+StructNet(MSFE, (1 · · · 5), ASPP) obtains a larger structure-
+level RMSE in the shadow regions, which implies that
+naively using ASPP is not good enough to fuse multi-
+scale features for shadow removal. Compared with
+the degraded version StructNet(MSFE, (1 · · · 5), MFRAv1),
+StructNet(MSFE, (1 · · · 5), MFRA) obtains lower RMSEs in
+both shadow and non-shadow regions, leading to a lower
+RMSE in “All” (i.e., 2.12 vs. 2.26), which demonstrates that
+combining multi-scale features with dynamically predictive
+weights via Eq. 8 indeed helps restore the structure better.
+ If we do not use the proposed strategy to extract the
+smallest scale features, the RMSEs in shadow and non-
+shadow regions increase from 4.20 to 4.42, and from 1.71
+to 1.82, respectively, which implies that our strategy avoids
+heavy information loss during down-sampling.
+Feature comparison. To further validate the MSFE and
+MFRA modules, we compare the modified convolution with
+the standard one in Fig. 4 by showing their processed
+features (See Fig. 4(e) vs. (c)). Clearly, the feature differ-
+ences between shadow and non-shadow regions after the
+modified convolution are much smaller than those after the
+standard convolution. In addition, we compute the absolute
+difference between the averages of shadow and shadow-
+free elements of the 2nd-layer features for each image in the
+ISTD+ test set. Compared with the output features of the
+standard convolution (i.e., vanilla UNet), those of the pro-
+posed operation present much smaller absolute differences
+(See Fig. 4(f)), which also demonstrates the effectiveness of
+the proposed MSFE and MFRA modules.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+14
+Input Shadow
+Shadow Mask
+MS-GAN [13] P+M+D-Net [12]
+G2R [29]
+DC-GAN [30]
+AEF [8]
+MStructNet
+GT
+Fig. 8: Quantitative comparison on the ISTD test set. Please zoom in to see the details.
+6.4
+Effectiveness of MStructNet
+In this section, we first compare MStructNet with existing
+state-of-the-art shadow removal algorithms on three bench-
+marks quantitatively and qualitatively in Sec. 6.4.1. We
+then perform ablated experiments on image-level shadow
+removal to demonstrate the rationality of the multi-level
+structure exploitation in Sec. 6.4.2.
+6.4.1
+Comparison with the State-of-the-art Methods
+Shadow removal evaluation on ISTD. As shown in Ta-
+ble 3, MStructNet achieves the lowest RMSE and highest
+PSNR among all shadow removal methods. In particular,
+MStructNet outperforms △SADC [17], the second best base-
+line, by 11.8%, 14.0%, and 13.5% on RMSE in the shadow
+regions, non-shadow regions, and “All”, respectively. Com-
+pared with BMNet [27], MStructNet obtains 16.6% and 6.8%
+RMSEs improvements in the shadow regions and “All”,
+respectively. In addition, the amount of parameters and
+computational cost (FLOPs) of MStructNet are far lower
+than most of the methods. Specifically, with only 15% of
+parameters and 20% of FLOPS, MStructNet delivers a PSNR
+improvement of 3.54 in “All” when compared with AEF. Al-
+though Mask-ShadowGAN [13] and DC-ShadowGAN [30]
+have fewer parameters than MStructNet, they have larger
+FLOPs (e.g., DC-ShadowGAN [30] has 105G FLOPs).
+Compared with our StructNet, MStructNet presents a
+much lower RMSE in the non-shadow regions (i.e., 4.35 vs.
+4.71) with similar results in the shadow regions (i.e., 6.34
+vs. 6.33), leading to better performance in “All” (i.e., 4.68
+vs. 4.98). More importantly, the model size of MStructNet
+is three times less than that of StructNet (i.e., 21.47MB vs.
+71.64MB), and the FLOPS is 1.5 times smaller.
+We show the visualization comparison between MStruct-
+Net and SOTA methods in Fig. 8. The proposed MStructNet
+can effectively complement low-level cues by integrating
+multi-level shadow-free structure features, thus facilitating
+maximum restoration of the original colors in the umbra and
+penumbra regions. In contrast, other methods either fail to
+restore the original colors (e.g., MS-ShadowGAN and DC-
+ShadowGAN), or cause obvious artifacts around the border
+(e.g., G2R and Param+M+D-Net).
+Shadow removal evaluation on ISTD+. We further con-
+duct the comparison on the ISTD+ [33] dataset to validate
+the effectiveness of our algorithm and report the results in
+Table 4. The first row shows the RMSE between the input
+shadow image and the corresponding GT image. Similar to
+the results on ISTD, MStructNet achieves the lowest RMSE.
+In particular, MStructNet outperforms SOTA SP+M+I-Net
+[34] by 11.7% and 13.9% on RMSE in the shadow regions
+and “All”. This confirms the effectiveness of the proposed
+method for employing shadow-free structure information.
+In addition, MstructNet achieves similar RMSEs in the non-
+shadow regions and “All”, but with better efficiency (fewer
+parameters) compared to StructNet.
+Shadow removal evaluation on SRD. As shown in Ta-
+ble 5, compared to the SOTA methods, MStructNet achieves
+the second highest RMSE and is only 0.08 lower than
+BMNet in the shadow regions. Although the RMSE of
+DC-ShadowGAN, aided by soft mask, outperforms ours
+MStructnNet in the non-shadow regions and “All”, MStruct-
+Net outperforms it by a large margin, i.e., 1.01, in the shadow
+regions. In comparison to AEF [8], MStructNet decreases the
+RMSE from 8.56 to 6.69 in the shadow regions, and from
+5.75 to 4.28 in the non-shadow regions. This also proves
+the superiority of MStructNet in recovering the original
+color and illumination of the shadow regions. Refer to the
+Supplemental for the visual comparison on SRD.
+6.4.2
+Efficacy of Different Numbers of Structure Levels
+Number of convolution layers within each block of
+the structure-aware encoder. In Sec. 5.1, we present the
+structure-aware encoder which is made up of several blocks
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+15
+1
+2
+3
+4
+5
+6
+7
+8
+9
+4
+Fig. 9: More visual results of natural images, obtained from the SBU dataset [57] for shadow detection. Note that SBU contains only shadow
+images and masks. So, we pair the inputs (left) and MStructNet prediction (right) here for visualization. Zoom in to see the details.
+TABLE 8: Ablation experiment of MStructNet on the ISTD+ dataset,
+with respect to different structure level utilization.
+Structure levels
+image-level RMSE ↓
+# Params
+(M: 106)
+# FLOPs
+(G:109)
+0.005 0.015 0.045 0.1 Shad. No.Shad.
+All
+✓
+5.46
+2.90
+3.32
+17.61
+19.35
+✓
+5.46
+2.78
+3.22
+✓
+5.61
+2.73
+3.20
+✓
+5.66
+2.74
+3.21
+✓
+✓
+5.46
+2.81
+3.24
+19.00
+24.21
+✓
+✓
+✓
+5.42
+2.79
+3.22
+20.24
+28.46
+✓
+✓
+✓
+✓
+5.29
+2.73
+3.15
+21.47
+32.72
+with each block representing one structure level and con-
+taining two convolution layers equipped with the MSFE and
+MFRA modules. Note that we set two convolution layers
+for each block due to the empirical results in Table 6 (as
+StructNet(MSFE, 2, Add/MFRA) achieves the lowest RMSE
+in “All”, among all single-convolution based variants).
+Number of blocks (or structure levels) in MStructNet.
+As discussed in Sec. 5.1, each block contains MSFE and
+MFRA modules to form a structure level, and the final
+MStructNet fuses structures of all different levels. Here,
+we study the effects of using different numbers of blocks
+in the structure-aware encoder to validate the advantages
+of exploiting the multi-level structures. Specifically, we
+may obtain four variants of MStructNet by using a
+single structure selected from {0.005, 0.015, 0.045, 0.1},
+and
+are
+denoted
+as:
+{MStructNet(l)|l
+∈
+{0.005, 0.015, 0.045, 0.1}}. We then add more structures
+to MStructNet(0.005) gradually to obtain three more
+variants,
+denoted
+as:
+MStructNet({0.005, 0.015}),
+MStructNet({0.005, 0.015, 0.045}),
+MStructNet({0.005, 0.015, 0.045, 0.1}).
+The
+last
+version
+denotes the final version of MStructNet. As reported in
+Table 8, we can see that: MStructNet with the structure
+level 0.015 shows the best results among all single structure
+Input Shadow
+MStructNet’s prediction
+Ground Truth
+Fig. 10: The visual illustration of the limitation of our method. The red
+box indicates that the object appears only in the shadow regions and
+no similar appearances found in the non-shadow regions.
+level variants. If we add more structure levels, the
+restoration quality gradually improves and MStructNet
+with all four structure levels achieves the lowest RMSE in
+the shadow and non-shadow regions, which demonstrates
+that the utilization of multi-level shadow structures at
+feature level can effectively benefit the image-level shadow
+removal.
+6.5
+Additional Results
+In this section, we provide more visual results of natural
+images in Sec. 6.5.1 and discuss limitations in Sec. 6.5.2.
+6.5.1
+Results on Wild Shadow Images
+We further validate the robustness of our method by test-
+ing it on real-world images outside the shadow removal
+datasets. Specifically, we conduct the inference on the SBU
+dataset [57] for shadow detection, using MStructNet trained
+on SRD. Fig. 9 shows prediction results of different types
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+16
+of shadows (e.g., stand-alone shadows and shadow with
+occluders) in diverse scenes. In cases of 1, 4, where not
+only the cast shadow but also the occluders are present,
+our method still achieves pleasing visual results with no
+obvious artifacts. In addition, our method can also handle
+cases that contain both cast shadows and self-shadows (e.g.,
+the legs of the dog in case 2 and the bricks in case 8).
+These results clearly demonstrate the superiority of our
+method in restoring intrinsic colours and details. Refer to
+the Supplemental for more visual comparisons.
+6.5.2
+Limitations and Failure Cases
+The proposed algorithms have the similar limitation as
+CANet [54]. This is because both methods heavily rely on
+the shadow-free information provided by the non-shadow
+regions to help shadow removal. In general, the structure
+information in the non-shadow regions can be used to
+propagate the consistent appearance features of shadow-
+free pixels to shadow regions. As shown in Fig. 10, if our
+model cannot find corresponding color cues (i.e., objects
+with similar appearances) from non-shadow regions, the
+results may present obvious color bias. One potential way
+to alleviate this issue is to borrow color cues from global se-
+mantics (i.e., from cross-samples) and ensure a deterministic
+color toning, which is included as a future work.
+7
+CONCLUSION
+In this paper, we have systematically investigated the uti-
+lization and efficacy of image structure for single-image
+shadow removal. First, we have built vanilla UNet-based
+networks to restore the shadow-free structure of the input
+shadow image, and demonstrated that image structure can
+help enhance the quality of shadow-removed images signif-
+icantly. We have also revealed the limitations of standard
+convolutions used by the vanilla UNet for structure-level
+shadow removal. Second, we have proposed a novel two-
+stage removal networks, named structure-informed shadow
+removal network (StructNet). It includes two new modules
+for the utilization of structure information, i.e., mask-guided
+shadow-free extraction (MSFE) module and multi-scale feature
+& residual aggregation (MFRA) module, to extract the image
+structural features and regularize the feature consistency,
+respectively. We have shown that StructNet can help im-
+prove the performances of three state-of-the-art methods.
+Third, based on StructNet, we have further proposed a self-
+contained shadow removal method to fully excavate the
+potential of multi-level structures at the feature level, named
+multi-level StructNets (MStructNet), which has fewer param-
+eters and low computational costs. The extensive results on
+three public datasets have demonstrated the advantages and
+effectiveness of StructNet and MStructNet.
+As a future work, we would like to investigate how to
+extend the structure-informed network to shadow editing
+tasks, where occluders are present, such as paired object-
+shadow segmentation [58] and shadow generation [59], [60].
+TABLE 9: Architecture of the vanilla UNet. Ei and Di are the ab-
+breviation of the i-th stage in encoder and decoder. ‘[.]’ denotes the
+concatenation layer along the channel aix. The k and c in conv[k×k, c]
+are the kernel size and number of output filters, respectively. ×j means
+the number of layers.
+Input
+Output
+Output size
+Archtectures
+E1
+[Sl,Ml]
+X1
+128 × 128
+Conv[4×4, 64]
+×1
+E2
+X1
+X2
+64 × 64
+Conv[4×4, 128]
+×1
+E3
+X2
+X3
+32 × 32
+Conv[4×4, 256]
+×1
+E4
+X3
+X4
+16 × 16
+Conv[4×4, 512]
+×1
+E5
+X4
+X5
+8 × 8
+Conv[4×4, 512]
+×1
+D5
+X5
+X6
+16 × 16
+Deconv[4×4, 512]
+×1
+D4
+[X6, X4]
+X7
+32 × 32
+Deconv[4×4, 256]
+×1
+D3
+[X7, X3]
+X8
+64 × 64
+Deconv[4×4, 128]
+×1
+D2
+[X8, X2]
+X9
+128 × 128
+Deconv[4×4, 64]
+×1
+D1
+[X9, X1]
+ˆSl
+256 × 256
+Deconv[4×4, 3]
+×1
+In this supplementary material, we first introduce the
+detailed configurations of the the vanilla UNet network
+in Sec. 8. Then, we provide more qualitative comparisons,
+including comparisons with the state-of-the-arts on SRD
+test set in Sec. 9.1 and on wild data in Sec. 9.2, and the
+comparisons of StructNet-variants in Sec. 9.3.
+8
+BASIC ARCHITECTURE OF VANILLA UNET
+We describe at length the architecture of the vanilla UNet
+φ(·) in Table 9, which consists of three components, i.e.,
+encoder, decoder and the skip connection between the en-
+coder and decoder. The encoder contains 5 blocks, each
+of which includes a convolution (i.e., E-i in Table 9) for
+down-sampling, an activation function (i.e., Leaky-ReLU
+[45]), and a normalization (i.e., Instance Norm [44]). The
+decoder has the similar composition as the encoder, where
+the convolution is replaced by its transposed counterpart
+(i.e., DeConv in Table 9) for up-sampling and the activation
+is changed to ReLU. We set the kernel size, padding and
+stride of convolution or deconvolution layers as 4, 2 and
+1, respectively. All skip connections are identical mapping.
+Note that the normalization function is not utilized in the
+first block of encoder and the last block of decoder.
+9
+MORE QUALITATIVE COMPARISONS
+9.1
+Comparisons of SRD test set
+We qualitatively compare the proposed MStructNet with
+existing SOTAs, including DSC [10], AR-GAN [53], DHAN
+[24], DC-ShadowGAN [30] and AEF [8]. As shown in Fig. 11,
+our method yield consistent colour and details along the
+shadow boundary, while the others either fail to recover the
+original colour (e.g., AR-GAN) or leave the obvious shadow
+traces (e.g., DSC, DC-ShadowGAN and AEF). Although DC-
+ShadowGAN has lower RMSE in non-shadow regions than
+ours, it has higher RMSE in the shadow regions and thus
+presents an obvious shadow artifact. In addition, in the
+2nd case where the shadow exists in over-saturated colour
+regions, almost all existing SOTAs fail and generate obvious
+colour shifts. On the contrary, our MStructNet borrows
+the shadow-free structure from the non-shadow regions
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+17
+and propagates it to the shadow regions, thus obtaining a
+consistent colour transition.
+9.2
+Comparisons of wild shadow images
+In addition to the comparison results on shadow removal
+test set, we also compare our MStructNet with SOTAs on
+wild shadow image that outside of the used dataset. As
+shown in Fig. 12, we show several visual comparison results
+on the shadow detection dataset SBU [57] 3. The first and
+the last cases contain only the shadow, while the other three
+contain both shadow and the corresponding occluders (e.g.,
+human and eaves). Our MStructNet achieves satisfactory re-
+sults without noticeable shadow artifacts in all cases, while
+others still produce easily observable shadow remnants.
+9.3
+Comparisons of StructNet-variants
+As shown in Fig. 13, Fig. 14, Fig. 15, we present the
+qualitative comparison of the different approaches (i.e., ST-
+CGAN [11], AEF [8], SADC [17]) before and after equipping
+the StructNet. The RMSE values for each sample are also
+given in red text in the corresponding results. We can
+clearly see that the colour shift in the baseline methods
+are adjusted, and thus the shadow trace gets significantly
+improved. Even though AEF and SADC have achieved ex-
+cellent performance (i.e., shadow residuals and colour shifts
+are slight), the enhanced StructNet-variants ((i.e., StructNet-
+AEF and StructNet-SADC)) that incorporate our predicted
+shadow-free structure (see the last column) still achieve
+better result enhancement. Overall, the results demonstrate
+that the restored shadow-free structure indeed benefits the
+image-level shadow removal and StructNet can enhance the
+state-of-the-art methods.
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+告
+告
+公告
+车及其他
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+Yuhao Liu received the B.Eng. and M.Sc de-
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+spectively. He is a first-year PhD candidate ma-
+joring in computer science at the City University
+of Hong Kong now. His current research inter-
+ests focus on low-level computer vision problems
+and image editing tasks.
+Qing Guo received Ph.D. degree in computer
+application technology from the School of Com-
+puter Science and Technology, Tianjin Univer-
+sity, China. He was a research fellow with the
+Nanyang Technology University, Singapore, from
+Dec. 2019 to Aug. 2020 and Dec. 2021 to
+Sep. 2022. He was assigned as the Wallenberg-
+NTU Presidential Postdoctoral Fellow with the
+Nanyang Technological University, Singapore,
+from Sep. 2020 to Dec. 2021. He is currently
+a research scientist at Center for Frontier AI
+Research (CFAR), Agency for Science, Technology, and Research
+(A*STAR), Singapore. His research interests include computer vision,
+AI security, and image processing. He is a member of IEEE.
+Lan Fu received the Ph.D. degree in Com-
+puter Science and Engineering from University
+of South Carolina, Columbia, SC, USA. Prior to
+that, she received the M.S. degree in Biomedi-
+cal Engineering from Tianjin University, Tianjin,
+China. Currently, she is a Senior Research En-
+gineer with the InnoPeak Technology Inc., Palo
+Alto, CA, USA. Her research interests include
+computer vision, deep learning, and image pro-
+cessing.
+Zhanghan Ke is currently a PhD candidate at
+the City University of Hong Kong. He obtained
+B.Eng. from Northeastern University (China). He
+serves as a reviewer for several computer vi-
+sion conferences (CVPR, ICCV, ECCV, etc.) and
+journals (TPAMI, IJCV, TCSVT, etc.). His current
+research interests include semi-/self-supervised
+learning and its applications in computer vision.
+Ke Xu is currently with the City University of
+Hong Kong. He received the dual Ph.D. degrees
+from Dalian University of Technology and City
+University of Hong Kong. He also served as a
+program committee member/reviewer for several
+CV and AI conferences and journals, including
+CVPR, ICCV, ECCV, NeurIPS, ICLR, ICML, IJCV
+and TIP. His research interests include deep
+learning and image enhancement.
+Wei Feng received the PhD degree in computer
+science from City University of Hong Kong in
+2008. From 2008 to 2010, he was a research
+fellow at the Chinese University of Hong Kong
+and City University of Hong Kong. He is now
+a full Professor at the School of Computer Sci-
+ence and Technology, College of Computing and
+Intelligence, Tianjin University, China. His major
+research interests are active robotic vision and
+visual intelligence, specifically including active
+camera relocalization and lighting recurrence,
+general Markov Random Fields modeling, energy minimization, active
+3D scene perception, SLAM, video analysis, and generic pattern recog-
+nition. Recently, he focuses on solving preventive conservation problems
+of cultural heritages via computer vision and machine learning. He is the
+Associate Editor of Neurocomputing and Journal of Ambient Intelligence
+and Humanized Computing.
+
+8JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+22
+Ivor W. Tsang is director of A*STAR Centre for
+Frontier AI Research (CFAR) since Jan 2022.
+Previously, he was a Professor of Artificial In-
+telligence, at University of Technology Sydney
+(UTS), and Research Director of the Australian
+Artificial Intelligence Institute (AAII), the largest
+AI institute in Australia, which is the key player
+to drive the University of Technology Sydney to
+rank 10th globally and 1st in Australia for AI
+research, in the latest AI Research Index. Prof
+Tsang is working at the forefront of big data an-
+alytics and Artificial Intelligence. His research focuses on transfer learn-
+ing, deep generative models, learning with weakly supervision, big data
+analytics for data with extremely high dimensions in features, samples
+and labels. In 2013, Prof Tsang received his ARC Future Fellowship for
+his outstanding research on big data analytics and large-scale machine
+learning. In 2019, he received the International Consortium of Chinese
+Mathematicians Best Paper Award. In 2020, he was recognized as the
+AI 2000 AAAI/IJCAI Most Influential Scholar in Australia between 2009
+and 2019. His research on transfer learning was awarded the Best
+Student Paper Award at CVPR 2010 and the 2014 IEEE TMM Prize
+Paper Award. In addition, he received the IEEE TNN Outstanding 2004
+Paper Award in 2007. Recently, Prof Tsang was conferred the IEEE
+Fellow for his outstanding contributions to large-scale machine learning
+and transfer learning. Besides these, Prof Tsang serves as the Editorial
+Board for the Journal of Machine Learning Research, Machine Learning,
+Journal of Artificial Intelligence Research, IEEE Transactions on Pattern
+Analysis and Machine Intelligence, IEEE Transactions on Artificial In-
+telligence, IEEE Transactions on Big Data, and IEEE Transactions on
+Emerging Topics in Computational Intelligence. He serves as a Senior
+Area Chair/Area Chair for NeurIPS, ICML, AAAI and IJCAI, and the
+steering committee of ACML.
+Rynson W.H. Lau received his Ph.D. degree
+from University of Cambridge. He was on the
+faculty of Durham University and is now with City
+University of Hong Kong.
+Rynson serves on the Editorial Board of the
+International Journal of Computer Vision (IJCV)
+and IET Computer Vision. He has served as the
+Guest Editor of a number of journal special is-
+sues, including ACM Trans. on Internet Technol-
+ogy, IEEE Trans. on Multimedia, IEEE Trans. on
+Visualization and Computer Graphics, and IEEE
+Computer Graphics & Applications. He has also served in the committee
+of a number of conferences, including Program Co-chair of ACM VRST
+2004, ACM MTDL 2009, IEEE U-Media 2010, and Conference Co-chair
+of CASA 2005, ACM VRST 2005, ACM MDI 2009, ACM VRST 2014.
+Rynson’s research interests include computer graphics and computer
+vision.
+
diff --git a/YtE1T4oBgHgl3EQfcQR-/content/tmp_files/load_file.txt b/YtE1T4oBgHgl3EQfcQR-/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf,len=2371
+page_content='JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 8, AUGUST 2015 1 Structure-Informed Shadow Removal Networks Yuhao Liu, Qing Guo, Lan Fu, Zhanghan Ke, Ke Xu, Wei Feng, Ivor W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Tsang, and Rynson W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Lau Abstract—Shadow removal is a fundamental task in computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Despite the success, existing deep learning-based shadow removal methods still produce images with shadow remnants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' These shadow remnants typically exist in homogeneous regions with low intensity values, making them untraceable in the existing image-to-image mapping paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We observe from our experiments that shadows mainly degrade object colors at the image structure level (in which humans perceive object outlines filled with continuous colors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Hence, in this paper, we propose to remove shadows at the image structure level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Based on this idea, we propose a novel structure-informed shadow removal network (StructNet) to leverage the image structure information to address the shadow remnant problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Specifically, StructNet first reconstructs the structure information of the input image without shadows, and then uses the restored shadow-free structure prior to guiding the image-level shadow removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' StructNet contains two main novel modules: (1) a mask-guided shadow-free extraction (MSFE) module to extract image structural features in a non-shadow to shadow directional manner, and (2) a multi-scale feature & residual aggregation (MFRA) module to leverage the shadow-free structure information to regularize feature consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In addition, we also propose to extend StructNet to exploit multi-level structure information (MStructNet), to further boost the shadow removal performance with minimum computational overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Extensive experiments on three shadow removal benchmarks demonstrate that our method outperforms existing shadow removal methods, and our StructNet can be integrated with existing methods to further boost their performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Index Terms—Single-image shadow removal, Image structure, Structure-level shadow removal, Structure-informed shadow removal network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 1 INTRODUCTION S HADOWS exist everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' They appear on surfaces where light cannot reach, due to occlusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The pres- ence of shadows causes color and texture inconsistency, which in turn poses challenges for many downstream tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', object tracking [1], detection [2], video segmentation [3], and face recognition [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Faithfully recovering the original color and textures of shadow regions helps facilitate the above tasks as well as other applications, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', game creation and relighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Hence, shadow removal is a long-standing problem in computer vision and graphics with many meth- ods proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Conventional shadow removal methods are typically based on modeling intensity inconsistency [5] and illumi- nation variations [6], or involving user interaction [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' These methods usually fail when the prior assumptions are not satisfied, or when the scene colors and textures are intricate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In recent years, deep learning-based shadow removal meth- Yuhao Liu, Zhanghan Ke, Ke Xu and Rynson W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Lau are with the Department of Computer Science, City University of Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' E- mail: yuhliu9-c@my.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='cityu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='hk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' zhanghake2-c@my.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='cityu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='hk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' kkang- wing@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Rynson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='Lau@cityu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Qing Guo and Ivor W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Tsang are with the Centre for Frontier AI Research, A*STAR, Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' E-mail: tsingqguo@ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='org, ivor tsang@ihpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='a- star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Lan Fu is with the InnoPeak Technology Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='. E-mail: lan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='fu@innopeaktech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Wei Feng is with the College of Intelligence and Computing, Tianjin University, China E-mail: wfeng@ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Yuhao Liu and Qing Guo are the joint first authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Qing Guo and Rynson W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Lau are the joint corresponding authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Manuscript received April 19, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' revised August 26, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' ods [8], [9], [10], [11], [12] achieve impressive performances, due to the high generalization capability of advanced neural networks as well as the availability of large-scale shadow removal datasets [9], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' These methods typically formu- late the shadow removal problem as an image-to-image mapping task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' For example, Qu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' [9] and Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' [10] use CNNs to extract shadow-related information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', location, appearance, and semantic information) and then predict the shadow matte/mask for shadow removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' [8] use CNNs to predict exposure parameters and then remove shadows by fusing shadow images of different exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' [13] propose a CycleGAN-based method to train a shadow removal network using unpaired train- ing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' However, these state-of-the-art methods may still produce unsatisfactory results with shadow remnants and color artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 1(a), we can see yellowish shadow remnants in the result from AEFNet [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' These remnants are usually internally homogeneous and of low intensity values, making them hard to detect by the existing image- level shadow removal paradigm represented by [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In this work, we propose to address the shadow remnant problem by incorporating the image structure information (which consists of object colors and outlines), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The structure of an image is the primary information perceived by the human vision system [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' It separates objects into homogeneous regions with similar intensities [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Hence, we speculate that it should be much easier to locate and much cleaner to remove shadows in the image structure layer, due to the absence of high- frequency texture details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' With the recovered shadow-free image structure layer as a guidance, it may then be possible to perform image-level shadow removal more effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' To verify our idea, we first construct a naive UNet- like model that performs structure-level shadow removal arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='03182v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='CV] 9 Jan 2023 JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 8, AUGUST 2015 2 Structure-level ShadowRemoval Structure Extraction Image-level ShadowRemoval Image-level ShadowRemoval (a) (b) Shadow Images Shadow Removal w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Structure Shadow Removal w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Structure Ground Truth (e) RMSE=18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='72 RMSE=14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='69 RMSE=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='14 RMSE=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='35 RMSE=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='01 RMSE=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='27 (c) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Structure (d) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Structure Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 1: (a) State-of-the-art shadow removal methods (such as AEFNet [8] used in here) typically learn an image-level pixel-to-pixel mapping directly, and may often produce shadow remnants with color artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' (b) We propose to incorporate image structure information into the shadow removal process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We visualize the features of approaches (a) and (b) in (c) and (d), respectively, which show that features of our method are structured according to region homogeneity, helping remove shadow remnants and their resulting color artifacts (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Two visual examples of original AEF and its structure-enhanced counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Red arrows indicate the region with shadow remnants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The RMSE in the lower right corner indicates the error of this sample compared to the shadow-free ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' and uses the restored shadow-free structure to guide the image-level shadow removal process (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' With this model, we show that structure-level shadow removal can help boost the performances of a state-of-the-art shadow removal method [8] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 1(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 1(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We visualize the feature maps of the traditional approach Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 1(a) and our proposed approach Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 1(b) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 1(c) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 1(d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We can see that features in (d) are structured based on region homogeneity, which helps alleviate the color bleeding artifact of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' However, we also note that the standard convolution used in our naive model (as well as in almost all existing methods) adopts spatially- shared weights to process both shadow and non-shadow regions, and neglects their distinct patterns, resulting in color shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Based on the above analysis, we propose the structure- informed shadow removal network (StructNet), which consists of the structure-level shadow removal step in stage-1 and the image-level shadow removal step in stage-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We propose two novel modules to help remove shadow in the structure- level: mask-guided shadow-free extraction (MSFE) module and multi-scale feature & residual aggregation (MFRA) module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The MSFE module aims to model non-shadow to shadow struc- ture information conditioned on the non-shadow regions, while the MFRA module focuses on incorporating the ex- tracted shadow-free structure information into the shadow removal process with feature consistency regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' They can dynamically extracts shadow-free structure in- formation and propagates them into shadow-regions for shadow removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We conduct extensive experiments on three standard benchmarks to evaluate the performances of our method, and show that StructNet outperforms state-of- the-art shadow removal methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Our results also show that StructNet can be incorporated into existing fully-supervised shadow removal methods to help enhance their perfor- mances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Finally, we propose to conduct the shadow removal task at multiple structure levels with a single architecture (named MStructNet), which is not only efficient but also further outperforms state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In summary, we make the following key contributions: We construct a naive model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', the vanilla UNet) for structure-level shadow removal and conduct exten- sive empirical studies on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We show that removing shadows at structure-level is more effective than at the image-level, and the restored shadow-free structures can greatly improve the quality of the output images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We propose the structure-informed shadow removal net- work (StructNet), which contains two novel modules for structure-level shadow removal: mask-guided shadow- free extraction (MSFE) module and multi-scale fea- ture & residual aggregation (MFRA) module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' MSFE learns directional shadow-free structure information from non-shadow to shadow regions, while MFRA regularizes feature consistency by dynamically fusing the output from MSFE with whole image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We further propose a self-contained shadow removal method, multi-level StructNet (MStructNet), which uti- lizes multi-level shadow structures at the feature-level with low parameters for high-quality shadow removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Extensive evaluations and ablation studies on three public datasets show that the proposed StructNet can help enhance the performances of three SOTA methods (STCGAN [11], AEF [8], SADC [17]) and MStructNet achieves high-quality image restoration, outperforming all SOTA shadow removal methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 2 RELATED WORK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1 Shadow Removal Traditional shadow removal methods [18], [19], [20], [21] mainly rely on image statistical priors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', image gradients Information 南京理三大学 现众入口 AudienceEntranc BENOA 训妹馆入口 Training Hall EntranceInformation 南京理三大学 观众入口 Audine ceEntrance Rond No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='3 四号门 Gate Nc 下 训练馆入口Information 南京理三大学 观众入口 AudienceEntrance Road No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='3 四号门 训练馆入口Information 南京理三大学 期众入口 MudienceEntrano BENOA 7 训练馆入口 Training Halli EntranceJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 8, AUGUST 2015 3 and colors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Finlayson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' [20], [22] solve shadow detection and removal via gradient consistency of illumination invari- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Shor and Lischinki [23] propose an illumination-based model in which pixel-wise relationship between shadow and shadow-free pixel intensities are modelled with shadow parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' [6] propose a relative illumination model based on paired data modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' However, conven- tional methods often do not work well when their hand- crafted features do not represent the real-world scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Deep learning-based approaches bring significant progress on the shadow removal task, with the help of large-scale datasets [9], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' DeShadowNet [9] is the first deep learning- based shadow removal method, which models the shadow removal as an image-to-image mapping process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' It uses a multi-branch CNNs to extract multi-level contexts for shadow removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Since then, many methods [10], [11], [24], [25], [26], [27], [28] have been proposed following this image-to-image mapping paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' They focus on design- ing intricate network architectures and exploiting distinctive properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', contexts, residuals and illuminations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Unsupervised methods [12], [13], [29], [30], [31], [32] have also been proposed to alleviate the labeling cost of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' They also fall into the category of image- to-image translation by generating pseudo ground truth images or using unpaired shadow-free images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Le et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' [33], [34] establish a physical shadow formation model and use linear illumination transformations to remove shadows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' [8] formulate shadow removal as an multi-exposure fusion problem, which lights the shadow regions by fus- ing images of multiple exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' BEDSR [35] focuses on document shadow removal and utilizes document-specific priori to develop a background color parameter estimation module and a text supplementation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' EMD-Net [36] removes shadows by using a shadow illumination model and formulates the shadow removal as a variable opti- misation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Although their formulations vary, these methods are still based on the image-to-image mapping paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Due to the abundant of details in images, such a learning paradigm may not remove shadows accurately, resulting in shadow remnants and color artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In this paper, we propose to exploit image structure to constrain the solution space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We demonstrate that our structure-to-image hierarchy can help improve the effective- ness of the shadow removal process significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2 Structure-aware Vision Tasks The use of structure [14], [15] has been receiving attention for its ability to reflect the primary data of the human visual system in processing visual signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Benefited from the evolution of two-layer separation [37], [38], structure has also been used for other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' For example, Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' [39] approaches image inpainting from the perspective of frequency differences, first using structure to generate information such as edges at low frequencies, and then supplementing it with high frequency details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In video interpolation, Gui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' [40] develop a structure-to-texture strategy by exploiting intermediate structure to maintain the smoothness of consecutive frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' [41] introduce structure into the cartoon representation to capture global structure information and sparse color blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' However, these methods are fundamentally different from our approach in the utilisation of structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Structure- aware inpainting [39] and video interpolation [40] rely on the structure to provide edge and contour information through the available background appearance and two con- secutive frames, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In cartoonization [41], on the other hand, the structure provides a globally consistent view of the entire image that can be processed directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' So, it still belongs to image-to-image translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In this paper, we focus on shadow images, where the patterns of shadow and non-shadow areas are highly variable and should be treated separately rather than uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Meanwhile, the in- formation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', color and illumination) in shadow and non- shadow regions of an image varies greatly, and restoring the shadow regions while ensuring consistency inside and outside of the shadow regions may be guided by shadow- free clues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Thus, this is a challenge that cannot be directly solved by existing structure-aware methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' To the best of our knowledge, we are the first to investigate how to utilize structure information in the shadow removal task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 3 STRUCTURE-LEVEL SHADOW REMOVAL In this section, we introduce our structure-level shadow removal approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Specifically, we first formulate the structure-level shadow removal problem (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We then investigate the application of structure information in shadow removal (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1 Formulation of Structure-Level Shadow Removal In structure-level shadow removal, we first use a struc- ture extraction method ϕ(·) to map the input image I ∈ RH×W ×3 to a structure image, in which image inherent col- ors and main outlines are preserved while detailed textures are removed (see examples in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 1(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 2), as Sl = ϕ(I, l), (1) where l > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='0 is a hyper-parameter determining the struc- ture level, and Sl ∈ RH×W ×3 is the structure image at the lth structure level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' A higher l will remove more detailed textures (see the first row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We follow the setups in [8], [33] to formulate the lth structure-level shadow removal: ˆSl = φl(Sl, M), (2) where φl(·) is the shadow removal model corresponding to Sl, and M ∈ RH×W is a binary mask that indicates shadow and non-shadow pixels with 1 and 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Note that the shadow mask is an input to the shadow removal task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The output ˆSl is a shadow-free structure at the lth structure level, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', the result of structure-level shadow removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2 Empirical Studies To study how the structure information affect shadow re- moval results, we employ the structure extraction model proposed by Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' [42] as ϕ(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We design a variant of vanilla UNet [43], which consists of an encoder with 5 convolution layers and a decoder with 5 de-convolution layers, as φl(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Each layer in our φl(·) is followed by an Instance Norm [44] function and a Leaky-ReLU [45] (for the encoder) or ReLU (for the decoder) function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 8, AUGUST 2015 4 Structure level=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='0 Structure level=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='005 Structure level=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='015 Structure level=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='045 Structure level=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1 Input Structure First-stage Shaodw Removal Second-stage Shadow Removal (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 2: Shadow removal results on different image structure levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The 1st row shows the original shadow image (a) and its structures (b)-(e) extracted by [42] at four different structure levels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', l ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='015, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='045, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The 2nd row shows the shadow removal results by feeding the shadow structures in the 1st row to respective vanilla UNets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Image (f) represents the result of the image-level shadow removal, while images (g)-(j) are the results of structure-level shadow removal with l > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The 3rd row shows restoration results of our naive two-stage shadow removal network by feeding the restored shadow-free structures (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', the images at 2nd row) into the second vanilla UNets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' set the kernel size, padding, and stride of each layer to 4, 2, and 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We provide more details on the used models in the Supplemental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Based on the above net- work configurations, we aim to answer the following three questions: \x82 how does the capability of shadow removal vary at different structure levels?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' \x83 whether the structure- level shadow removal results could guide the image-level shadow removal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' \x84 whether existing model architectures are suitable for structure-level shadow removal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1 Shadow Removal at Different Structure Levels Since the shadow removal results may vary at different structure levels (lth), we train and test φl(·) at five structure levels l ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='015, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='045, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Note that l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='0 is equivalent to image-level shadow removal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 1 with l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='0 is an identity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' To avoid the possible influence of elaborately designed loss functions, we only optimize the prediction ˆSl = φl(ϕ(I, l), M) via the mean ab- solute error L1(ˆSl, S∗ l ) = ∥ˆSl − S∗ l ∥1, where S∗ l = ϕ(I∗, l) is the ground truth structure generated from the shadow-free image I∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' On the validation set, we calculate the root mean square error (RMSE) between ˆSl and S∗ l after converting them into the LAB color space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' A smaller RMSE indicates a better shadow removal result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We conduct evaluations on two widely used datasets, including ISTD+ [34] and SRD [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Based on the results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 3, we observe that \x82 the RMSE on shadow regions decreases continuously as l increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' This suggests that it is easier to obtain high quality shadow removal results at the structure level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', l > 0) than at the image level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', l = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Such a phenomenon is also reflected in visual results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 2, in which there are obvious artifacts in the image-level shadow removal result (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 2(f)), but such artifacts are greatly reduced at the structure-level shadow removal results (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 2(g)-(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' \x83 The RMSE curves ISTD+ SRD Structure level (l) Structure level (l) RMSE RMSE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 3: Comparison of the image-level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='0) and four structure- level shadow removal process with l ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='015, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='045, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1} on two public datasets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', ISTD+ [34] and SRD [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We employ the root mean square error (RMSE) in the LAB color space as metric to evaluate the shadow-removal performances in the non-shadow regions, shadow regions, and the whole (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='All) image, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' in the non-shadow regions descend at the beginning then become flat when l increases to reach a certain level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The RMSE curves of the whole images have similar shapes to those of non-shadow regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' For non-shadow regions of the ISTD+ dataset, l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1 has even worse RMSE than that of l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='045 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 3 left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' These experiments show that a higher structure level l generally facilitates shadow removal by making the shadow removal network focus more on color and structure infor- mation instead of texture information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' However, if l is too large, it may lead to shadow spreading, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', similar shadow visual patterns may appear in neighboring non-shadow regions (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 2(e)), which in turn causes a higher error in the non-shadow regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2 Shadow Removal with Structure-level Guidance We would like to investigate if structure-level shadow re- moval is beneficial to image-level shadow removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 8, AUGUST 2015 5 (c) (b) (d) (e) (f) (a) (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Shadow Structure (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Results of Vanilla UNet (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Features in Vanilla UNet (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Results of StructNet (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Quantitative Comparision (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Features in StructNet Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 4: Visualization and quantitative comparison of vanilla UNet and StructNet for structure-level shadow removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' (a) is the input shadow structure, which is fed to the vanilla UNet and StructNet to obtain (b) and (d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Images (c) and (e) show the randomly sampled three feature channels produced by the 2nd convolutional layer of the two networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In addition, we also extract the features from the 2nd convolution layer of the vanilla UNet and StructNet of all images in the ISTD+ test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' For each image, we calculate the absolute difference between the shadow and non-shadow regions in each feature channel and obtain the average difference across all channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Image (f) shows the average feature differences of all images using the vanilla UNet (green points) and StructNet (blue points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' formulate a two-stage pipeline of which the first stage com- bines Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 1 and 2 for structure-level shadow removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The second stage uses a new model ψl(·), which additionally takes the predicted shadow-free structures ˆSl as inputs for image-level shadow removal, as: ˆIl = ψl(I, ˆSl, M), (3) where ˆIl denotes the image-level shadow removal results guided by ˆSl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Theoretically, ψl can be an arbitrary image- level shadow removal method (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', ST-CGAN [11] or AEF [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' For simplicity, we simply assume ψl to have the same architecture as φl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' When training the pipeline corresponding to l ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='015, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='045, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1}, we fix φl optimized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1 and learn ψl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We apply the same L1 loss function and RMSE metric as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Table 1 shows the results (on the ISTD+ dataset) of the two-stage shadow removal pipeline with different l and a single-stage image-level shadow removal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Note that the pipeline with l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='0 can be regarded as a stack of two vanilla UNet models for image-level shadow removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We observe that \x82 two-stage image-level shadow removal does not yield better performance, compared to single-stage image-level shadow removal, and shadow remnants cannot be eliminated by simply adding more CNN parameters as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 2(a,f,k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' \x83 two-stage shadow removal with l > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='0 achieves lower RMSE than that of image-level shadow removal (either two-stage shadow removal with l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='0 or single-stage shadow removal), which shows that the restored shadow-free structures can help image-level shadow removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We also observe from the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 2 that the artifacts in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 2(f) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', the result of single-stage image-level shadow removal) are eliminated by the two- stage shadow removal with l > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='0, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 2(l-o)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='3 Limitations of using the Vanilla UNet In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2, we have demonstrated that the structure-level shadow removal results can benefit image-level shadow removal to some degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Here, we would like to know if the vanilla UNet is good enough for this two-stage shadow removal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', first at structure-level and then at image- level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We observe that the standard convolution operations TABLE 1: Comparison between direct single-stage image-level shadow removal and four variants of two-stage structure-level shadow removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' All experiments are conducted on the ISTD+ dataset with the vanilla UNet and L1 loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Structure level l for the first stage Shadow ↓ Non-shadow ↓ All ↓ Two-stage shadow removal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='99 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='005 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='98 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='55 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='015 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='89 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='49 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='045 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='57 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='56 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='15 Single-stage Image-level shadow removal 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='33 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='78 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='36 used in the vanilla UNet process shadow and non-shadow regions uniformly, and ignore the distinctions between them (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', color-bias).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In other words, the standard convolution used in the vanilla UNet attempts to map shadow and non- shadow regions that have very different appearances to the same pattern, which makes the learning of the convolution weights challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' As a result, the vanilla UNet may produce obvious color shifts between shadow and non- shadow regions in the output image, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' To support the above analysis, we visualize three ran- domly selected feature channels of the 2nd convolution layer in the vanilla UNet in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We can see that the features of the shadow and non-shadow regions show obvious divergences, although we expect them to be consis- tent in order to recover colors and textures of the shadow regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We further conduct a quantitative experiment on the test set of ISTD+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' For each sample, we first extract the features output by the 2nd convolution layer 1 of the vanilla UNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We then compute the means of it feature maps in the shadow and non-shadow regions separately, and show the absolute difference between the two with a single point in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 4(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We can see that there are huge differences between shadow and non-shadow regions in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Such feature differences are caused by the uniform processing of standard convolutions used in the vanilla UNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' As a result, the vanilla UNet produces results with color shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' This motivates us to design a novel solution to overcome 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The difference between shadow and non-shadow regions in deeper layers is minimal and indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Thus, we choose conv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 8, AUGUST 2015 6 (a) MSFE Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' MFRA Dalition Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Constant Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' for Mask MSFE( ) (c) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' (d) MFRA( ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Image-level shadow removal (b) I I Structure-level shadow removal B B B 1 B w1 w2 wS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 5: Pipeline of the proposed StructNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' (a) shows the structure-level shadow removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' (b) shows the image-level shadow removal with the assistance of predicted shadow-free structure from (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' (c) shows the architecture of the mask-guided shadow-free extraction (MSFE) module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' (d) shows the multi-scale feature & residual aggregation (MFRA) module for the fusion function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' the problems of applying the vanilla UNet to structure-level shadow removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 4 STRUCTNET In this section, we propose a novel two-stage model, named structure-informed shadow removal network (StructNet), to better utilize the structure-level shadow removal results to guide the image-level shadow removal step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' StructNet con- tains two novel designs, including a mask-guided shadow- free extraction (MSFE) module as detailed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1, and a multi-scale feature & residual aggregation (MFRA) module as detailed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The configuration details of StructNet are described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='3, using a vanilla UNet in the first stage cannot differentiate between shadow and non- shadow regions properly, which in turn can exacerbate the differences between them and lead to artifacts, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' To address such a spatial-invariant problem caused by the standard convolution operations in the vanilla UNet, we propose to make the convolution layer shadow-aware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Normally, pixels in the shadow regions are not only associ- ated with their neighboring pixels, but also with the distant non-shadow pixels of the same characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Hence, we propose to add a directional bridge to the standard con- volution operations, which are guided by the non-shadow regions to make the elements of the shadow features similar to those of the non-shadow features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Specifically, given the input features Xj in ∈ RHj in×W j in×Cj in at the jth layer, we propose to process the features as: Xj out = Fusion(Xj in ∗ Wj, Bj), (4) where Xj out ∈ RHj out×W j out×Cj out are the output features, Wj are the learnable weights, and Bj ∈ RHj out×W j out×Cj out is a learned shifting tensor aiming to reduce the feature difference between the non-shadow and shadow regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Fusion(·) is a function to fuse the shifting information in Bj and the features Xj in ∗ Wj effectively, thus regularizing the output features to be consistent between the shadow and non-shadow regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Bj is computed by Bridge(·), as: Bj = Bridge(Xj in, Bj−1, Mj in), (5) where Mj in ∈ RHj in×W j in is a binary map that indicates the shadow regions with 1’s and non-shadow regions with 0’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Bj−1 is the shifting tensor of the previous layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Intuitively, Bridge(·) is trained to extract directional shifting from non- shadow to shadow regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Note that such a solution has two benefits: \x82 The advantages of the standard convolution are preserved via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 4, which can extract perception across the whole scene/image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' \x83 The potential shifting between shadow and non-shadow regions is supplemented via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' With the above formulation, we propose the structure- informed shadow removal network (StructNet), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' StructNet consists of two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The first stage performs structure-level shadow removal, while the second stage conducts image-level shadow removal guided by the results from the first stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In the first stage, we propose the two novel modules, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', MSFE and MFRA, to extensively exploit the structure information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' As for the second stage, it can be one existing supervised shadow removal method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1 The MSFE Module Inspired by the segmentation-aware convolution [46], we propose to embed the shadow mask in the convolution operation explicitly and formulate Bridge(·) as: Bj[p] = αp � q∈Np Bj−1[q](1 − Mj in[q])Wj R[q − p], (6) where Wj R ∈ RKj×Kj×Cj in×Cj out are the weights of a convo- lution layer, p and q are the coordinates of elements in Xj in, Mj, Bj, or Wj R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The set Np contains neighboring elements of p, and its size is equal to the kernel size of Wj B (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', K2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The normalization term αp is defined as 1 � q∈Np Mj in[q].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The mask (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', Mj in) is obtained by convoluting the mask from the previous layer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', Mj−1 in ) with a constant weight (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', W1 whose elements are one) through Mj in = Mj−1 in ∗ Wj 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Intuitively, with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6, the output tensor Bj is only depen- dent on the non-shadow regions due to the guidance of the mask Mj and is able to fill the gap across shadow and non- shadow regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' As the examples shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6, the features of the whole scene from the standard convolution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', Xj in ∗ Wj) present obvious shadow regions while the shifting (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', Bj) JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 8, AUGUST 2015 7 B Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6: Feature visualization of the global perception (Xj in ∗ Wj), the offset (Bj), and output features by fusing the former two (Xj out).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' contains the shifting information predicted from the non- shadow regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' As a result, compared with Xj in ∗ Wj, the final fusion result (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', Xj out) shows similar and consistent appearances across shadow and non-shadow regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In addition, instead of training the weight Wj B for all examples, we propose to make it dynamically changed according to different input features, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', Wj R = η(Xj in), where η(·) is a sub-network having two convolution layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Specifically, we set the stride of the first layer as 2, the kernel with the size of Kj, which halves the spatial resolution of Xj in and maintains low computation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The output size becomes Hj in 2 × W j in 2 × Cj in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' For the second layer, we set the size of convolution kernel as 1∗1, which produces a tensor with size Hj in 2 × W j in 2 × � Cj in × Kj × Kj� , and is then reshaped to produce the weight Wj R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Our method is similar to the partial convolution [47] but has the following differences: \x82 The convolution weights of the proposed bridge function are conditional on the input whole scene features, while those of the partial convolution are fixed after training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' \x83 The operations with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 4 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 5 are a combination of standard and dynamic partial convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The former aims to extract the perception of the whole image, while the latter is to bridge the shifting between shadow and non-shadow regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2 The MFRA Module Beyond the element-wise additive fusion of the convolu- tional perception (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', Xj in ∗ Wj) and the shifting Bj, we propose to conduct the fusion at multiple scales with a dynamic aggregation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The main motivation stems from the fact that the desired output features should be consistent across shadow and non-shadow regions, and then each desired element in the output features is dependent on the context of the shifting bridge and the convolutional per- ception around its position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' However, the effective context range for different elements and examples are different, and how to make the fusion adaptive to this change is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Specifically, given the convolutional perception (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', Xj in∗ Wj) and the shifting Bj in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 4, we first conduct multiple atrous convolutions with different dilation rates to obtain multi-scale features, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', Xj s = σ([Xj in ∗ Wj, Bj] ∗ Dj s), (7) where σ(·) is the ReLU function, Dj s is the weight of an atrous convolution with dilation rate s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Here, we consider s ∈ S = {1, 24, 12, 6} and get the first three features {Xj s|s ∈ {1, 24, 12}} via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The size of the weights of all Dj s is 3 × 3 × Cj in × Cj out with strides {1, 1, 2, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' For the last and smallest scale features (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', s = 6), we do not extract from Xj in like Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 7, but feed Xj 12 to a dilation convolution to obtain Xj 6 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 5(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Such an implementation can alleviate heavy information loss caused by directly down- sampling the input features two times, and reduce the computation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Then, the key problem is how to combine the four sets of features according to different inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' To this end, we propose to estimate the combination parameters dynamically according to the inputs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', Xj out = S � s wj s ⊙ Xj s, with {wj s|s ∈ S} = Φ([Xj in ∗ Wj, wj]), (8) where wj s ∈ RHj in×W j in×Cj in assigns weights for each element in Xj s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Note that the elements at the same positions but dif- ferent channels share the same weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Φ(·) is a subnetwork containing two convolution layers and a softmax layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Each convolution is followed by a ReLU layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='3 Configuration Details In this subsection, we detail the configuration of the first stage of StructNet, which contains three branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The first branch takes the shadow structure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', Sl) and the shadow mask (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', M) as inputs, and aims to estimate the shadow- free structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Its encoder contains five convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' It receives shifting predicted by the second branch and con- ducts fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The second branch is to estimate the shifting (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', Bj) for the convolutional layer in the first branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Each layer of the second branch (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', the jth layer) takes the shifting from the previous layer (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', Bj−1), the shadow mask from the third branch (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', Mj in), and the jth features from the first branch (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', Xj) as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The third branch is to produce the binary masks for all layers along the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The ten convolution layers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', 5 for the encoder and 5 for the decoder) in the first branch share the same settings with the vanila UNet in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In terms of the fusion function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', MFRA), we set the kernel size of all convo- lutional layers to 3, and the number of kernels/filters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', Cj out) is {64, 128, 256, 512, 512} except for the second layer of weight generation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', Φ(·)) where the number of kernels is equal to the number of parallel branches (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' For the second branch, we have a total of five convolutional layers, each corresponding to one of the encoder layers in the first branch and having the same strides and number of kernels as the encoder layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The kernel size of each convolutional layer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', Kj) is set to {7, 5, 3, 3, 3}, and each layer is followed by a Batch-Norm (except for the first layer) and a ReLU function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' For the third branch, we fix the constant convolutional kernels Wj 1 of size Kj and with stride 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' As for the production of mask Mj, when the kernel JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 8, AUGUST 2015 8 MSFE Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' MFRA Constant Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' for Mask (a) (b) Block l Structure-aware encoder Fusion-oriented encoder Decoder B Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 7: Pipeline of the multi-level StructNet (MStructNet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' (a) presents the whole pipeline, while (b) shows the detail of the blue blocks in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Wj 1 is sliding on Mj−1 in , the updated mask value of that window is set to 0 in Mj in if the convolved value is smaller than the sum of kernels Wj 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' otherwise, it is set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In addition, as shown in Table 1, the maximum performance gain is delivered when the structure level is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='015, which also coincides with the observation of trade-off between the smoothness and performance of two-stage shadow removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Thus, unless otherwise stated, we assume l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='015 in the proposed StructNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 5 MULTI-LEVEL STRUCTNETS (MSTRUCTNET) Although our StructNet presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 4 is able to re- store the shadow structure effectively and performs better than the vanilla UNet, benefiting the image-level shadow removal step significantly, such a two-stage solution leads to large computational overheads due to the naive com- bination of two networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' To address this problem, we further propose a self-contained shadow removal method that utilizes multi-level structures at the feature level with only a small increase in the parameter numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Specifically, we omits the step for predicting the shadow-free structure image through the first stage of StructNet, but use the non-shadow structure information directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We refer to this method as MStructNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 7 shows the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1 Pipeline Given a shadow image I, we extract structures via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 1 and consider four levels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', l ∈ L = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='015, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='045, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Thus, we obtain four levels of structure, {Sl|l ∈ L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' MStructNet takes the original shadow image, all the struc- tures, and the shadow mask as inputs to predict the shadow- free image directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The whole pipeline contains three com- ponents, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', structure-aware encoder, fusion-oriented en- coder, and decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The structure-aware encoder contains |L| blocks to address |L| structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Note that each block follows the design of StructNet and makes the shadow elements of the output features similar to the shadow- free elements under the guidance of the input shadow image and structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The fusion-oriented encoder consists of standard convolutions and is to further extract deep feature embedding from the structure-aware features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The decoder is to map the feature embedding to the shadow- free image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We show the whole pipeline in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' As the main difference between this pipeline and the vanilla UNet lies in the structure-aware encoder as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 7(b), we discuss the design of this encoder in detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In terms of the lth block in the structure-aware encoder, we have the original shadow image I, the lth structure Sl, and shadow mask M as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We then feed them to the block having two convolutional layers equipped with the proposed MSFE and MFRA modules (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 7(b)), which produce two features denoted as X1 l and X2 l corresponding to the outputs of the first and second convolution layers, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' For the four levels of structures (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', {Sl|l ∈ L}), we obtain eight output features, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', {X1 l }l∈L and {X2 l }l∈L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We combine the four sets of features in {X1 l }l∈L or {X2 l }l∈L via an element-wise additive operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The combined fea- tures are fed to the fusion-oriented encoder and decoder to estimate the shadow-free image ˆI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2 Configuration Details Same as the first stage of StructNet in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 4, each block l in the structure-aware encoder in MStructNet also has three branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Take the lth block as an example, the inputs to the first branch include shadow image I, shadow structure Sl with level l and shadow mask M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The three inputs are concatenated along the channel axis and further fed to the standard convolution to perceive the global scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The inputs to the second branch are shadow structure Sl and shadow mask M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Then, with the global perceptual features of the first branch as the guiding weights, the shifting features can be obtained by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The third branch updates the shadow mask Mj in the same way as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Regarding the fusion-oriented encoder and the decoder, they contain only standard convolutional layers, instance- norm and activation function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', Leaky-ReLU or ReLU), and all the settings are the same as those in the vanilla UNet in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6 EXPERIMENT 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1 Implementation Details 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1 Loss Functions Instead of using just the L1 loss function in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2 as the re- construction loss, we add the perceptual loss to train Struct- Net and MStructNet for high restoration quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Specifically, given a restored image ˆI and its ground truth I∗, we have L(ˆI, I∗) = λ1L1(ˆI, I∗) + λ2Lperc(ˆI, I∗), (9) where L1(ˆI, I∗) is the L1-norm distance to ensure pixel- level visual consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Lperc is the perceptual loss [48], which aims to ensure the restored image to have the same perception as the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' It is formulated as: Lperc(ˆI, I∗) = 3 � i=1 ∥VGG16i(ˆI) − VGG16i(I∗)∥1, (10) where VGG16i(·) represents the activation map of the ith max-pooling layer in the VGG16 [49] pretrained on the JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 8, AUGUST 2015 9 ImageNet [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The two trade-off parameters in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 9, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', λ1 and λ2, ensure the numerical and gradient equivalence during training, and we empirically set λ1 = 1 and λ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1 for all the experiments on StructNet and MStructNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We employ L(ˆI, I∗) to end-to-end train MStructNet directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' For StructNet, we use the same loss but with < ˆSl, S∗ l > to train the first stage, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', L(ˆSl, S∗ l ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' After that, we fix the parameters of the first-stage network and use L(ˆI, I∗) to train the second-stage network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2 Training and Testing Configurations We have implemented the proposed methods and all vari- ants in PyTorch on a single NVIDIA TITAN GPU with 12G memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We optimize all networks by Adam [51] optimizer with the initial learning rate, β1 and β2 as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='0002, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='9, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='999, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We adopt the warmup for the first 2k it- erations and cosine decay strategy [52] to adjust the learning rate during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' For the training schedule, we train the first-stage network of StructNet for 100k iterations and the second-stage network for another 100k iterations to reach convergence on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' For MStructNet, we train it for 300k iterations to reach convergence for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Besides, we perform data augmentation to prevent over-fitting for both methods, which includes random left-right flipping, random rotation of angles ranging between -20o to 20o, random cropping, and resolution adjustment (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', all inputs are resized to 256×256).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We then conduct a normalization to transform the inputs into a range of -1 to 1, and feed them to the networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' During inference, for a fair comparison and accommodating the hardware limitation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', GPU memory usage), we follow [8], [11], [30], [53] and directly resize the inputs to 256×256, and obtain the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The batch sizes for training and testing are 6 and 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2 Method Settings 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1 Baseline Methods Baseline methods enhanced by StructNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Our StructNet proposed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 4 is able to enhance existing shadow removal methods by first conducting structure-level shadow removal, and then regarding the restored shadow-free struc- ture as an auxiliary prior for the baseline method to predict the shadow-free image in the second stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We use four baseline methods for comparison: the vanilla UNet in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 3, and three state-of-the-art methods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', STCGAN [11], AEF [8] and SADC [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We regard these baseline methods as the second-stage networks in StructNet, resulting in four vari- ants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Note that we select these three SOTA methods as they have very different frameworks, which can demonstrate the high extensibility and flexibility of StructNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' For the three SOTA baseline methods to receive the structure input from the first stage, we need to make small modifications to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' (The modification of the vanilla UNet is already discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=') STCGAN [11]: It takes the shadow image as input, and performs shadow detection and removal with two successive sub-networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' It can be formulated as: ˆI = Γrem (Γdet (I) , I) , (11) where Γdet(·) and Γrem(·) refer to the shadow detec- tion and removal sub-networks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We first use Γdet (I) to get the shadow mask M and compute the structure Sl of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We then feed M and Sl to our structure-level shadow removal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', first stage of StructNet) to obtain ˆSl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We insert ˆSl to Γrem(·) by: ˆI = Γrem � Γdet (I) , I, ˆSl � (12) AEF [8]: It concatenates shadow image and shadow mask at inputs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', I and M) and conducts image-level shadow removal first by automatically adjusting the illumination of shadow region via multiple exposure combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' It then performs boundary refinement (Γedge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' It can be formulated as: ˆI = Γedge �Γexp (I, M) , I, M � , (13) where Γexp(·) denotes the exposure-fusion-based shadow removal function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' To enhance AEF with our StructNet, we plug in the predicted shadow-free struc- ture ˆSl as an input to Γexp(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The updated method is formulated as: ˆI = Γedge � Γexp � I, M, ˆSl � , I, M � , (14) SADC [17]: It uses a two-branch network to extract fea- tures of shadow image I and mask M separately, and then decodes them to predict the shadow-free image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' For simplicity, we formulate it as: ˆI = Γrem (Γfeat (I) , M) , (15) where Γfeat(·) is the pre-trained VGG16 [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' It is to extract the input image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We implement SADC- based StructNet by embedding the restored structure ˆSl to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 15, as: ˆI = Γrem � Γfeat (I) , M, ˆSl � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' (16) Based on above modifications, we obtain three vari- ants of StructNet denoted as StructNet-STCGAN, StructNet- AEF, and StructNet-SADC, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Note that the three versions share the same structure-level shadow removal network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We fix the first-stage network and retrain only the second-stage networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Since the code and model weights of ST-CGAN are not publicly available, we re-implement this method and re-train it with the same training strategy and hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' For AEF, we use the publicly released model weights for the ISTD and ISTD+ datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We strictly follow their original settings and parameters to train the en- hanced model that incorporates the predicted shadow-free structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' SADC only provides model weights for ISTD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' For ISTD+, we strictly follow its code to train the corresponding models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content=', original and enhanced ISTD+ models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Baseline methods for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' To further demon- strate the advantages of the structure-informed shadow removal networks, we compare the StructNet variants and MStructNet with state-of-the-art methods including Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content=' [7], DeshadwoNet [9], STCGAN [11], DSC [10], Mask-ShadowGAN [13], AR-GAN [53], SP+M- Net [33], CLA-GAN [25], RIS-GAN [26], Param+M+D-Net [12], DHAN [24], G2R [29], AEF [8], DC-ShadowGAN [30], SP+M+I-Net [34], BMNet [27], SADC [17], EMD-Net [36], and SG-ShadowNet [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content=', test environment and inference resolution) JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 8, AUGUST 2015 10 TABLE 2: Validation results of StructNet-equipped shadow removal methods on ISTD and ISTD+ datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content=', vanilla UNet, STCGAN [11], AEF [8] and SADC [17], in our StructNet framework as four variants, and compare them with the original methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content='961 StructNet-SADC 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content='04 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='40 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='27 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='989 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='963 may not be the same, we list the ways that we obtain the quantitative results (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', metrics values) for a better evaluation, as follows: △ - We retrain their models strictly according to their hyper-parameters, but at a resolution of 256, and then compute the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' ♦ - We down-sample their publicly released predictions to a resolution of 256, and then compute the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' ⋆ - We replicate their reported data from their papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2 Datasets We conduct our experiments on three shadow removal benchmark datasets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', SRD [9], ISTD [11] and ISTD+ [34], to evaluate the effectiveness of the proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' SRD [9] is the first large-scale shadow removal dataset, consisting of 3,088 paired shadow and shadow-free images, of which 2,680 are for training and 408 for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Since shadow masks are not available in SRD, we follow AEF [8] to utilize the Otsu’s algorithm to extract the shadow masks from the difference between the shadow and shadow- free images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We adopt the extracted masks for network usage (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', training and testing), and use the available masks from DHAN [24] for metric evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' ISTD [11] is another benchmark dataset that comprises 1,870 triplets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', shadow image, shadow mask and shadow-free image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' This dataset is collected under 135 scenes and split into 1,330 for training and 540 triplets for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' As the images in ISTD suffers from the color consistency problem in the non-shadow re- gions, Le et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', [33] corrected this problem to form the ISTD+ dataset, which has the same data setting as the original ISTD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' For ISTD and ISTD+ datasets, we follow AEF [8] to use the ground-truth shadow masks for training, and extracted masks from Otsu’s algorithm for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='3 Evaluation Metrics We follow methods [8], [10], [12], [34] to compute the root mean square error (RMSE) between shadow-removed im- age and ground-truth shadow-free image in the LAB color space, which is also named as image-level RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' When evaluating structure-level shadow removal, we compute RMSE between the predicted shadow-free and ground truth structures as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2, which is denoted as structure-level RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Following [27], [29], we also report the peak signal-to-noise ratio (PSNR) and structural similar- ity index (SSIM) to measure the visual quality scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Note that all metrics are computed in the shadow region (Shad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' ), non-shadow regions (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Shad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' ), and the whole image (All), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='3 Evaluation of StructNet In this section, we validate the effectiveness of StructNet in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1 by using it to enhance four existing baseline methods and comparing with their original versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We then conduct extensive ablation studies in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2 and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='3 to analyze and validate the effectiveness of the proposed MSFE and MFRA modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1 Validation Results Quantitative comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' With the implementations dis- cussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1, we evaluate four StructNet variants (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', StructNet-UNet/-STCGAN/-AEF/-SADC) on ISTD+ and ISTD, and compare them with their original versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We show the results in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We can see that the proposed StructNet improves all four baselines with a significant mar- gin on RSME in the shadow region over the two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' For example, the RMSE of STCGAN decreases from 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='39 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='25 (an improvement of 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='4%) on ISTD+, and from 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='11 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='52 (an improvement of 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='6%) on ISTD, which leads to obvious RMSE reduction in the whole image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' As StructNet- UNet obtains the best results across all counterparts, for convenience, we refer to StructNet-UNet as StructNet in all subsequent experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 8, AUGUST 2015 11 TABLE 3: Quantitative comparison with the SOTA methods on the ISTD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content='72 We also compare StructNet (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content=', StructNet-UNet) with the state-of-the-art methods in Table 3, Table 4, and Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We can see from Table 3 and Table 4 that the proposed StructNet outperforms all baseline methods on both ISTD+ and ISTD, achieving the lowest RMSE, demonstrating the advantages of our structure-informed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In partic- ular, on ISTD as shown in Table 3, StructNet obtains 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='33 in the shadow regions, an improvement of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='0% over the second best method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', SADC△).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Although StructNet has larger model size with more parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content=', 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='64 M), its FLOPs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content=', 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='23 G) is much lower than the baseline meth- ods (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', SADC and AEF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Although Mash-ShadowGAN has a smaller model size, it requires more computations for the cycle process and thus has a higher FLOPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' AEF involves multiple sub-networks, and hence more parameters and higher FLOPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' SADC, which processes features at high resolution by reducing the number of downsampling steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Hence, the FLOPS is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2 Effectiveness of the MSFE Module Here, we study the effectiveness of the mask-guided shadow- free extraction (MSFE) module on the ISTD+ dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Based on StructNet (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', StructNet-UNet), we construct different StructNet variants by using different structure-level shadow removal networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We then evaluate the quality of the restored structures (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', structure-level RMSEs) from the first stage as well as the quality of the restored images (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', image-level RMSEs) from the second stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Adding MSFE to a single convolution layer in the vanilla UNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We add the MSFE module to different convo- lution layers in the encoder of the vanilla UNet for structure- level shadow removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' To avoid the influence of the fusion function carried out by the MFRA module, we replace it with a naive element-wise additive operation instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We TABLE 4: Quantitative comparison with the SOTA methods on the ISTD+ dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' ‘-’ indicates values that are not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The best and second best results are highlighted in red and green, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Method \\ RMSE ↓ Shad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='Shad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' All Input Images 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content='0 ⋆G2R (w sup) [29] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content='6 ⋆AEF [8] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content='2 ⋆DC-ShadowGAN [30] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content='6 ⋆SP+M+I-Net [34] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content='6 ⋆BMNet [27] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content='5 △SADC [17] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='6 ⋆SG-ShadowNet [28] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='4 StructNet 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='0 MStructNet 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1 use StructNet(MSFE, j, Add) to denote the StructNet whose first-stage network uses the MSFE as Bridge() at the jth layer and the element-wise additive operation as Fusion().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We then obtain five variants, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', {StructNet(MSFE, j, Add)|j ∈ {1, 2, 3, 4, 5}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Table 6 shows the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' From these re- sults, we observe: \x82 Compared with the naive two-stage shadow removal method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', vanilla UNet), StructNets with a single MSFE achieves lower structure-level and image- level RMSEs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', StructNet(MSFE, 1/2/3/4/5, Add) in Ta- ble 6 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' two-stage shadow removal in Table 1) in the JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 8, AUGUST 2015 12 TABLE 5: Quantitative comparison with the SOTA methods on the SRD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' ‘-’ indicates values that are not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The best and second best results are highlighted in red and green, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Method \\ RMSE ↓ Shad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='Shad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' All ⋆Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' [6] 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='89 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='47 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='60 ⋆DeShadowNet [9] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='78 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='84 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='64 ⋆DSC [10] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='89 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='99 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='23 ⋆Mask-ShadowGAN [13] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='32 ⋆AR-GAN [53] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='71 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='74 ⋆DHAN [24] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='39 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='67 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='46 ⋆RIS-GAN [26] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='22 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='78 ⋆CLA-GAN [25] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='10 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='01 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='59 ⋆AEF [8] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='56 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='75 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='51 ⋆DC-ShadowGAN [30] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='70 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='39 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='66 ⋆CANet [54] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='82 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='88 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='98 ⋆BMNet [27] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='61 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='61 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='46 △SADC [17] (256) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='68 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='45 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='89 ⋆EMD-Net [36] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='44 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='74 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='79 ⋆SG-ShadowNet [28] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='53 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='97 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='23 StructNet 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='93 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='94 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='81 MStructNet 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='69 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='28 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='97 shadow regions, which demonstrates that the MSFE does benefit the structure-level shadow removal and enhance the image-level shadow removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' \x83 In general, if we embed MSFE in a deeper convolution layer, we get lower RMSEs in the shadow regions while higher RMSEs in the non- shadow regions at the structure level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' For example, the structure-level RMSE of the shadow region decreases from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='10 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='73 if we add MSFE from the 1st to the 5th layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We have similar observations on the image-level RMSEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The above observations hint that bridging the difference between shadow and non-shadow regions at higher level helps enhance restoration quality of shadow regions but slightly harms the restoration quality of non-shadow re- gions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Adding MSFE to multiple convolutions in the vanilla UNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We further add MSFE to more layers to study the change in RMSEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' As shown in Table 6, we de- note the variant as StructNet(MSFE, (1 · · · 5), Add), where we add MSFE to all convolutions between the 1st layer and 5th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Compared with StructNet(MSFE, 5, Add), StructNet(MSFE, (1 · · · 5), Add) has a lower structure- level RMSE (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='87) in the non-shadow regions but a slightly higher structure-level RMSE (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='82) in the shadow regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The overall RMSE becomes 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='35, which is smaller than that of StructNet(MSFE, 5, Add).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In contrast, compared with StructNet(MSFE, 1, Add), StructNet(MSFE, (1 · · · 5), Add) has a much lower structure- level RMSE in the shadow regions and the same RSME in the non-shadow regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Such observations imply that equipping more convolutions with MSFE can balance the restoration quality in the non-shadow and shadow regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Comparison with the convolutional skip function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' To validate the necessity of MSFE, we implement a variant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', StructNet(CONV, (1 · · · 5), Add), which formulates the fusion function as an additive operation and the bridge as a convolution layer like the convolutional skip connection in the residual network [55], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', Xj out = Xj in ∗ Wj + Bj, where Bj = Conv([Xj in, Bj−1, Mj in]), (17) However, such a solution is not easy to address the limitations of the standard convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' With skip connections, although the spatial information is embedded via the mask Mj in, skip connections with the convolution layer inherit the spatial-invariant property and cannot address the shadow and non-shadow shifting explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Similarly, the element-wise addition neglects the relationship between different shadow and non-shadow regions and does not benefit the shifting prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' As reported in Table 6, StructNet(CONV, (1 · · · 5), Add) obtains lower RMSEs in the shadow regions, which shows that it also benefits shadow removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' However, compared with StructNet(MSFE, (1 · · · 5), Add), StructNet(CONV, (1 · · · 5), Add) has much higher RMSEs in both shadow and non-shadow regions, even leading to a higher RMSE in “All” than the vanilla UNet (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='48 vs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' This demonstrates the advantages of our MSFE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Model size comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Compared with the naive two- stage shadow removal, other variants have similar model size (See the “Params” column in Table 6) since the em- bedded module (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', MSFE or CONV) does not introduce a large amount of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Hence, the improvement with MSFE mainly lies in the utilization of shadow-free structure information rather than the increase of model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Note that the computation of Params and FLOPs covers two sub-networks, structure-level shadow removal at stage- 1 and image-level shadow removal at stage-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' By default, we resort to THOP2 to conduct the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='3 Effectiveness of the MFRA Module Adding MFRA to the convolution in the MSFE-based UNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' To validate the effectiveness of the MFRA module, we replace the element-wise additive fusion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', “Add”) of the variants in Table 6 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', StructNet(MSFE, ⋆, Add)) with the proposed MFRA to obtain new variants, StructNet(MSFE, ⋆, MFRA), where ‘⋆’ denotes specific layer indexes used by StructNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We show the results in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We have the following observations: \x82 All single-MSFE- based variants with MFRA (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', StructNet(MSFE, ⋆, MFRA)) outperform the variants with the element-wise addition operation, which demonstrates that the proposed aggre- gation function does enhance shadow removal signifi- cantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' For example, StructNet(MSFE, 2, MFRA) achieves 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='32 structure-level RMSE in the shadow regions, outper- forming StructNet(MSFE, 2, Add) by 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='0%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In addition, we also see improvements on image-level RMSEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' \x82 When we embed MSFE to multiple convolutions with MFRA, we find that StructNet(MSFE, (1 · · · 5), MFRA) achieves much better restoration quality in both shadow and non- shadow regions than StructNet(MSFE, (1 · · · 5), Add).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' As a result, in “All”, StructNet(MSFE, (1 · · · 5), MFRA) obtains 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='12 on structure-level RMSE and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='97 on image-level RMSE, which are 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='7% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='3% higher than those of StructNet(MSFE, (1 · · · 5), Add).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='com/Lyken17/pytorch-OpCounter JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 8, AUGUST 2015 13 TABLE 6: Comparison between StructNet variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The comparisons are conducted on the ISTD+ dataset from two aspects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', structure-level and image-level shadow removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We denote all variants with StructNet(Factor1, Factor2, Factor3) where ‘Factor1’ represents the function for the Bridge(·), ‘Factor2’ means the positions to embed the ‘Factor1’, and ‘Factor3’ is the function for the Fusion(·) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Methods for the first stage Structure-level RMSE ↓ Image-level RMSE ↓ # Params (M:106) # FLOPs (G:109) Shad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='Shad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' All Shad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='Shad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' All Two-stage shadow removal in Table 1 with l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='015 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='54 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='86 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='46 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='89 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='49 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='05 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='59 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='44 StructNet(MSFE, 1, Add) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='87 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='41 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='72 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='59 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='10 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='60 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='59 StructNet(MSFE, 2, Add) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='36 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='55 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='59 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='07 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='80 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='43 StructNet(MSFE, 3, Add) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='76 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='46 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='56 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='58 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='07 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='62 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='27 StructNet(MSFE, 4, Add) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='73 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='49 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='62 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='61 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='11 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='80 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='57 StructNet(MSFE, 5, Add) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='74 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='97 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='43 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='55 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='57 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='06 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='16 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='72 StructNet(MSFE, 1, MFRA) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content='36 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='68 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='59 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='09 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='78 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='60 StructNet(MSFE, 2, MFRA) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content='05 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content='43 StructNet(MSFE, 3, MFRA) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='43 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='54 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='58 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='07 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='57 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='27 StructNet(MSFE, 4, MFRA) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='33 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='57 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='52 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='02 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='60 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='57 StructNet(MSFE, 5, MFRA) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='58 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='92 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='35 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='43 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='52 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='00 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='97 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='22 StructNet(CONV, (1 · · · 5), Add) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='97 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='48 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='61 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='14 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='53 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='38 StructNet(MSFE, (1 · · · 5), Add) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='82 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='87 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='35 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='60 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='07 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='16 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='72 StructNet(MSFE, (1 · · · 5), MFRA) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='71 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='31 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='52 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='97 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='64 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='23 TABLE 7: Ablation study on the proposed MFRA module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' StructNet(MSFE, (1 · · · 5), MFRAv1) is the degraded MFRA by remov- ing the dynamic weights Bj s in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 8 and adding different scale fea- tures directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We include another degraded variant of MFRA (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', StructNet(MSFE, (1 · · · 5), MFRAv2)) by computing all four scale fea- tures through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 7 directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Variants \\ Structure-level RMSE ↓ Shad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='Shad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' All StructNet(MSFE, (1 · · · 5), Add) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='82 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='87 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='35 StructNet(MSFE, (1 · · · 5), ASPP) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='81 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='37 StructNet(MSFE, (1 · · · 5), MFRAv1) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='79 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='26 StructNet(MSFE, (1 · · · 5), MFRAv2) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='42 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='25 StructNet(MSFE, (1 · · · 5), MFRA) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='71 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='12 Comparison with alternative fusion solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We fur- ther compare the proposed MFRA with three potential fusion approaches to validate its advantages and effec- tiveness, by comparing the structure restoration quality (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', structure-level RMSE) on the ISTD+ dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' First, we substitute MFRA with the well-known ASPP [56] that parallels atrous convolution layers with different rates to capture multi-scale information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We denote this variant as StructNet(MSFE, (1 · · · 5), ASPP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Second, we degrade MFRA by removing the dynamic weights Bj s in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 8 and adding different scale features directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We denote this vari- ant as StructNet(MSFE, (1 · · · 5), MFRAv1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Third, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2, we extract four scale features where the first three-scale features are calculated by feeding the input features to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 7 and the fourth scale features (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', the smallest scale features) are extracted by feeding the third-scale features instead of the input features to a dilation convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' To validate this specific design, we construct a degraded variant of MFRA, that is, we calculate all four scale features through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 7 directly, and we name the corresponding StructNet as StructNet(MSFE, (1 · · · 5), MFRAv2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We report the comparison results in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We have the following observations: \x82 Compared with the baseline fusion strategy StructNet(MSFE, (1 · · · 5), Add), StructNet(MSFE, (1 · · · 5), ASPP) obtains a larger structure- level RMSE in the shadow regions, which implies that naively using ASPP is not good enough to fuse multi- scale features for shadow removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' \x83 Compared with the degraded version StructNet(MSFE, (1 · · · 5), MFRAv1), StructNet(MSFE, (1 · · · 5), MFRA) obtains lower RMSEs in both shadow and non-shadow regions, leading to a lower RMSE in “All” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='12 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='26), which demonstrates that combining multi-scale features with dynamically predictive weights via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 8 indeed helps restore the structure better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' \x84 If we do not use the proposed strategy to extract the smallest scale features, the RMSEs in shadow and non- shadow regions increase from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='20 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='42, and from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='71 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='82, respectively, which implies that our strategy avoids heavy information loss during down-sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Feature comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' To further validate the MSFE and MFRA modules, we compare the modified convolution with the standard one in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 4 by showing their processed features (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 4(e) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Clearly, the feature differ- ences between shadow and non-shadow regions after the modified convolution are much smaller than those after the standard convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In addition, we compute the absolute difference between the averages of shadow and shadow- free elements of the 2nd-layer features for each image in the ISTD+ test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Compared with the output features of the standard convolution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', vanilla UNet), those of the pro- posed operation present much smaller absolute differences (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 4(f)), which also demonstrates the effectiveness of the proposed MSFE and MFRA modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 8, AUGUST 2015 14 Input Shadow Shadow Mask MS-GAN [13] P+M+D-Net [12] G2R [29] DC-GAN [30] AEF [8] MStructNet GT Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 8: Quantitative comparison on the ISTD test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Please zoom in to see the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='4 Effectiveness of MStructNet In this section, we first compare MStructNet with existing state-of-the-art shadow removal algorithms on three bench- marks quantitatively and qualitatively in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We then perform ablated experiments on image-level shadow removal to demonstrate the rationality of the multi-level structure exploitation in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1 Comparison with the State-of-the-art Methods Shadow removal evaluation on ISTD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' As shown in Ta- ble 3, MStructNet achieves the lowest RMSE and highest PSNR among all shadow removal methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In particular, MStructNet outperforms △SADC [17], the second best base- line, by 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='8%, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='0%, and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='5% on RMSE in the shadow regions, non-shadow regions, and “All”, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Com- pared with BMNet [27], MStructNet obtains 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='6% and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='8% RMSEs improvements in the shadow regions and “All”, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In addition, the amount of parameters and computational cost (FLOPs) of MStructNet are far lower than most of the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Specifically, with only 15% of parameters and 20% of FLOPS, MStructNet delivers a PSNR improvement of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='54 in “All” when compared with AEF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Al- though Mask-ShadowGAN [13] and DC-ShadowGAN [30] have fewer parameters than MStructNet, they have larger FLOPs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', DC-ShadowGAN [30] has 105G FLOPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Compared with our StructNet, MStructNet presents a much lower RMSE in the non-shadow regions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='35 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='71) with similar results in the shadow regions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='34 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='33), leading to better performance in “All” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='68 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='98).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' More importantly, the model size of MStructNet is three times less than that of StructNet (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='47MB vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='64MB), and the FLOPS is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='5 times smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We show the visualization comparison between MStruct- Net and SOTA methods in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The proposed MStructNet can effectively complement low-level cues by integrating multi-level shadow-free structure features, thus facilitating maximum restoration of the original colors in the umbra and penumbra regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In contrast, other methods either fail to restore the original colors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', MS-ShadowGAN and DC- ShadowGAN), or cause obvious artifacts around the border (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', G2R and Param+M+D-Net).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Shadow removal evaluation on ISTD+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We further con- duct the comparison on the ISTD+ [33] dataset to validate the effectiveness of our algorithm and report the results in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The first row shows the RMSE between the input shadow image and the corresponding GT image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Similar to the results on ISTD, MStructNet achieves the lowest RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In particular, MStructNet outperforms SOTA SP+M+I-Net [34] by 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='7% and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='9% on RMSE in the shadow regions and “All”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' This confirms the effectiveness of the proposed method for employing shadow-free structure information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In addition, MstructNet achieves similar RMSEs in the non- shadow regions and “All”, but with better efficiency (fewer parameters) compared to StructNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Shadow removal evaluation on SRD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' As shown in Ta- ble 5, compared to the SOTA methods, MStructNet achieves the second highest RMSE and is only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='08 lower than BMNet in the shadow regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Although the RMSE of DC-ShadowGAN, aided by soft mask, outperforms ours MStructnNet in the non-shadow regions and “All”, MStruct- Net outperforms it by a large margin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='01, in the shadow regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In comparison to AEF [8], MStructNet decreases the RMSE from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='56 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='69 in the shadow regions, and from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='75 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='28 in the non-shadow regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' This also proves the superiority of MStructNet in recovering the original color and illumination of the shadow regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Refer to the Supplemental for the visual comparison on SRD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2 Efficacy of Different Numbers of Structure Levels Number of convolution layers within each block of the structure-aware encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1, we present the structure-aware encoder which is made up of several blocks JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 8, AUGUST 2015 15 1 2 3 4 5 6 7 8 9 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 9: More visual results of natural images, obtained from the SBU dataset [57] for shadow detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Note that SBU contains only shadow images and masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' So, we pair the inputs (left) and MStructNet prediction (right) here for visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Zoom in to see the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' TABLE 8: Ablation experiment of MStructNet on the ISTD+ dataset, with respect to different structure level utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Structure levels image-level RMSE ↓ # Params (M: 106) # FLOPs (G:109) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1 Shad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='Shad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' All ✓ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content='35 ✓ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='78 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content='46 ✓ ✓ ✓ ✓ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='29 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='73 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='15 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='47 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='72 with each block representing one structure level and con- taining two convolution layers equipped with the MSFE and MFRA modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Note that we set two convolution layers for each block due to the empirical results in Table 6 (as StructNet(MSFE, 2, Add/MFRA) achieves the lowest RMSE in “All”, among all single-convolution based variants).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Number of blocks (or structure levels) in MStructNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1, each block contains MSFE and MFRA modules to form a structure level, and the final MStructNet fuses structures of all different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Here, we study the effects of using different numbers of blocks in the structure-aware encoder to validate the advantages of exploiting the multi-level structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Specifically, we may obtain four variants of MStructNet by using a single structure selected from {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='015, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='045, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1}, and are denoted as: {MStructNet(l)|l ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='015, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='045, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We then add more structures to MStructNet(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='005) gradually to obtain three more variants, denoted as: MStructNet({0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='015}), MStructNet({0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='015, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='045}), MStructNet({0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='015, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='045, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The last version denotes the final version of MStructNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' As reported in Table 8, we can see that: \x82 MStructNet with the structure level 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='015 shows the best results among all single structure Input Shadow MStructNet’s prediction Ground Truth Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 10: The visual illustration of the limitation of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The red box indicates that the object appears only in the shadow regions and no similar appearances found in the non-shadow regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' level variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' \x83 If we add more structure levels, the restoration quality gradually improves and MStructNet with all four structure levels achieves the lowest RMSE in the shadow and non-shadow regions, which demonstrates that the utilization of multi-level shadow structures at feature level can effectively benefit the image-level shadow removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='5 Additional Results In this section, we provide more visual results of natural images in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1 and discuss limitations in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1 Results on Wild Shadow Images We further validate the robustness of our method by test- ing it on real-world images outside the shadow removal datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Specifically, we conduct the inference on the SBU dataset [57] for shadow detection, using MStructNet trained on SRD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 9 shows prediction results of different types JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 8, AUGUST 2015 16 of shadows (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', stand-alone shadows and shadow with occluders) in diverse scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In cases of 1, 4, where not only the cast shadow but also the occluders are present, our method still achieves pleasing visual results with no obvious artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In addition, our method can also handle cases that contain both cast shadows and self-shadows (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', the legs of the dog in case 2 and the bricks in case 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' These results clearly demonstrate the superiority of our method in restoring intrinsic colours and details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Refer to the Supplemental for more visual comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2 Limitations and Failure Cases The proposed algorithms have the similar limitation as CANet [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' This is because both methods heavily rely on the shadow-free information provided by the non-shadow regions to help shadow removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In general, the structure information in the non-shadow regions can be used to propagate the consistent appearance features of shadow- free pixels to shadow regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 10, if our model cannot find corresponding color cues (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', objects with similar appearances) from non-shadow regions, the results may present obvious color bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' One potential way to alleviate this issue is to borrow color cues from global se- mantics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', from cross-samples) and ensure a deterministic color toning, which is included as a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 7 CONCLUSION In this paper, we have systematically investigated the uti- lization and efficacy of image structure for single-image shadow removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' First, we have built vanilla UNet-based networks to restore the shadow-free structure of the input shadow image, and demonstrated that image structure can help enhance the quality of shadow-removed images signif- icantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We have also revealed the limitations of standard convolutions used by the vanilla UNet for structure-level shadow removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Second, we have proposed a novel two- stage removal networks, named structure-informed shadow removal network (StructNet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' It includes two new modules for the utilization of structure information, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', mask-guided shadow-free extraction (MSFE) module and multi-scale feature & residual aggregation (MFRA) module, to extract the image structural features and regularize the feature consistency, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We have shown that StructNet can help im- prove the performances of three state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Third, based on StructNet, we have further proposed a self- contained shadow removal method to fully excavate the potential of multi-level structures at the feature level, named multi-level StructNets (MStructNet), which has fewer param- eters and low computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The extensive results on three public datasets have demonstrated the advantages and effectiveness of StructNet and MStructNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' As a future work, we would like to investigate how to extend the structure-informed network to shadow editing tasks, where occluders are present, such as paired object- shadow segmentation [58] and shadow generation [59], [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' TABLE 9: Architecture of the vanilla UNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Ei and Di are the ab- breviation of the i-th stage in encoder and decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' ‘[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' ]’ denotes the concatenation layer along the channel aix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The k and c in conv[k×k, c] are the kernel size and number of output filters, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' ×j means the number of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Input Output Output size Archtectures E1 [Sl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='Ml] X1 128 × 128 Conv[4×4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 64] ×1 E2 X1 X2 64 × 64 Conv[4×4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 128] ×1 E3 X2 X3 32 × 32 Conv[4×4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 256] ×1 E4 X3 X4 16 × 16 Conv[4×4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 512] ×1 E5 X4 X5 8 × 8 Conv[4×4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 512] ×1 D5 X5 X6 16 × 16 Deconv[4×4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 512] ×1 D4 [X6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' X4] X7 32 × 32 Deconv[4×4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 256] ×1 D3 [X7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' X3] X8 64 × 64 Deconv[4×4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 128] ×1 D2 [X8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' X2] X9 128 × 128 Deconv[4×4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 64] ×1 D1 [X9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' X1] ˆSl 256 × 256 Deconv[4×4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 3] ×1 In this supplementary material,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' we first introduce the detailed configurations of the the vanilla UNet network in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Then, we provide more qualitative comparisons, including comparisons with the state-of-the-arts on SRD test set in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1 and on wild data in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2, and the comparisons of StructNet-variants in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 8 BASIC ARCHITECTURE OF VANILLA UNET We describe at length the architecture of the vanilla UNet φ(·) in Table 9, which consists of three components, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', encoder, decoder and the skip connection between the en- coder and decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The encoder contains 5 blocks, each of which includes a convolution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', E-i in Table 9) for down-sampling, an activation function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', Leaky-ReLU [45]), and a normalization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', Instance Norm [44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The decoder has the similar composition as the encoder, where the convolution is replaced by its transposed counterpart (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', DeConv in Table 9) for up-sampling and the activation is changed to ReLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We set the kernel size, padding and stride of convolution or deconvolution layers as 4, 2 and 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' All skip connections are identical mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Note that the normalization function is not utilized in the first block of encoder and the last block of decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 9 MORE QUALITATIVE COMPARISONS 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='1 Comparisons of SRD test set We qualitatively compare the proposed MStructNet with existing SOTAs, including DSC [10], AR-GAN [53], DHAN [24], DC-ShadowGAN [30] and AEF [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 11, our method yield consistent colour and details along the shadow boundary, while the others either fail to recover the original colour (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', AR-GAN) or leave the obvious shadow traces (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', DSC, DC-ShadowGAN and AEF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Although DC- ShadowGAN has lower RMSE in non-shadow regions than ours, it has higher RMSE in the shadow regions and thus presents an obvious shadow artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In addition, in the 2nd case where the shadow exists in over-saturated colour regions, almost all existing SOTAs fail and generate obvious colour shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' On the contrary, our MStructNet borrows the shadow-free structure from the non-shadow regions JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 8, AUGUST 2015 17 and propagates it to the shadow regions, thus obtaining a consistent colour transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='2 Comparisons of wild shadow images In addition to the comparison results on shadow removal test set, we also compare our MStructNet with SOTAs on wild shadow image that outside of the used dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 12, we show several visual comparison results on the shadow detection dataset SBU [57] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The first and the last cases contain only the shadow, while the other three contain both shadow and the corresponding occluders (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', human and eaves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Our MStructNet achieves satisfactory re- sults without noticeable shadow artifacts in all cases, while others still produce easily observable shadow remnants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='3 Comparisons of StructNet-variants As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 13, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 14, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 15, we present the qualitative comparison of the different approaches (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', ST- CGAN [11], AEF [8], SADC [17]) before and after equipping the StructNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' The RMSE values for each sample are also given in red text in the corresponding results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' We can clearly see that the colour shift in the baseline methods are adjusted, and thus the shadow trace gets significantly improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Even though AEF and SADC have achieved ex- cellent performance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', shadow residuals and colour shifts are slight), the enhanced StructNet-variants ((i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', StructNet- AEF and StructNet-SADC)) that incorporate our predicted shadow-free structure (see the last column) still achieve better result enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Overall, the results demonstrate that the restored shadow-free structure indeed benefits the image-level shadow removal and StructNet can enhance the state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', SBU [57] dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content=' Please zoom in to see the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Input Shadow AEF [8] StructNet-AEF GT Predicted Structure Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 14: Quantitative comparison of StructNet-AEF on ISTD+ test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Please zoom in to see the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content='60 RMSE=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content='38 RMSE=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='14 RMSE=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='58 RMSE=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content=' Yu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Dong, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Xiao, “Arshad- owgan: Shadow generative adversarial network for augmented reality in single light scenes,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content=' [60] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+page_content=' Niu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Zhang, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Zhang, “Shadow generation for composite image in real-world scenes,” AAAI, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Yuhao Liu received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='Sc de- grees from Zhengzhou University and the Dalian University of Technology in 2019 and 2022, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' He is a first-year PhD candidate ma- joring in computer science at the City University of Hong Kong now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' His current research inter- ests focus on low-level computer vision problems and image editing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Qing Guo received Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' degree in computer application technology from the School of Com- puter Science and Technology, Tianjin Univer- sity, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' He was a research fellow with the Nanyang Technology University, Singapore, from Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 2019 to Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 2020 and Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 2021 to Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' He was assigned as the Wallenberg- NTU Presidential Postdoctoral Fellow with the Nanyang Technological University, Singapore, from Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 2020 to Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' He is currently a research scientist at Center for Frontier AI Research (CFAR), Agency for Science, Technology, and Research (A*STAR), Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' His research interests include computer vision, AI security, and image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' He is a member of IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Lan Fu received the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' degree in Com- puter Science and Engineering from University of South Carolina, Columbia, SC, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Prior to that, she received the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' degree in Biomedi- cal Engineering from Tianjin University, Tianjin, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Currently, she is a Senior Research En- gineer with the InnoPeak Technology Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=', Palo Alto, CA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Her research interests include computer vision, deep learning, and image pro- cessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Zhanghan Ke is currently a PhD candidate at the City University of Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' He obtained B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' from Northeastern University (China).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' He serves as a reviewer for several computer vi- sion conferences (CVPR, ICCV, ECCV, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=') and journals (TPAMI, IJCV, TCSVT, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' His current research interests include semi-/self-supervised learning and its applications in computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Ke Xu is currently with the City University of Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' He received the dual Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' degrees from Dalian University of Technology and City University of Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' He also served as a program committee member/reviewer for several CV and AI conferences and journals, including CVPR, ICCV, ECCV, NeurIPS, ICLR, ICML, IJCV and TIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' His research interests include deep learning and image enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Wei Feng received the PhD degree in computer science from City University of Hong Kong in 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' From 2008 to 2010, he was a research fellow at the Chinese University of Hong Kong and City University of Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' He is now a full Professor at the School of Computer Sci- ence and Technology, College of Computing and Intelligence, Tianjin University, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' His major research interests are active robotic vision and visual intelligence, specifically including active camera relocalization and lighting recurrence, general Markov Random Fields modeling, energy minimization, active 3D scene perception, SLAM, video analysis, and generic pattern recog- nition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Recently, he focuses on solving preventive conservation problems of cultural heritages via computer vision and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' He is the Associate Editor of Neurocomputing and Journal of Ambient Intelligence and Humanized Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 8JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' 8, AUGUST 2015 22 Ivor W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Tsang is director of A*STAR Centre for Frontier AI Research (CFAR) since Jan 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Previously, he was a Professor of Artificial In- telligence, at University of Technology Sydney (UTS), and Research Director of the Australian Artificial Intelligence Institute (AAII), the largest AI institute in Australia, which is the key player to drive the University of Technology Sydney to rank 10th globally and 1st in Australia for AI research, in the latest AI Research Index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Prof Tsang is working at the forefront of big data an- alytics and Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' His research focuses on transfer learn- ing, deep generative models, learning with weakly supervision, big data analytics for data with extremely high dimensions in features, samples and labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In 2013, Prof Tsang received his ARC Future Fellowship for his outstanding research on big data analytics and large-scale machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In 2019, he received the International Consortium of Chinese Mathematicians Best Paper Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In 2020, he was recognized as the AI 2000 AAAI/IJCAI Most Influential Scholar in Australia between 2009 and 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' His research on transfer learning was awarded the Best Student Paper Award at CVPR 2010 and the 2014 IEEE TMM Prize Paper Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' In addition, he received the IEEE TNN Outstanding 2004 Paper Award in 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Recently, Prof Tsang was conferred the IEEE Fellow for his outstanding contributions to large-scale machine learning and transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Besides these, Prof Tsang serves as the Editorial Board for the Journal of Machine Learning Research, Machine Learning, Journal of Artificial Intelligence Research, IEEE Transactions on Pattern Analysis and Machine Intelligence, IEEE Transactions on Artificial In- telligence, IEEE Transactions on Big Data, and IEEE Transactions on Emerging Topics in Computational Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' He serves as a Senior Area Chair/Area Chair for NeurIPS, ICML, AAAI and IJCAI, and the steering committee of ACML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Rynson W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Lau received his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' degree from University of Cambridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' He was on the faculty of Durham University and is now with City University of Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Rynson serves on the Editorial Board of the International Journal of Computer Vision (IJCV) and IET Computer Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' He has served as the Guest Editor of a number of journal special is- sues, including ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' on Internet Technol- ogy, IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' on Multimedia, IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' on Visualization and Computer Graphics, and IEEE Computer Graphics & Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' He has also served in the committee of a number of conferences, including Program Co-chair of ACM VRST 2004, ACM MTDL 2009, IEEE U-Media 2010, and Conference Co-chair of CASA 2005, ACM VRST 2005, ACM MDI 2009, ACM VRST 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
+page_content=' Rynson’s research interests include computer graphics and computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE1T4oBgHgl3EQfcQR-/content/2301.03182v1.pdf'}
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+1
+Federated Multi-Agent Deep Reinforcement
+Learning Approach via Physics-Informed Reward
+for Multi-Microgrid Energy Management
+Yuanzheng Li, Member IEEE, Shangyang He, Yang Li, Senior Member IEEE,
+Yang Shi, Fellow IEEE, and Zhigang Zeng, Fellow IEEE
+Abstract—The utilization of large-scale distributed renewable
+energy promotes the development of the multi-microgrid (MMG),
+which raises the need of developing an effective energy man-
+agement method to minimize economic costs and keep self
+energy-sufficiency. The multi-agent deep reinforcement learning
+(MADRL) has been widely used for the energy management
+problem because of its real-time scheduling ability. However, its
+training requires massive energy operation data of microgrids
+(MGs), while gathering these data from different MGs would
+threaten their privacy and data security. Therefore, this paper
+tackles this practical yet challenging issue by proposing a
+federated multi-agent deep reinforcement learning (F-MADRL)
+algorithm via the physics-informed reward. In this algorithm, the
+federated learning (FL) mechanism is introduced to train the F-
+MADRL algorithm thus ensures the privacy and the security
+of data. In addition, a decentralized MMG model is built, and
+the energy of each participated MG is managed by an agent,
+which aims to minimize economic costs and keep self energy-
+sufficiency according to the physics-informed reward. At first,
+MGs individually execute the self-training based on local energy
+operation data to train their local agent models. Then, these
+local models are periodically uploaded to a server and their
+parameters are aggregated to build a global agent, which will be
+broadcasted to MGs and replace their local agents. In this way,
+the experience of each MG agent can be shared and the energy
+operation data is not explicitly transmitted, thus protecting the
+privacy and ensuring data security. Finally, experiments are
+conducted on Oak Ridge national laboratory distributed energy
+control communication lab microgrid (ORNL-MG) test system,
+and the comparisons are carried out to verify the effectiveness of
+introducing the FL mechanism and the outperformance of our
+proposed F-MADRL.
+Index Terms—Multi-microgrid, multi-agent deep reinforce-
+ment learning, federated learning, proximal policy optimization.
+This work is supported in part by the National Natural Science Foundation
+of China (Grant 62073148), in part by Key Project of National Natural Science
+Foundation of China (Grant 62233006), and in part by Key Scientific and
+Technological Research Project of State Grid Corporation of China (Grant
+No. 1400-202099523A-0-0-00). (Corresponding author: Yang Li)
+Y. Z. Li and Z. G. Zeng are with School of Artificial Intelligence and
+Automation, Key Laboratory on Image Information Processing and Intelligent
+Control of Ministry of Education, Huazhong University of Science and Tech-
+nology, Wuhan 430074, and also with China-Belt and Road Joint Laboratory
+on Measurement and Control Technology, Wuhan, China, 430074 (Email:
+Yuanzheng Li@hust.edu.cn, zgzeng@hust.edu.cn).
+S. Y. He is with China-EU Institute for Clean and Renewable Energy,
+Huazhong University of Science and Technology, Wuhan 430074, China
+(Email: heshangyang10@hust.edu.cn).
+Y. Li is with School of Electrical Engineering, Northeast Electric Power
+University, Jilin 132012, China (Email:liyang@neepu.edu.cn).
+Y. Shi is with the Department of Mechanical Engineering, University of
+Victoria, Victoria, BC V8P 5C2, Canada (E-mail:yshi@uvic.ca).
+I. INTRODUCTION
+In recent years, renewable energy (RE) has been widely
+deployed, such as wind power and photovoltaic. Unlike tra-
+ditional power plants, RE resources are usually distributed.
+Therefore, microgrids (MGs) have been paid much attention
+to utilize the RE. Note that the MG usually works in a local
+area, and provides the required electricity for a small entity,
+such as a school, a hospital, or a community [1]–[4].
+Normally, the main target of MG is to achieve the self-
+sufficiency of energy via the utilization of RE. However,
+due to its limited capacity, the MG has to take the risk of
+power shortage. Specifically, since the user demand and RE
+are dependent on the user behavior and weather condition,
+the power demand may exceed the capacity of MG while
+RE generation may be insufficient, thus causing the power
+shortage [5]. For this reason, numerous adjacent MGs are
+interconnected to form a multi-microgrid (MMG) system.
+Compared with an isolated MG, the MMG system is more
+capable of utilizing RE because of its larger capacity. Besides,
+although these MGs belong to different entities, the energy
+is allowed to be traded among different MGs, i.e., each MG
+can actively sell its surplus power when its power generation
+exceeds the demand, or purchase power from other MGs when
+the generation is insufficient [6]. Therefore, the MMG is more
+promising to achieve energy self-sufficiency compared with an
+isolated MG.
+However, because of the complexity of energy management
+of the MMG, it is essential to adopt an effective scheme. The
+present studies of MMG energy management can be mainly
+categorized into two types, i.e., the centralized and decen-
+tralized schemes. The former one is based on a centralized
+energy management center, which could get access to the
+related energy information of all MGs in the MMG system
+[7], [8]. Then, this center can well make decisions to achieve
+the energy self-sufficiency of the MMG system. However, note
+that the multiple MGs usually belong to different entities, and
+it is difficult for the centralized management center to acquire
+operation data of all MGs due to the increasing awareness of
+privacy protection.
+Therefore, a more popular research direction is the decen-
+tralized MMG management scheme. For instance, Ng et al.
+have proposed the concept of MMG control, which uses the
+multi-agent approach to achieve the decentralized control of
+each MG [9]. In addition, Yang et al. have adopted multiple
+arXiv:2301.00641v1 [eess.SY] 29 Dec 2022
+
+2
+self-decision agents replacing the energy management center
+for the energy self-sufficiency of participated MGs [10]. Liu
+et al. have treated the MMG system as a fully distributed
+optimization model, which is solved by a robust optimal
+scheduling algorithm [11]. Moreover, Ref. [12] has proposed
+a multi-agent MMG system, where the individual agent of
+each MG collects the data from local units and performs
+optimization separately. In addition, Ref. [13] has proposed the
+MG agents to well utilize the partially observed information,
+for achieving the optimal energy management of MMG.
+The aforementioned literature focuses on building accurate
+optimization models, which can be summarized as the model-
+based approach. However, there exists an essential drawback,
+i.e., the model-based approach is merely suited for the prede-
+termined scheduling solution rather than a real-time decision.
+In other words, the predetermined scheduling is difficult to
+handle emergencies or the unexpected change in the load
+demand occurring in the MMG system.
+To tackle this problem, the learning-based approach has
+been developed in recent years [14], [15]. Benefiting from
+the development of the physics-informed deep learning tech-
+niques, the outputs of the black-box model are more gen-
+eralized and interpretable [16]. One of the most representa-
+tive approaches is multi-agent deep reinforcement learning
+(MADRL), which is widely deployed in the MMG energy
+management problem due to its nature of interacting with
+the physical characteristic of the real world [17], [18]. For
+instance, The MADRL used in Ref. [19] observes the tem-
+perature, energy generation and other physical parameters to
+control soft load and transaction effectively for MMG. The
+experiments demonstrate the convergence of these algorithms
+and emphasize the outperformance of the actor-critic algo-
+rithm. Ref. [20] proposes an energy management approach
+that takes advantage of a multi-agent model-free reinforcement
+learning algorithm. This distributed and hierarchical decision
+mechanism effectively increases the energy self-sufficiency of
+MMG. Besides, a MADRL method is adopted in Ref. [21] to
+realize the post-disaster resilience of distributed MG system.
+Aiming to increase the income of the system, the MADRL
+shows its strong adaptability in different conditions through
+experiments. Moreover, the implementation of MADRL would
+significantly increase the autonomy of each MG [22] [23].
+For instance, a MADRL framework based on the deep neural
+network is proposed in Ref. [22] to improve the operational
+performance and autonomy of each participant MG. Ref. [23]
+sets the agents in different MGs for the distributed control
+and achieve higher MG autonomy. To balance the benefits of
+the MMG participants and guarantee the efficiency, Ref. [24]
+proposes an equilibrium selection multi-agent reinforcement
+learning algorithm based on Q-learning to promote the auton-
+omy of MG operation.
+However, since the MADRL technology requires massive
+data to train the MG agent, the concern of user privacy is
+raised. The data of the users can be utilized to analyze their
+habits and even their life tracks. In the case of MMG energy
+management, to train an effective agent with a high generaliza-
+tion, massive energy operation data should be collected from
+different MGs. However, although each MG aims to pursure a
+better performance through experiences sharing, they may be
+not willing to submit their processing data because of privacy
+awareness [22]. On the other hand, the security during data
+transmission cannot be guaranteed.
+To tackle the above issues, we introduce an emerging
+distributed learning approach, namely federated learning (FL),
+for training MADRL in the MMG energy management via
+physics-informed reward [25]. In other words, we apply the
+FL to protect user privacy and guarantee data security while
+ensuring the generalization of each MG agent in the MMG
+system. Specifically, each MG is controlled by an agent, which
+deploys a recent deep reinforcement model, namely proximal
+policy optimization (PPO) [26]. Each agent firstly executes
+the self-training according to the local energy operation data
+of each MG to maximize the physics-informed reward, i.e.,
+the economic operation and self energy-sufficiency. Then,
+the agents upload their local model parameters, such as the
+weights and biases of the model, to a server. After that, these
+parameters are aggregated by the server to construct a global
+model, which will be broadcasted to each MG and replace the
+local model. In this way, agents share their experiences through
+the FL mechanism, which thus enhances their generalization1
+compared with the local training. Moreover, the FL mechanism
+only requires model parameters, and the operation data of each
+MG would stay locally. Therefore, the user privacy and data
+security can be guaranteed.
+The main contributions of this paper are presented as
+follows.
+(1) A MMG system model is developed for the deployment
+of FL, where each MG contains conventional generators
+(CGs), batteries (BAs), renewable energy generators (REGs),
+load and the energy management center. Then, a server is
+introduced to implement the FL mechanism which can com-
+municate with MGs and aggregates the parameters of the MG
+agent, such as the weight and bias of the neural network
+models. Since the server would not perform as a center of
+MMG that guides the decisions of each MG, MGs would
+endow a high autonomy and suffer from less risk of privacy
+leakage.
+(2) A federated multi-agent deep reinforcement learning
+algorithm (F-MADRL) is proposed for the energy management
+of the MMG system. Each MG has an agent that collects the
+operation data for self-training. Then, the agent parameters
+are uploaded to the server and aggregated to a global agent.
+Afterwards, the agent of each MG is replaced by the global
+one. In this way, the privacy of each MG user can be protected.
+(3) A physics-informed reward is developed by orienting
+targets of the MG agent, i.e., the economic operation and the
+self energy-sufficiency. The MG agents trained through the
+physics-informed reward would be endowed with a better in-
+terpretation of action because of the consideration of physical
+targets.
+1Since the MGs in the MMG belong to different kinds of entities, their local
+operation data manifest the perference of local users. Thus the agent trained
+by local data would be confronted of performances decline when operating
+in other MGs, and this phenoma is termed as the generalization decrease of
+the MG agent.
+
+3
+(4) Case studies conducted on the Oak Ridge national labo-
+ratory distributed energy control communication lab microgrid
+(ORNL-MG) test system [27] demonstrate that our proposed
+F-MADRL algorithm is effective under different demands
+and renewable energy scenarios. Moreover, we verify that F-
+MADRL outperforms other state-of-the-art DRL algorithms
+under the distributed MMG model.
+The remainder of this paper is organized as follows. Sec-
+tion II introduces the theoretical basis of the reinforcement
+learning. In Section III, a decentralized MMG model is built.
+Section IV proposes the F-MADRL algorithm, and provides
+its overall structure and technical details. In Section V, com-
+prehensive case studies are conducted. Finally, Section VI
+concludes this paper.
+II. THEORETICAL BASIS OF REINFORCEMENT LEARNING
+Normally, the Markov decision process (MDP) is defined by
+a five-tuple ⟨S, A, P, R, γ⟩, where S is the finite state space
+that stands for all valid states and A represents the finite set of
+actions. P = {p(st+1|st, at)} stands for the set of transition
+probability from state st to st+1, and R = r(st, at), R ∈
+R; S × A → R is termed as the reward function, which is
+normally the metric to evaluate the action. γ ∈ [0, 1] indicates
+the discount factor, which represents the importance of the
+present reward [18], [28].
+To solve the MDP, a policy π should be developed to
+provide the probability of executing action a when observing
+the state s, i.e. π(a|s) = P[At = a|St = s]. The aim of π
+is to maximize the discounted cumulative reward during the
+finite time T, which is termed as the return function:
+Ut =
+T
+�
+k=t
+γk−tr(sk, ak)
+(1)
+where r(sk, ak) is the reward function, which calculates the
+reward value under state sk with action ak; γ ∈ [0, 1] is
+the discount factor, representing the importance of the future
+reward [29]. Then, two kinds of value functions are defined
+based on Ut to help the policy make decisions. The first is the
+state value function Vπ(s) and the other is the action value
+function Qπ(a, s), which are formulated as follows:
+Vπ(s) = Eπ[Ut|St = s]
+=
+�
+a
+π(a|s)
+�
+s′
+P a
+ss′[r(s, a) + γVπ(s
+′)]
+(2)
+Qπ(a, s) = Eπ[Ut|St = s, At = a]
+=
+�
+s′
+P a
+ss′[r(s, s′|a) + γ
+�
+a′
+Qπ(a′, s′)]
+(3)
+where Vπ(s) stands for the expectation of future reward at
+the state s, and the Qπ(a, s) represents the future expected
+reward when selecting an action a at state s. s′ and a′ stand
+for the possible reaching state and action at state s. P a
+ss′ is
+the transition probability from s to s′ under a. In fact, Vπ(s)
+and Qπ(a, s) are used to evaluate the quality of the state s
+and the action-state pair (a, s), respectively. They are updated
+according to above two equations and help the policy π decide
+whether reaching the state or executing the action.
+III. THE DECENTRALIZED MULTI-MICROGRID ENERGY
+MANAGEMENT MODEL
+The decentralized MMG system includes numerous MGs
+that are connected to a distribution power network. Usually,
+an energy management center is set in each MG, which
+performs as an agent to conduct self-training and control the
+dispatchable elements, such as conventional generators (CGs),
+batteries (BAs), etc. In this section, to describe the energy
+management model of the MMG system more clearly, we
+firstly introduce the isolated MG model with the MDP format
+before developing the MMG model.
+A. The Isolated Microgrid Energy Management Model
+Fig. 1 illustrates the structure of the isolated MG model
+and a real-world MG system case. Normally, as shown in Fig.
+1(a), a MG is constructed by five types of elements: renewable
+power generators, BA, CG, conventional load (CL) and energy
+management center. Note that BAs and CGs are dispatchable
+since their outputs are controlled by the management center.
+On the contrary, because of the high uncertainties of RE, the
+outputs of REG cannot be controlled. Additionally, the energy
+management center is termed as the agent that controls these
+dispatchable elements by observing the state of MG operation.
+Following this structure, the Oak Ridge national laboratory
+distributed energy control communication lab microgrid test
+system (ORNL-MG) is selected as the real-world case in this
+paper, which is illustrated in Fig. 1(b).
+Fig. 1. The structure of (a) an isolated MG and (b) the ORNL-MG [27].
+
+(a)
+Micro Turbine
+Conventional Generator
+Diesel Engine
+Battery
+Energy
+Management
+Center
+Wind Turbine
+Load
+RenewablePower Generator
+Photovoltaic
+(b)
+ORNLSubstation
+(161to13.8kV)
+LVA
+FromTVA
+erORNL13.8kV
+10Substation
+From Circuit #2
+3.8 to 2.4kV)
+，4000
+3000Substation
+(13.8 to 2.4kV)
+ElectricalServicefrom750kVATransformers
+Static
+From Circuit #4
+Rotating
+estArea4
+1) Conventional Generator: It can be seen from Fig. 1 that
+the CG includes diesel engine generator and micro turbine,
+which generate power through fossil fuels. The cost functions
+of CGs can be represented as follows:
+C(PCG,i) = aCG,iP 2
+CG,i + bCG,iPCG,i + cCG,i
+(4)
+P min
+CG,i ≤ PCG,i ≤ P max
+CG,i
+(5)
+where C(PCG,i) represents the generation cost of ith CG, and
+PCG,i is its generation power. aCG,i, bCG,i and cCG,i denote the
+cost coefficients of the ith CG. P min
+CG,i and P max
+CG,i are the lower
+and upper bounds of the ith CG.
+2) Renewable Energy Generator: Fig. 1 presents two kinds
+of REGs, namely wind turbine and photovoltaic. The gener-
+ation of REG normally depends on the natural environment
+such as wind speed, temperature, weather and solar irradiance
+[30]. Since the REGs do not consume any fossil fuels, their
+generation costs are not considered in this paper.
+3) Battery: As one of the most commonly used energy
+storage devices, BA can store energy generated by CGs and
+REGs, and release it when needed. Thus, the BA has two
+operation states, namely charging and discharging, which are
+represented by the transition of its state of charge, and can be
+formulated as follows:
+SOCt+1 = (1 − δ)SOCt −
+P t
+BA
+ηchCBA
+(6)
+SOCt+1 = (1 − δ)SOCt − ηdchP t
+BA
+CBA
+(7)
+where SOCt and SOCt+1 denote the charging state of BA
+at time t and t + 1. PBA is the charging-discharging power of
+BA. Here, we assume PBA > 0 when the BA is discharging,
+and PBA < 0 when the BA is charging. The ηch and ηdch
+are the charging and discharging efficiencies. δ denotes the
+discharging rate, which is set as 0.2%. CBA represents the
+capacity of BA.
+The operation of BA would bring about the costs due to the
+amortized purchase and maintenance, which is formulated by
+the following equation [31]:
+C(PBA,j) =aBA,j(PBA,j + 3P max
+BA,j(1 − SOC))2
++ bBA,j(PBA,j + 3P max
+BA,j(1 − SOC)) + cBA,j
+(8)
+P max
+BA,j < PBA,j < P max
+BA,j
+(9)
+where C(PBA,j) represents the cost of the jth BA. aBA,j,
+bBA,j and cBA,j are cost coefficients of the jth BA. P max
+BA,j and
+P min
+BA,j are the upper and lower bounds of BA output power.
+4) Network Power Loss of MG: Practically, there exists the
+power loss because of the operation of generators and the
+transmission of energy in the MG. The power loss usually
+corresponds to the active generation power and can be esti-
+mated as follows [32]:
+λCG = ∂Ploss
+∂PCG
+, λREG = ∂Ploss
+∂PREG
+, λBA = ∂Ploss
+∂PBA
+(10)
+where λCG, λREG and λBA represent the power loss coefficients
+of CG, REG and BA, respectively. According to Ref. [32],
+λCG, λREG and λBA are recommended to be set in [0.01, 0.02].
+Therefore, they are set as 0.02 in this paper.
+Then, the power loss Ploss can be given by the following
+equation [32]:
+Ploss =
+nCG
+�
+i=1
+λCGPCG,i +
+nREG
+�
+j=1
+λREGPREG,j +
+nBA
+�
+k=1
+λBAPBA,k
+(11)
+where nCG, nREG and nBA are the numbers of CGs, REGs and
+BAs in the isolated MG, respectively.
+B. Isolated MG Energy Management Model via MDP and
+Physics-Informed Reward
+Since the energy management center of the MG is an agent
+which is trained by the DRL algorithm, the above isolated MG
+model should be reformulated as the MDP model. In addition,
+considering the physical feasibility of the agent, the definition
+of reward is designed to integrate the physical-informed rules,
+which are presented as follows.
+1) State: In this paper, we consider a 24-hour scheduling
+of the MG, and each hour is denoted by t ∈ {1, 2, ..., 24}.
+The state of MG at time t includes the energy operation
+information, which is defined as follows:
+st = {P t−1
+L
+, P t−1
+REG,1, ..., P t−1
+REG,nREG, SOCt−1, Et−1
+λ
+}
+(12)
+where st indicates the state of MG at time t; P t−1
+L
+and P t−1
+REG,i
+stand for the load demand and the ith REG at time t − 1. In
+addition, the Et−1
+λ
+is the electricity price in the transaction
+between the MG and the distribution power network.
+2) Action: The action at is generated by the agent, which
+controls the power outputs of the CGs and BAs at each time t,
+according to the state st. In this study, it is defined as follows:
+at = {P t
+CG,1, ..., P t
+CG,nCG, P t
+BA,1, ..., P t
+BA,nBA}
+(13)
+In DRL, the agent is normally a neural network, which is
+difficult to produce consistent and feasible in the early training
+stage. Therefore, the actions are enforced to fulfill the output
+constraints provided in Eqs (5) and (9)
+P t
+CG,i = clip(P t
+CG,i, P min
+CG,i, P max
+CG,i), i ∈ [1, nCG]
+(14)
+P t
+BA,j = clip(P t
+BA,j, P min
+BA,j, P max
+BA,j), j ∈ [1, nBA]
+(15)
+where clip(t, tmin, tmax) is the clip function, which returns tmax
+if t > tmax, and tmin if t < tmin.
+3) Reward: The design of the reward significantly impacts
+the performance of the DRL training. A specific physical task-
+oriented reward would endow interpretability to the strategy
+of the agent [16]. However, in some classical reinforcement
+learning tasks, such as CartPole [33] and Atari Games [34],
+the design of their rewards is independent of the physical
+characteristic of the problem. For instance, in CartPole, the
+reward is set as 0 if the action is available. Such the intuitive
+design of reward may mislead the agent thus slowing down the
+training process and decreasing the interpretability of the agent
+strategy, it is not suitable for the MG energy management. Nor-
+mally, the MG agent is expected to operate economically while
+ensuring the self energy-sufficiency. Therefore, considering
+the physical characteristic of the MG, the reward is designed
+as physics-informed to satisfy the two explicit targets, i.e.,
+the training of the agent and realizing the requirements of
+
+5
+operation cost and self energy-sufficiency, simultaneously. The
+reward function is defined as follows.
+rt = − wC
+�nCG
+�
+i=1
+C(P t
+CG,i) +
+nBA
+�
+j=1
+C(P t
+BA,j)
+�
+− wdeEl(t) × abs(P t
+de)
+(16)
+where rt is the reward value at time t, and El(t) ≥ 0 indicates
+the price of purchasing electricity from the distribution power
+grid. wC ∈ [0, 1] and wde ∈ [0, 1] indicate the weights to limit
+the order of magnitude of reward. abs(·) stands for the absolute
+function. P t
+de evaluates the deviation between load demand and
+real generation, which is formulated by:
+P t
+de = P t
+L −
+�nCG
+�
+i=1
+P t
+CG,i +
+nREG
+�
+j=1
+P t
+REG,j +
+nBA
+�
+k=1
+P t
+BA,k − P t
+loss
+�
+(17)
+In
+this
+study,
+the
+physics-informed
+reward
+is
+com-
+posed
+of
+two
+physical
+targets
+of
+MG,
+i.e.,
+operation
+costs and self energy-sufficiency. They are formulated as
+��nCG
+i=1 C(P t
+CG,i) + �nBA
+j=1 C(P t
+BA,j)
+�
+, and El(t) × abs(P t
+de),
+respectively. To keep the order of magnitude of the reward
+consistent, the self energy-sufficiency is designed as P t
+de times
+El(t). Since the reward is related to the physical valuables, i.e.,
+P t
+CG,i, P t
+BA,j and P t
+de, it can be endowed the physical meaning.
+In this way, the reward is able to guide the agent to produce
+a series of actions that minimize the generation costs of CGs
+and BAs while ensuring self energy-sufficiency.
+C. Decentralized Multi-Microgrid Energy Management Model
+As shown in Fig. 2, a decentralized MMG model that
+contains np MGs is considered in this paper. These MGs
+are connected to the distribution power network, and the
+energy transaction between MGs is also allowed. Each MG
+is controlled by an agent, which observes the state st of MG
+and provides the action at.
+Fig. 2. The structure of MMG system.
+Since the MG is encouraged to maximize the physics-
+informed reward rt for achieving energy self-sufficiency and
+economic operation, the target of the MMG should be the
+maximum of the systematic rewards rsys,t, which can be
+represented by the sum of rewards obtained by all the MG
+agents. The rsys,t is given by
+rsys,t =
+np
+�
+i=1
+ri
+t =
+np
+�
+i=1
+−ϵi × abs(P t
+i,de)
+(18)
+where ri
+t represents the reward obtained by the ith MG agent at
+time t. ϵi and P t
+i,de are the shrinkage coefficient and deviation
+of MG i.
+Besides, since the load demand of MG cannot be known
+in advance, excessive or insufficient power generation of an
+isolated MG is unavoidable, thus the energy transaction in the
+MMG system is inevitable. Therefore, the energy transaction
+mechanism between different MGs is developed, which is
+given below.
+That is, MG is allowed to conduct energy transactions with
+the distribution power network and other MGs, as shown in
+Fig. 2. If the generated power of MG i exceeds its load demand
+at time t, the excess energy will be sold to other MGs with
+a price Ei(t). If the demand of MG i cannot be satisfied, the
+MG will purchase electricity from MG j, which has the lowest
+price of the whole participated MGs.
+j = arg min
+l
+El(t) × Ll, l ∈ [1, 2, ..., np]
+(19)
+where Ll indicates whether the generation of MG l exceeds
+its demand. The Ll is set as 1 if the demand is exceeded or
+set as infinite if not. The MGs will preferentially purchase
+the surplus power generated by other MGs. When the MG
+generations are fully consumed, the distribution power network
+will provide power with price Edpn(t), which is usually higher
+than El(t), l ∈ [1, 2, ..., np].
+IV. FEDERATED MULTI-AGENT DEEP REINFORCEMENT
+LEARNING ALGORITHM
+As discussed above, the MMG is decentralized, thus each
+MG agent has a high autonomy. However, the decentralized
+structure threatens the generalization performance of the agent,
+because the diversity of the data in isolated MG is limited,
+which may make the agent getting trap into a local optima.
+To tackle this issue, we propose a federated multi-agent deep
+reinforcement learning (F-MADRL) algorithm. The FL is used
+to improve the generalization of the agent during training
+while ensuring data privacy.
+There are two characteristics in the FL, one is called
+participant and the other is termed as the collaborator. The
+participant j, j ∈ [1, np], is denoted as a neural network
+model f j
+wj. It conducts self-training at the local and uploads
+its parameters wj to the collaborator periodically, where np is
+the number of participants which are processed in parallel.
+Constrained by the data privacy, the participant f j
+wj only
+trains on the local dataset, which may cause the insufficient
+training since the capacity and diversity of the data are limited.
+The FL could tackle this problem through the following
+steps. First, at the training epoch e, e ∈ [1, Ne], the model
+of jth participant is defined as f j
+we
+j , which conducts self-
+training to obtain the parameters we
+j , where Ne is the total
+number of training epoches. Then, each participant uploads
+its parameters to the collaborator and constructs a parameter
+list we =
+�
+we
+1, we
+2, ..., we
+np
+�
+. The collaborator calculates
+the weight average of we to estimate a global model f e+1
+G
+with parameters we+1
+G
+. After aggregation, the collaborator
+broadcasts we+1
+G
+to all the participants and replaces their own
+parameters, i.e., we+1
+G
+= we+1
+1
+= we+1
+2
+... = we+1
+np . The
+
+Distribution
+power
+network
+Energy and price
+MG2
+MG n
+MG1
+MA
+INA
+INA6
+aggregating mechanism of FL is formulated by the following
+equations:
+we+1
+j
+= we
+G − η∇Fj(we
+j ), ∀j
+(20)
+we+1
+G
+=
+np
+�
+j=1
+1
+np
+we+1
+j
+(21)
+where η and Fj(·) are the learning rate and local loss function
+of the jth participant, respectively.
+In this paper, the participant can be considered as the agent
+in each MG and the collaborator is a server that takes the
+responsibility for aggregating and broadcasting the parame-
+ters. The F-MADRL aims to solve the following distributed
+optimization model:
+min
+we
+G
+F(we
+G) =
+np
+�
+j=1
+pjFj(we
+j )
+(22)
+where F(·) is the global loss function. pj represents the
+relative weight of each MG agent on the global model, and
+pj > 0, �np
+k=1 pj = 1. We set pj = |Dj|/ �np
+j=1 |Dj|, where
+|Dj| is the data size used for the local training of jth MG. Note
+that F(·) cannot be directly computed without the information
+sharing of each participant.
+The overall structure of F-MADRL is illustrated in Fig. 3.
+At the epoch e, the agent in three MGs are firstly replaced
+by the global agent in the (e − 1)th epoch. Then, the three
+MG agents conduce self-training to obtain parameters, which
+are uploaded to the server for aggregation. Next, the global
+agent would be built on the server, and the parameters will be
+broadcasted to the MG agents for the (e + 1)th epoch.
+Fig. 3.
+The proposed federated multi-agent deep reinforcement learning
+algorithm.
+It can be learned from the figure that the F-MADRL
+contains two parts. One is executed on server, which can be
+considered as the collaborator and the other is executed on MG
+agents, can be considered as the participant. The procedures
+running on the server and MG agent are provided in the
+following subsections, respectively.
+A. The F-MADRL: Server Part
+The F-MADRL proceed on the server mainly focus on the
+aggregating and broadcasting of the agent parameters, and its
+procedure is shown in Algorithm 1.
+At the beginning of the training epoch of F-MADRL, the
+server would build a global agent with the parameter w0
+G,
+which is then broadcasted to each MG agent for self-training.
+Since the agents update their parameters in parallel, the server
+aggregates the parameters list we = [we
+1, we
+2, ..., we
+np] by Eq.
+(21). Furthermore, the aggregated parameters we+1
+G
+are used
+to update the global model parameters and broadcast to the
+MG agents for the training of epoch e + 1.
+Algorithm 1 The federated multi-agent deep reinforcement
+learning algorithm on the server.
+1: Execute on the server:
+2: Initialize the model parameters w0
+G and broadcast them
+to the MG agents.
+3: for Global epoch e = 1 to Ne do
+4:
+for MG agent j = 1 to np parallelly do
+5:
+Update the MG parameter we
+j at the local agent.
+6:
+Store the we
+j .
+7:
+Upload the we
+j to the server
+8:
+end for
+9:
+Receive the parameters from each MG agent and
+construct
+10:
+we ← [we
+1, we
+2, ..., we
+np]
+11:
+Aggregating the model parameters through
+12:
+we+1
+G
+= �np
+j=1
+1
+np we+1
+j
+.
+13:
+Broadcast the we+1
+G
+to other MG agents.
+14: end for
+B. The F-MADRL: MG Agent
+On the other hand, the MG agent adopts the self-training in
+the procedure of F-MADRL and cooperates with the server.
+When the MG agents receive the parameter we
+G from global
+model at the epoch e, their parameters are replaced by we
+G,
+i.e., we
+j = we
+G. Then, each MG agent executes Ni individual
+self-training epochs in parallel. Afterwards, the parameters of
+the MG agent at the last self-training epoch, namely θNi, µNi
+are stored and uploaded to the server.
+In this paper, each MG agent performs self-training with a
+famous deep reinforcement learning algorithm, namely PPO,
+to obtain the optimal policy π. There are two types of deep
+neural networks, namely, actor and critic, defined by the
+MG agent. Actor πθ is parameterized by θ, which aims to
+produce the action, and the critic is denoted as V µ, which is
+parameterized by µ.
+The overall training process of the self-training pro-
+cedure
+during
+one
+episode
+is
+illustrated
+in
+Fig.
+4.
+First
+of
+all,
+the
+experiment
+tuples
+T
+are
+sampled
+T = {⟨s0, a0, r0, s1⟩, ⟨s1, a1, r1, s2⟩, ..., ⟨sU, aU, rU, sU+1⟩}
+where U indicates the length of T. Then, the loss function
+of the actor at kth episodes is calculated, which is defined as
+follows:
+LC = Es,a∼T
+�
+min( πθ
+k(a | s)
+πθ
+k−1(a | s)A
+πθ
+k
+s,a,
+clip( πθ
+k(a | s)
+πθ
+k−1(a | s), 1 − ϵ, 1 + ϵ)A
+πθ
+k
+s,a)
+�
+(23)
+where Es,a∼T [·] represents the empirical average over the
+sampled experiment tuples T. The πk−1 and πk stand for
+
+Upload w
+MG1
+AgentofMGl
+Aggregation
+Upload we
+Global Agent
+Server
+MG2
+Agent of MG2
+M
+Upload w
+e+1
+Broadcast
+e+1
+G
+MG3
+AgentofMG37
+Fig. 4. The procedure of the self-training of MG agent.
+the previous and new policy, respectively. The ϵ is the clip
+parameter. Aπk
+s,a stands for the advantage, which measures if
+the action is worth taking by comparing the action value and
+state value:
+Aπk
+st,at = Eτ[U|s0 = st, a0 = at]−V (st) = Q(st, at)−V (st)
+(24)
+However, the Aπk
+s,a cannot be directly obtained since
+Q(st, at) is difficult to be determined. In this way, the gen-
+eralized advantage estimation method is implemented in this
+study:
+Aπθ
+k
+s,a = δV
+0 + (γλ)δV
+1 + (γλ)2δV
+2 , ..., +(γλ)U−t+1δV
+U−1 (25)
+where γ ∈ [0, 1] and λ ∈ [0, 1] represent the discount factor
+and a hyperparameter that adjusts the tradeoff between bias
+and variance of the estimation. Note that the variance would
+be increased when raising λ while the bias is decreased
+accordingly. According to the recommendation of Ref. [26],
+λ is set as 0.95. δV
+k is calculated by:
+δV
+k = rk + γV µ
+k (st+1) − V µ
+k (st)
+(26)
+where V µ
+k (st+1) and V µ
+k (st) are given by the critic, which is
+trained by the loss function LV :
+LV = Es,a∼T
+�
+(γV µ
+k (st+1) + r(st, at) − V µ
+k (st))2�
+.
+(27)
+With the above equations, the parameter θ and µ of the actor
+and the critic can be updated by the following equations:
+θk+1 = θk + ηπ∇θkLC
+(28)
+µk+1 = µk + ηV ∇µkLV
+(29)
+where ηπ and ηV are the learning rates of actor and critic.
+Since both LC and LV are optimized in each MG agent of
+the proposed F-MADRL algorithm, they are the local loss
+functions which construct the global loss following Eq. (22).
+Overall, the F-MADRL algorithm applied on the server can
+be summarised in the Algorithm 2.
+C. The Theoretical Convergence Analysis of the F-MADRL
+In this section, the convergence of the F-MADRL is eval-
+uated. At first, the following assumptions considering the
+function Fk, k ∈ [1, np] are made, by referring to Ref. [35].
+Assumption 1: The Fk is L-smooth, ∀w, w′, ∥∆Fk(w) −
+∆Fk(w′)∥2
+≤
+L∥w − w′∥;
+Fk(w)
+≤
+Fk(w′) +
+∇Fk(w′)T (w − w′) + L
+2 ∥w′ − w∥2
+2.
+Algorithm 2 The federated multi-agent deep reinforcement
+learning algorithm on the MG agent.
+1: Execute on each MG agent:
+2: Parallel running on j, j ∈ [1, np] agent at global epoch e
+3: Receive the parameters from the server we
+j ← we
+G
+4: for Individual training epoch i = 1 to Ni do
+5:
+Collect the experience tuple T = {⟨s0, a0, r0, s1⟩,
+⟨s1, a1, r1, s2⟩, ..., ⟨sU, aU, rU, sU+1⟩}
+6:
+Compute the discounted factor:
+7:
+δV
+k ← rk + γV µ
+k (st+1) − V µ
+k (st)
+8:
+Estimate the advantage:
+9:
+Aπθ
+k
+s,a ← δV
+0 +(γλ)δV
+1 +(γλ)2δV
+2 , ..., +(γλ)U−t+1δV
+U−1
+10:
+Calculate the loss function of the actor:
+11:
+LC ← Es,a∼T
+�
+min( πθ
+i (a|s)
+πθ
+i−1(a|s)Aπθ
+i
+s,a, clip( πθ
+i (a|s)
+πθ
+i−1(a|s), 1−
+12:
+ϵ, 1 + ϵ)Aπθ
+i
+s,a)
+�
+13:
+Update the actor parameter:
+14:
+θi+1 ← θi + ηπ∇s,a∼T LC
+15:
+Calculate the loss function of the critic:
+16:
+LV ← Es,a∼T
+�
+(γV µ
+i (st+1) + r(st, at) − V µ
+i (st))2�
+17:
+Update the critic parameter:
+18:
+µi+1 ← µi + ηV ∇µiLV
+19: end for
+20: Store the network parameters. we
+j ← {θNi, µNi}
+21: Upload the parameters we
+j to the server.
+Assumption
+2:
+The
+Fk
+is
+µ-strongly
+convex,
+∀w,
+w′,
+Fk(w)− µ
+2 ∥Fk∥2 is convex;
+Fk(w)
+≥
+Fk(w′) +
+∇Fk(w′)T (w − w′) + µ
+2 ∥w′ − w∥2
+2.
+Based on the above Assumptions, we have the following
+Lemmas.
+Lemma 1: F is µ-strongly convex and L-smooth.
+Proof: Straightforwardly from Assumption 1 and Assump-
+tion 2, in line with the definition of convex, F is the finite-sum
+of the Fk, thus it is µ-strongly convex and L-smooth as well.
+Lemma 2: ∀w, w′ ∈ Rn and wt = w + t(w − w′) for
+t ∈ [0, 1]. Then,
+F(w) − F(w′) =
+� 1
+0
+∇F(wt)T (w′ − w)dt
+(30)
+and
+F(w) − F(w′) − ∇F(w)T (w′ − w)
+=
+� 1
+0
+(∇F(wt) − ∇F(w))T (w′ − w)dt
+(31)
+Proof:
+Eq. (30) follows the fundamental theorem of
+calculus. Eq. (31) follows from Eq. (30) by subtracting
+∇F(w)T (w′ − w) from both sides of the equation.
+Lemma 3: If F is smooth and µ-strongly convex for µ > 0,
+then for the w∗ = arg min
+w
+F(w),
+1
+2µ∥∇F(w)∥2
+2 ≥ F(w) − w∗ ≥ µ
+2 ∥w − w∗∥2
+2
+(32)
+
+sample
+Environment
+Experiments Tuple
+action
+ 0 represents the learning rate. Then, we yield
+F(wk+1) − (F(wk) + η∥∇F(wk)∥2
+2) ≤ η2L
+2 ∥∇F(wk)∥2
+2 (42)
+if we pick η = 1
+L, then F(wk+1) ≤ F(wk)− 1
+2L∥∇F(wk)∥2
+2.
+From Lemma 3, ∥∇F(w)∥2
+2 ≥ 2µ(F(w) − F(w∗)). When
+putting them together,
+F(wk+1) − F(w∗) ≤ F(wk) − F(w∗) − 1
+2L∥∇F(wk)∥2
+2
+≤ F(wk) − F(w∗)
+− µ
+L(F(wk) − F(w∗))
+= (1 − µ
+L)(F(wk) − F(w∗))
+(43)
+Repeatedly applying this bound yields
+F(wk) − F(w∗) ≤
+�
+1 − µ
+L
+�k
+(F(w0) − F(w∗))
+(44)
+Using the fact that 1+x ≤ ex, the convergence rate is given
+by picking k ≥
+L
+µ log( F (w0)−F (w∗)
+ϵ
+), where ϵ = F(wk) −
+F(w∗), denoting the error between the loss at epoch k and
+the optimal one.
+V. CASE STUDY
+A. Experiment Setup
+In this section, we conduct case studies based on the
+modified Oak Ridge National Laboratory Distributed Energy
+Control Communication lab microgrid test system to demon-
+strate the effectiveness of the proposed F-MADRL algorithm.
+Without loss of generality, three MGs are applied to form the
+MMG system and the parameters of elements in each MG
+are provided in Table.I. A wind turbine and a PV panel are
+set as the REGs and the corresponding power data is referred
+from Ref. [36]. The forecast errors of wind and PV power
+is assumed to be independent of Gaussian distribution with a
+15% standard variation [36]. In addition, the time horizon of
+the experiment is set as 24-hour schedule, and the time interval
+is set to be 1h. The forecast total load demands and the day-
+ahead market price of each MG are provided in the Table. II
+and the forecast error of the load is assumed to follow the
+Gaussian distribution with a 3% standard variation [37] [38].
+It can be learned from the load data that the power demands
+of MG1 surpass its maximum capacity, which means the MG1
+operates in the energy self-insufficient state. On the contrary,
+the MG2 and MG3 operate in the energy self-sufficient state.
+As for the F-MADRL, the training epoch is set as 1500.
+Moreover, γ and λ are set as 0.99 and 0.95, receptively [29].
+A famous neural network optimizer, Adam [26], is used to
+update the F-MADRL, and the learning rate of actor ηπ and
+critic ηV are set as 0.0001 and 0.001. Note that all simulation
+studies are conducted using Python 3.6.8 with PyTorch 1.7.1.
+TABLE I
+THE PARAMETERS SETTING OF EACH MG
+a($/kW)
+b($/kW)
+c($/kW)
+Pmin(kW)
+Pmax(kW)
+MG1
+CG
+0.0081
+5.72
+63
+0
+200
+BA
+0.0153
+5.54
+26
+-50
+50
+MG2
+CG
+0.0076
+5.68
+365
+0
+280
+BA
+0.0163
+5.64
+32
+-50
+50
+MG3
+CG
+0.0095
+5.81
+108
+0
+200
+BA
+0.0173
+5.74
+38
+-50
+50
+B. Analysis of the F-MADRL algorithm
+In this section, the proposed F-MADRL is applied to the
+MMG system and its performance evaluation is reported. Fig.
+5 presents the reward curve of the three MG agents, respec-
+tively. During the whole training epochs, the FL mechanism
+is applied every 500 epochs and the training process can be
+separated into three phases. Since the FL mechanism averages
+the parameters of the MG agents, the reward value of each
+
+9
+TABLE II
+THE FORECASTING WIND AND PV POWER, TRADING PRICE AND LOADS OF THE THREE MGS.
+Hour
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+Wind power (kW)
+51.48
+38.37
+43.56
+40.75
+27.74
+30.15
+28.65
+23.38
+21.75
+34.82
+27.17
+30.20
+PV outputs (kW)
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+0.16
+1.77
+5.30
+11.60
+36.64
+42.68
+Trading price among MG and
+distribution network ($/kW)
+8.65
+8.11
+8.25
+8.10
+8.14
+8.13
+8.34
+9.35
+12.00
+9.19
+12.30
+20.70
+Trading price among MGs ($/kW)
+4.33
+4.06
+4.13
+4.05
+4.07
+4.07
+4.17
+4.68
+6.00
+4.60
+6.15
+10.35
+Load of MG1 (kW)
+457.70
+336.50
+274.90
+272.60
+245.30
+233.70
+274.60
+291.00
+315.70
+362.40
+320.00
+350.00
+Load of MG2 (kW)
+110.50
+109.85
+112.45
+110.50
+113.75
+120.25
+130.00
+157.95
+165.10
+169.00
+173.55
+168.35
+Load of MG3 (kW)
+124.71
+123.98
+126.91
+124.71
+128.38
+135.43
+146.72
+178.26
+186.33
+190.73
+195.87
+190.00
+Hour
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24
+Wind power (kW)
+23.52
+39.48
+35.74
+18.06
+24.27
+26.26
+26.77
+26.22
+32.84
+36.02
+37.23
+44.12
+PV outputs (kW)
+35.22
+35.46
+34.83
+23.62
+14.18
+4.67
+0.18
+0.00
+0.00
+0.00
+0.00
+0.00
+Trading price among MG and
+distribution network ($/kW)
+26.82
+27.35
+13.81
+17.31
+16.42
+9.83
+8.63
+8.87
+8.35
+16.44
+16.19
+8.87
+Trading price among MGs ($/kW)
+13.41
+13.68
+6.91
+8.66
+8.21
+4.92
+4.32
+4.44
+4.18
+8.22
+8.10
+4.44
+Load of MG1 (kW)
+345.20
+320.60
+333.20
+316.80
+291.30
+413.80
+539.80
+557.20
+557.10
+535.00
+437.80
+447.30
+Load of MG2 (kW)
+168.35
+165.75
+170.30
+172.25
+165.75
+164.25
+162.50
+165.75
+169.00
+161.20
+148.00
+119.60
+Load of MG3 (kW)
+190.00
+187.07
+192.20
+194.40
+187.07
+185.60
+183.40
+187.07
+190.73
+181.93
+161.39
+134.98
+Fig. 5.
+The reward curve of (a) MG1 agent (b) MG2 agent and (c) MG3
+agent during the training.
+agent would dramatically change in the two adjacent phases
+and all three MG agents receive benefits from this.
+For instance, depending on the parameter setting of each
+MG, the MG1 agent wouldn’t converge in the first phase.
+However, after applying the FL mechanism, its reward raises
+at the 500th epoch and converges at -15059.85 at the end of
+phase 2. Besides, although the MG2 agent converges in phase
+1, the FL mechanism substantially renews its parameters and
+further raises the reward to -3524.99 in the second phase. As
+for the MG3 agent, the benefit of the FL mechanism is mainly
+shown in the third phase, which helps the agent escape from
+the local optimal converged at phase 2 and finally achieve
+a higher reward at the end of the training. In summary, the
+FL mechanism can be used to get rid of the local optimum,
+which is caused by insufficient training data due to privacy
+constraints.
+Then, the policies obtained by the F-MADRL are applied to
+determine the scheduling of each MG. Specifically, Figs. 6∼8
+denote the scheduling of MG1, MG2 and MG3, respectively.
+Each figure includes two kinds of graphs, where the above
+graph is the scheduling solution and the lower graph shows
+Fig. 6. The scheduling of MG1 obtained by the F-MADRL algorithm.
+the unbalanced demands, namely the difference between gen-
+eration and load demands of the MG. The positive unbalanced
+demands indicate the generation of the MG surpasses its load
+demands, which means the demands are satisfied while the
+negative one means the demands are unsatisfied.
+As shown in Fig. 6, since the demands of MG1 are
+higher than its capacity, the MG1 operates in the energy
+self-insufficient state. Therefore, the demands of MG1 are all
+unsatisfied during the 24 hours. Moreover, the scheduling of
+BA generation obtained by the agent mainly considers the
+power balance between its charging and discharging. In the 1st
+hour, the power demand of MG1 is 457.7 kW, which is higher
+than the capacity of MG1, thus the agent chooses to discharge
+the battery for demand supply. Since MG1 operates in the
+energy self-insufficient state, it requires external power from
+other MGs and distribution power system, which is achieved
+by the transaction mechanism in the MMG system. Even the
+unbalanced demands of MG1 are as high as -402 kW at 20:00,
+these power shortages can be supplied by other MGs and the
+distribution power network. This is the reason why the MG1
+agent does not fully operate its CG and BA all the time. The
+agent learns that the energy transaction is more economic than
+generating power by itself.
+
+Phase 1
+Phase 2
+Phase3
+(a)
+-18000
+MG1
+MG2
+MG3
+-36000
+Iteration:472
+Reward: -71536.09
+-54000
+Iteration: 956
+Reward:-15059.85
+Iteration:1328Reward:-15059.85
+-72000
+(b)
+-8000
+lard
+16000
+ew
+Iteration:1437
+Reward:-3415.94
+Iteration: 942
+Reward: -3524.99
+-24000
+Iteration: 480
+Reward: -5664.51
+-32000
+[()
+-5000
+-10000
+Iteration:1406
+Reward:-3743.82
+Iteration: 938
+Reward: -6164.79
+-15000
+Iteration:482
+Reward: -5952.39
+-20000
+0
+200
+400
+600
+800
+1000
+1200
+1400
+IterationPCG
+PBA10
+Fig. 7. The scheduling of MG2 obtained by the F-MADRL algorithm.
+Fig. 8. The scheduling of MG3 obtained by the F-MADRL algorithm.
+The scheduling policies of the MG2 agent and MG3 agent
+are similar but different from that of MG1. As illustrated in
+Fig. 7 and Fig. 8, since MG2 and MG3 work in an energy self-
+sufficient state, their power generations of CG could almost
+satisfy their demand. In this way, to shrink the generation
+costs, the two agents perform a similar strategy in the operation
+of BA. In the most time of the 24 hours, the power outputs
+of BA are almost 0. Alternatively, the power of CGs would
+change with the demands of MGs. For example, in the 15th
+and 16th hours, the generation and demand of MG2 are
+nearly equal, thus only causing minor unbalanced demands in
+these two hours. Besides, the absolute values of unbalanced
+demands during MG2 and MG3 operation are lower than 50
+kW. Those unbalanced demands can be eliminated according
+to the transaction mechanism in the MMG. In this way, the
+excess powers are sold to other MGs that work in a self-
+insufficient state and the power shortages can be supplied by
+the power from other MGs or the distribution power network.
+The above experiments show the efficiency of introducing a
+federated learning mechanism in the proposed F-MADRL al-
+gorithm. The MG agents trained by F-MADRL make efficient
+scheduling solutions regardless of whether the MG operates
+in the energy self-sufficient or self-insufficient state.
+Fig. 9.
+The changes of the decomposition of physics-informed reward
+according to the training iterations.
+C. The Interpretation of the Agents Performance
+It should be noted that the physics-informed reward would
+bring about the interpretation of the strategy of agents, to
+some degree. The three aspects of the reward function, i.e., the
+costs of CG and BA and the balanced demand, are illustrated
+along with iterations to study their changes. In this section,
+six iterations located at a different phase of the training are
+selected, namely the 1st, 50th, 500th, 700th, 900th and 1400th
+iterations. Wherein, the 1st and 50th iterations would present
+the training performance in the early stages, the 500th iteration
+is located at the end of the first federated phase and the other
+three iterations are set at the convergent state of the reward
+value. Note that the values of the reward increase along with
+the increase of the six iterations.
+Fig. 9 illustrates the changes of the unbalanced demands,
+costs of CG and cost of BA in the subfigure (a), (b) and (c),
+respectively. As shown in Fig. 9(a), the unbalanced demand
+of the MG1 decreases from about 5500kW to 3000kW at
+the 1400th iteration. Besides, the figures for MG2 and MG3
+also present a downward trend, they start at 3000kW and
+1000kW at 1st iteration and descend to lower than 500kW after
+1400 iterations. This means the agents could learn strategies
+for energy self-sufficiency as much as possible. Besides, Fig.
+9(b) and (c) present the different strategies of MG agents
+when operating in the energy self-sufficient and energy self-
+insufficient state. As illustrated in Fig. 9(b), the CG costs
+of MG2 and MG3 decrease whereas MG1 raises. A similar
+situation can also be observed in Fig. 9(c), the costs of BA
+for MG1 increase from around $6000 to $25000. The BA costs
+for MG2 and MG3 continuously decrease from about $10000
+and dramatically increase at 900th and 1400th iterations.
+Overall, the MG agents trained by our proposed F-MADRL
+algorithm could satisfy the target of the MMG system, namely,
+the energy self-sufficient with the designed physics-informed
+reward. The algorithm has endowed the explainability by
+analyzing the performance of the MG agents from the angle
+of the physics-informed reward.
+
+PBA
+PcG
+iuPCG
+PBA6000
+ (kW
+(a)
+MG1
+-MG2
+Demand
+4500
+3000
+Unbalanced j
+1500
+50
+500
+700
+900
+1400
+60000
+(b)
+50000
+G
+40000
+C
+JO
+30000
+Cost
+20000
+10000
+50
+500
+700
+900
+1400
+L(c)
+25000
+20000
+BA
+15000
+JO1
+Cost
+10000
+5000
+1
+50
+500
+700
+900
+1400
+Iteration11
+D. Performance Comparison
+In this section, the effectiveness of introducing the FL
+mechanism is demonstrated by comparing the performance of
+the proposed F-MADRL with other algorithms that merely
+implement the self-training of MG agents. These algorithms
+include numerous well-known deep reinforcement learning
+algorithms, namely, PPO, A2C and TRPO. Since they are
+conducted on the multiple MG agents, these comparative algo-
+rithms are termed PPO-MADRL, A2C-MADRL and TRPO-
+MADRL, respectively.
+The comparisons are conducted from two aspects, i.e., the
+convergence and the generalization. First, the convergences of
+the MG agents are illustrated by their reward curves. Besides,
+to compare the generalization of the MG agents, the testing
+rewards are set as the metric, which are obtained by testing the
+agents in both the energy self-sufficient and self-insufficient
+states. The MG agents are easy to get trapped in local optima
+since they are trained by the local operation data, which merely
+contain the perference of the local user. Conesequently, the
+testing reward is applied to well verify the generalization of
+the F-MADRL. Note that the physics-informed reward is a
+sufficient index for the performance comparison, since the
+economic cost and power balance are measured in the reward,
+simultaneously.
+Fig. 10. The training reward of each MG agent.
+Fig. 10 compares the convergence of each algorithm, and
+the subplots (a), (b) and (c) represent the training process of
+the MG1 agent, MG2 agent and MG3 agent, respectively.
+As shown in this figure, the A2C-MADRL performs the
+worst, under the same learning rate and training epoch, the
+agents trained by A2C-MADRL cannot be well converged.
+In addition, although the PPO-MADRL and TRPO-MADRL
+are able to train the converge agents, their rewards are lower
+than that of F-MADRL because of lacking the mechanism of
+sharing information.
+In addition to the reward curves, the generalization of F-
+MADRL is verified in Table. III, which presents the test
+rewards of each MG agent in the energy self-sufficient and
+TABLE III
+THE TEST REWARDS OF THE MG AGENTS
+Working State
+Agent
+F-MADRL
+PPO-MADRL
+A2C-MADRL
+TRPO-MADRL
+Energy self-sufficient
+MG1
+-29017
+-32727
+-56732
+-46231
+MG2
+-9529
+-12361
+-19757
+-21818
+MG3
+-2935
+-5832
+-9782
+-5691
+Energy self-insufficient
+MG1
+-28650
+-54881
+-54834
+-29517
+MG2
+-10276
+-23854
+-22151
+-32756
+MG3
+-3172
+-5900
+-5005
+-3842
+self-insufficient state. In this figure, the value of the best test
+reward under each algorithm is bold. It can be learnt that the
+test rewards obtained by F-MADRL are -29017, -9529 and
+-2935 for the three MG agents under an energy self-sufficient
+state, which surpasses other comparative algorithms. Besides,
+the test rewards of three MG agents trained by F-MADRL are -
+28650, -10276 and -3172, which perform the best in the energy
+self-insufficient state along with the comparative algorithms as
+well. In addition, since the performance of the F-MADRL is
+better than those of comparative algorithms in both energy
+self-sufficient and self-insufficient states, it can be concluded
+that the F-MADRL has a better generalization performance.
+The comparisons clarify the introduction of the FL mech-
+anism leads to performance diversity between the proposed
+F-MADRL and the other three algorithms. Since the MG
+agents trained by PPO-MADRL, A2C-MADRL and TRPO-
+MADRL are only based on the local operation data of MG
+due to the limitation of privacy and thus causing lower diver-
+sity of the training data. Consequently, the decision-making
+ability of MG agents would decline. The introduction of the
+FL mechanism alleviates this drawback. By using FL, the
+experiences of MG agents can be shared without threatening
+user privacy and data security. In this way, the generalization
+ability of the agent would be improved, as verified in the
+above experiments. Therefore, the comparisons conducted in
+this section demonstrate the effectiveness of introducing the
+FL mechanism in the MADRL algorithm and also reveal a
+better generalization of F-MADRL.
+VI. CONCLUSION
+This paper proposes a federated multi-agent deep rein-
+forcement learning algorithm for the multi-microgrids system
+energy management. A decentralized MMG model is built
+first, which includes numerous isolated MGs and an agent
+is used to control the dispatchable elements of each MG
+to achieve the physics-informed reward. Due to the privacy
+protection and data security, the F-MADRL is implemented
+to train the agents. First, each agent adopts the self-training.
+Then, the FL mechanism is introduced to build a global
+agent that aggregates the parameters of all local agents on
+the server and replaces the local MG agent with the global
+one. Therefore, the experiences of each agent can be shared
+without threatening the privacy and data security.
+The case studies are conducted on a MMG with three
+isolated MGs. The convergence and the performance of F-
+MADRL are illustrated first. Then, explanations of the strat-
+egy of the three MG agents are presented by decomposing
+the physics-informed reward under different iterations. After-
+wards, by comparing with PPO-MADRL, A2C-MADRL and
+
+1e4
+(a)
+-2
+-3
+-4
+M
+-5
+F-MADRI
+-6
+PPO-MADRL
+A2C-MADRL
+-7
+TRPO-MADRL
+-8
+(b)
+-0.5
+-1.0
+-1.5
+-2.0
+-2.5
+3.0
+-3.5
+-4.0
+(c)
+0.5
+-1.0
+-1.5
+-2.0
+-2.5
+400
+600
+800
+1200
+0
+200
+1000
+1400
+Iterations12
+TRPO-MADRL, the F-MADRL achieves higher test rewards,
+which means a better generalization. Therefore, it indicates the
+performance enhancement of introducing the FL mechanism
+in the MADRL and also demonstrates the effectiveness of
+our proposed F-MADRL. In this paper, the uncertainty of
+renewable energy is not considered because its complexity
+will cause difficulties in the training of F-MADRL and reduce
+the accuracy of the MG agent strategy. This issue is worth
+investigating in our future work.
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf,len=1058
+page_content='1 Federated Multi-Agent Deep Reinforcement Learning Approach via Physics-Informed Reward for Multi-Microgrid Energy Management Yuanzheng Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Member IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Shangyang He,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Yang Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Senior Member IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Yang Shi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Fellow IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' and Zhigang Zeng,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Fellow IEEE Abstract—The utilization of large-scale distributed renewable energy promotes the development of the multi-microgrid (MMG),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' which raises the need of developing an effective energy man- agement method to minimize economic costs and keep self energy-sufficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The multi-agent deep reinforcement learning (MADRL) has been widely used for the energy management problem because of its real-time scheduling ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' However, its training requires massive energy operation data of microgrids (MGs), while gathering these data from different MGs would threaten their privacy and data security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Therefore, this paper tackles this practical yet challenging issue by proposing a federated multi-agent deep reinforcement learning (F-MADRL) algorithm via the physics-informed reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' In this algorithm, the federated learning (FL) mechanism is introduced to train the F- MADRL algorithm thus ensures the privacy and the security of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' In addition, a decentralized MMG model is built, and the energy of each participated MG is managed by an agent, which aims to minimize economic costs and keep self energy- sufficiency according to the physics-informed reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' At first, MGs individually execute the self-training based on local energy operation data to train their local agent models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Then, these local models are periodically uploaded to a server and their parameters are aggregated to build a global agent, which will be broadcasted to MGs and replace their local agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' In this way, the experience of each MG agent can be shared and the energy operation data is not explicitly transmitted, thus protecting the privacy and ensuring data security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Finally, experiments are conducted on Oak Ridge national laboratory distributed energy control communication lab microgrid (ORNL-MG) test system, and the comparisons are carried out to verify the effectiveness of introducing the FL mechanism and the outperformance of our proposed F-MADRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Index Terms—Multi-microgrid, multi-agent deep reinforce- ment learning, federated learning, proximal policy optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' This work is supported in part by the National Natural Science Foundation of China (Grant 62073148), in part by Key Project of National Natural Science Foundation of China (Grant 62233006), and in part by Key Scientific and Technological Research Project of State Grid Corporation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 1400-202099523A-0-0-00).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' (Corresponding author: Yang Li) Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Li and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Zeng are with School of Artificial Intelligence and Automation, Key Laboratory on Image Information Processing and Intelligent Control of Ministry of Education, Huazhong University of Science and Tech- nology, Wuhan 430074, and also with China-Belt and Road Joint Laboratory on Measurement and Control Technology, Wuhan, China, 430074 (Email: Yuanzheng Li@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='cn, zgzeng@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' He is with China-EU Institute for Clean and Renewable Energy, Huazhong University of Science and Technology, Wuhan 430074, China (Email: heshangyang10@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Li is with School of Electrical Engineering, Northeast Electric Power University, Jilin 132012, China (Email:liyang@neepu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Shi is with the Department of Mechanical Engineering, University of Victoria, Victoria, BC V8P 5C2, Canada (E-mail:yshi@uvic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='ca).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' INTRODUCTION In recent years, renewable energy (RE) has been widely deployed, such as wind power and photovoltaic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Unlike tra- ditional power plants, RE resources are usually distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Therefore, microgrids (MGs) have been paid much attention to utilize the RE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Note that the MG usually works in a local area, and provides the required electricity for a small entity, such as a school, a hospital, or a community [1]–[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Normally, the main target of MG is to achieve the self- sufficiency of energy via the utilization of RE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' However, due to its limited capacity, the MG has to take the risk of power shortage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Specifically, since the user demand and RE are dependent on the user behavior and weather condition, the power demand may exceed the capacity of MG while RE generation may be insufficient, thus causing the power shortage [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' For this reason, numerous adjacent MGs are interconnected to form a multi-microgrid (MMG) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Compared with an isolated MG, the MMG system is more capable of utilizing RE because of its larger capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Besides, although these MGs belong to different entities, the energy is allowed to be traded among different MGs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=', each MG can actively sell its surplus power when its power generation exceeds the demand, or purchase power from other MGs when the generation is insufficient [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Therefore, the MMG is more promising to achieve energy self-sufficiency compared with an isolated MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' However, because of the complexity of energy management of the MMG, it is essential to adopt an effective scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The present studies of MMG energy management can be mainly categorized into two types, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=', the centralized and decen- tralized schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The former one is based on a centralized energy management center, which could get access to the related energy information of all MGs in the MMG system [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Then, this center can well make decisions to achieve the energy self-sufficiency of the MMG system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' However, note that the multiple MGs usually belong to different entities, and it is difficult for the centralized management center to acquire operation data of all MGs due to the increasing awareness of privacy protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Therefore, a more popular research direction is the decen- tralized MMG management scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' For instance, Ng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' have proposed the concept of MMG control, which uses the multi-agent approach to achieve the decentralized control of each MG [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' In addition, Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' have adopted multiple arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='00641v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='SY] 29 Dec 2022 2 self-decision agents replacing the energy management center for the energy self-sufficiency of participated MGs [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' have treated the MMG system as a fully distributed optimization model, which is solved by a robust optimal scheduling algorithm [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Moreover, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' [12] has proposed a multi-agent MMG system, where the individual agent of each MG collects the data from local units and performs optimization separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' In addition, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' [13] has proposed the MG agents to well utilize the partially observed information, for achieving the optimal energy management of MMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The aforementioned literature focuses on building accurate optimization models, which can be summarized as the model- based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' However, there exists an essential drawback, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=', the model-based approach is merely suited for the prede- termined scheduling solution rather than a real-time decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' In other words, the predetermined scheduling is difficult to handle emergencies or the unexpected change in the load demand occurring in the MMG system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' To tackle this problem, the learning-based approach has been developed in recent years [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Benefiting from the development of the physics-informed deep learning tech- niques, the outputs of the black-box model are more gen- eralized and interpretable [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' One of the most representa- tive approaches is multi-agent deep reinforcement learning (MADRL), which is widely deployed in the MMG energy management problem due to its nature of interacting with the physical characteristic of the real world [17], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' For instance, The MADRL used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' [19] observes the tem- perature, energy generation and other physical parameters to control soft load and transaction effectively for MMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The experiments demonstrate the convergence of these algorithms and emphasize the outperformance of the actor-critic algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' [20] proposes an energy management approach that takes advantage of a multi-agent model-free reinforcement learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' This distributed and hierarchical decision mechanism effectively increases the energy self-sufficiency of MMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Besides, a MADRL method is adopted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' [21] to realize the post-disaster resilience of distributed MG system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Aiming to increase the income of the system, the MADRL shows its strong adaptability in different conditions through experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Moreover, the implementation of MADRL would significantly increase the autonomy of each MG [22] [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' For instance, a MADRL framework based on the deep neural network is proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' [22] to improve the operational performance and autonomy of each participant MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' [23] sets the agents in different MGs for the distributed control and achieve higher MG autonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' To balance the benefits of the MMG participants and guarantee the efficiency, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' [24] proposes an equilibrium selection multi-agent reinforcement learning algorithm based on Q-learning to promote the auton- omy of MG operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' However, since the MADRL technology requires massive data to train the MG agent, the concern of user privacy is raised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The data of the users can be utilized to analyze their habits and even their life tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' In the case of MMG energy management, to train an effective agent with a high generaliza- tion, massive energy operation data should be collected from different MGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' However, although each MG aims to pursure a better performance through experiences sharing, they may be not willing to submit their processing data because of privacy awareness [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' On the other hand, the security during data transmission cannot be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' To tackle the above issues, we introduce an emerging distributed learning approach, namely federated learning (FL), for training MADRL in the MMG energy management via physics-informed reward [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' In other words, we apply the FL to protect user privacy and guarantee data security while ensuring the generalization of each MG agent in the MMG system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Specifically, each MG is controlled by an agent, which deploys a recent deep reinforcement model, namely proximal policy optimization (PPO) [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Each agent firstly executes the self-training according to the local energy operation data of each MG to maximize the physics-informed reward, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=', the economic operation and self energy-sufficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Then, the agents upload their local model parameters, such as the weights and biases of the model, to a server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' After that, these parameters are aggregated by the server to construct a global model, which will be broadcasted to each MG and replace the local model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' In this way, agents share their experiences through the FL mechanism, which thus enhances their generalization1 compared with the local training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Moreover, the FL mechanism only requires model parameters, and the operation data of each MG would stay locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Therefore, the user privacy and data security can be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The main contributions of this paper are presented as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' (1) A MMG system model is developed for the deployment of FL, where each MG contains conventional generators (CGs), batteries (BAs), renewable energy generators (REGs), load and the energy management center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Then, a server is introduced to implement the FL mechanism which can com- municate with MGs and aggregates the parameters of the MG agent, such as the weight and bias of the neural network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Since the server would not perform as a center of MMG that guides the decisions of each MG, MGs would endow a high autonomy and suffer from less risk of privacy leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' (2) A federated multi-agent deep reinforcement learning algorithm (F-MADRL) is proposed for the energy management of the MMG system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Each MG has an agent that collects the operation data for self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Then, the agent parameters are uploaded to the server and aggregated to a global agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Afterwards, the agent of each MG is replaced by the global one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' In this way, the privacy of each MG user can be protected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' (3) A physics-informed reward is developed by orienting targets of the MG agent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=', the economic operation and the self energy-sufficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The MG agents trained through the physics-informed reward would be endowed with a better in- terpretation of action because of the consideration of physical targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 1Since the MGs in the MMG belong to different kinds of entities, their local operation data manifest the perference of local users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Thus the agent trained by local data would be confronted of performances decline when operating in other MGs, and this phenoma is termed as the generalization decrease of the MG agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 3 (4) Case studies conducted on the Oak Ridge national labo- ratory distributed energy control communication lab microgrid (ORNL-MG) test system [27] demonstrate that our proposed F-MADRL algorithm is effective under different demands and renewable energy scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Moreover, we verify that F- MADRL outperforms other state-of-the-art DRL algorithms under the distributed MMG model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Sec- tion II introduces the theoretical basis of the reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' In Section III, a decentralized MMG model is built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Section IV proposes the F-MADRL algorithm, and provides its overall structure and technical details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' In Section V, com- prehensive case studies are conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Finally, Section VI concludes this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' THEORETICAL BASIS OF REINFORCEMENT LEARNING Normally, the Markov decision process (MDP) is defined by a five-tuple ⟨S, A, P, R, γ⟩, where S is the finite state space that stands for all valid states and A represents the finite set of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' P = {p(st+1|st, at)} stands for the set of transition probability from state st to st+1, and R = r(st, at), R ∈ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' S × A → R is termed as the reward function, which is normally the metric to evaluate the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' γ ∈ [0, 1] indicates the discount factor, which represents the importance of the present reward [18], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' To solve the MDP, a policy π should be developed to provide the probability of executing action a when observing the state s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' π(a|s) = P[At = a|St = s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The aim of π is to maximize the discounted cumulative reward during the finite time T, which is termed as the return function: Ut = T � k=t γk−tr(sk, ak) (1) where r(sk, ak) is the reward function, which calculates the reward value under state sk with action ak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' γ ∈ [0, 1] is the discount factor, representing the importance of the future reward [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Then, two kinds of value functions are defined based on Ut to help the policy make decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The first is the state value function Vπ(s) and the other is the action value function Qπ(a, s), which are formulated as follows: Vπ(s) = Eπ[Ut|St = s] = � a π(a|s) � s′ P a ss′[r(s, a) + γVπ(s ′)] (2) Qπ(a, s) = Eπ[Ut|St = s, At = a] = � s′ P a ss′[r(s, s′|a) + γ � a′ Qπ(a′, s′)] (3) where Vπ(s) stands for the expectation of future reward at the state s, and the Qπ(a, s) represents the future expected reward when selecting an action a at state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' s′ and a′ stand for the possible reaching state and action at state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' P a ss′ is the transition probability from s to s′ under a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' In fact, Vπ(s) and Qπ(a, s) are used to evaluate the quality of the state s and the action-state pair (a, s), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' They are updated according to above two equations and help the policy π decide whether reaching the state or executing the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' THE DECENTRALIZED MULTI-MICROGRID ENERGY MANAGEMENT MODEL The decentralized MMG system includes numerous MGs that are connected to a distribution power network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Usually, an energy management center is set in each MG, which performs as an agent to conduct self-training and control the dispatchable elements, such as conventional generators (CGs), batteries (BAs), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' In this section, to describe the energy management model of the MMG system more clearly, we firstly introduce the isolated MG model with the MDP format before developing the MMG model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The Isolated Microgrid Energy Management Model Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 1 illustrates the structure of the isolated MG model and a real-world MG system case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Normally, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 1(a), a MG is constructed by five types of elements: renewable power generators, BA, CG, conventional load (CL) and energy management center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Note that BAs and CGs are dispatchable since their outputs are controlled by the management center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' On the contrary, because of the high uncertainties of RE, the outputs of REG cannot be controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Additionally, the energy management center is termed as the agent that controls these dispatchable elements by observing the state of MG operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Following this structure, the Oak Ridge national laboratory distributed energy control communication lab microgrid test system (ORNL-MG) is selected as the real-world case in this paper, which is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The structure of (a) an isolated MG and (b) the ORNL-MG [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' (a) Micro Turbine Conventional Generator Diesel Engine Battery Energy Management Center Wind Turbine Load RenewablePower Generator Photovoltaic (b) ORNLSubstation (161to13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='8kV) LVA FromTVA erORNL13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='8kV 10Substation From Circuit #2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='8 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='4kV) ，4000 3000Substation (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='8 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='4kV) ElectricalServicefrom750kVATransformers Static From Circuit #4 Rotating estArea4 1) Conventional Generator: It can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 1 that the CG includes diesel engine generator and micro turbine, which generate power through fossil fuels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The cost functions of CGs can be represented as follows: C(PCG,i) = aCG,iP 2 CG,i + bCG,iPCG,i + cCG,i (4) P min CG,i ≤ PCG,i ≤ P max CG,i (5) where C(PCG,i) represents the generation cost of ith CG, and PCG,i is its generation power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' aCG,i, bCG,i and cCG,i denote the cost coefficients of the ith CG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' P min CG,i and P max CG,i are the lower and upper bounds of the ith CG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 2) Renewable Energy Generator: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 1 presents two kinds of REGs, namely wind turbine and photovoltaic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The gener- ation of REG normally depends on the natural environment such as wind speed, temperature, weather and solar irradiance [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Since the REGs do not consume any fossil fuels, their generation costs are not considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 3) Battery: As one of the most commonly used energy storage devices, BA can store energy generated by CGs and REGs, and release it when needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Thus, the BA has two operation states, namely charging and discharging, which are represented by the transition of its state of charge, and can be formulated as follows: SOCt+1 = (1 − δ)SOCt − P t BA ηchCBA (6) SOCt+1 = (1 − δ)SOCt − ηdchP t BA CBA (7) where SOCt and SOCt+1 denote the charging state of BA at time t and t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' PBA is the charging-discharging power of BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Here, we assume PBA > 0 when the BA is discharging, and PBA < 0 when the BA is charging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The ηch and ηdch are the charging and discharging efficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' δ denotes the discharging rate, which is set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' CBA represents the capacity of BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The operation of BA would bring about the costs due to the amortized purchase and maintenance, which is formulated by the following equation [31]: C(PBA,j) =aBA,j(PBA,j + 3P max BA,j(1 − SOC))2 + bBA,j(PBA,j + 3P max BA,j(1 − SOC)) + cBA,j (8) P max BA,j < PBA,j < P max BA,j (9) where C(PBA,j) represents the cost of the jth BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' aBA,j, bBA,j and cBA,j are cost coefficients of the jth BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' P max BA,j and P min BA,j are the upper and lower bounds of BA output power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 4) Network Power Loss of MG: Practically, there exists the power loss because of the operation of generators and the transmission of energy in the MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The power loss usually corresponds to the active generation power and can be esti- mated as follows [32]: λCG = ∂Ploss ∂PCG , λREG = ∂Ploss ∂PREG , λBA = ∂Ploss ∂PBA (10) where λCG, λREG and λBA represent the power loss coefficients of CG, REG and BA, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' According to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' [32], λCG, λREG and λBA are recommended to be set in [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='02].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Therefore, they are set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='02 in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Then, the power loss Ploss can be given by the following equation [32]: Ploss = nCG � i=1 λCGPCG,i + nREG � j=1 λREGPREG,j + nBA � k=1 λBAPBA,k (11) where nCG, nREG and nBA are the numbers of CGs, REGs and BAs in the isolated MG, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Isolated MG Energy Management Model via MDP and Physics-Informed Reward Since the energy management center of the MG is an agent which is trained by the DRL algorithm, the above isolated MG model should be reformulated as the MDP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' In addition, considering the physical feasibility of the agent, the definition of reward is designed to integrate the physical-informed rules, which are presented as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 1) State: In this paper, we consider a 24-hour scheduling of the MG, and each hour is denoted by t ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=', 24}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The state of MG at time t includes the energy operation information, which is defined as follows: st = {P t−1 L , P t−1 REG,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=', P t−1 REG,nREG, SOCt−1, Et−1 λ } (12) where st indicates the state of MG at time t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' P t−1 L and P t−1 REG,i stand for the load demand and the ith REG at time t − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' In addition, the Et−1 λ is the electricity price in the transaction between the MG and the distribution power network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 2) Action: The action at is generated by the agent, which controls the power outputs of the CGs and BAs at each time t, according to the state st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' In this study, it is defined as follows: at = {P t CG,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=', P t CG,nCG, P t BA,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=', P t BA,nBA} (13) In DRL, the agent is normally a neural network, which is difficult to produce consistent and feasible in the early training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Therefore, the actions are enforced to fulfill the output constraints provided in Eqs (5) and (9) P t CG,i = clip(P t CG,i, P min CG,i, P max CG,i), i ∈ [1, nCG] (14) P t BA,j = clip(P t BA,j, P min BA,j, P max BA,j), j ∈ [1, nBA] (15) where clip(t, tmin, tmax) is the clip function, which returns tmax if t > tmax, and tmin if t < tmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 3) Reward: The design of the reward significantly impacts the performance of the DRL training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' A specific physical task- oriented reward would endow interpretability to the strategy of the agent [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' However, in some classical reinforcement learning tasks, such as CartPole [33] and Atari Games [34], the design of their rewards is independent of the physical characteristic of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' For instance, in CartPole, the reward is set as 0 if the action is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Such the intuitive design of reward may mislead the agent thus slowing down the training process and decreasing the interpretability of the agent strategy, it is not suitable for the MG energy management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Nor- mally, the MG agent is expected to operate economically while ensuring the self energy-sufficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Therefore, considering the physical characteristic of the MG, the reward is designed as physics-informed to satisfy the two explicit targets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=', the training of the agent and realizing the requirements of 5 operation cost and self energy-sufficiency, simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The reward function is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' rt = − wC �nCG � i=1 C(P t CG,i) + nBA � j=1 C(P t BA,j) � − wdeEl(t) × abs(P t de) (16) where rt is the reward value at time t, and El(t) ≥ 0 indicates the price of purchasing electricity from the distribution power grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' wC ∈ [0, 1] and wde ∈ [0, 1] indicate the weights to limit the order of magnitude of reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' abs(·) stands for the absolute function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' P t de evaluates the deviation between load demand and real generation, which is formulated by: P t de = P t L − �nCG � i=1 P t CG,i + nREG � j=1 P t REG,j + nBA � k=1 P t BA,k − P t loss � (17) In this study, the physics-informed reward is com- posed of two physical targets of MG, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=', operation costs and self energy-sufficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' They are formulated as ��nCG i=1 C(P t CG,i) + �nBA j=1 C(P t BA,j) � , and El(t) × abs(P t de), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' To keep the order of magnitude of the reward consistent, the self energy-sufficiency is designed as P t de times El(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Since the reward is related to the physical valuables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=', P t CG,i, P t BA,j and P t de, it can be endowed the physical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' In this way, the reward is able to guide the agent to produce a series of actions that minimize the generation costs of CGs and BAs while ensuring self energy-sufficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Decentralized Multi-Microgrid Energy Management Model As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 2, a decentralized MMG model that contains np MGs is considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' These MGs are connected to the distribution power network, and the energy transaction between MGs is also allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Each MG is controlled by an agent, which observes the state st of MG and provides the action at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The structure of MMG system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Since the MG is encouraged to maximize the physics- informed reward rt for achieving energy self-sufficiency and economic operation, the target of the MMG should be the maximum of the systematic rewards rsys,t, which can be represented by the sum of rewards obtained by all the MG agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The rsys,t is given by rsys,t = np � i=1 ri t = np � i=1 −ϵi × abs(P t i,de) (18) where ri t represents the reward obtained by the ith MG agent at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' ϵi and P t i,de are the shrinkage coefficient and deviation of MG i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Besides, since the load demand of MG cannot be known in advance, excessive or insufficient power generation of an isolated MG is unavoidable, thus the energy transaction in the MMG system is inevitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Therefore, the energy transaction mechanism between different MGs is developed, which is given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' That is, MG is allowed to conduct energy transactions with the distribution power network and other MGs, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' If the generated power of MG i exceeds its load demand at time t, the excess energy will be sold to other MGs with a price Ei(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' If the demand of MG i cannot be satisfied, the MG will purchase electricity from MG j, which has the lowest price of the whole participated MGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' j = arg min l El(t) × Ll, l ∈ [1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=', np] (19) where Ll indicates whether the generation of MG l exceeds its demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The Ll is set as 1 if the demand is exceeded or set as infinite if not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The MGs will preferentially purchase the surplus power generated by other MGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' When the MG generations are fully consumed, the distribution power network will provide power with price Edpn(t), which is usually higher than El(t), l ∈ [1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=', np].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' FEDERATED MULTI-AGENT DEEP REINFORCEMENT LEARNING ALGORITHM As discussed above, the MMG is decentralized, thus each MG agent has a high autonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' However, the decentralized structure threatens the generalization performance of the agent, because the diversity of the data in isolated MG is limited, which may make the agent getting trap into a local optima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' To tackle this issue, we propose a federated multi-agent deep reinforcement learning (F-MADRL) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The FL is used to improve the generalization of the agent during training while ensuring data privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' There are two characteristics in the FL, one is called participant and the other is termed as the collaborator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The participant j, j ∈ [1, np], is denoted as a neural network model f j wj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' It conducts self-training at the local and uploads its parameters wj to the collaborator periodically, where np is the number of participants which are processed in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Constrained by the data privacy, the participant f j wj only trains on the local dataset, which may cause the insufficient training since the capacity and diversity of the data are limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The FL could tackle this problem through the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' First, at the training epoch e, e ∈ [1, Ne], the model of jth participant is defined as f j we j , which conducts self- training to obtain the parameters we j , where Ne is the total number of training epoches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Then, each participant uploads its parameters to the collaborator and constructs a parameter list we = � we 1, we 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=', we np � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The collaborator calculates the weight average of we to estimate a global model f e+1 G with parameters we+1 G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' After aggregation, the collaborator broadcasts we+1 G to all the participants and replaces their own parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=', we+1 G = we+1 1 = we+1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' = we+1 np .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The Distribution power network Energy and price MG2 MG n MG1 MA INA INA6 aggregating mechanism of FL is formulated by the following equations: we+1 j = we G − η∇Fj(we j ), ∀j (20) we+1 G = np � j=1 1 np we+1 j (21) where η and Fj(·) are the learning rate and local loss function of the jth participant, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' In this paper, the participant can be considered as the agent in each MG and the collaborator is a server that takes the responsibility for aggregating and broadcasting the parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The F-MADRL aims to solve the following distributed optimization model: min we G F(we G) = np � j=1 pjFj(we j ) (22) where F(·) is the global loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' pj represents the relative weight of each MG agent on the global model, and pj > 0, �np k=1 pj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' We set pj = |Dj|/ �np j=1 |Dj|, where |Dj| is the data size used for the local training of jth MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Note that F(·) cannot be directly computed without the information sharing of each participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The overall structure of F-MADRL is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' At the epoch e, the agent in three MGs are firstly replaced by the global agent in the (e − 1)th epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Then, the three MG agents conduce self-training to obtain parameters, which are uploaded to the server for aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Next, the global agent would be built on the server, and the parameters will be broadcasted to the MG agents for the (e + 1)th epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The proposed federated multi-agent deep reinforcement learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' It can be learned from the figure that the F-MADRL contains two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' One is executed on server, which can be considered as the collaborator and the other is executed on MG agents, can be considered as the participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The procedures running on the server and MG agent are provided in the following subsections, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The F-MADRL: Server Part The F-MADRL proceed on the server mainly focus on the aggregating and broadcasting of the agent parameters, and its procedure is shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' At the beginning of the training epoch of F-MADRL, the server would build a global agent with the parameter w0 G, which is then broadcasted to each MG agent for self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Since the agents update their parameters in parallel, the server aggregates the parameters list we = [we 1, we 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=', we np] by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Furthermore, the aggregated parameters we+1 G are used to update the global model parameters and broadcast to the MG agents for the training of epoch e + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Algorithm 1 The federated multi-agent deep reinforcement learning algorithm on the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 1: Execute on the server: 2: Initialize the model parameters w0 G and broadcast them to the MG agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 3: for Global epoch e = 1 to Ne do 4: for MG agent j = 1 to np parallelly do 5: Update the MG parameter we j at the local agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 6: Store the we j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 7: Upload the we j to the server 8: end for 9: Receive the parameters from each MG agent and construct 10: we ← [we 1, we 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=', we np] 11: Aggregating the model parameters through 12: we+1 G = �np j=1 1 np we+1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 13: Broadcast the we+1 G to other MG agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 14: end for B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The F-MADRL: MG Agent On the other hand, the MG agent adopts the self-training in the procedure of F-MADRL and cooperates with the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' When the MG agents receive the parameter we G from global model at the epoch e, their parameters are replaced by we G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=', we j = we G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Then, each MG agent executes Ni individual self-training epochs in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Afterwards, the parameters of the MG agent at the last self-training epoch, namely θNi, µNi are stored and uploaded to the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' In this paper, each MG agent performs self-training with a famous deep reinforcement learning algorithm, namely PPO, to obtain the optimal policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' There are two types of deep neural networks, namely, actor and critic, defined by the MG agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Actor πθ is parameterized by θ, which aims to produce the action, and the critic is denoted as V µ, which is parameterized by µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The overall training process of the self-training pro- cedure during one episode is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' First of all, the experiment tuples T are sampled T = {⟨s0, a0, r0, s1⟩, ⟨s1, a1, r1, s2⟩, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=', ⟨sU, aU, rU, sU+1⟩} where U indicates the length of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Then, the loss function of the actor at kth episodes is calculated, which is defined as follows: LC = Es,a∼T � min( πθ k(a | s) πθ k−1(a | s)A πθ k s,a, clip( πθ k(a | s) πθ k−1(a | s), 1 − ϵ, 1 + ϵ)A πθ k s,a) � (23) where Es,a∼T [·] represents the empirical average over the sampled experiment tuples T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The πk−1 and πk stand for Upload w MG1 AgentofMGl Aggregation Upload we Global Agent Server MG2 Agent of MG2 M Upload w e+1 Broadcast e+1 G MG3 AgentofMG37 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The procedure of the self-training of MG agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' the previous and new policy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The ϵ is the clip parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Aπk s,a stands for the advantage, which measures if the action is worth taking by comparing the action value and state value: Aπk st,at = Eτ[U|s0 = st, a0 = at]−V (st) = Q(st, at)−V (st) (24) However, the Aπk s,a cannot be directly obtained since Q(st, at) is difficult to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' In this way, the gen- eralized advantage estimation method is implemented in this study: Aπθ k s,a = δV 0 + (γλ)δV 1 + (γλ)2δV 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=', +(γλ)U−t+1δV U−1 (25) where γ ∈ [0, 1] and λ ∈ [0, 1] represent the discount factor and a hyperparameter that adjusts the tradeoff between bias and variance of the estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Note that the variance would be increased when raising λ while the bias is decreased accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' According to the recommendation of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' [26], λ is set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' δV k is calculated by: δV k = rk + γV µ k (st+1) − V µ k (st) (26) where V µ k (st+1) and V µ k (st) are given by the critic, which is trained by the loss function LV : LV = Es,a∼T � (γV µ k (st+1) + r(st, at) − V µ k (st))2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' (27) With the above equations, the parameter θ and µ of the actor and the critic can be updated by the following equations: θk+1 = θk + ηπ∇θkLC (28) µk+1 = µk + ηV ∇µkLV (29) where ηπ and ηV are the learning rates of actor and critic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Since both LC and LV are optimized in each MG agent of the proposed F-MADRL algorithm, they are the local loss functions which construct the global loss following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Overall, the F-MADRL algorithm applied on the server can be summarised in the Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' The Theoretical Convergence Analysis of the F-MADRL In this section, the convergence of the F-MADRL is eval- uated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' At first, the following assumptions considering the function Fk, k ∈ [1, np] are made, by referring to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Assumption 1: The Fk is L-smooth, ∀w, w′, ∥∆Fk(w) − ∆Fk(w′)∥2 ≤ L∥w − w′∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Fk(w) ≤ Fk(w′) + ∇Fk(w′)T (w − w′) + L 2 ∥w′ − w∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Algorithm 2 The federated multi-agent deep reinforcement learning algorithm on the MG agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 1: Execute on each MG agent: 2: Parallel running on j, j ∈ [1, np] agent at global epoch e 3: Receive the parameters from the server we j ← we G 4: for Individual training epoch i = 1 to Ni do 5: Collect the experience tuple T = {⟨s0, a0, r0, s1⟩, ⟨s1, a1, r1, s2⟩, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=', ⟨sU, aU, rU, sU+1⟩} 6: Compute the discounted factor: 7: δV k ← rk + γV µ k (st+1) − V µ k (st) 8: Estimate the advantage: 9: Aπθ k s,a ← δV 0 +(γλ)δV 1 +(γλ)2δV 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=', +(γλ)U−t+1δV U−1 10: Calculate the loss function of the actor: 11: LC ← Es,a∼T � min( πθ i (a|s) πθ i−1(a|s)Aπθ i s,a, clip( πθ i (a|s) πθ i−1(a|s), 1− 12: ϵ, 1 + ϵ)Aπθ i s,a) � 13: Update the actor parameter: 14: θi+1 ← θi + ηπ∇s,a∼T LC 15: Calculate the loss function of the critic: 16: LV ← Es,a∼T � (γV µ i (st+1) + r(st, at) − V µ i (st))2� 17: Update the critic parameter: 18: µi+1 ← µi + ηV ∇µiLV 19: end for 20: Store the network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' we j ← {θNi, µNi} 21: Upload the parameters we j to the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Assumption 2: The Fk is µ-strongly convex, ∀w, w′, Fk(w)− µ 2 ∥Fk∥2 is convex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Fk(w) ≥ Fk(w′) + ∇Fk(w′)T (w − w′) + µ 2 ∥w′ − w∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Based on the above Assumptions, we have the following Lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Lemma 1: F is µ-strongly convex and L-smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Proof: Straightforwardly from Assumption 1 and Assump- tion 2, in line with the definition of convex, F is the finite-sum of the Fk, thus it is µ-strongly convex and L-smooth as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Lemma 2: ∀w, w′ ∈ Rn and wt = w + t(w − w′) for t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Then, F(w) − F(w′) = � 1 0 ∇F(wt)T (w′ − w)dt (30) and F(w) − F(w′) − ∇F(w)T (w′ − w) = � 1 0 (∇F(wt) − ∇F(w))T (w′ − w)dt (31) Proof: Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' (30) follows the fundamental theorem of calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' (31) follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' (30) by subtracting ∇F(w)T (w′ − w) from both sides of the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' Lemma 3: If F is smooth and µ-strongly convex for µ > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' then for the w∗ = arg min w F(w),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQfv_mu/content/2301.00641v1.pdf'}
+page_content=' 1 2µ∥∇F(w)∥2 2 ≥ F(w) − w∗ ≥ µ 2 ∥w − w∗∥2 2 (32) sample Environment Experiments Tuple action 1}
+admitting quasiconformal extensions across the boundary unit circle S1 = ∂D∗, hence to
+the whole Riemann sphere �C = C ∪ {∞}. To have compactness in the topology of locally
+Date: February 1, 2023
+(SobolBel1.tex).
+1
+
+2
+Samuel L. Krushkal
+uniform convergence on C, one must add the third normalization condition, for example,
+f(0) = 0.
+The Beltrami coefficients of extensions are supported in the unit disk D = {|z| < 1} and
+run over the unit ball
+Belt(D)1 = {µ ∈ L∞(C) : µ(z)|D∗ = 0,
+∥µ∥∞ < 1}.
+Each µ ∈ Belt(D)1 determines a unique homeomorphic solution to the Beltrami equation
+∂w = µ∂w on C (quasiconformal automorphism of �C) normalized by the assumptions wµ ∈
+ΣQ, wµ(0) = 0.
+The Schwarzian derivatives of these functions
+Sw(z) =
+�w′′(z)
+w′(z)
+�′
+− 1
+2
+�w′′(z)
+w′(z)
+�2
+,
+z ∈ D∗
+belong to the complex Banach space B = B(D∗) of hyperbolically bounded holomorphic
+functions in the disk D∗ with norm
+∥ϕ∥B = sup
+D
+(|z|2 − 1)2|ϕ(z)|
+and run over a bounded domain in B modeling the universal Teichm¨uller space T. The
+space B is dual to the Bergman space A1(D∗), a subspace of L1(D∗) formed by integrable
+holomorphic functions (quadratic differentials ϕ(z)dz2) on D∗. Note that ϕ(z) = O(z−4)
+near z = ∞. The needed results from Teichm¨uller space theory see, e.g., in [7], [8], [10].
+One defines for any f ∈ ΣQ its Grunsky coefficients αmn from the expansion
+log f(z) − f(ζ)
+z − ζ
+=
+∞
+�
+m,n=1
+αmnz−mζ−n,
+(z, ζ) ∈ (D∗)2,
+(1)
+where the principal branch of the logarithmic function is chosen. These coefficients satisfy
+the inequality
+���
+∞
+�
+m,n=1
+√mn αmnxmxn
+��� ≤ 1
+(2)
+for any sequence x = (xn) from the unit sphere S(l2) of the Hilbert space l2 with norm
+∥x∥ =
+� ∞
+�
+1
+|xn|2�1/2; conversely, the inequality (2) also is sufficient for univalence of a locally
+univalent function in D∗ (cf. [9], [23]).
+The minimum k(f) of dilatations k(wµ) = ∥µ∥∞ among all quasiconformal extensions
+wµ(z) of f onto the whole plane �C (forming the equivalence class of f) is called the Te-
+ichm¨uller norm of this function. Hence,
+k(f) = tanh dT(0, Sf),
+where dT denotes the Teichm¨uller-Kobayashi metric on T. This quantity dominates the
+Grunsky norm
+κ(f) = sup
+����
+∞
+�
+m,n=1
+√mn αmn(f)xmxn
+��� : x = (xn) ∈ S(l2)
+�
+by κ(f) ≤ k(f). For most functions f, we have the strong inequality κ(f) < k(f) (moreover,
+the functions satisfying this inequality form a dense subset of Σ), while the functions with
+the equal norms play a crucial role in many applications.
+
+Extremal properties of Sobolev’s Beltrami coefficients
+3
+These norms coincide only when any extremal Beltrami coefficient µ0 for f (i.e., with
+∥µ0∥∞ = k(f)) satisfies
+∥µ0∥∞ = sup
+����
+��
+D
+µ0(z)ψ(z)dxdy
+��� : ψ ∈ A2
+1(D), ∥ψ∥A1(D) = 1
+�
+= κ(f)
+(z = x + iy).
+(3)
+Here A1(D) denotes the subspace in L1(D) formed by integrable holomorphic functions (qua-
+dratic differentials ψ(z)dz2 on D, and A2
+1(D) is its subset consisting of ψ with zeros of even
+order on D, i.e., of the squares of holomorphic functions (see, e.g., [11], [13], [16], [20]). Note
+that every ψ ∈ A2
+1(D) has the form
+ψ(z) = 1
+π
+∞
+�
+m+n=4
+√mn xmxnzm+n−2
+(4)
+and ∥ψ∥A1(D) = ∥x∥l2 = 1, x = (xn).
+1.3. Generalization. The method of Grunsky inequalities was generalized in several direc-
+tions, even to bordered Riemann surfaces X with a finite number of boundary components
+(see, e.g., [23], [26]). We shall consider these inequalities in unbounded simply connected
+hyperbolic domains.
+Let L ⊂ C be an oriented bounded quasicircle separating the points 0 and ∞. Denote its
+interior and exterior domains by D and D∗ (so 0 ∈ D, ∞ ∈ D∗). Then, if δD(z) denotes
+the Euclidean distance of z from the boundary of D and λD(z)|dz| is its hyperbolic metric
+of Gaussian curvature −4, we have
+1
+4 ≤ λD(z)δD(z) ≤ 1.
+(5)
+The right hand inequality follows from the Schwarz lemma and the left from the Koebe
+one-quoter theorem.
+For such a domain D∗ ∋ ∞, the expansion (1) assumes the form
+− log f(z) − f(ζ)
+z − ζ
+=
+∞
+�
+m,n=1
+βmn
+√mn χ(z)m χ(ζ)n,
+where χ denotes a conformal map of D∗ onto the disk D∗ so that χ(∞) = ∞, χ′(∞) > 0
+(cf. [23]).
+Accordingly, the generalized Grunsky norm is defined by
+κD∗(f) = sup
+����
+∞
+�
+m,n=1
+βmn xmxn
+��� : x = (xn) ∈ S(l2)
+�
+.
+We now consider the class ΣQ(D∗) of univalent functions in domain D∗ with expansions
+f(z) = b0 + b1z−1 + . . . near z = ∞, admitting quasiconformal extensions onto the com-
+plementary domain D. Similar to above, we subject these extensions to f(0) = 0. Their
+Beltrami coefficients run over the ball
+Belt(D)1 = {µ ∈ L∞(C) : µ(z)|D∗ = 0,
+∥µ∥∞ < 1}.
+A coefficient µ0 ∈ Belt(D∗)1 is extremal in its class if and only if
+∥µ0∥∞ = sup
+����
+��
+D∗ µ0(z)ψ(z)dxdy
+��� : ψ ∈ A1(D∗), ∥ψ∥A1 = 1
+�
+,
+
+4
+Samuel L. Krushkal
+and similar to (3), the equality κD∗(f µ) = k(f µ) is valid if and only if
+∥µ∥∞ = sup
+����
+��
+D
+µ(z)ψ(z)dxdy
+��� : ψ ∈ A2
+1(D), ∥ψ∥A1(D) = 1
+�
+= κD(f).
+If additionally the equivalence class of f is a Strebel point of the space T with base point
+D∗, which means that this class contains the Teichm¨uller extremal extension f k|ψ0|/ψ0 with
+ψ0 ∈ A1(D), then necessarily ψ0 = ω2 ∈ A2
+1 (cf. [11], [16], [21]), [28]). The Strebel points
+are dense in any Teichm¨uller space, see [8].
+For arbitrary quasidisks D, the corresponding set A2
+1(D) is characterized similar to (4),
+but in more complicated way (see [17]).
+Assume that µ0 ∈ Belt(D)1 is extremal in its class but not of Teichm¨uller type. A point
+z0 ∈ ∂D is called substantial (or essential) for µ0 if for any ε > 0 there exists a neighborhood
+U0 of z0 such that
+sup
+D∗\U0
+|µ0(z)| < ∥µ0∥∞ − ε;
+so the maximal dilatation k(wµ0) = ∥µ∥∞ is attained on D by approaching this point.
+In addition, there exists a sequence {ψn} ⊂ A1(D) such that ψn(z) → 0 locally uniformly
+on D but ∥ψn∥ = 1 for any n, and
+lim
+n→∞
+��
+D
+µ0(z)ψn(z)dxdy = ∥µ0∥∞.
+Such sequences are called degenerated.
+The image of a substantial point is a common point of two quasiconformal arcs, which
+can be of spiral type.
+1.4. Fredholm eigenvalues and quasireflections. The Teichm¨uller and Grunsky norms
+are intrinsically connected with quasiconformal reflections, Fredholm eigenvalues and other
+quasiinvariants of quasiconformal curves. We outline briefly the main notions; the details
+see, e.g., in [2], [12], [17], [22].
+The quasiconformal reflections (or quasireflections) are the orientation reversing qua-
+siconformal homeomorphisms of the sphere �C which preserve point-wise some (oriented)
+quasicircle L ⊂ �C and interchange its interior and exterior domains. One defines for L its
+reflection coefficient
+qL = inf k(f) = inf ∥∂zf/∂zf∥∞,
+taking the infimum over all quasireflections across L. Due to [2], [22], the dilatation
+QL = (1 + qL)/(1 − qL) ≥ 1
+is equal to the quantity
+QL = (1 + kL)2/(1 − kL)2,
+(6)
+where kL is the minimal dilatation among all orientation preserving quasiconformal auto-
+morphisms f∗ of �C carrying the unit circle onto L, and k(f∗) = ∥∂zf∗/∂zf∗∥∞.
+The reflection with dilatation QL is extremal. A remarkable and very useful fact estab-
+lished by Ahlfors is that any quasicircle also admits a Lipschitz continuous quasireflection
+with some coefficient C(qL) (see [2]).
+
+Extremal properties of Sobolev’s Beltrami coefficients
+5
+The Fredholm eigenvalues ρn of an oriented smooth closed Jordan curve L ⊂ �C are the
+eigenvalues of its double-layer potential. These values are crucial in many applications.
+The least positive eigenvalue ρL = ρ1 plays is naturally connected with conformal and
+quasiconformal maps and can be defined for any oriented closed Jordan curve L by
+1
+ρL
+= sup |DG(u) − DG∗(u)|
+DG(u) + DG∗(u) ,
+where G and G∗ are, respectively, the interior and exterior of L; D denotes the Dirichlet
+integral, and the supremum is taken over all functions u continuous on �C and harmonic on
+G ∪ G∗.
+A rough upper bound for ρL is given by Ahlfors’ inequality
+1
+ρL
+≤ qL,
+where qL denotes the minimal dilatation of quasireflections across L [1].
+One of the basic tools in quantitative estimating the Freholm eigenvalues ρL of quasicircles
+is given by the K¨uhnau-Schiffer theorem [20], [25], which states that the value ρL is reciprocal
+to the Grunsky norm κ(f) of the Riemann mapping function of the exterior domain of L.
+For all functions f ∈ SQ (i.e., univalent in the disk D∗) with k(f) = κ(f), we have the
+exact explicit values
+qf(S1) =
+1
+ρf(S1)
+= κ(f).
+(7)
+1.5.
+Harmonic and pseudo-harmonic Beltrami coefficients. By the Ahlfors-Weill
+theorem strengthening the classical Nehari’s result on univalence in terms of the Schwarzians,
+every function ϕ ∈ B(D) with ∥ϕ∥B < 2 is the Schwarzian derivative of a univalent function
+f(z) in the unit disk D, and the function f has quasiconformal extension onto the disk D∗
+with Beltrami coefficient
+µϕ(z) = −1
+2(|z|2 − 1)2ϕ(1/z)(1/z4),
+z ∈ D∗;
+(8)
+see [3], [4], [8].
+This deep fact is extended to arbitrary quasidisks in much weaker form. For example,
+we have the following result of Bers, which is a special case of his more general extension
+theorem given in [5].
+Lemma 1. [5] Let L be a quasicircle on �C with the interior DL and exterior D∗
+L. Then,
+for some ε > 0, there exists an anti-holomorphic homeomorphism τ (with τ(0) = 0) of the
+ball Vε = {ϕ ∈ B(D∗
+L) : ∥ϕ∥} < ε into B(DL) such that every ϕ in Vε is the Schwarzian
+derivative of some univalent function f which is the restriction to D∗
+L of a quasiconformal
+automorphism �f of Riemann sphere �C. This �f can be chosen in such a way that its Beltrami
+coefficient on DL has the form
+µ �f(z) = λ−2
+D (z)ψ(z),
+ψ = τ(ϕ).
+(9)
+The Beltrami coefficients of such form are called now harmonic in view of their connection
+with the Kodaira-Spencer deformation theory of complex structures.
+
+6
+Samuel L. Krushkal
+In this paper we shall use the notion of harmonicity in its original sense, i.e., only for
+solutions of the Laplace equation ∆u = 0, and regard the coefficients of type (9) as pseudo-
+harmonic.
+2. GENERAL THEOREM AND ITS CONSEQUENCES
+Our goal is to describe the features of extremal Beltramic oefficients of non-Teichm¨uller
+type maximizing the Grunsky norm.
+We shall consider the univalent functions f(z) ∈ SQ(D∗) whose restrictions to the bound-
+ary quasicircle L = ∂D∗ have substantional points, hence do not have the Teichm¨uller
+extremal extensions across L. Denote the collection of such f by S0
+Q(D∗). For L = S1, we
+use the notation S0
+Q.
+Fix p > 1 and consider the subset Mp(D) of Belt(D)1, which consists of the Beltrami
+coefficients µ defining the maps f µ ∈ S0
+Q(D) and satisfying:
+(i) µ ∈ L∞(D) � W 1,p(D),where W 1,p(D) is the Sobolev space of functions µ in D having
+the first distributional derivatives which belong to Lp(D);
+(ii) the value ∥µ∥∞ = ess supD |µ(z)| is attained by approaching z the boundary of D;
+(iii) there is a subarc γ ⊂ ∂D depending on µ such that µ(z) → 0 as z approaches γ from
+inside D.
+The boundary values µ(z0) for z0 ∈ ∂D must be understand as lim
+z→z0 µ(z). Note that the
+value ∥µ∥∞ also can be attained by approaching the inner points of domain D and that the
+arc γ is locally Cα smooth with α > 0 depending on ∥µ∥∞, in accordance with the H¨older
+continuity of quasiconformal automorphisms of �C.
+The main result of this paper is the following general theorem.
+Theorem 1.
+For any p > 2, every Beltrami coefficient µ ∈ Mp(D) is extremal in its
+equivalence class [µ], and the corresponding quasiconformal automorphism f µ of �C satisfies
+k(f µ) = κD∗(f µ) = ∥µ∥∞.
+(10)
+In the case p > 1, there is a weakened version of this theorem, presented in Section 4.
+Until now, there were no results in geometric function theory giving the explicit expression
+of the generalized Grunsky norm for the general quasiconformal domains different from the
+disk. Theorem 1 implies the first general result in this direction.
+In view of the assumptions on the set Mp(D), every boundary function f µ|∂D must have
+at least one substantial point, at which ∥µ∥∞ is attained. This yields that the coefficient µ
+is not uniquely extremal in its equivalence class.
+It follows from what was indicated above that the situation described by Theorem 1
+does not appear in the case of sufficiently high boundary regularity of univalent functions f
+(and of ∂D) because, for example, for any C2+α smooth µ the map f µ is C2+α on C), and
+by Strebel’s frame mapping criterion [8], [28]) the equivalence class of f µ contains unique
+Teichm¨uller coefficient µ0 = k|ψ0|/ψ0 with ψ0 ∈ A1(D), which cannot have the substantial
+boundary points.
+The assumption on the set Mp are natural and cannot be replaced in terms of smoothness
+or non-smoothness of µ on the boundary points.
+
+Extremal properties of Sobolev’s Beltrami coefficients
+7
+In the case, when the domain D is the unit disk D, Theorem 1 can be completed by
+the quantitative results on Fredholm eigenvalues and reflection coefficients indicated in the
+previous section (see the relations (6), (7)).
+Theorem 2.
+For any p > 2, every Beltrami coefficient µ ∈ Mp(D) is extremal in its
+equivalence class; in addition, the reflection coefficient and the Fredholm eigenvalue of the
+curve L1 = f µ(S1) are explicitly given by
+qL1 = 1/ρL1 = ∥µ∥∞.
+(11)
+Here also µ can be regular only in admissible bounds. The assumption for µ to belong to
+the Sobolev space W p
+1 with p > 2 is essential for the proof.
+In the case of pseudo-harmonic Beltrami coefficients, the assumptions of the above theo-
+rems can be weakened.
+Theorem 3. The equalities (9) and (10) are valid for any admissible for univalence pseudo-
+harmonic Beltrami coefficient µϕ(z) of the form (9) or (8) with ϕ = Sf such that the maxi-
+mum of the function λ−2
+D (z)Sf(z) is attained at some boundary point and Sf is bounded on
+some subarc γ of ∂D∗ .
+For such coefficients, the required smoothness of µ is provided by the representations (8),
+(9), while vanishing on γ follows from (5). A special case of Theorem 3 has been obtained
+in [18].
+3. PROOF OF THEOREM 1
+To prove that the Beltrami coefficient µ satisfying the prescribed conditions (i), (ii), (iii)
+is extremal in its equivalence class and provides the equalities (10), we establish that it must
+satisfy
+∥µ∥∞ =
+sup
+∥ψ∥A2
+1(D)=1
+���
+��
+D
+µ(z)ψ(z)dxdy
+���.
+(12)
+This equality means that ∥µ∥∞ is attained in L1(D) on the set of abelian quadratic differ-
+entials intrinsically connected with the Grunsky coefficients.
+By the Sobolev embedding theorem, the function µ(z) is extended to a continuous function
+on the closed domain D. The assumption (ii) implies that there is a point z0 ∈ ∂D at which
+|µ(z0)| = ∥µ∥∞
+(13)
+(in the general case, |µ(z0)| = lim sup
+z→∂D
+|µ(z)|).
+We take the conformal map z = χ(ζ) of the half-strip
+Π+ = {ζ = ξ + iη : ξ > 0, 0 < η < 1}
+onto D such that χ−1(z0) = ∞ and the pre-images of the endpoints of the arc γ are the
+points 0 and 1, and pull back the coefficient µ to Beltrami coefficient
+ν(ζ) := χ∗µ(z) = (µ ◦ χ)(ζ) χ′(ζ)/χ′(ζ)
+(14)
+on Π+. The horizontal lines
+lη = {ζ = ξ + iη : 0 < ξ < ∞},
+0 < η < 1,
+
+8
+Samuel L. Krushkal
+are moving under this map into some analytic curves in D with endpoints on ∂D. The
+infinite point ζ = ∞ is substantial for the composite map F ν = f µ ◦ χ, and by assumptions
+on µ and from (14), we have
+lim
+ξ→∞ |ν(ξ + iη)| = ∥ν∥∞ = ∥µ∥∞.
+The assumptions on the coefficient µ(z) and the smoothness of conformal map χ imply that
+ν ∈ L∞
+� W 2
+p (Π+) and that the limit function
+ν(ζ0) =
+lim
+ζ→ζ0∈∂Π+ ν(ζ),
+is defined at all ζ0 ∈ ∂Π+ different from the points 0, i, ∞. Hence, ν(ζ) is bounded on the
+interval [0, i] of the imaginary axes, and moreover,
+ν(iη) = 0,
+0 ≤ η ≤ 1.
+(15)
+Now we pick the sequence
+ωm(ζ) = 1
+me−ζ/m,
+ζ ∈ Π+ (m = 1, 2, ...);
+all these ωm belong to A2
+1(Π+) and ωm(ζ) → 0 uniformly on Π+
+�{|ζ| < M} for any M < ∞.
+Also, ∥ωm∥A1(Π+) = 1 (moreover,
+����
+Π+ ωmdξdη
+�� = 1 − O(1/m)), which shows that {ωm} is
+a degenerating sequence for the affine horizontal stretching of Π+.
+We prove that this sequence is also degenerated for ν, estimating the integrals
+Im =
+��
+Π+
+ν(ζ)ωm(ζ)dξdη
+for large m. We have
+Im =
+1
+�
+0
+e−iη/mdη
+� 1
+m
+∞
+�
+0
+ν(ξ + iη)e−ξ/mdξ
+�
+.
+(16)
+The inner integral in (16) represents the values of the Laplace transform
+Lν =
+∞
+�
+0
+ν(t)e−stdt
+of ν(ξ + iη) in the points s = 1/m, and the assumptions of the theorem imply the existence
+of this transform also for the derivative ∂ν(ξ + iη)/∂ξ, which is integrable over Π+.
+Integrating by parts and using (15), one obtains
+∞
+�
+0
+∂ν(ξ + iη)
+∂ξ
+e−ξ/mdξ = 1
+m
+∞
+�
+0
+ν(ξ + iη)e−ξ/mdξ − ν(iη) = 1
+m
+∞
+�
+0
+ν(ξ + iη)e−ξ/mdξ
+(17)
+Since for any s > 0,
+����
+∂ν(ξ + iη)
+∂ξ
+���� e−sξ <
+����
+∂ν(ξ + iη)
+∂ξ
+���� ,
+
+Extremal properties of Sobolev’s Beltrami coefficients
+9
+the Lebesgue theorem on dominated convergence yields
+lim
+s→0
+∞
+�
+0
+∂ν(ξ + iη)
+∂ξ
+e−sξdξ =
+∞
+�
+0
+∂ν(ξ + iη)
+∂ξ
+dξ.
+Together with (15) and (17), this implies
+lim
+m→∞
+1
+m
+���
+∞
+�
+0
+ν(ξ + iη)e−ξ/mdξ
+��� = |ν(∞) − ν(iη)| = |ν(∞)|,
+(18)
+where
+ν(∞) = lim
+ξ→−∞ ν(ξ + iη).
+It follows that the integral (16) has the limit value
+lim
+m→∞
+��
+Π+
+ν(ζ)ωm(ζ)dξdη =
+1
+�
+0
+dη
+lim
+m→∞
+1
+m
+∞
+�
+0
+ν(ξ + iη)e−ξ/mdξ = ν(∞).
+(19)
+Now we return to the initial domain D by applying the inverse conformal map χ−1(z) :
+D → Π+. The corresponding sequence
+ψm = (ωm ◦ χ−1)(χ′)−2,
+m = 1, 2, . . . ,
+is a degenerating sequence for the initial Beltrami coefficient µ on D, and by (19),
+lim
+m→∞
+���
+��
+D
+µ(z)ψm(z)dxdy
+��� = lim
+m→∞
+���
+��
+Π+
+ν(ζ)ωm(ζ)dξdη
+��� = |ν(∞)|.
+(20)
+In view of the assumption (13), all terms in (20) are equal to ∥ν∥∞ = ∥µ∥∞. This proves
+the equality (12), which implies that µ is extremal in its class and that the map f µ obeys
+the relations (10). The theorem follows.
+Note that under the assumption p > 2, the conformal map χ of Π+ onto D is C1,α-smooth
+in the closed domain Π+, excluding the points 0, 1, ∞, which yields that the restrictions
+νη of the Beltrami coefficient (14) to the lines lη belong to W 1,p(lη). Then the embedding
+theorem for W n,p functions giving the continuity of these restrictions implies the existence
+of the limits
+ν(0) = νη(0) = lim
+ξ→0 νη(ξ)
+for all η,
+and νη(∞) = limξ→∞ ν(ξ + iη).
+This remark indicates the way in which the above proof can be extended to p > 1 (see
+remark 4.3.
+4. ADDITIONAL REMARKS TO THEOREMS 1 AND 2
+4.1. Theorem 1 implies that for every Beltrami coefficient µ0 ∈ Mp(D) the disk
+{tµ0/∥µ0∥∞ : |t| < 1}
+is geodesic in the ball Belt(D)1 simultaneously for the Teichm¨uller, Kobayashi and Carath´eodory
+metrics. It is pushed down under the projection φT : µ → Swµ ∈ B(D∗) onto a holomorphic
+
+10
+Samuel L. Krushkal
+disk in universal Teichm¨uller space T, on which all these invariant distances also coincide,
+though this coefficient does not be necessarily uniquely extremal in its equivalence class (the
+images of two different t′µ0/∥µ0∥∞ and t′′µ0/∥µ0∥∞ in T can be the same; as well as, there
+are µ ∈ [µ0] with ∥µ∥∞ = ∥µ0∥∞).
+4.2. The equality (18) is equivalent to the Tauberian theorem for Laplace transform. All
+known theorems of this type also require some smoothness of the original functions.
+This nice connection with the Laplace transform implies simultaneously the extremality
+and equality of the Teichm¨uller and Grunsky norms norms for a broad set of Beltrami
+coefficients given by Theorems 1 and 2.
+4.3. An extension of Theorem 1 to p > 1 is possible under the additional assumptions on D
+and µ, for example, when the boundary of domain D is C1,α-smooth (α > 0) and in (ii) the
+value ∥µ∥∞ = ess supD |µ(z)| is attained at some point z0 ∈ ∂D as z → z0 along any way in
+D, then the above prove can be modified for µ ∈ L∞(D) � W 1,p(D) with p > 1 as follows.
+Since now the conformal map χ of Π+ onto D also is C1,α-smooth in the closed domain
+Π+, excluding the points 0, 1, ∞, we have that the restrictions νη of the Beltrami coefficient
+(14) to the lines lη belong to W 1,p(lη). Applying the embedding theorem in dimension n = 1,
+one obtains for p > 1 the existence of limits
+νη(0) = lim
+ξ→0 νη(ξ) = 0
+and νη(∞) = lim
+ξ→∞ ν(ξ + iη).
+The indicated change of (ii) yields that all values νη(∞) must coincide and are equal to
+∥ν∥∞. Therefore, one can again apply the relations (18)-(20) and derive the assertions of
+Theorem 1.
+4.4. The assertion of Theorem 1 can be extended to arbitrary quasiconformal domains,
+even with fractal boundaries L in the following form: one can take in the half-strip Π+
+the Beltrami coefficients µ obeying the Tauberian theorem for the Laplace transform, i.e.,
+for which the arguments of the proof of Theorem 1 are valid, and pull back these µ to the
+interior of L.
+4.5. The canonical quasiconformal extensions of univalent functions (with Teichm¨uller or
+pseudo-harmonic of type (9) coefficients µ) are unique; the uniqueness intrinsically relates
+to their analyticity.
+Theorem 1 involving Beltrami coefficients from Sobolev’s space provides a possibility to
+distinguish some subsets in Mp admitting uniqueness. For example, it holds for µ which are
+(even weak) solutions of the Dirichlet problem for uniformly elliptic differential equations
+of the second order Lu = 0 on D with prescribed values µ on the boundary curve ∂D, in
+particular, for harmonic µ(z) (with ∆µ = 0). In addition, in this case the assumption (ii) is
+trivially fulfilled by the maximum principle.
+In particular, all harmonic µ (with ∆µ = 0) vanishing on some boundary subarcs γ are
+extremal, except when the boundary function f µ|∂D satisfies the Strebel frame mapping
+condition (for example, it is C2+α smooth).
+4.6. The theory of extremal quasiconformal maps originated in [29] plays now a crucial
+role in quasiconformal analysis and in its deep applications to geometric function the-
+ory, Teichm¨uller space theory and other fields of mathematics and mathematical physics.
+The canonical extremal Beltrami differentials of Teichm¨uller type µ(z)dz/dz with µ(z) =
+
+Extremal properties of Sobolev’s Beltrami coefficients
+11
+|ψ(z)|/ψ(z) generated by integrable holomorphic quadratic differentials ψ(z)dz2 naturally
+arise in many problems.
+As was mentioned above, any such differential is unique in its
+equivalence class, and such maps f µ are dense in SQ(D∗). On unique extremality see also
+[6].
+The extremal quasiconformal extensions of univalent functions with equal Teichm¨uller
+and Grunsky norms (hence, determined by the squares of abelian differentials) have similar
+features.
+The first example of extremal quasiconformal maps of non-Teichm¨uller type was given by
+Strebel [27]. Recently, the author found in [13] an important application of such coefficients
+to geometric problems of Teichm¨uller space theory. Other new types of not canonical ex-
+tremal Beltrami coefficients are given in [18]. The present paper continues this line, and
+Theorem 1 provides, in particular, that the structure of such coefficients can be rather
+pathological.
+4.7. Since the quantities k(f µ) and κD∗(f µ) depend continuously on µ and on the Schwarzians
+Sfµ (respectively, in L∞ and B norms), all assertions of the above theorems are valid for the
+limit functions of sequences µn → µ0 in the indicated norms.
+5. GENERALIZATION
+5.1. Improvement of Theorem 1. In the case, when the domain D is the unit disk D, we
+have the following strengthening of Theorems 1 and 2. Its proof is much more complicated.
+It is based on an important theorem from [17] whose proof involves the deep results on the
+Gaussian curvature, Grunsky inequalities and complex geometry of universal Teichm¨uller
+space.
+Theorem 4. Let the Beltrami coefficient µ ∈ Belt(D)1 satisfy µ ∈ L∞(D) � W 1,p(D) with
+some p > 2, and suppose that the value ∥µ∥∞ = ess supD |µ(z)| is attained by approaching z
+the unit circle S1 and on some subarc γ of S1, we have
+µ(z) ≡ q = const
+with |q| < ∥µ∥∞.
+Then µ is extremal in its class and the corresponding quasiconformal automorphism f µ of �C
+admits the equalities
+k(f µ) = κ(f µ) = qfµ(S1) = 1/ρfµ(S1) = ∥µ∥∞.
+(21)
+Proof. Denote D = f µ(D), D∗ = f µ(D∗) and consider the maps gc, which are conformal
+in D∗ and have in D a constant quasiconformal dilatation c. We regard such maps as the
+affine-like deformations of domain D and the collection of images gc(D) as the affine
+class of D. Each map gc has on D the same Beltrami coefficient as the affine map
+ωc(w) = c1w + c2w + c3
+whose Beltrami coefficient with µωc(w) = c2/c1 = c on C (so gc and ωc differ on ω ◦ f µ(D)
+by a conformal map).
+Consider the map
+fc0(z) = g−q ◦ f µ(z),
+where c0 = −q coincides with the value of µ on the subarc γ0 and is the Beltrami coefficient
+of the map g−q(ω) (inverse to affine deformation gq).
+
+12
+Samuel L. Krushkal
+By the chain rule for Beltrami coefficients, fq has the Beltrami coefficient
+µfq(z) = µ(z) − q
+1 − q µ(z)
+∂zf µ
+∂zf µ.
+It vanishes on the arc f µ(γ) ⊂ ∂D, and therefore the map fq satisfies the assumptions of
+Theorems 1 and 2 on the disk D, which imply for fq the corresponding equalities (10) and
+(11).
+Now we apply the following theorem from [17].
+Theorem A. For any function f ∈ ΣQ with κ(f) = k(f) < 1 mapping the disk D∗ onto
+the complement of a bounded quasidisk D and any affine-like deformation gc of this domain
+(with |c| < 1), we have the equality
+κ(gc ◦ f) = k(gc ◦ f).
+Taking f = fq|D∗ and c = q, one obtains by Theorem A the equalities (21), completing
+the proof of Theorem 4.
+5.2. Geometric features. If a Beltrami coefficient µ ∈ Mp(D), p > 2, is harmonic, then
+also all tµ with |t| < 1 are harmonic. In view of their unique extremality, the image
+Dh(µ0) = {φT(tµ) : |t| < 1}
+is a holomorphic disk (without singular points) in the universal Teichm¨uller space T. By
+Theorem 4 this disk is geodesic in the Teichm¨uller, Kobayashi and Carath´eodory metrics on
+T. This improves the assertion of Remark 4.1.
+It was established in [14] that the Grunsky coefficients of univalent functions generate a
+Finsler structure GT(ϕ, v) on the tangent bundle of the space T, which is dominated by its
+canonical Finsler structure FT(ϕ, v) generating the Teichm¨uller metric of this space. The
+structure GT(ϕ, v) canonically generates the corresponding measurable infinitesimal Finsler
+metric λκ, and due to [14], on any extremal Teichm¨uller disk D(µ0) = {φT(tµ0) : t ∈ D}
+and its isometric images in T, we have the equality
+tanh−1[κ(f rµ0)] =
+r
+�
+0
+λκ(t)dt.
+The arguments applied in [14] remain in force also for harmonic geodesic disks Dh(µ).
+6. ILLUSTRATING EXAMPLES
+We illustrate the above theorems by the following examples.
+Example 1: rectangles. Even this case has been unsolved a long time. Take a rectangle
+P4 with vertices
+A = 0,
+B = a > 0,
+C = a + ib (b > 0),
+D = ib
+and define on segments 0 ≤ x ≤ a and 0 ≤ y ≤ b two absolutely continuous functions h1(x)
+and h2(y) with the derivatives h′
+1(x), h′
+2(y) ∈ Lp, p > 1, and such that h1(0) = 0 and both
+
+Extremal properties of Sobolev’s Beltrami coefficients
+13
+functions are not decreasing. So, h1(x)h2(y) vanishes on the left side of P4. Assume also
+that h1(x) < 1, h2(y) < 1 and define
+µ(z) = (1 + i)h1(x)h2(y),
+z = x + iy.
+Any such µ satisfies the assumption of Theorem 1. The corresponding solution f µ(z) =
+z + a2z2 + . . . , |z| < 1, of the Beltrami equation ∂zw = µ(z)∂zw on C obeys the property
+(10).
+Composing this function with conformal map of the disk D onto P4 given by the Schwarz-
+Christoffel integral
+g(z) =
+z
+�
+0
+dt
+�
+(t2 − 1)(t − i)(t − α)
+,
+where the points 1, i, −1, α are the preimages of the vertices of P4 on S1, one obtains that
+f µ ◦ χ satisfies (10) and (11).
+This strengthens the results of [12] on Fredholm eigenvalues of rectangles obtained by
+applying the Finsler geometry of universal Teichm¨uller space.
+Example 2: ellipses. Let D is the ellipse with the foci at −1, 1 and semiaxes a, b (a > b).
+An orthonormal basis in the Hilbert space A2(D) of the square integrable holomorphic
+functions on D is formed by the polynomials
+Pn(z) = 2
+�
+n + 1
+π
+(rn+1 − r−n−1) Un(z),
+where r = (a + b)2 and Un(z) are the Chebyshev polynomials of the second kind,
+Un(z) =
+1
+√
+1 − z2 sin[(n + 1) arccos z],
+n = 0, 1, . . .
+(see [24]). Then for any Beltramu coefficient µ satisfying the conditions (i), (ii), (iii), we
+have the equalities:
+∥µ∥∞ = sup
+����
+��
+D
+µ(z)
+∞
+�
+0
+cnPn(z)dxdy
+��� :
+�����
+∞
+�
+0
+cnPn(z)
+�����
+A1(D)
+= 1
+�
+= k(f µ) = κD∗(f µ).
+Example 3:
+Analytic curvelinear polygons.
+To simplify the formulas, we pass to
+quasiconformal automorphisms f µ of �C conformal on the lower half-plane H∗ = {z : Im z <
+0} (instead of the disk). Their Beltrami coefficients µ are supported in the upper half-plane
+H = {z : Im z > 0} and run over the ball
+Belt(H)1 = {µ ∈ L∞, µ(z)|H∗ = 0, ∥µ∥∞ < 1},
+and the Schwarzian derivatives Sfµ belong to the space B(H∗) formed by holomorphic qua-
+dratic differentials ϕ(z)dz2 on H with norm ∥ϕ∥B = supH∗ |z − z|2|ϕ(z)|.
+Pick unbounded convex rectilinear polygon Pn with finite vertices A1, . . . , An−1 and An =
+∞. Denoting its exterior angles at Aj by παj so that π < αj < 2π, j = 1, . . . , n − 1, one
+
+14
+Samuel L. Krushkal
+obtains that the conformal map fn of the lower half-plane H∗ = {z : Im z < 0} onto the
+complementary polygon P ∗
+n = �C \ Pn is realized by the Schwarz-Christoffel integral
+fn(z) = d1
+z
+�
+0
+(ξ − a1)α1−1(ξ − a2)α2−1...(ξ − an−1)αn−1−1dξ + d0,
+with aj = f −1
+n (Aj) ∈ R and complex constants d0, d1; here f −1
+n (∞) = ∞. Its Schwarzian
+derivative equals
+Sfn(z) = b′
+fn(z) − 1
+2b2
+fn(z) =
+n−1
+�
+1
+Cj
+(z − aj)2 −
+n−1
+�
+j,l=1
+Cjl
+(z − aj)(z − al),
+where bf = f ′′/f ′,
+Cj = −(αj − 1) − (αj − 1)2/2 < 0,
+Cjl = (αj − 1)(αl − 1) > 0.
+Denote by r0 the positive root of the equation
+1
+2
+�n−1
+�
+1
+(αj − 1)2 +
+n−1
+�
+j,l=1
+(αj − 1)(αl − 1)
+�
+r2 −
+n−1
+�
+1
+(αj − 1) r − 2 = 0,
+and define
+Sfn,t = tb′
+fn − b2
+fn/2, t > 0.
+By Theorem 3 and Ahlfors-Weill, every Schwarzian rSfn,r0 with 0 < r < r0 generates
+a univalent function wr : H∗ → C whose pseudo-harmonic Beltrami coefficient νr(z) =
+−(r/2)y2Sfn,r0z) in H is extremal in its equivalence class, and
+k(wr) = κ(wr◦σ) = r
+2∥Sfn,r0∥B(H∗),
+where σ is the appropriate Moebius map of D∗ onto H∗.
+The point is that in view of extremality of pseudo-harmonic coefficients νr following from
+Theorem 3, the Schwarzians Sfνr with r > r0 close to r0 cannot lie in the space T modelled
+by the Schwarzians; this relates to the well-known problem on starlikness of Teichm¨uller
+spaces in Bers’ embedding, cf. [14] and the references therein.
+The images f νr(H) with 0 < r < r0 are curvilinear polygons with piecewise analytic
+boundaries (in particular, spirals).
+Example 4: Quasiconformal polygons with affine-like sides. We fix on the unit circle
+some points a1, a2, . . . , an following counterclockwise and regard the disk D as a polygon with
+vertices at these points. Take a function u(θ) = c1eiθ + c2e−iθ + c3 for arg a1 ≤ θ ≤ arg a2,
+with complex c1, c2, c3 and |u(θ)| < 1 and extend it to a function �u(θ) on [−π, π] defining
+by Poisson integral a harmonic function µ(z) on the disk D with |µ(z)| < 1, which belongs
+to W 1,p, p > 2, and such that its pull-back to the half-strip Π+ is compatible with the
+Tauberian theorem for the corresponding Laplace transform, giving the equality (20) with
+ν(∞) = ∥µ∥∞. We continue this µ by zero to D∗.
+By Theorem 4, the coefficient µ is extremal in its class, and the corresponding quasicon-
+formal homeomorphism f µ of �C obeys the relations (21).
+
+Extremal properties of Sobolev’s Beltrami coefficients
+15
+References
+[1] L. Ahlfors, Remarks on the Neumann-Poincar´e integral equation, Pacific J. Math. 2 (1952),
+271-280.
+[2] L.V. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand, Princeton, 1966.
+[3] L.V. Ahlfors and G. Weill, A uniqueness theorem for Beltrami equations, Proc. Amer.
+Math. Soc. 13 (1962), 975-978.
+[4] J. Becker, Conformal mappings with quasiconformal extensions, Aspects of Contemporary
+Complex Analysis, Proc. Confer. Durham 1979 (D.A. Brannan and J.G. Clunie, eds.),
+Academic Press, New York, 1980, pp. 37-77.
+[5] L. Bers, A non-standard integral equation with applications to quasiconformal mappings,
+Acta Math. 116 (1966), 113-134.
+[6] V. Bo˘zin, N. Lakic, V. Markovi´c, M. Mateljevi´c, Unique extremality, J. Anal. Math. 75
+(1998), 299-338.
+[7] C.J. Earle, I. Kra and S.L. Krushkal, Holomorphic motions and Teichm¨uller spaces, Trans.
+Amer. Math. Soc. 944 (1994), 927-948.
+[8] F.P. Gardiner and N. Lakic, Quasiconformal Teichm¨uller Theory, Amer. Math. Soc., Prov-
+idence, RI, 2000.
+[9] H. Grunsky, Koeffizientenbedingungen f¨ur schlicht abbildende meromorphe Funktionen,
+Math. Z. 45 (1939), 29-61.
+[10] S.L. Krushkal, Quasiconformal Mappings and Riemann Surfaces, Wiley, New York, 1979.
+[11] S.L. Krushkal, Grunsky coefficient inequalities, Carath´eodory metric and extremal quasi-
+conformal mappings, Comment. Math. Helv. 64 (1989), 650-660.
+[12] S.L. Krushkal, Quasireflections, Fredholm eigenvalues and Finsler metrics, Doklady Math-
+ematics 69 (2004), 221-224.
+[13] S.L. Krushkal, Quasiconformal extensions and reflections, Ch. 11 in: Handbook of Com-
+plex Analysis: Geometric Function Theory, Vol. II (R. K¨uhnau, ed.), Elsevier Science,
+Amsterdam, 2005, pp. 507-553.
+[14] S.L. Krushkal, Strengthened Moser’s conjecture, geometry of Grunsky coefficients and Fred-
+holm eigenvalues, Central European J. Math 5(3) (2007), 551-580.
+[15] S.L. Krushkal, On shape of Teichm¨uller spaces, Journal of Analysis 22 (2014), 69-76.
+[16] S.L. Krushkal, Strengthened Grunsky and Milin inequalities, Contemp. Mathematics 667
+(2016), 159-179.
+[17] S.L. Krushkal, Fredholm eigenvalues and quasiconformal geometry of polygons, J. Math.
+Sci. 252(4) (2021), 472-501. DOI: 10.1007/s10958-020-05175-4
+[18] S.L. Krushkal, On extremality of harmonic Beltrami coefficients, Mathematics 10 (14)
+(2022), 2460. https://doi.org/10.3390/math10142460
+[19] R. K¨uhnau, Verzerrungss¨atze und Koeffizientenbedingungen vom Grunskyschen Typ f¨ur
+quasikonforme Abbildungen, Math. Nachr. 48 (1971), 77-105.
+[20] R. K¨uhnau, Quasikonforme Fortsetzbarkeit, Fredholmsche Eigenwerte und Grunskysche
+Koeffizientenbedingungen, Ann. Acad. Sci. Fenn. Ser. AI. Math. 7 (1982), 383-391.
+[21] R. K¨uhnau, Wann sind die Grunskyschen Koeffizientenbedingungen hinreichend f¨ur Q-
+quasikonforme Fortsetzbarkeit?, Comment. Math. Helv. 61 (1986), 290-307.
+[22] R. K¨uhnau, M¨oglichst konforme Spiegelung an einer Jordankurve, Jber. Deutsch. Math.
+Verein. 90 (1988), 90-109.
+[23] I.M. Milin, Univalent Functions and Orthonormal Systems, Transl. of Mathematical Mono-
+graphs, vol. 49, Transl. of Odnolistnye funktcii i normirovannie systemy, Amer. Math. Soc.,
+Providence, RI, 1977.
+[24] Z. Nehari, Conformal Mapping, McGraw-Hill, NY, 1952.
+[25] M. Schiffer, Fredholm eigenvalues and Grunsky matrices, Ann. Polon. Math. 39 (1981),
+149-164.
+
+16
+Samuel L. Krushkal
+[26] M. Schiffer and D.S. Spencer, Functionals of Finite Riemann Surfaces, Princeton Univer-
+sity Press, Princeton, 1954.
+[27] K. Strebel, Zur Frage der Eindeutigkeit extremaler quasikonformer Abbildungen des Ein-
+heitskreises, Comment. Math. Helv. 36 (1962), 306-323.
+[28] K. Strebel, On the existence of extremal Teichmueller mappings, J. Analyse Math. 30
+(1976), 464-480.
+[29] O. Teichm¨uller, Extremale quasikonforme Abbildungen und quadratische Differentiale,
+Abh. Preuss. Akad. Wiss., Math. Naturw. Kl., 1939 22 (1940), 1-197.
+Department of Mathematics, Bar-Ilan University, 5290002 Ramat-Gan, Israel
+and Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA
+
diff --git a/edFQT4oBgHgl3EQfkDbq/content/tmp_files/load_file.txt b/edFQT4oBgHgl3EQfkDbq/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..b8b7e42e4d6b0b9bbb3db62c848c386cc1830062
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@@ -0,0 +1,486 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf,len=485
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='13357v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='CV] 31 Jan 2023 EXTREMAL PROPERTIES OF SOBOLEV’S BELTRAMI COEFFICIENTS AND DISTORTION OF CURVELINEAR FUNCTIONALS SAMUEL L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' KRUSHKAL Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' An important problem in applications of quasiconformal analysis and in its numerical aspect is to establish algorithms for explicit or approximate determination of the basic quasiinvariant curvelinear and analytic functionals intrinsically connected with con- formal and quasiconformal maps, such as their Teichm¨uller and Grunsky norms, Fredholm eigenvalues and the quasireflection coefficients of associated quasicircles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' We prove a general theorem of new type answering this question for univalent functions in arbitrary quasiconformal domains and provide its applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The results are strengthened in the case of maps of the disk and give rise to extremal Beltrami coefficients of a new type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' 2020 Mathematics Subject Classification: Primary: 30C62, 30C75;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Secondary: 30F60, 32F45,32G15, 46G20 Key words and phrases: Univalent function, Grunsky operator, quasiconformal extension, the Sobolev spaces, harmonic Beltrami coefficients, quasicircles, quasireflections, universal Teichm¨uller space 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' PRELIMINARIES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Preamble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' An important still open problem in geometric complex analysis is to es- tablish algorithms for explicit or approximate determination of the basic curvilinear and analytic functionals intrinsically connected with conformal and quasiconformal maps, such as their Teichm¨uller and Grunsky norms, Fredholm eigenvalues and the quasireflection co- efficients of associated quasicircles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' It is important also for the potential theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' However, this has not been solved completely even for convex polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The problem has intrinsic interest also in view of its connection with geometry of Te- ichm¨uller spaces and with the approximation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' It is crucial also for numerical aspect of quasiconformal analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' This paper deals with functions not admitting the canonical Teichm¨uller extensions and solves explicitly the indicated problem for a natural rather broad class of Beltrami coefficients supported in generic quasiconformal domains, in other words, for univalent functions with quasiconformal extension on arbitrary quasidisks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The results give rise to extremal Beltrami coefficients of a new type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The Teichm¨uller and Grunsky norms of univalent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' We start with the class ΣQ of univalent functions f(z) = b0 + b1z−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' in the disk D∗ = {z ∈ �C : |z| > 1} admitting quasiconformal extensions across the boundary unit circle S1 = ∂D∗, hence to the whole Riemann sphere �C = C ∪ {∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' To have compactness in the topology of locally Date: February 1, 2023 (SobolBel1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='tex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' 1 2 Samuel L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Krushkal uniform convergence on C, one must add the third normalization condition, for example, f(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The Beltrami coefficients of extensions are supported in the unit disk D = {|z| < 1} and run over the unit ball Belt(D)1 = {µ ∈ L∞(C) : µ(z)|D∗ = 0, ∥µ∥∞ < 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Each µ ∈ Belt(D)1 determines a unique homeomorphic solution to the Beltrami equation ∂w = µ∂w on C (quasiconformal automorphism of �C) normalized by the assumptions wµ ∈ ΣQ, wµ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The Schwarzian derivatives of these functions Sw(z) = �w′′(z) w′(z) �′ − 1 2 �w′′(z) w′(z) �2 , z ∈ D∗ belong to the complex Banach space B = B(D∗) of hyperbolically bounded holomorphic functions in the disk D∗ with norm ∥ϕ∥B = sup D (|z|2 − 1)2|ϕ(z)| and run over a bounded domain in B modeling the universal Teichm¨uller space T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The space B is dual to the Bergman space A1(D∗), a subspace of L1(D∗) formed by integrable holomorphic functions (quadratic differentials ϕ(z)dz2) on D∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Note that ϕ(z) = O(z−4) near z = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The needed results from Teichm¨uller space theory see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=', in [7], [8], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' One defines for any f ∈ ΣQ its Grunsky coefficients αmn from the expansion log f(z) − f(ζ) z − ζ = ∞ � m,n=1 αmnz−mζ−n, (z, ζ) ∈ (D∗)2, (1) where the principal branch of the logarithmic function is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' These coefficients satisfy the inequality ��� ∞ � m,n=1 √mn αmnxmxn ��� ≤ 1 (2) for any sequence x = (xn) from the unit sphere S(l2) of the Hilbert space l2 with norm ∥x∥ = � ∞ � 1 |xn|2�1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' conversely, the inequality (2) also is sufficient for univalence of a locally univalent function in D∗ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' [9], [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The minimum k(f) of dilatations k(wµ) = ∥µ∥∞ among all quasiconformal extensions wµ(z) of f onto the whole plane �C (forming the equivalence class of f) is called the Te- ichm¨uller norm of this function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Hence, k(f) = tanh dT(0, Sf), where dT denotes the Teichm¨uller-Kobayashi metric on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' This quantity dominates the Grunsky norm κ(f) = sup ���� ∞ � m,n=1 √mn αmn(f)xmxn ��� : x = (xn) ∈ S(l2) � by κ(f) ≤ k(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' For most functions f, we have the strong inequality κ(f) < k(f) (moreover, the functions satisfying this inequality form a dense subset of Σ), while the functions with the equal norms play a crucial role in many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Extremal properties of Sobolev’s Beltrami coefficients 3 These norms coincide only when any extremal Beltrami coefficient µ0 for f (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=', with ∥µ0∥∞ = k(f)) satisfies ∥µ0∥∞ = sup ���� �� D µ0(z)ψ(z)dxdy ��� : ψ ∈ A2 1(D), ∥ψ∥A1(D) = 1 � = κ(f) (z = x + iy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' (3) Here A1(D) denotes the subspace in L1(D) formed by integrable holomorphic functions (qua- dratic differentials ψ(z)dz2 on D, and A2 1(D) is its subset consisting of ψ with zeros of even order on D, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=', of the squares of holomorphic functions (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=', [11], [13], [16], [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Note that every ψ ∈ A2 1(D) has the form ψ(z) = 1 π ∞ � m+n=4 √mn xmxnzm+n−2 (4) and ∥ψ∥A1(D) = ∥x∥l2 = 1, x = (xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The method of Grunsky inequalities was generalized in several direc- tions, even to bordered Riemann surfaces X with a finite number of boundary components (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=', [23], [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' We shall consider these inequalities in unbounded simply connected hyperbolic domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Let L ⊂ C be an oriented bounded quasicircle separating the points 0 and ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Denote its interior and exterior domains by D and D∗ (so 0 ∈ D, ∞ ∈ D∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Then, if δD(z) denotes the Euclidean distance of z from the boundary of D and λD(z)|dz| is its hyperbolic metric of Gaussian curvature −4, we have 1 4 ≤ λD(z)δD(z) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' (5) The right hand inequality follows from the Schwarz lemma and the left from the Koebe one-quoter theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' For such a domain D∗ ∋ ∞, the expansion (1) assumes the form − log f(z) − f(ζ) z − ζ = ∞ � m,n=1 βmn √mn χ(z)m χ(ζ)n, where χ denotes a conformal map of D∗ onto the disk D∗ so that χ(∞) = ∞, χ′(∞) > 0 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Accordingly, the generalized Grunsky norm is defined by κD∗(f) = sup ���� ∞ � m,n=1 βmn xmxn ��� : x = (xn) ∈ S(l2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' We now consider the class ΣQ(D∗) of univalent functions in domain D∗ with expansions f(z) = b0 + b1z−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' near z = ∞, admitting quasiconformal extensions onto the com- plementary domain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Similar to above, we subject these extensions to f(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Their Beltrami coefficients run over the ball Belt(D)1 = {µ ∈ L∞(C) : µ(z)|D∗ = 0, ∥µ∥∞ < 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' A coefficient µ0 ∈ Belt(D∗)1 is extremal in its class if and only if ∥µ0∥∞ = sup ���� �� D∗ µ0(z)ψ(z)dxdy ��� : ψ ∈ A1(D∗), ∥ψ∥A1 = 1 � , 4 Samuel L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Krushkal and similar to (3), the equality κD∗(f µ) = k(f µ) is valid if and only if ∥µ∥∞ = sup ���� �� D µ(z)ψ(z)dxdy ��� : ψ ∈ A2 1(D), ∥ψ∥A1(D) = 1 � = κD(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' If additionally the equivalence class of f is a Strebel point of the space T with base point D∗, which means that this class contains the Teichm¨uller extremal extension f k|ψ0|/ψ0 with ψ0 ∈ A1(D), then necessarily ψ0 = ω2 ∈ A2 1 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' [11], [16], [21]), [28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The Strebel points are dense in any Teichm¨uller space, see [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' For arbitrary quasidisks D, the corresponding set A2 1(D) is characterized similar to (4), but in more complicated way (see [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Assume that µ0 ∈ Belt(D)1 is extremal in its class but not of Teichm¨uller type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' A point z0 ∈ ∂D is called substantial (or essential) for µ0 if for any ε > 0 there exists a neighborhood U0 of z0 such that sup D∗\\U0 |µ0(z)| < ∥µ0∥∞ − ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' so the maximal dilatation k(wµ0) = ∥µ∥∞ is attained on D by approaching this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' In addition, there exists a sequence {ψn} ⊂ A1(D) such that ψn(z) → 0 locally uniformly on D but ∥ψn∥ = 1 for any n, and lim n→∞ �� D µ0(z)ψn(z)dxdy = ∥µ0∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Such sequences are called degenerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The image of a substantial point is a common point of two quasiconformal arcs, which can be of spiral type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Fredholm eigenvalues and quasireflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The Teichm¨uller and Grunsky norms are intrinsically connected with quasiconformal reflections, Fredholm eigenvalues and other quasiinvariants of quasiconformal curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' We outline briefly the main notions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' the details see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=', in [2], [12], [17], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The quasiconformal reflections (or quasireflections) are the orientation reversing qua- siconformal homeomorphisms of the sphere �C which preserve point-wise some (oriented) quasicircle L ⊂ �C and interchange its interior and exterior domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' One defines for L its reflection coefficient qL = inf k(f) = inf ∥∂zf/∂zf∥∞, taking the infimum over all quasireflections across L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Due to [2], [22], the dilatation QL = (1 + qL)/(1 − qL) ≥ 1 is equal to the quantity QL = (1 + kL)2/(1 − kL)2, (6) where kL is the minimal dilatation among all orientation preserving quasiconformal auto- morphisms f∗ of �C carrying the unit circle onto L, and k(f∗) = ∥∂zf∗/∂zf∗∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The reflection with dilatation QL is extremal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' A remarkable and very useful fact estab- lished by Ahlfors is that any quasicircle also admits a Lipschitz continuous quasireflection with some coefficient C(qL) (see [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Extremal properties of Sobolev’s Beltrami coefficients 5 The Fredholm eigenvalues ρn of an oriented smooth closed Jordan curve L ⊂ �C are the eigenvalues of its double-layer potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' These values are crucial in many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The least positive eigenvalue ρL = ρ1 plays is naturally connected with conformal and quasiconformal maps and can be defined for any oriented closed Jordan curve L by 1 ρL = sup |DG(u) − DG∗(u)| DG(u) + DG∗(u) , where G and G∗ are, respectively, the interior and exterior of L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' D denotes the Dirichlet integral, and the supremum is taken over all functions u continuous on �C and harmonic on G ∪ G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' A rough upper bound for ρL is given by Ahlfors’ inequality 1 ρL ≤ qL, where qL denotes the minimal dilatation of quasireflections across L [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' One of the basic tools in quantitative estimating the Freholm eigenvalues ρL of quasicircles is given by the K¨uhnau-Schiffer theorem [20], [25], which states that the value ρL is reciprocal to the Grunsky norm κ(f) of the Riemann mapping function of the exterior domain of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' For all functions f ∈ SQ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=', univalent in the disk D∗) with k(f) = κ(f), we have the exact explicit values qf(S1) = 1 ρf(S1) = κ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' (7) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Harmonic and pseudo-harmonic Beltrami coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' By the Ahlfors-Weill theorem strengthening the classical Nehari’s result on univalence in terms of the Schwarzians, every function ϕ ∈ B(D) with ∥ϕ∥B < 2 is the Schwarzian derivative of a univalent function f(z) in the unit disk D, and the function f has quasiconformal extension onto the disk D∗ with Beltrami coefficient µϕ(z) = −1 2(|z|2 − 1)2ϕ(1/z)(1/z4), z ∈ D∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' (8) see [3], [4], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' This deep fact is extended to arbitrary quasidisks in much weaker form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' For example, we have the following result of Bers, which is a special case of his more general extension theorem given in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' [5] Let L be a quasicircle on �C with the interior DL and exterior D∗ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Then, for some ε > 0, there exists an anti-holomorphic homeomorphism τ (with τ(0) = 0) of the ball Vε = {ϕ ∈ B(D∗ L) : ∥ϕ∥} < ε into B(DL) such that every ϕ in Vε is the Schwarzian derivative of some univalent function f which is the restriction to D∗ L of a quasiconformal automorphism �f of Riemann sphere �C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' This �f can be chosen in such a way that its Beltrami coefficient on DL has the form µ �f(z) = λ−2 D (z)ψ(z), ψ = τ(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' (9) The Beltrami coefficients of such form are called now harmonic in view of their connection with the Kodaira-Spencer deformation theory of complex structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' 6 Samuel L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Krushkal In this paper we shall use the notion of harmonicity in its original sense, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=', only for solutions of the Laplace equation ∆u = 0, and regard the coefficients of type (9) as pseudo- harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' GENERAL THEOREM AND ITS CONSEQUENCES Our goal is to describe the features of extremal Beltramic oefficients of non-Teichm¨uller type maximizing the Grunsky norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' We shall consider the univalent functions f(z) ∈ SQ(D∗) whose restrictions to the bound- ary quasicircle L = ∂D∗ have substantional points, hence do not have the Teichm¨uller extremal extensions across L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Denote the collection of such f by S0 Q(D∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' For L = S1, we use the notation S0 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Fix p > 1 and consider the subset Mp(D) of Belt(D)1, which consists of the Beltrami coefficients µ defining the maps f µ ∈ S0 Q(D) and satisfying: (i) µ ∈ L∞(D) � W 1,p(D),where W 1,p(D) is the Sobolev space of functions µ in D having the first distributional derivatives which belong to Lp(D);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' (ii) the value ∥µ∥∞ = ess supD |µ(z)| is attained by approaching z the boundary of D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' (iii) there is a subarc γ ⊂ ∂D depending on µ such that µ(z) → 0 as z approaches γ from inside D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The boundary values µ(z0) for z0 ∈ ∂D must be understand as lim z→z0 µ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Note that the value ∥µ∥∞ also can be attained by approaching the inner points of domain D and that the arc γ is locally Cα smooth with α > 0 depending on ∥µ∥∞, in accordance with the H¨older continuity of quasiconformal automorphisms of �C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The main result of this paper is the following general theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' For any p > 2, every Beltrami coefficient µ ∈ Mp(D) is extremal in its equivalence class [µ], and the corresponding quasiconformal automorphism f µ of �C satisfies k(f µ) = κD∗(f µ) = ∥µ∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' (10) In the case p > 1, there is a weakened version of this theorem, presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Until now, there were no results in geometric function theory giving the explicit expression of the generalized Grunsky norm for the general quasiconformal domains different from the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Theorem 1 implies the first general result in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' In view of the assumptions on the set Mp(D), every boundary function f µ|∂D must have at least one substantial point, at which ∥µ∥∞ is attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' This yields that the coefficient µ is not uniquely extremal in its equivalence class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' It follows from what was indicated above that the situation described by Theorem 1 does not appear in the case of sufficiently high boundary regularity of univalent functions f (and of ∂D) because, for example, for any C2+α smooth µ the map f µ is C2+α on C), and by Strebel’s frame mapping criterion [8], [28]) the equivalence class of f µ contains unique Teichm¨uller coefficient µ0 = k|ψ0|/ψ0 with ψ0 ∈ A1(D), which cannot have the substantial boundary points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The assumption on the set Mp are natural and cannot be replaced in terms of smoothness or non-smoothness of µ on the boundary points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Extremal properties of Sobolev’s Beltrami coefficients 7 In the case, when the domain D is the unit disk D, Theorem 1 can be completed by the quantitative results on Fredholm eigenvalues and reflection coefficients indicated in the previous section (see the relations (6), (7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' For any p > 2, every Beltrami coefficient µ ∈ Mp(D) is extremal in its equivalence class;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' in addition, the reflection coefficient and the Fredholm eigenvalue of the curve L1 = f µ(S1) are explicitly given by qL1 = 1/ρL1 = ∥µ∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' (11) Here also µ can be regular only in admissible bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The assumption for µ to belong to the Sobolev space W p 1 with p > 2 is essential for the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' In the case of pseudo-harmonic Beltrami coefficients, the assumptions of the above theo- rems can be weakened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The equalities (9) and (10) are valid for any admissible for univalence pseudo- harmonic Beltrami coefficient µϕ(z) of the form (9) or (8) with ϕ = Sf such that the maxi- mum of the function λ−2 D (z)Sf(z) is attained at some boundary point and Sf is bounded on some subarc γ of ∂D∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' For such coefficients, the required smoothness of µ is provided by the representations (8), (9), while vanishing on γ follows from (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' A special case of Theorem 3 has been obtained in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' PROOF OF THEOREM 1 To prove that the Beltrami coefficient µ satisfying the prescribed conditions (i), (ii), (iii) is extremal in its equivalence class and provides the equalities (10), we establish that it must satisfy ∥µ∥∞ = sup ∥ψ∥A2 1(D)=1 ��� �� D µ(z)ψ(z)dxdy ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' (12) This equality means that ∥µ∥∞ is attained in L1(D) on the set of abelian quadratic differ- entials intrinsically connected with the Grunsky coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' By the Sobolev embedding theorem, the function µ(z) is extended to a continuous function on the closed domain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The assumption (ii) implies that there is a point z0 ∈ ∂D at which |µ(z0)| = ∥µ∥∞ (13) (in the general case, |µ(z0)| = lim sup z→∂D |µ(z)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' We take the conformal map z = χ(ζ) of the half-strip Π+ = {ζ = ξ + iη : ξ > 0, 0 < η < 1} onto D such that χ−1(z0) = ∞ and the pre-images of the endpoints of the arc γ are the points 0 and 1, and pull back the coefficient µ to Beltrami coefficient ν(ζ) := χ∗µ(z) = (µ ◦ χ)(ζ) χ′(ζ)/χ′(ζ) (14) on Π+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The horizontal lines lη = {ζ = ξ + iη : 0 < ξ < ∞}, 0 < η < 1, 8 Samuel L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Krushkal are moving under this map into some analytic curves in D with endpoints on ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The infinite point ζ = ∞ is substantial for the composite map F ν = f µ ◦ χ, and by assumptions on µ and from (14), we have lim ξ→∞ |ν(ξ + iη)| = ∥ν∥∞ = ∥µ∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The assumptions on the coefficient µ(z) and the smoothness of conformal map χ imply that ν ∈ L∞ � W 2 p (Π+) and that the limit function ν(ζ0) = lim ζ→ζ0∈∂Π+ ν(ζ), is defined at all ζ0 ∈ ∂Π+ different from the points 0, i, ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Hence, ν(ζ) is bounded on the interval [0, i] of the imaginary axes, and moreover, ν(iη) = 0, 0 ≤ η ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' (15) Now we pick the sequence ωm(ζ) = 1 me−ζ/m, ζ ∈ Π+ (m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=');' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' all these ωm belong to A2 1(Π+) and ωm(ζ) → 0 uniformly on Π+ �{|ζ| < M} for any M < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Also, ∥ωm∥A1(Π+) = 1 (moreover, ���� Π+ ωmdξdη �� = 1 − O(1/m)), which shows that {ωm} is a degenerating sequence for the affine horizontal stretching of Π+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' We prove that this sequence is also degenerated for ν, estimating the integrals Im = �� Π+ ν(ζ)ωm(ζ)dξdη for large m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' We have Im = 1 � 0 e−iη/mdη � 1 m ∞ � 0 ν(ξ + iη)e−ξ/mdξ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' (16) The inner integral in (16) represents the values of the Laplace transform Lν = ∞ � 0 ν(t)e−stdt of ν(ξ + iη) in the points s = 1/m, and the assumptions of the theorem imply the existence of this transform also for the derivative ∂ν(ξ + iη)/∂ξ, which is integrable over Π+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Integrating by parts and using (15), one obtains ∞ � 0 ∂ν(ξ + iη) ∂ξ e−ξ/mdξ = 1 m ∞ � 0 ν(ξ + iη)e−ξ/mdξ − ν(iη) = 1 m ∞ � 0 ν(ξ + iη)e−ξ/mdξ (17) Since for any s > 0, ���� ∂ν(ξ + iη) ∂ξ ���� e−sξ < ���� ∂ν(ξ + iη) ∂ξ ���� , Extremal properties of Sobolev’s Beltrami coefficients 9 the Lebesgue theorem on dominated convergence yields lim s→0 ∞ � 0 ∂ν(ξ + iη) ∂ξ e−sξdξ = ∞ � 0 ∂ν(ξ + iη) ∂ξ dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Together with (15) and (17), this implies lim m→∞ 1 m ��� ∞ � 0 ν(ξ + iη)e−ξ/mdξ ��� = |ν(∞) − ν(iη)| = |ν(∞)|, (18) where ν(∞) = lim ξ→−∞ ν(ξ + iη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' It follows that the integral (16) has the limit value lim m→∞ �� Π+ ν(ζ)ωm(ζ)dξdη = 1 � 0 dη lim m→∞ 1 m ∞ � 0 ν(ξ + iη)e−ξ/mdξ = ν(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' (19) Now we return to the initial domain D by applying the inverse conformal map χ−1(z) : D → Π+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The corresponding sequence ψm = (ωm ◦ χ−1)(χ′)−2, m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' , is a degenerating sequence for the initial Beltrami coefficient µ on D, and by (19), lim m→∞ ��� �� D µ(z)ψm(z)dxdy ��� = lim m→∞ ��� �� Π+ ν(ζ)ωm(ζ)dξdη ��� = |ν(∞)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' (20) In view of the assumption (13), all terms in (20) are equal to ∥ν∥∞ = ∥µ∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' This proves the equality (12), which implies that µ is extremal in its class and that the map f µ obeys the relations (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The theorem follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Note that under the assumption p > 2, the conformal map χ of Π+ onto D is C1,α-smooth in the closed domain Π+, excluding the points 0, 1, ∞, which yields that the restrictions νη of the Beltrami coefficient (14) to the lines lη belong to W 1,p(lη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Then the embedding theorem for W n,p functions giving the continuity of these restrictions implies the existence of the limits ν(0) = νη(0) = lim ξ→0 νη(ξ) for all η, and νη(∞) = limξ→∞ ν(ξ + iη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' This remark indicates the way in which the above proof can be extended to p > 1 (see remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' ADDITIONAL REMARKS TO THEOREMS 1 AND 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Theorem 1 implies that for every Beltrami coefficient µ0 ∈ Mp(D) the disk {tµ0/∥µ0∥∞ : |t| < 1} is geodesic in the ball Belt(D)1 simultaneously for the Teichm¨uller, Kobayashi and Carath´eodory metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' It is pushed down under the projection φT : µ → Swµ ∈ B(D∗) onto a holomorphic 10 Samuel L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Krushkal disk in universal Teichm¨uller space T, on which all these invariant distances also coincide, though this coefficient does not be necessarily uniquely extremal in its equivalence class (the images of two different t′µ0/∥µ0∥∞ and t′′µ0/∥µ0∥∞ in T can be the same;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' as well as, there are µ ∈ [µ0] with ∥µ∥∞ = ∥µ0∥∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The equality (18) is equivalent to the Tauberian theorem for Laplace transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' All known theorems of this type also require some smoothness of the original functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' This nice connection with the Laplace transform implies simultaneously the extremality and equality of the Teichm¨uller and Grunsky norms norms for a broad set of Beltrami coefficients given by Theorems 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' An extension of Theorem 1 to p > 1 is possible under the additional assumptions on D and µ, for example, when the boundary of domain D is C1,α-smooth (α > 0) and in (ii) the value ∥µ∥∞ = ess supD |µ(z)| is attained at some point z0 ∈ ∂D as z → z0 along any way in D, then the above prove can be modified for µ ∈ L∞(D) � W 1,p(D) with p > 1 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Since now the conformal map χ of Π+ onto D also is C1,α-smooth in the closed domain Π+, excluding the points 0, 1, ∞, we have that the restrictions νη of the Beltrami coefficient (14) to the lines lη belong to W 1,p(lη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Applying the embedding theorem in dimension n = 1, one obtains for p > 1 the existence of limits νη(0) = lim ξ→0 νη(ξ) = 0 and νη(∞) = lim ξ→∞ ν(ξ + iη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The indicated change of (ii) yields that all values νη(∞) must coincide and are equal to ∥ν∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Therefore, one can again apply the relations (18)-(20) and derive the assertions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The assertion of Theorem 1 can be extended to arbitrary quasiconformal domains, even with fractal boundaries L in the following form: one can take in the half-strip Π+ the Beltrami coefficients µ obeying the Tauberian theorem for the Laplace transform, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=', for which the arguments of the proof of Theorem 1 are valid, and pull back these µ to the interior of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The canonical quasiconformal extensions of univalent functions (with Teichm¨uller or pseudo-harmonic of type (9) coefficients µ) are unique;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' the uniqueness intrinsically relates to their analyticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Theorem 1 involving Beltrami coefficients from Sobolev’s space provides a possibility to distinguish some subsets in Mp admitting uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' For example, it holds for µ which are (even weak) solutions of the Dirichlet problem for uniformly elliptic differential equations of the second order Lu = 0 on D with prescribed values µ on the boundary curve ∂D, in particular, for harmonic µ(z) (with ∆µ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' In addition, in this case the assumption (ii) is trivially fulfilled by the maximum principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' In particular, all harmonic µ (with ∆µ = 0) vanishing on some boundary subarcs γ are extremal, except when the boundary function f µ|∂D satisfies the Strebel frame mapping condition (for example, it is C2+α smooth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The theory of extremal quasiconformal maps originated in [29] plays now a crucial role in quasiconformal analysis and in its deep applications to geometric function the- ory, Teichm¨uller space theory and other fields of mathematics and mathematical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The canonical extremal Beltrami differentials of Teichm¨uller type µ(z)dz/dz with µ(z) = Extremal properties of Sobolev’s Beltrami coefficients 11 |ψ(z)|/ψ(z) generated by integrable holomorphic quadratic differentials ψ(z)dz2 naturally arise in many problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' As was mentioned above, any such differential is unique in its equivalence class, and such maps f µ are dense in SQ(D∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' On unique extremality see also [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The extremal quasiconformal extensions of univalent functions with equal Teichm¨uller and Grunsky norms (hence, determined by the squares of abelian differentials) have similar features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The first example of extremal quasiconformal maps of non-Teichm¨uller type was given by Strebel [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Recently, the author found in [13] an important application of such coefficients to geometric problems of Teichm¨uller space theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Other new types of not canonical ex- tremal Beltrami coefficients are given in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The present paper continues this line, and Theorem 1 provides, in particular, that the structure of such coefficients can be rather pathological.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Since the quantities k(f µ) and κD∗(f µ) depend continuously on µ and on the Schwarzians Sfµ (respectively, in L∞ and B norms), all assertions of the above theorems are valid for the limit functions of sequences µn → µ0 in the indicated norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' GENERALIZATION 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Improvement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' In the case, when the domain D is the unit disk D, we have the following strengthening of Theorems 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Its proof is much more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' It is based on an important theorem from [17] whose proof involves the deep results on the Gaussian curvature, Grunsky inequalities and complex geometry of universal Teichm¨uller space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Let the Beltrami coefficient µ ∈ Belt(D)1 satisfy µ ∈ L∞(D) � W 1,p(D) with some p > 2, and suppose that the value ∥µ∥∞ = ess supD |µ(z)| is attained by approaching z the unit circle S1 and on some subarc γ of S1, we have µ(z) ≡ q = const with |q| < ∥µ∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Then µ is extremal in its class and the corresponding quasiconformal automorphism f µ of �C admits the equalities k(f µ) = κ(f µ) = qfµ(S1) = 1/ρfµ(S1) = ∥µ∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' (21) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Denote D = f µ(D), D∗ = f µ(D∗) and consider the maps gc, which are conformal in D∗ and have in D a constant quasiconformal dilatation c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' We regard such maps as the affine-like deformations of domain D and the collection of images gc(D) as the affine class of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Each map gc has on D the same Beltrami coefficient as the affine map ωc(w) = c1w + c2w + c3 whose Beltrami coefficient with µωc(w) = c2/c1 = c on C (so gc and ωc differ on ω ◦ f µ(D) by a conformal map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Consider the map fc0(z) = g−q ◦ f µ(z), where c0 = −q coincides with the value of µ on the subarc γ0 and is the Beltrami coefficient of the map g−q(ω) (inverse to affine deformation gq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' 12 Samuel L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Krushkal By the chain rule for Beltrami coefficients, fq has the Beltrami coefficient µfq(z) = µ(z) − q 1 − q µ(z) ∂zf µ ∂zf µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' It vanishes on the arc f µ(γ) ⊂ ∂D, and therefore the map fq satisfies the assumptions of Theorems 1 and 2 on the disk D, which imply for fq the corresponding equalities (10) and (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Now we apply the following theorem from [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' For any function f ∈ ΣQ with κ(f) = k(f) < 1 mapping the disk D∗ onto the complement of a bounded quasidisk D and any affine-like deformation gc of this domain (with |c| < 1), we have the equality κ(gc ◦ f) = k(gc ◦ f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Taking f = fq|D∗ and c = q, one obtains by Theorem A the equalities (21), completing the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Geometric features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' If a Beltrami coefficient µ ∈ Mp(D), p > 2, is harmonic, then also all tµ with |t| < 1 are harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' In view of their unique extremality, the image Dh(µ0) = {φT(tµ) : |t| < 1} is a holomorphic disk (without singular points) in the universal Teichm¨uller space T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' By Theorem 4 this disk is geodesic in the Teichm¨uller, Kobayashi and Carath´eodory metrics on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' This improves the assertion of Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' It was established in [14] that the Grunsky coefficients of univalent functions generate a Finsler structure GT(ϕ, v) on the tangent bundle of the space T, which is dominated by its canonical Finsler structure FT(ϕ, v) generating the Teichm¨uller metric of this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The structure GT(ϕ, v) canonically generates the corresponding measurable infinitesimal Finsler metric λκ, and due to [14], on any extremal Teichm¨uller disk D(µ0) = {φT(tµ0) : t ∈ D} and its isometric images in T, we have the equality tanh−1[κ(f rµ0)] = r � 0 λκ(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The arguments applied in [14] remain in force also for harmonic geodesic disks Dh(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' ILLUSTRATING EXAMPLES We illustrate the above theorems by the following examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Example 1: rectangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Even this case has been unsolved a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Take a rectangle P4 with vertices A = 0, B = a > 0, C = a + ib (b > 0), D = ib and define on segments 0 ≤ x ≤ a and 0 ≤ y ≤ b two absolutely continuous functions h1(x) and h2(y) with the derivatives h′ 1(x), h′ 2(y) ∈ Lp, p > 1, and such that h1(0) = 0 and both Extremal properties of Sobolev’s Beltrami coefficients 13 functions are not decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' So, h1(x)h2(y) vanishes on the left side of P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Assume also that h1(x) < 1, h2(y) < 1 and define µ(z) = (1 + i)h1(x)h2(y), z = x + iy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Any such µ satisfies the assumption of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The corresponding solution f µ(z) = z + a2z2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' , |z| < 1, of the Beltrami equation ∂zw = µ(z)∂zw on C obeys the property (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Composing this function with conformal map of the disk D onto P4 given by the Schwarz- Christoffel integral g(z) = z � 0 dt � (t2 − 1)(t − i)(t − α) , where the points 1, i, −1, α are the preimages of the vertices of P4 on S1, one obtains that f µ ◦ χ satisfies (10) and (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' This strengthens the results of [12] on Fredholm eigenvalues of rectangles obtained by applying the Finsler geometry of universal Teichm¨uller space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Example 2: ellipses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Let D is the ellipse with the foci at −1, 1 and semiaxes a, b (a > b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' An orthonormal basis in the Hilbert space A2(D) of the square integrable holomorphic functions on D is formed by the polynomials Pn(z) = 2 � n + 1 π (rn+1 − r−n−1) Un(z), where r = (a + b)2 and Un(z) are the Chebyshev polynomials of the second kind, Un(z) = 1 √ 1 − z2 sin[(n + 1) arccos z], n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' (see [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Then for any Beltramu coefficient µ satisfying the conditions (i), (ii), (iii), we have the equalities: ∥µ∥∞ = sup ���� �� D µ(z) ∞ � 0 cnPn(z)dxdy ��� : ����� ∞ � 0 cnPn(z) ����� A1(D) = 1 � = k(f µ) = κD∗(f µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Example 3: Analytic curvelinear polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' To simplify the formulas, we pass to quasiconformal automorphisms f µ of �C conformal on the lower half-plane H∗ = {z : Im z < 0} (instead of the disk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Their Beltrami coefficients µ are supported in the upper half-plane H = {z : Im z > 0} and run over the ball Belt(H)1 = {µ ∈ L∞, µ(z)|H∗ = 0, ∥µ∥∞ < 1}, and the Schwarzian derivatives Sfµ belong to the space B(H∗) formed by holomorphic qua- dratic differentials ϕ(z)dz2 on H with norm ∥ϕ∥B = supH∗ |z − z|2|ϕ(z)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Pick unbounded convex rectilinear polygon Pn with finite vertices A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' , An−1 and An = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Denoting its exterior angles at Aj by παj so that π < αj < 2π, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' , n − 1, one 14 Samuel L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Krushkal obtains that the conformal map fn of the lower half-plane H∗ = {z : Im z < 0} onto the complementary polygon P ∗ n = �C \\ Pn is realized by the Schwarz-Christoffel integral fn(z) = d1 z � 0 (ξ − a1)α1−1(ξ − a2)α2−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content='(ξ − an−1)αn−1−1dξ + d0, with aj = f −1 n (Aj) ∈ R and complex constants d0, d1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' here f −1 n (∞) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Its Schwarzian derivative equals Sfn(z) = b′ fn(z) − 1 2b2 fn(z) = n−1 � 1 Cj (z − aj)2 − n−1 � j,l=1 Cjl (z − aj)(z − al), where bf = f ′′/f ′, Cj = −(αj − 1) − (αj − 1)2/2 < 0, Cjl = (αj − 1)(αl − 1) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Denote by r0 the positive root of the equation 1 2 �n−1 � 1 (αj − 1)2 + n−1 � j,l=1 (αj − 1)(αl − 1) � r2 − n−1 � 1 (αj − 1) r − 2 = 0, and define Sfn,t = tb′ fn − b2 fn/2, t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' By Theorem 3 and Ahlfors-Weill, every Schwarzian rSfn,r0 with 0 < r < r0 generates a univalent function wr : H∗ → C whose pseudo-harmonic Beltrami coefficient νr(z) = −(r/2)y2Sfn,r0z) in H is extremal in its equivalence class, and k(wr) = κ(wr◦σ) = r 2∥Sfn,r0∥B(H∗), where σ is the appropriate Moebius map of D∗ onto H∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The point is that in view of extremality of pseudo-harmonic coefficients νr following from Theorem 3, the Schwarzians Sfνr with r > r0 close to r0 cannot lie in the space T modelled by the Schwarzians;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' this relates to the well-known problem on starlikness of Teichm¨uller spaces in Bers’ embedding, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' [14] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' The images f νr(H) with 0 < r < r0 are curvilinear polygons with piecewise analytic boundaries (in particular, spirals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Example 4: Quasiconformal polygons with affine-like sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' We fix on the unit circle some points a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' , an following counterclockwise and regard the disk D as a polygon with vertices at these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Take a function u(θ) = c1eiθ + c2e−iθ + c3 for arg a1 ≤ θ ≤ arg a2, with complex c1, c2, c3 and |u(θ)| < 1 and extend it to a function �u(θ) on [−π, π] defining by Poisson integral a harmonic function µ(z) on the disk D with |µ(z)| < 1, which belongs to W 1,p, p > 2, and such that its pull-back to the half-strip Π+ is compatible with the Tauberian theorem for the corresponding Laplace transform, giving the equality (20) with ν(∞) = ∥µ∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' We continue this µ by zero to D∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' By Theorem 4, the coefficient µ is extremal in its class, and the corresponding quasicon- formal homeomorphism f µ of �C obeys the relations (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
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+page_content=' Analyse Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' 30 (1976), 464-480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' [29] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Teichm¨uller, Extremale quasikonforme Abbildungen und quadratische Differentiale, Abh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Preuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Wiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=', Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Naturw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Kl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=', 1939 22 (1940), 1-197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
+page_content=' Department of Mathematics, Bar-Ilan University, 5290002 Ramat-Gan, Israel and Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFQT4oBgHgl3EQfkDbq/content/2301.13357v1.pdf'}
diff --git a/ftE4T4oBgHgl3EQfRQzA/content/tmp_files/2301.04989v1.pdf.txt b/ftE4T4oBgHgl3EQfRQzA/content/tmp_files/2301.04989v1.pdf.txt
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+arXiv:2301.04989v1 [quant-ph] 12 Jan 2023
+UMTG–315
+Qudit Dicke state preparation
+Rafael I. Nepomechie
+Physics Department, P.O. Box 248046, University of Miami
+Coral Gables, FL 33124 USA
+Abstract
+Qudit Dicke states are higher-dimensional analogues of an important class of highly-
+entangled quantum states known as (qubit) Dicke states. A circuit for preparing ar-
+bitrary qudit Dicke states deterministically is formulated. For the case of qutrits, an
+explicit decomposition of the circuit in terms of elementary gates is presented.
+nepomechie@miami.edu
+
+1
+Introduction
+The (qubit) Dicke state |Dn
+k⟩ is an equal-weight superposition of all n-qubit states with k
+|1⟩’s and n − k |0⟩’s. For example,
+|D4
+2⟩ = 1
+√
+6 (|1100⟩ + |1010⟩ + |1001⟩ + |0110⟩ + |0101⟩ + |0011⟩) ,
+where the tensor product is understood, e.g. |1100⟩ = |1⟩ ⊗ |1⟩ ⊗ |0⟩ ⊗ |0⟩. These highly-
+entangled states have long been studied and exploited in quantum information and com-
+putation, see e.g. [1–22]. Efficient quantum circuits for preparing qubit Dicke states have
+been considered in [3, 23–28]. Such quantum circuits have recently been used as the starting
+point for preparing exact eigenstates of the Heisenberg spin chain [29–31] via coordinate
+Bethe ansatz [32, 33].1
+Higher-dimensional analogues of qubit Dicke states, namely qudit Dicke states (also called
+generalized Dicke states, or symmetric basis states), have also received attention over many
+years, see e.g. [37–41]2.
+Let us consider d-dimensional qudits, with computational basis
+vectors |0⟩, |1⟩, . . ., |d − 1⟩ that span a vector space V . In order to specify an n-qudit Dicke
+state, similarly to [41], we introduce a d-dimensional vector ⃗k such that
+⃗k = (k0, k1, . . . , kd−1)
+with
+kj ∈ {0, 1, . . . , n}
+and
+d−1
+�
+j=0
+kj = n ,
+(1.1)
+and we define B(⃗k) as the set of all permutations of n symbols such that kj symbols are equal
+to j for j = 0, 1, . . . , d − 1. The number of such permutations is given by the multinomial
+|B(⃗k)| =
+n!
+�d−1
+j=0 kj!
+.
+(1.2)
+We denote the corresponding n-qudit Dicke state by
+|Dn(⃗k)⟩ =
+1
+�
+|B(⃗k)|
+�
+u∈B(⃗k)
+|u⟩ ,
+(1.3)
+where |u⟩ is the n-qudit state corresponding to the permutation u. An example with qutrits
+(d = 3) is
+|D4(2, 1, 1)⟩ =
+1
+√
+12
+�
+|0012⟩ + |1002⟩ + |0102⟩ + |0021⟩ + |0201⟩ + |2001⟩
++ |0210⟩ + |0120⟩ + |1020⟩ + |1200⟩ + |2010⟩ + |2100⟩
+�
+.
+1Alternative approaches for preparing such eigenstates [34, 35] via algebraic Bethe ansatz [36] do not
+make use of Dicke states.
+2A recent review of qudits is given in [42].
+1
+
+For the special case of qubits (d = 2), by setting ⃗k = (k0, k1) = (n − k, k), we see that
+|Dn(⃗k)⟩ reduces to the familiar Dicke state |Dn
+k⟩.
+The main goal of this paper is to formulate a circuit for preparing arbitrary qudit Dicke
+states deterministically. Such a quantum circuit will be useful for generalizing the many
+applications of (qubit) Dicke states to qudits. In particular, it will be needed in order to
+extend the algorithm [29] for the spin-1/2 Heisenberg spin chain to higher-spin integrable
+spin chains [43, 44].
+The outline of the remainder of this paper is as follows. In Sec. 2, following the approach
+of B¨artschi and Eidenbenz [25] for the qubit case, we introduce a qudit Dicke operator
+that generates an arbitrary qudit Dicke state from a simple initial state, and we obtain an
+expression (2.10) for this operator as a product of certain W operators (2.4). In Sec. 3,
+focussing on the case of qutrits, we determine an explicit decomposition of the W operators
+in terms of elementary qutrit gates (3.17). These results are briefly discussed in Sec. 4.
+Matrix representations of the required qutrit gates and notational details are presented in
+Appendix A.
+2
+Qudit Dicke operators as products of W operators
+In order to formulate a circuit for preparing arbitrary qudit Dicke states deterministically,
+we generalize the approach used in [25] for the case d = 2. We therefore look for a unitary
+operator Un,K acting on V ⊗n, which we call the qudit Dicke operator, that generates an
+arbitrary n-qudit Dicke state (1.3) from a simple initial state
+|Dn(⃗k)⟩ = Un,K |0⟩⊗k0|1⟩⊗k1 . . . |d − 1⟩⊗kd−1 ,
+d−1
+�
+j=0
+kj = n ,
+d−1
+�
+j=1
+kj ≤ K ≤ n .
+(2.1)
+The qudit Dicke state (1.3) satisfies a recursion relation
+|Dn(k0, . . . , kd−1)⟩ =
+d−1
+�
+j=0
+�
+kj
+n |Dn−1(k0, . . . kj − 1, . . . , kd−1)⟩ ⊗ |j⟩ ,
+d−1
+�
+j=0
+kj = n ,
+(2.2)
+which generalizes the d = 2 result noted in [16, 21, 25]. Substituting (2.1) into both sides of
+(2.2), we obtain after some algebra
+Un,K |0⟩⊗k0 . . . |d − 1⟩⊗kd−1 = (Un−1,K ⊗ I)
+�
+I⊗(n−1−K) ⊗ Wn,K
+�
+|0⟩⊗k0 . . . |d − 1⟩⊗kd−1 , (2.3)
+2
+
+where Wn,K is an operator acting on V ⊗(K+1) such that
+Wn,K |0⟩⊗(K+1−�d−1
+i=1 ki)|1⟩⊗k1 . . . |d − 1⟩⊗kd−1
+=
+�
+k0
+n |0⟩⊗(K−�d−1
+i=1 ki)|1⟩⊗k1 . . . |d − 1⟩⊗kd−1|0⟩
++
+d−1
+�
+j=1
+�
+kj
+n |0⟩⊗(K+1−�d−1
+i=1 ki)|1⟩⊗k1 . . . |j⟩⊗kj−1 . . . |d − 1⟩⊗kd−1|j⟩ ,
+1 ≤
+d−1
+�
+j=1
+kj ≤ K ,
+d−1
+�
+j=0
+kj = n .
+(2.4)
+Moreover, Wn,K |j⟩⊗(K+1) = |j⟩⊗(K+1) for j = 0, 1, . . . , d − 1.
+We see from (2.3) that the qudit Dicke operator satisfies the recursion relation
+Un,K = (Un−1,K ⊗ I)
+�
+I⊗(n−1−K) ⊗ Wn,K
+�
+.
+(2.5)
+Telescoping the recursions in (2.5), we obtain
+Un,K =
+�
+UK,K ⊗ I⊗(n−K)�
+�
+n
+�
+l=K+1
+I⊗(l−K−1) ⊗ Wl,K ⊗ I⊗(n−l)
+�
+.
+(2.6)
+We can derive an expression for UK,K in (2.6) by starting from the recursion relation
+(2.2) with n = K
+|DK(l0, . . . , ld−1)⟩ =
+d−1
+�
+j=0
+�
+lj
+K |DK−1(l0, . . . lj − 1, . . . , ld−1)⟩ ⊗ |j⟩ ,
+d−1
+�
+j=0
+lj = K ,
+(2.7)
+which leads, in a similar way as before, to a recursion relation for UK,K
+UK,K = (UK−1,K−1 ⊗ I) WK,K−1 .
+(2.8)
+Telescoping the recursions in (2.8), we obtain
+UK,K =
+K
+�
+l=2
+�
+Wl,l−1 ⊗ I⊗(K−l)�
+,
+(2.9)
+where we have used U1,1 = I. Substituting this result into (2.6), we arrive at an expression
+for the qudit Dicke operator as a product of W operators
+Un,K =
+� K
+�
+l=2
+Wl,l−1 ⊗ I⊗(n−l)
+� �
+n
+�
+l=K+1
+I⊗(l−K−1) ⊗ Wl,K ⊗ I⊗(n−l)
+�
+.
+(2.10)
+This result is evidently a direct generalization of Lemma 2 in [25], albeit with significantly
+more complicated operators Wn,K (2.4) if d > 2; if d = 2, then Wn,K is equivalent to the
+3
+
+“Split & Cyclic Shift” operator SCSn,K in [25]. While the number of terms in the qudit
+Dicke state |Dn(⃗k)⟩ generated by Un,K is given by |B(⃗k)| (1.2) (which can be large), the
+number of terms in the state (2.4) generated by Wn,K is at most d.
+Two simple examples of (2.10) are
+U4,2 =
+�
+W2,1 ⊗ I⊗2�
+(W3,2 ⊗ I) (I ⊗ W4,2) ,
+(2.11)
+U4,3 =
+�
+W2,1 ⊗ I⊗2�
+(W3,2 ⊗ I) W4,3 .
+(2.12)
+The corresponding circuit diagrams are shown in Figs. 1a and 1b, respectively.3
+U4,2
+=
+W4,2
+W3,2
+W2,1
+(a) Circuit diagram for U4,2
+U4,3
+=
+W4,3
+W3,2
+W2,1
+(b) Circuit diagram for U4,3
+3
+Gate decompositions of the W operators
+We now turn to the problem of finding explicit decompositions of the W operators in terms
+of elementary qudit gates. For simplicity, we restrict here to qutrits (d = 3), for which case
+(2.4) reduces to
+Wn,K |0⟩⊗(K+1−k1−k2)|1⟩⊗k1|2⟩⊗k2 =
+�
+k0
+n |0⟩⊗(K−k1−k2)|1⟩⊗k1|2⟩⊗k2|0⟩
++
+�
+k1
+n |0⟩⊗(K+1−k1−k2)|1⟩⊗k1−1|2⟩⊗k2|1⟩ +
+�
+k2
+n |0⟩⊗(K+1−k1−k2)|1⟩⊗k1|2⟩⊗k2 ,
+1 ≤ k1 + k2 ≤ K ,
+k0 + k1 + k2 = n .
+(3.1)
+3.1
+Elementary qutrit gates
+We shall see that the W operators can be decomposed entirely in terms of certain NOT gates,
+Ry rotation gates, and controlled versions thereof. Following [46] (see also [42] and references
+therein), we denote by X(ij) the (1-qutrit) NOT gate that performs the interchange |i⟩ ↔ |j⟩
+3The circuit diagrams in this paper were generated using quantikz [45].
+4
+
+and leaves unchanged the remaining basis vector, where i, j ∈ {0, 1, 2} and i < j; that is,
+X(01)|0⟩ = |1⟩ ,
+X(01)|1⟩ = |0⟩ ,
+X(01)|2⟩ = |2⟩ ,
+X(02)|0⟩ = |2⟩ ,
+X(02)|2⟩ = |0⟩ ,
+X(02)|1⟩ = |1⟩ ,
+X(12)|1⟩ = |2⟩ ,
+X(12)|2⟩ = |1⟩ ,
+X(12)|0⟩ = |0⟩ .
+(3.2)
+We similarly denote by R(ij)(θ) the (1-qutrit) gate that performs an Ry(θ) rotation in the
+subspace spanned by |i⟩ and |j⟩; hence,
+R(ij)(θ)|i⟩ = cos(θ/2)|i⟩ + sin(θ/2)|j⟩ ,
+R(ij)(θ)|j⟩ = − sin(θ/2)|i⟩ + cos(θ/2)|j⟩ ,
+(3.3)
+with (i, j) ∈ {(0, 1), (0, 2), (1, 2)}.
+We denote by C[n1]
+q1 X(ij)
+q0
+the (2-qutrit) controlled-X(ij) gate, which acts as X(ij) on the
+“target” qutrit in vector space q0 if the “control” qutrit in vector space q1 is in the state
+|n1⟩, and otherwise acts as the identity operator. That is,
+C[n1]
+q1 X(ij)
+q0 |x1⟩q1|x0⟩q0 =
+�
+|x1⟩q1X(ij)|x0⟩q0
+if
+x1 = n1
+|x1⟩q1|x0⟩q0
+if
+x1 ̸= n1
+,
+(3.4)
+where x0, x1, n1 ∈ {0, 1, 2}, and q0, q1 ∈ {0, 1, . . . , n − 1}. The corresponding circuit diagram
+is shown in Fig. 2a.
+n1
+q0
+X(i,j)
+q1
+(a) Circuit diagram for C[n1]
+q1 X(ij)
+q0
+n1
+n2
+q0
+X(i,j)
+q1
+q2
+(b) Circuit diagram for C[n2 n1]
+q2 q1
+X(ij)
+q0
+Similarly, C[n2 n1]
+q2 q1 X(ij)
+q0
+denotes the (3-qutrit) double-controlled-X(ij) gate, with control qutrits
+in vector spaces q1 and q2, which must be in the states |n1⟩ and |n2⟩, respectively, in order
+for the gate to act nontrivially on the target qutrit in vector space q0, see Fig. 2b; and
+similarly for higher multiple-controlled-X(ij) gates. Controlled Ry rotation gates are defined
+in a similar way, with X(ij) replaced by R(ij)(θ).
+Matrix representations of these gates and further notational details are presented in
+Appendix A.
+3.2
+Special case
+Let us begin with the simpler special case that one (and only one) of the k’s is zero, i.e.
+either
+k0 = 0 ,
+or
+k1 = 0 ,
+or
+k2 = 0 ,
+(3.5)
+5
+
+in which case (3.1) takes the form
+Wn,K |i⟩⊗(K+1−l)|j⟩⊗l =
+�
+n − l
+n
+|i⟩⊗(K−l)|j⟩⊗l|i⟩ +
+�
+l
+n |i⟩⊗(K+1−l)|j⟩⊗l ,
+1 ≤ l ≤ K ,
+(3.6)
+where i < j; that is, there are 3 such possibilities, namely (i, j) ∈ {(0, 1), (0, 2), (1, 2)}. Let
+us denote by �
+Wn,K the restriction of Wn,K to this special case (3.5). We observe that �
+Wn,K
+acts similarly to the “Split & Cyclic Shift” operator SCSn,k for the qubit case [25], except the
+latter involves only the single possibility (i, j) = (0, 1). We will see, following similar logic,
+that �
+Wn,K is given by a product of operators I depending on wire label l ∈ {1, 2, . . . , K}.
+Three cases of l-values must be treated separately, and it is convenient to distinguish these
+cases with separate names (a,b,c), as indicated in Table 1.
+l
+case
+1
+a
+2
+b
+...
+...
+K − 1
+b
+K
+c
+Table 1: Three cases (a,b,c) of l-values
+That is, �
+Wn,K is given by
+�
+Wn,K = Ic
+n,K
+�K−1
+�
+l=2
+Ib
+n,K,l
+�
+Ia
+n,K ,
+(3.7)
+where the I-operators are defined below.
+3.2.1
+Ia
+n,K
+The circuit diagram for Ia
+n,K with K ≥ 2 is given in Fig. 3.
+1
+1
+2
+2
+1
+1
+1
+2
+1
+...
+1
+0
+R(0,1)(θ)
+R(0,2)(θ)
+R(1,2)(θ)
+1
+X(0,1)
+X(0,1)
+X(0,2)
+X(0,2)
+X(1,2)
+X(1,2)
+K
+1
+2
+3
+Figure 3: Circuit diagram for Ia
+n,K with K ≥ 2, where θ = −2 arccos(
+�
+1/n)
+6
+
+Indeed, in Fig. 3, the part of the circuit up to slice 1 performs the transformation
+|0⟩1|1⟩0 �→
+�
+1
+n|0⟩1|1⟩0 +
+�
+n − 1
+n
+|1⟩1|0⟩0 ;
+the part of the circuit between slice 1 and slice 2 performs the transformation
+|0⟩1|2⟩0 �→
+�
+1
+n|0⟩1|2⟩0 +
+�
+n − 1
+n
+|2⟩1|0⟩0 ;
+and the part of the circuit between slice 2 and slice 3 performs the transformation
+|1⟩1|2⟩0 �→
+�
+1
+n|1⟩1|2⟩0 +
+�
+n − 1
+n
+|2⟩1|1⟩0 ,
+provided that the “last” wire (labeled K) is in state |1⟩ (i.e., not |0⟩). The latter condition
+ensures that the special case (3.5) holds, since the operator acts on ordered states
+|xK⟩K|xK−1⟩K−1 . . . |x0⟩0
+where
+x0, . . . , xK ∈ {0, 1, 2}
+with
+xK ≤ xK−1 ≤ . . . ≤ x0 .
+(3.8)
+For the particular case K = 1, the circuit is similarly given by Fig. 4.
+1
+1
+2
+2
+1
+1
+1
+2
+1
+0
+R(0,1)(θ)
+R(0,2)(θ)
+R(1,2)(θ)
+1
+X(0,1)
+X(0,1)
+X(0,2)
+X(0,2)
+X(1,2)
+X(1,2)
+1
+2
+3
+Figure 4: Circuit diagram for Ia
+n,K with K = 1, where θ = −2 arccos(
+�
+1/n)
+We see that the operator Ia
+n,K implements (3.6) for l = 1.
+3.2.2
+Ib
+n,K,l
+The circuit diagram for Ib
+n,K,l with 2 ≤ l ≤ K − 1 and K > 2 is given in Fig. 5
+Indeed, in Fig. 5, the part of the circuit up to slice 1 performs the transformation
+|0⟩l|1⟩0 �→
+�
+l
+n|0⟩l|1⟩0 +
+�
+n − l
+n
+|1⟩l|0⟩0 ,
+provided that wire l − 1 is in state |1⟩; the part of the circuit between slice 1 and slice 2
+performs the transformation
+|0⟩l|2⟩0 �→
+�
+l
+n|0⟩l|2⟩0 +
+�
+n − l
+n
+|2⟩l|0⟩0 ,
+7
+
+1
+1
+2
+2
+2
+2
+...
+1
+2
+2
+1
+2
+2
+...
+1
+0
+R(0,1)(θ)
+R(0,2)(θ)
+R(1,2)(θ)
+l − 1
+l
+X(0,1)
+X(0,1)
+X(0,2)
+X(0,2)
+X(1,2)
+X(1,2)
+K
+1
+2
+3
+Figure 5:
+Circuit diagram for Ib
+n,K,l with 2 ≤ l ≤ K − 1 and K > 2, where θ =
+−2 arccos(
+�
+l/n)
+provided that wire l − 1 is in state |2⟩ ; and the part of the circuit between slice 2 and slice
+3 performs the transformation
+|1⟩l|2⟩0 �→
+�
+l
+n|1⟩l|2⟩0 +
+�
+n − l
+n
+|2⟩l|1⟩0 ,
+provided that wire l − 1 is in state |2⟩ and wire K is in state |1⟩ (i.e., not |0⟩ ); again, the
+latter condition ensures that the special case (3.5) holds, see (3.8).
+We see that the operators Ib
+n,K,l implement (3.6) for l = 2, . . . , K − 1.
+3.2.3
+Ic
+n,K
+The Ic
+n,K circuit diagram with K ≥ 2 is similarly given by Fig. 6.
+1
+1
+2
+2
+2
+2
+...
+1
+2
+2
+1
+2
+2
+0
+R(0,1)(θ)
+R(0,2)(θ)
+R(1,2)(θ)
+K − 1
+K
+X(0,1)
+X(0,1)
+X(0,2)
+X(0,2)
+X(1,2)
+X(1,2)
+1
+2
+3
+Figure 6: Circuit diagram for Ic
+n,K with K ≥ 2, where θ = −2 arccos(
+�
+l/n)
+We see that the operator Ic
+n,K implements (3.6) for l = K; and therefore (3.7) indeed
+implements �
+Wn,K.
+8
+
+3.3
+Generic case
+In Sec. 3.2 we focused on the special case that one of the k’s is zero (3.5), for which case Wn,K
+generates only 2 terms (3.6), and therefore only one rotation angle (θ) is necessary. Let us
+now consider the generic case that all of the k’s are nonzero, for which case Wn,K generates 3
+terms (3.1), and therefore two rotation angles (θ1, θ2) are necessary. Let us denote by �
+�
+W n,K
+the restriction of Wn,K to this generic case. We will see that �
+�
+W n,K is given by a product of
+operators II depending on wire labels l2 ∈ {0, 1, 2, . . .} and l1 ∈ {l2 + 1 , l2 + 2 , . . .}, where
+l2 is the wire in the initial state with the “last” |2⟩, and l1 is the wire with the “last” |1⟩,
+going from right to left:
+K
+↓
+|0⟩ · · ·
+l1+1
+↓
+|0⟩
+l1↓
+|1⟩ · · ·
+l2+1
+↓
+|1⟩
+l2↓
+|2⟩ · · ·
+0
+↓
+|2⟩ .
+(3.9)
+In terms of the k’s in (3.1), we have
+l2 = k2 − 1 ,
+l1 = k1 + k2 − 1 .
+(3.10)
+Four cases of possible (l2, l1)-values must be treated separately, and it is convenient to dis-
+tinguish these cases with separate names (a,b,c,d), as indicated in Table 2.
+l2
+l1
+1
+2
+3
+4
+· · ·
+0
+a
+b
+b
+b
+· · ·
+1
+-
+c
+d
+d
+· · ·
+2
+-
+-
+c
+d
+· · ·
+3
+-
+-
+-
+c
+· · ·
+...
+...
+...
+...
+...
+...
+Table 2: Four cases (a,b,c,d) of (l2, l1)-values
+Specifically, we will argue that �
+�
+W n,K is given by
+�
+�
+W n,K =
+�K−3
+�
+l2=1
+K−1
+�
+l1=l2+2
+IId
+n,K,l1,l2
+� �K−2
+�
+l2=1
+IIc
+n,K,l2
+� �K−1
+�
+l1=2
+IIb
+n,K,l1
+�
+IIa
+n,K ,
+(3.11)
+where the II-operators are defined below.
+3.3.1
+IIa
+n,K
+The circuit diagram for IIa
+n,K with K ≥ 2 is given in Fig. 7.
+The part of the circuit up to slice 1 performs the transformation
+|0⟩2|1⟩1|2⟩0 �→ cos(θ1/2)|0⟩2|1⟩1|2⟩0 − sin(θ1/2)|0⟩2|2⟩1|1⟩0 ;
+(3.12)
+9
+
+2
+2
+1
+1
+2
+2
+0
+1
+...
+0
+R(1,2)(θ1)
+R(0,1)(θ2)
+1
+X(1,2)
+X(1,2)
+2
+X(0,1)
+X(0,1)
+K
+1
+2
+Figure 7:
+Circuit diagram for IIa
+n,K with K ≥ 2, where θ1 = −2 arccos(
+�
+1/n) , θ2 =
+−2 arccos(
+�
+1/(n − 1))
+and at slice 2, the state (3.12) is further transformed to
+cos(θ1/2)|0⟩2|1⟩1|2⟩0−sin(θ1/2) cos(θ2/2)|0⟩2|2⟩1|1⟩0+sin(θ1/2) sin(θ2/2)|1⟩2|2⟩1|0⟩0 . (3.13)
+For general values of l2 and l1 (see IId
+n,K,l1,l2 below), we demand
+cos(θ1/2) =
+�
+l2 + 1
+n
+,
+sin(θ1/2) cos(θ2/2) = −
+�
+l1 − l2
+n
+,
+sin(θ1/2) sin(θ2/2) =
+�
+n − l1 − 1
+n
+,
+(3.14)
+in order to match with (3.1) and (3.10). Hence, we assign to the rotation angles the values
+θ1 = −2 arccos
+��
+l2 + 1
+n
+�
+,
+θ2 = −2 arccos
+��
+l1 − l2
+n − l2 − 1
+�
+.
+(3.15)
+In particular, we see from (3.13) and (3.14) that the operator IIa
+n,K indeed implements (3.1)
+for the case k1 = k2 = 1, which corresponds to (l2, l1) = (0, 1), see (3.10).
+3.3.2
+IIb
+n,K,l1
+The circuit diagram for IIb
+n,K,l1 with l1 ≥ 2 and K ≥ l1 + 1 ≥ 3, which corresponds to the
+case l2 = 0, is given in Fig. 8.
+This circuit is a generalization of the one for IIa
+n,K in Fig. 7, with the additional require-
+ment that wire l1 be in state |1⟩.
+3.3.3
+IIc
+n,K,l2
+The circuit diagram for IIc
+n,K,l2 with l2 ≥ 1 and K ≥ 3, which corresponds to the case
+l1 = l2 + 1, is given in Fig. 9.
+This circuit is also a generalization of the one for IIa
+n,K in Fig. 7, with the additional
+requirement that wire l2 be in state |2⟩.
+10
+
+2
+2
+1
+1
+2
+2
+...
+1
+1
+0
+1
+...
+0
+R(1,2)(θ1)
+R(0,1)(θ2)
+1
+X(1,2)
+X(1,2)
+l1
+l1 + 1
+X(0,1)
+X(0,1)
+K
+Figure 8: Circuit diagram for IIb
+n,K,l1 with l1 ≥ 2 and K ≥ l1 + 1 ≥ 3, where θ1 =
+−2 arccos(
+�
+1/n) , θ2 = −2 arccos(
+�
+l1/(n − 1))
+2
+2
+1
+1
+...
+2
+2
+2
+2
+0
+1
+...
+0
+R(1,2)(θ1)
+R(0,1)(θ2)
+l2
+l1 = l2 + 1
+X(1,2)
+X(1,2)
+l1 + 1 = l2 + 2
+X(0,1)
+X(0,1)
+K
+Figure 9:
+Circuit diagram for IIc
+n,K,l2 with l2
+≥
+1 and K
+≥
+3,
+where θ1
+=
+−2 arccos(
+�
+(l2 + 1)/n) , θ2 = −2 arccos(
+�
+1/(n − l2 − 1))
+3.3.4
+IId
+n,K,l1,l2
+The circuit diagram for IId
+n,K,l1,l2 with l2 ≥ 1, l1 ≥ l2 + 2 ≥ 3 and K ≥ l1 + 1 ≥ 4 is given in
+Fig. 10.
+This circuit performs the transformation
+|0⟩l1+1|1⟩l2+1|2⟩0 �→ cos(θ1/2)|0⟩l1+1|1⟩l2+1|2⟩0 − sin(θ1/2) cos(θ2/2)|0⟩l1+1|2⟩l2+1|1⟩0
++ sin(θ1/2) sin(θ2/2)|1⟩l1+1|2⟩l2+1|0⟩0 ,
+(3.16)
+similarly to (3.13), provided that wire l2 is in state |2⟩ and wire l1 is in state |1⟩ . With the
+angles as in (3.14)-(3.15), we see from (3.16) and (3.10) that IId
+n,K,l1,l2 indeed implements
+(3.1) for generic values of (l2, l1). For special cases of (l2, l1), this circuit reduces to those in
+Figs. 7, 8 and 9.
+11
+
+2
+2
+1
+1
+...
+2
+2
+2
+2
+...
+1
+1
+0
+1
+...
+0
+R(1,2)(θ1)
+R(0,1)(θ2)
+l2
+l2 + 1
+X(1,2)
+X(1,2)
+l1
+l1 + 1
+X(0,1)
+X(0,1)
+K
+Figure 10: Circuit diagram for IId
+n,K,l1,l2 with l2 ≥ 1, l1 ≥ l2 + 2 ≥ 3 and K ≥ l1 + 1 ≥ 4,
+where θ1 = −2 arccos(
+�
+(l2 + 1)/n) , θ2 = −2 arccos(
+�
+(l1 − l2)/(n − l2 − 1))
+3.4
+Final result
+For the general case (no conditions on the k’s, apart from those in (3.1)), we arrive at our
+final result for an operator Wn,K that satisfies (3.1), namely
+Wn,K = �
+�
+W n,K �
+Wn,K ,
+(3.17)
+where �
+Wn,K is given by (3.7), and �
+�
+W n,K is given by (3.11).
+We have implemented the circuit for Un,K (2.10), with the W operators given by (3.17),
+in Mathematica using the matrix representations in Appendix A, and we have explicitly
+checked that the key result (2.1) is satisfied up to n = 6 and K = 6.
+4
+Discussion
+Our main results are the expression (2.10) for the qudit Dicke operator Un,K as a product
+of W operators (2.4), and (for the qutrit case) the decomposition of the W operators in
+terms of elementary gates (3.17). The algorithm is deterministic, and does not use ancillary
+qutrits.
+Whereas the preparation of arbitrary qubit Dicke states has been considered in a number
+of works (see [3, 23–28] and references therein), ours is the first work (to our knowledge)
+to consider the preparation of arbitrary qudit Dicke states. We have seen that, already for
+the qutrit case, the algorithm entails a nontrivial generalization of [25]. For the explicit
+gate implementation of the W operators in Section 3, we have aimed for clarity rather than
+12
+
+economy; it is likely that alternative implementations with reduced gate counts can be found.
+We have performed the explicit gate implementation only for qutrits (d = 3); for the general
+qudit case, we expect that d − 1 rotation angles θ1, . . . , θd−1 would be required, and many
+edge cases would need to be treated separately.
+Having in hand a way to prepare qudit Dicke states, it will be interesting to go further
+and formulate an algorithm for preparing eigenstates of spin s > 1/2 integrable spin chains
+based on coordinate Bethe ansatz [43, 44], thereby extending the approach [29] for preparing
+eigenstates of the s = 1/2 Heisenberg spin chain.
+Time will tell whether current efforts to build practical quantum computers will be ex-
+tended from qubits to qutrits, or to even higher-dimensional qudits. If so, then the generation
+of Dicke states (1.3), which are highly entangled yet relatively simple, will be a natural early
+goal.
+Acknowledgments
+We thank Hai-Rui Wei and Huangjun Zhu for helpful correspondence. This research was
+supported in part by the National Science Foundation under Grant No. NSF PHY-1748958,
+and by a Cooper fellowship.
+A
+Matrix representations of qutrit gates
+A 1-qutrit state lives in the complex vector space V spanned by |0⟩, |1⟩, |2⟩. Let us set
+|0⟩ =
+
+
+1
+0
+0
+
+ ,
+|1⟩ =
+
+
+0
+1
+0
+
+ ,
+|2⟩ =
+
+
+0
+0
+1
+
+ .
+(A.1)
+The NOT gates X(ij) (3.2) are represented by the 3 × 3 matrices [46]
+X(01) =
+
+
+0
+1
+0
+1
+0
+0
+0
+0
+1
+
+ ,
+X(02) =
+
+
+0
+0
+1
+0
+1
+0
+1
+0
+0
+
+ ,
+X(12) =
+
+
+1
+0
+0
+0
+0
+1
+0
+1
+0
+
+ .
+(A.2)
+Similarly, the rotation gates R(ij)(θ) (3.3) are represented by
+R(01)(θ) =
+
+
+cos( θ
+2)
+− sin( θ
+2)
+0
+sin( θ
+2)
+cos( θ
+2)
+0
+0
+0
+1
+
+ ,
+R(02)(θ) =
+
+
+cos( θ
+2)
+0
+− sin( θ
+2)
+0
+1
+0
+sin( θ
+2)
+0
+cos( θ
+2)
+
+ ,
+R(12)(θ) =
+
+
+1
+0
+0
+0
+cos( θ
+2)
+− sin( θ
+2)
+0
+sin( θ
+2)
+cos( θ
+2)
+
+ .
+(A.3)
+13
+
+An n-qutrit state lives in V ⊗n. We label these vector spaces from 0 to n − 1, going from
+right to left
+n−1
+↓
+V ⊗ · · · ⊗
+1
+↓
+V ⊗
+0
+↓
+V .
+(A.4)
+In circuit diagrams, the n vector spaces are represented by n horizontal wires, which are
+labeled from 0 to n − 1, going from top (0) to bottom (n − 1). We use subscripts to indicate
+the vector spaces on which operators act nontrivially. For example, if A is a 1-qutrit operator,
+then
+Aq = I⊗(n−1−q) ⊗ A ⊗ I⊗q
+(A.5)
+is an operator on V ⊗n acting nontrivially on the qth vector space q ∈ {0, 1, . . . , n − 1}. The
+vector space on which an operator acts nontrivially can be changed using the permutation
+operator Pqq′, for example
+Aq′ = Pq q′ Aq Pq q′ ,
+(A.6)
+where
+Pq q′ =
+3
+�
+i,j=1
+ei,j
+q ej,i
+q′ ,
+(A.7)
+and ei,j is the elementary 3 × 3 matrix whose (i, j) matrix element is 1, and all others are 0;
+that is, (ei,j)a,b = δi,a δj,b.
+For the controlled-X(ij) gates (3.4), we have the 9 × 9 block-diagonal matrices
+C[2]
+1 X(ij)
+0
+=
+�16
+X(ij)
+�
+,
+C[1]
+1 X(ij)
+0
+=
+
+
+13
+X(ij)
+13
+
+ ,
+C[0]
+1 X(ij)
+0
+=
+�
+X(ij)
+16
+�
+,
+(A.8)
+where 1n denotes the n × n identity matrix. The controlled gates are related by NOT gates
+on the controls, for example
+C[1]
+1 X(ij)
+0
+= X(12)
+1
+�
+C[2]
+1 X(ij)
+0
+�
+X(12)
+1
+,
+C[0]
+1 X(ij)
+0
+= X(02)
+1
+�
+C[2]
+1 X(ij)
+0
+�
+X(02)
+1
+.
+(A.9)
+In terms of circuit diagrams, these identities are shown in Figs. 11a and 11b, respectively.
+1
+0
+X(i,j)
+1
+=
+2
+0
+X(i,j)
+1
+X(1,2)
+X(1,2)
+(a) Identity for C[1]
+1 X(ij)
+0
+0
+0
+X(i,j)
+1
+=
+2
+0
+X(i,j)
+1
+X(0,2)
+X(0,2)
+(b) Identity for C[0]
+1 X(ij)
+0
+Double-controlled-X(ij) gates are given by 33 × 33 block-diagonal matrices
+C[22]
+21 X(ij)
+0
+=
+�124
+X(ij)
+�
+,
+C[11]
+21 X(ij)
+0
+=
+
+
+112
+X(ij)
+112
+
+ ,
+C[00]
+21 X(ij)
+0
+=
+�
+X(ij)
+124
+�
+,
+(A.10)
+and similarly for higher-controlled-X(ij) gates.
+Matrices corresponding to controlled Ry
+rotation gates are defined in a similar way, with X(ij) replaced by R(ij)(θ).
+14
+
+References
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+
diff --git a/ftE4T4oBgHgl3EQfRQzA/content/tmp_files/load_file.txt b/ftE4T4oBgHgl3EQfRQzA/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf,len=815
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='04989v1 [quant-ph] 12 Jan 2023 UMTG–315 Qudit Dicke state preparation Rafael I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Nepomechie Physics Department, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Box 248046, University of Miami Coral Gables, FL 33124 USA Abstract Qudit Dicke states are higher-dimensional analogues of an important class of highly- entangled quantum states known as (qubit) Dicke states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' A circuit for preparing ar- bitrary qudit Dicke states deterministically is formulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' For the case of qutrits, an explicit decomposition of the circuit in terms of elementary gates is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' nepomechie@miami.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='edu 1 Introduction The (qubit) Dicke state |Dn k⟩ is an equal-weight superposition of all n-qubit states with k |1⟩’s and n − k |0⟩’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' For example, |D4 2⟩ = 1 √ 6 (|1100⟩ + |1010⟩ + |1001⟩ + |0110⟩ + |0101⟩ + |0011⟩) , where the tensor product is understood, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' |1100⟩ = |1⟩ ⊗ |1⟩ ⊗ |0⟩ ⊗ |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' These highly- entangled states have long been studied and exploited in quantum information and com- putation, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' [1–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Efficient quantum circuits for preparing qubit Dicke states have been considered in [3, 23–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Such quantum circuits have recently been used as the starting point for preparing exact eigenstates of the Heisenberg spin chain [29–31] via coordinate Bethe ansatz [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='1 Higher-dimensional analogues of qubit Dicke states, namely qudit Dicke states (also called generalized Dicke states, or symmetric basis states), have also received attention over many years, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' [37–41]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Let us consider d-dimensional qudits, with computational basis vectors |0⟩, |1⟩, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=', |d − 1⟩ that span a vector space V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' In order to specify an n-qudit Dicke state, similarly to [41], we introduce a d-dimensional vector ⃗k such that ⃗k = (k0, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' , kd−1) with kj ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' , n} and d−1 � j=0 kj = n , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='1) and we define B(⃗k) as the set of all permutations of n symbols such that kj symbols are equal to j for j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' , d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' The number of such permutations is given by the multinomial |B(⃗k)| = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' �d−1 j=0 kj!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='2) We denote the corresponding n-qudit Dicke state by |Dn(⃗k)⟩ = 1 � |B(⃗k)| � u∈B(⃗k) |u⟩ , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='3) where |u⟩ is the n-qudit state corresponding to the permutation u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' An example with qutrits (d = 3) is |D4(2, 1, 1)⟩ = 1 √ 12 � |0012⟩ + |1002⟩ + |0102⟩ + |0021⟩ + |0201⟩ + |2001⟩ + |0210⟩ + |0120⟩ + |1020⟩ + |1200⟩ + |2010⟩ + |2100⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 1Alternative approaches for preparing such eigenstates [34, 35] via algebraic Bethe ansatz [36] do not make use of Dicke states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 2A recent review of qudits is given in [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 1 For the special case of qubits (d = 2), by setting ⃗k = (k0, k1) = (n − k, k), we see that |Dn(⃗k)⟩ reduces to the familiar Dicke state |Dn k⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' The main goal of this paper is to formulate a circuit for preparing arbitrary qudit Dicke states deterministically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Such a quantum circuit will be useful for generalizing the many applications of (qubit) Dicke states to qudits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' In particular, it will be needed in order to extend the algorithm [29] for the spin-1/2 Heisenberg spin chain to higher-spin integrable spin chains [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' The outline of the remainder of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 2, following the approach of B¨artschi and Eidenbenz [25] for the qubit case, we introduce a qudit Dicke operator that generates an arbitrary qudit Dicke state from a simple initial state, and we obtain an expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='10) for this operator as a product of certain W operators (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 3, focussing on the case of qutrits, we determine an explicit decomposition of the W operators in terms of elementary qutrit gates (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' These results are briefly discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Matrix representations of the required qutrit gates and notational details are presented in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 2 Qudit Dicke operators as products of W operators In order to formulate a circuit for preparing arbitrary qudit Dicke states deterministically, we generalize the approach used in [25] for the case d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' We therefore look for a unitary operator Un,K acting on V ⊗n, which we call the qudit Dicke operator, that generates an arbitrary n-qudit Dicke state (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='3) from a simple initial state |Dn(⃗k)⟩ = Un,K |0⟩⊗k0|1⟩⊗k1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' |d − 1⟩⊗kd−1 , d−1 � j=0 kj = n , d−1 � j=1 kj ≤ K ≤ n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='1) The qudit Dicke state (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='3) satisfies a recursion relation |Dn(k0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' , kd−1)⟩ = d−1 � j=0 � kj n |Dn−1(k0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' kj − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' , kd−1)⟩ ⊗ |j⟩ , d−1 � j=0 kj = n , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='2) which generalizes the d = 2 result noted in [16, 21, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='1) into both sides of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='2), we obtain after some algebra Un,K |0⟩⊗k0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' |d − 1⟩⊗kd−1 = (Un−1,K ⊗ I) � I⊗(n−1−K) ⊗ Wn,K � |0⟩⊗k0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' |d − 1⟩⊗kd−1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='3) 2 where Wn,K is an operator acting on V ⊗(K+1) such that Wn,K |0⟩⊗(K+1−�d−1 i=1 ki)|1⟩⊗k1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' |d − 1⟩⊗kd−1 = � k0 n |0⟩⊗(K−�d−1 i=1 ki)|1⟩⊗k1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' |d − 1⟩⊗kd−1|0⟩ + d−1 � j=1 � kj n |0⟩⊗(K+1−�d−1 i=1 ki)|1⟩⊗k1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' |j⟩⊗kj−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' |d − 1⟩⊗kd−1|j⟩ , 1 ≤ d−1 � j=1 kj ≤ K , d−1 � j=0 kj = n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='4) Moreover, Wn,K |j⟩⊗(K+1) = |j⟩⊗(K+1) for j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' , d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' We see from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='3) that the qudit Dicke operator satisfies the recursion relation Un,K = (Un−1,K ⊗ I) � I⊗(n−1−K) ⊗ Wn,K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='5) Telescoping the recursions in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='5), we obtain Un,K = � UK,K ⊗ I⊗(n−K)� � n � l=K+1 I⊗(l−K−1) ⊗ Wl,K ⊗ I⊗(n−l) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='6) We can derive an expression for UK,K in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='6) by starting from the recursion relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='2) with n = K |DK(l0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' , ld−1)⟩ = d−1 � j=0 � lj K |DK−1(l0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' lj − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' , ld−1)⟩ ⊗ |j⟩ , d−1 � j=0 lj = K , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='7) which leads, in a similar way as before, to a recursion relation for UK,K UK,K = (UK−1,K−1 ⊗ I) WK,K−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='8) Telescoping the recursions in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='8), we obtain UK,K = K � l=2 � Wl,l−1 ⊗ I⊗(K−l)� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='9) where we have used U1,1 = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Substituting this result into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='6), we arrive at an expression for the qudit Dicke operator as a product of W operators Un,K = � K � l=2 Wl,l−1 ⊗ I⊗(n−l) � � n � l=K+1 I⊗(l−K−1) ⊗ Wl,K ⊗ I⊗(n−l) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='10) This result is evidently a direct generalization of Lemma 2 in [25], albeit with significantly more complicated operators Wn,K (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='4) if d > 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' if d = 2, then Wn,K is equivalent to the 3 “Split & Cyclic Shift” operator SCSn,K in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' While the number of terms in the qudit Dicke state |Dn(⃗k)⟩ generated by Un,K is given by |B(⃗k)| (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='2) (which can be large), the number of terms in the state (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='4) generated by Wn,K is at most d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Two simple examples of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='10) are U4,2 = � W2,1 ⊗ I⊗2� (W3,2 ⊗ I) (I ⊗ W4,2) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='11) U4,3 = � W2,1 ⊗ I⊗2� (W3,2 ⊗ I) W4,3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='12) The corresponding circuit diagrams are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 1a and 1b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='3 U4,2 = W4,2 W3,2 W2,1 (a) Circuit diagram for U4,2 U4,3 = W4,3 W3,2 W2,1 (b) Circuit diagram for U4,3 3 Gate decompositions of the W operators We now turn to the problem of finding explicit decompositions of the W operators in terms of elementary qudit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' For simplicity, we restrict here to qutrits (d = 3), for which case (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='4) reduces to Wn,K |0⟩⊗(K+1−k1−k2)|1⟩⊗k1|2⟩⊗k2 = � k0 n |0⟩⊗(K−k1−k2)|1⟩⊗k1|2⟩⊗k2|0⟩ + � k1 n |0⟩⊗(K+1−k1−k2)|1⟩⊗k1−1|2⟩⊗k2|1⟩ + � k2 n |0⟩⊗(K+1−k1−k2)|1⟩⊗k1|2⟩⊗k2 , 1 ≤ k1 + k2 ≤ K , k0 + k1 + k2 = n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='1 Elementary qutrit gates We shall see that the W operators can be decomposed entirely in terms of certain NOT gates, Ry rotation gates, and controlled versions thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Following [46] (see also [42] and references therein), we denote by X(ij) the (1-qutrit) NOT gate that performs the interchange |i⟩ ↔ |j⟩ 3The circuit diagrams in this paper were generated using quantikz [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 4 and leaves unchanged the remaining basis vector, where i, j ∈ {0, 1, 2} and i < j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' that is, X(01)|0⟩ = |1⟩ , X(01)|1⟩ = |0⟩ , X(01)|2⟩ = |2⟩ , X(02)|0⟩ = |2⟩ , X(02)|2⟩ = |0⟩ , X(02)|1⟩ = |1⟩ , X(12)|1⟩ = |2⟩ , X(12)|2⟩ = |1⟩ , X(12)|0⟩ = |0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='2) We similarly denote by R(ij)(θ) the (1-qutrit) gate that performs an Ry(θ) rotation in the subspace spanned by |i⟩ and |j⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' hence, R(ij)(θ)|i⟩ = cos(θ/2)|i⟩ + sin(θ/2)|j⟩ , R(ij)(θ)|j⟩ = − sin(θ/2)|i⟩ + cos(θ/2)|j⟩ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='3) with (i, j) ∈ {(0, 1), (0, 2), (1, 2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' We denote by C[n1] q1 X(ij) q0 the (2-qutrit) controlled-X(ij) gate, which acts as X(ij) on the “target” qutrit in vector space q0 if the “control” qutrit in vector space q1 is in the state |n1⟩, and otherwise acts as the identity operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' That is, C[n1] q1 X(ij) q0 |x1⟩q1|x0⟩q0 = � |x1⟩q1X(ij)|x0⟩q0 if x1 = n1 |x1⟩q1|x0⟩q0 if x1 ̸= n1 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='4) where x0, x1, n1 ∈ {0, 1, 2}, and q0, q1 ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' , n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' The corresponding circuit diagram is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' n1 q0 X(i,j) q1 (a) Circuit diagram for C[n1] q1 X(ij) q0 n1 n2 q0 X(i,j) q1 q2 (b) Circuit diagram for C[n2 n1] q2 q1 X(ij) q0 Similarly, C[n2 n1] q2 q1 X(ij) q0 denotes the (3-qutrit) double-controlled-X(ij) gate, with control qutrits in vector spaces q1 and q2, which must be in the states |n1⟩ and |n2⟩, respectively, in order for the gate to act nontrivially on the target qutrit in vector space q0, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 2b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' and similarly for higher multiple-controlled-X(ij) gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Controlled Ry rotation gates are defined in a similar way, with X(ij) replaced by R(ij)(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Matrix representations of these gates and further notational details are presented in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='2 Special case Let us begin with the simpler special case that one (and only one) of the k’s is zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' either k0 = 0 , or k1 = 0 , or k2 = 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='5) 5 in which case (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='1) takes the form Wn,K |i⟩⊗(K+1−l)|j⟩⊗l = � n − l n |i⟩⊗(K−l)|j⟩⊗l|i⟩ + � l n |i⟩⊗(K+1−l)|j⟩⊗l , 1 ≤ l ≤ K , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='6) where i < j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' that is, there are 3 such possibilities, namely (i, j) ∈ {(0, 1), (0, 2), (1, 2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Let us denote by � Wn,K the restriction of Wn,K to this special case (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' We observe that � Wn,K acts similarly to the “Split & Cyclic Shift” operator SCSn,k for the qubit case [25], except the latter involves only the single possibility (i, j) = (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' We will see, following similar logic, that � Wn,K is given by a product of operators I depending on wire label l ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' , K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Three cases of l-values must be treated separately, and it is convenient to distinguish these cases with separate names (a,b,c), as indicated in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' l case 1 a 2 b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' K − 1 b K c Table 1: Three cases (a,b,c) of l-values That is, � Wn,K is given by � Wn,K = Ic n,K �K−1 � l=2 Ib n,K,l � Ia n,K , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='7) where the I-operators are defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='1 Ia n,K The circuit diagram for Ia n,K with K ≥ 2 is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 1 1 2 2 1 1 1 2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 1 0 R(0,1)(θ) R(0,2)(θ) R(1,2)(θ) 1 X(0,1) X(0,1) X(0,2) X(0,2) X(1,2) X(1,2) K 1 2 3 Figure 3: Circuit diagram for Ia n,K with K ≥ 2, where θ = −2 arccos( � 1/n) 6 Indeed, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 3, the part of the circuit up to slice 1 performs the transformation |0⟩1|1⟩0 �→ � 1 n|0⟩1|1⟩0 + � n − 1 n |1⟩1|0⟩0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' the part of the circuit between slice 1 and slice 2 performs the transformation |0⟩1|2⟩0 �→ � 1 n|0⟩1|2⟩0 + � n − 1 n |2⟩1|0⟩0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' and the part of the circuit between slice 2 and slice 3 performs the transformation |1⟩1|2⟩0 �→ � 1 n|1⟩1|2⟩0 + � n − 1 n |2⟩1|1⟩0 , provided that the “last” wire (labeled K) is in state |1⟩ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=', not |0⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' The latter condition ensures that the special case (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='5) holds, since the operator acts on ordered states |xK⟩K|xK−1⟩K−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' |x0⟩0 where x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' , xK ∈ {0, 1, 2} with xK ≤ xK−1 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' ≤ x0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='8) For the particular case K = 1, the circuit is similarly given by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 1 1 2 2 1 1 1 2 1 0 R(0,1)(θ) R(0,2)(θ) R(1,2)(θ) 1 X(0,1) X(0,1) X(0,2) X(0,2) X(1,2) X(1,2) 1 2 3 Figure 4: Circuit diagram for Ia n,K with K = 1, where θ = −2 arccos( � 1/n) We see that the operator Ia n,K implements (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='6) for l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='2 Ib n,K,l The circuit diagram for Ib n,K,l with 2 ≤ l ≤ K − 1 and K > 2 is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 5 Indeed, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 5, the part of the circuit up to slice 1 performs the transformation |0⟩l|1⟩0 �→ � l n|0⟩l|1⟩0 + � n − l n |1⟩l|0⟩0 , provided that wire l − 1 is in state |1⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' the part of the circuit between slice 1 and slice 2 performs the transformation |0⟩l|2⟩0 �→ � l n|0⟩l|2⟩0 + � n − l n |2⟩l|0⟩0 , 7 1 1 2 2 2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 1 2 2 1 2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 1 0 R(0,1)(θ) R(0,2)(θ) R(1,2)(θ) l − 1 l X(0,1) X(0,1) X(0,2) X(0,2) X(1,2) X(1,2) K 1 2 3 Figure 5: Circuit diagram for Ib n,K,l with 2 ≤ l ≤ K − 1 and K > 2, where θ = −2 arccos( � l/n) provided that wire l − 1 is in state |2⟩ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' and the part of the circuit between slice 2 and slice 3 performs the transformation |1⟩l|2⟩0 �→ � l n|1⟩l|2⟩0 + � n − l n |2⟩l|1⟩0 , provided that wire l − 1 is in state |2⟩ and wire K is in state |1⟩ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=', not |0⟩ );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' again, the latter condition ensures that the special case (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='5) holds, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' We see that the operators Ib n,K,l implement (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='6) for l = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' , K − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='3 Ic n,K The Ic n,K circuit diagram with K ≥ 2 is similarly given by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 1 1 2 2 2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 1 2 2 1 2 2 0 R(0,1)(θ) R(0,2)(θ) R(1,2)(θ) K − 1 K X(0,1) X(0,1) X(0,2) X(0,2) X(1,2) X(1,2) 1 2 3 Figure 6: Circuit diagram for Ic n,K with K ≥ 2, where θ = −2 arccos( � l/n) We see that the operator Ic n,K implements (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='6) for l = K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' and therefore (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='7) indeed implements � Wn,K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='3 Generic case In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='2 we focused on the special case that one of the k’s is zero (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='5), for which case Wn,K generates only 2 terms (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='6), and therefore only one rotation angle (θ) is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Let us now consider the generic case that all of the k’s are nonzero, for which case Wn,K generates 3 terms (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='1), and therefore two rotation angles (θ1, θ2) are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Let us denote by � � W n,K the restriction of Wn,K to this generic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' We will see that � � W n,K is given by a product of operators II depending on wire labels l2 ∈ {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='} and l1 ∈ {l2 + 1 , l2 + 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' }, where l2 is the wire in the initial state with the “last” |2⟩, and l1 is the wire with the “last” |1⟩, going from right to left: K ↓ |0⟩ · · · l1+1 ↓ |0⟩ l1↓ |1⟩ · · · l2+1 ↓ |1⟩ l2↓ |2⟩ · · · 0 ↓ |2⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='9) In terms of the k’s in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='1), we have l2 = k2 − 1 , l1 = k1 + k2 − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='10) Four cases of possible (l2, l1)-values must be treated separately, and it is convenient to dis- tinguish these cases with separate names (a,b,c,d), as indicated in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' l2 l1 1 2 3 4 · · 0 a b b b · · 1 c d d · · 2 c d · · 3 c · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Table 2: Four cases (a,b,c,d) of (l2, l1)-values Specifically, we will argue that � � W n,K is given by � � W n,K = �K−3 � l2=1 K−1 � l1=l2+2 IId n,K,l1,l2 � �K−2 � l2=1 IIc n,K,l2 � �K−1 � l1=2 IIb n,K,l1 � IIa n,K , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='11) where the II-operators are defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='1 IIa n,K The circuit diagram for IIa n,K with K ≥ 2 is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' The part of the circuit up to slice 1 performs the transformation |0⟩2|1⟩1|2⟩0 �→ cos(θ1/2)|0⟩2|1⟩1|2⟩0 − sin(θ1/2)|0⟩2|2⟩1|1⟩0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='12) 9 2 2 1 1 2 2 0 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 0 R(1,2)(θ1) R(0,1)(θ2) 1 X(1,2) X(1,2) 2 X(0,1) X(0,1) K 1 2 Figure 7: Circuit diagram for IIa n,K with K ≥ 2, where θ1 = −2 arccos( � 1/n) , θ2 = −2 arccos( � 1/(n − 1)) and at slice 2, the state (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='12) is further transformed to cos(θ1/2)|0⟩2|1⟩1|2⟩0−sin(θ1/2) cos(θ2/2)|0⟩2|2⟩1|1⟩0+sin(θ1/2) sin(θ2/2)|1⟩2|2⟩1|0⟩0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='13) For general values of l2 and l1 (see IId n,K,l1,l2 below), we demand cos(θ1/2) = � l2 + 1 n , sin(θ1/2) cos(θ2/2) = − � l1 − l2 n , sin(θ1/2) sin(θ2/2) = � n − l1 − 1 n , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='14) in order to match with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Hence, we assign to the rotation angles the values θ1 = −2 arccos �� l2 + 1 n � , θ2 = −2 arccos �� l1 − l2 n − l2 − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='15) In particular, we see from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='13) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='14) that the operator IIa n,K indeed implements (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='1) for the case k1 = k2 = 1, which corresponds to (l2, l1) = (0, 1), see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='2 IIb n,K,l1 The circuit diagram for IIb n,K,l1 with l1 ≥ 2 and K ≥ l1 + 1 ≥ 3, which corresponds to the case l2 = 0, is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' This circuit is a generalization of the one for IIa n,K in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 7, with the additional require- ment that wire l1 be in state |1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='3 IIc n,K,l2 The circuit diagram for IIc n,K,l2 with l2 ≥ 1 and K ≥ 3, which corresponds to the case l1 = l2 + 1, is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' This circuit is also a generalization of the one for IIa n,K in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 7, with the additional requirement that wire l2 be in state |2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 10 2 2 1 1 2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 1 1 0 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 0 R(1,2)(θ1) R(0,1)(θ2) 1 X(1,2) X(1,2) l1 l1 + 1 X(0,1) X(0,1) K Figure 8: Circuit diagram for IIb n,K,l1 with l1 ≥ 2 and K ≥ l1 + 1 ≥ 3, where θ1 = −2 arccos( � 1/n) , θ2 = −2 arccos( � l1/(n − 1)) 2 2 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 2 2 2 2 0 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 0 R(1,2)(θ1) R(0,1)(θ2) l2 l1 = l2 + 1 X(1,2) X(1,2) l1 + 1 = l2 + 2 X(0,1) X(0,1) K Figure 9: Circuit diagram for IIc n,K,l2 with l2 ≥ 1 and K ≥ 3, where θ1 = −2 arccos( � (l2 + 1)/n) , θ2 = −2 arccos( � 1/(n − l2 − 1)) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='4 IId n,K,l1,l2 The circuit diagram for IId n,K,l1,l2 with l2 ≥ 1, l1 ≥ l2 + 2 ≥ 3 and K ≥ l1 + 1 ≥ 4 is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' This circuit performs the transformation |0⟩l1+1|1⟩l2+1|2⟩0 �→ cos(θ1/2)|0⟩l1+1|1⟩l2+1|2⟩0 − sin(θ1/2) cos(θ2/2)|0⟩l1+1|2⟩l2+1|1⟩0 + sin(θ1/2) sin(θ2/2)|1⟩l1+1|2⟩l2+1|0⟩0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='16) similarly to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='13), provided that wire l2 is in state |2⟩ and wire l1 is in state |1⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' With the angles as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='14)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='15), we see from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='16) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='10) that IId n,K,l1,l2 indeed implements (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='1) for generic values of (l2, l1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' For special cases of (l2, l1), this circuit reduces to those in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 7, 8 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 11 2 2 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 2 2 2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 1 1 0 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 0 R(1,2)(θ1) R(0,1)(θ2) l2 l2 + 1 X(1,2) X(1,2) l1 l1 + 1 X(0,1) X(0,1) K Figure 10: Circuit diagram for IId n,K,l1,l2 with l2 ≥ 1, l1 ≥ l2 + 2 ≥ 3 and K ≥ l1 + 1 ≥ 4, where θ1 = −2 arccos( � (l2 + 1)/n) , θ2 = −2 arccos( � (l1 − l2)/(n − l2 − 1)) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='4 Final result For the general case (no conditions on the k’s, apart from those in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='1)), we arrive at our final result for an operator Wn,K that satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='1), namely Wn,K = � � W n,K � Wn,K , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='17) where � Wn,K is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='7), and � � W n,K is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' We have implemented the circuit for Un,K (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='10), with the W operators given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='17), in Mathematica using the matrix representations in Appendix A, and we have explicitly checked that the key result (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='1) is satisfied up to n = 6 and K = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 4 Discussion Our main results are the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='10) for the qudit Dicke operator Un,K as a product of W operators (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='4), and (for the qutrit case) the decomposition of the W operators in terms of elementary gates (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' The algorithm is deterministic, and does not use ancillary qutrits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Whereas the preparation of arbitrary qubit Dicke states has been considered in a number of works (see [3, 23–28] and references therein), ours is the first work (to our knowledge) to consider the preparation of arbitrary qudit Dicke states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' We have seen that, already for the qutrit case, the algorithm entails a nontrivial generalization of [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' For the explicit gate implementation of the W operators in Section 3, we have aimed for clarity rather than 12 economy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' it is likely that alternative implementations with reduced gate counts can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' We have performed the explicit gate implementation only for qutrits (d = 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' for the general qudit case, we expect that d − 1 rotation angles θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' , θd−1 would be required, and many edge cases would need to be treated separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Having in hand a way to prepare qudit Dicke states, it will be interesting to go further and formulate an algorithm for preparing eigenstates of spin s > 1/2 integrable spin chains based on coordinate Bethe ansatz [43, 44], thereby extending the approach [29] for preparing eigenstates of the s = 1/2 Heisenberg spin chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Time will tell whether current efforts to build practical quantum computers will be ex- tended from qubits to qutrits, or to even higher-dimensional qudits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' If so, then the generation of Dicke states (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='3), which are highly entangled yet relatively simple, will be a natural early goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Acknowledgments We thank Hai-Rui Wei and Huangjun Zhu for helpful correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' This research was supported in part by the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' NSF PHY-1748958, and by a Cooper fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' A Matrix representations of qutrit gates A 1-qutrit state lives in the complex vector space V spanned by |0⟩, |1⟩, |2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Let us set |0⟩ = \uf8eb \uf8ed 1 0 0 \uf8f6 \uf8f8 , |1⟩ = \uf8eb \uf8ed 0 1 0 \uf8f6 \uf8f8 , |2⟩ = \uf8eb \uf8ed 0 0 1 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='1) The NOT gates X(ij) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='2) are represented by the 3 × 3 matrices [46] X(01) = \uf8eb \uf8ed 0 1 0 1 0 0 0 0 1 \uf8f6 \uf8f8 , X(02) = \uf8eb \uf8ed 0 0 1 0 1 0 1 0 0 \uf8f6 \uf8f8 , X(12) = \uf8eb \uf8ed 1 0 0 0 0 1 0 1 0 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='2) Similarly, the rotation gates R(ij)(θ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='3) are represented by R(01)(θ) = \uf8eb \uf8ed cos( θ 2) − sin( θ 2) 0 sin( θ 2) cos( θ 2) 0 0 0 1 \uf8f6 \uf8f8 , R(02)(θ) = \uf8eb \uf8ed cos( θ 2) 0 − sin( θ 2) 0 1 0 sin( θ 2) 0 cos( θ 2) \uf8f6 \uf8f8 , R(12)(θ) = \uf8eb \uf8ed 1 0 0 0 cos( θ 2) − sin( θ 2) 0 sin( θ 2) cos( θ 2) \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='3) 13 An n-qutrit state lives in V ⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' We label these vector spaces from 0 to n − 1, going from right to left n−1 ↓ V ⊗ · · · ⊗ 1 ↓ V ⊗ 0 ↓ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='4) In circuit diagrams, the n vector spaces are represented by n horizontal wires, which are labeled from 0 to n − 1, going from top (0) to bottom (n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' We use subscripts to indicate the vector spaces on which operators act nontrivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' For example, if A is a 1-qutrit operator, then Aq = I⊗(n−1−q) ⊗ A ⊗ I⊗q (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='5) is an operator on V ⊗n acting nontrivially on the qth vector space q ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' , n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' The vector space on which an operator acts nontrivially can be changed using the permutation operator Pqq′, for example Aq′ = Pq q′ Aq Pq q′ , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='6) where Pq q′ = 3 � i,j=1 ei,j q ej,i q′ , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='7) and ei,j is the elementary 3 × 3 matrix whose (i, j) matrix element is 1, and all others are 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' that is, (ei,j)a,b = δi,a δj,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' For the controlled-X(ij) gates (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='4), we have the 9 × 9 block-diagonal matrices C[2] 1 X(ij) 0 = �16 X(ij) � , C[1] 1 X(ij) 0 = \uf8eb \uf8ed 13 X(ij) 13 \uf8f6 \uf8f8 , C[0] 1 X(ij) 0 = � X(ij) 16 � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='8) where 1n denotes the n × n identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' The controlled gates are related by NOT gates on the controls, for example C[1] 1 X(ij) 0 = X(12) 1 � C[2] 1 X(ij) 0 � X(12) 1 , C[0] 1 X(ij) 0 = X(02) 1 � C[2] 1 X(ij) 0 � X(02) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='9) In terms of circuit diagrams, these identities are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 11a and 11b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 1 0 X(i,j) 1 = 2 0 X(i,j) 1 X(1,2) X(1,2) (a) Identity for C[1] 1 X(ij) 0 0 0 X(i,j) 1 = 2 0 X(i,j) 1 X(0,2) X(0,2) (b) Identity for C[0] 1 X(ij) 0 Double-controlled-X(ij) gates are given by 33 × 33 block-diagonal matrices C[22] 21 X(ij) 0 = �124 X(ij) � , C[11] 21 X(ij) 0 = \uf8eb \uf8ed 112 X(ij) 112 \uf8f6 \uf8f8 , C[00] 21 X(ij) 0 = � X(ij) 124 � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='10) and similarly for higher-controlled-X(ij) gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Matrices corresponding to controlled Ry rotation gates are defined in a similar way, with X(ij) replaced by R(ij)(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 14 References [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
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+page_content=' van Handel, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Mabuchi, “Deterministic Dicke-state preparation with continuous measurement and control,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
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+page_content=' A 70 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
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+page_content=' Wei, “Synthesis of multivalued quantum logic circuits by elementary gates,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' A 87 (2013) 012325, arXiv:1105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content='5485 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
+page_content=' 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfRQzA/content/2301.04989v1.pdf'}
diff --git a/g9E3T4oBgHgl3EQfIQkV/content/tmp_files/2301.04331v1.pdf.txt b/g9E3T4oBgHgl3EQfIQkV/content/tmp_files/2301.04331v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..0108d817f9190cdd423f58fc60d4aff8237b0a39
--- /dev/null
+++ b/g9E3T4oBgHgl3EQfIQkV/content/tmp_files/2301.04331v1.pdf.txt
@@ -0,0 +1,1925 @@
+Nucleon Electric Dipole Moment from the θ Term with Lattice Chiral Fermions
+Jian Liang,1, 2, ∗ Andrei Alexandru,3 Terrence Draper,4
+Keh-Fei Liu,4 Bigeng Wang,4 Gen Wang,5 and Yi-Bo Yang6, 7, 8, 9
+1Guangdong Provincial Key Laboratory of Nuclear Science,
+Institute of Quantum Matter, South China Normal University, Guangzhou 51006, China
+2Guangdong-Hong Kong Joint Laboratory of Quantum Matter,
+Southern Nuclear Science Computing Center, South China Normal University, Guangzhou 51006, China
+3Department of Physics, The George Washington University, Washington, DC 20052, USA
+4Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506, USA
+5Aix-Marseille Université, Université de Toulon, CNRS, CPT, Marseille, France
+6CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
+Chinese Academy of Sciences, Beijing 100190, China
+7School of Fundamental Physics and Mathematical Sciences,
+Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China
+8International Centre for Theoretical Physics Asia-Pacific, Beijing/Hangzhou, China
+9University of Chinese Academy of Sciences, School of Physical Sciences, Beijing 100049, China
+We calculate the nucleon electric dipole moment (EDM) from the θ term with overlap fermions
+on three domain wall lattices with different sea pion masses at lattice spacing 0.11 fm.
+Due to
+the chiral symmetry conserved by the overlap fermions, we have well defined topological charge and
+chiral limit for the EDM. Thus, the chiral extrapolation can be carried out reliably at nonzero lattice
+spacings. We use three to four different partially quenched valence pion masses for each sea pion
+mass and find that the EDM dependence on the valence and sea pion masses behaves oppositely,
+which can be described by partially quenched chiral perturbation theory.
+With the help of the
+cluster decomposition error reduction (CDER) technique, we determine the neutron and proton
+EDM at the physical pion mass to be dn = −0.00148 (14) (31) ¯θ e·fm and dp = 0.0038 (11) (8) ¯θ e·fm.
+This work is a clear demonstration of the advantages of using chiral fermions in the nucleon EDM
+calculation and paves the road to future precise studies of the strong CP violation effects.
+Introduction: Symmetries and their breaking are es-
+sential topics in modern physics, among which the dis-
+crete symmetries C (charge conjugation), P (parity), and
+T (time reversal) are of special importance. This is par-
+tially because the violation of the combined C and P
+symmetries is one of the three Sakharov conditions [1]
+that are necessary to give rise to the baryon asymmetry
+of the universe (BAU). However, despite the great suc-
+cess of the standard model (SM), the weak baryogenesis
+mechanism from the CP violation (��
+CP) within the SM
+contributes negligibly (∼ 16 orders of magnitude smaller
+than the observed BAU [2–6]). This poses a hint that,
+besides the possible θ term in QCD, there could exist
+beyond-standard-model (BSM) sources of ��
+CP and thus
+the study of ��
+CP plays an important role in the efforts of
+searching for BSM physics.
+The electric dipole moment of nucleons (NEDM) serves
+as an important observable to study��
+CP. The first exper-
+imental upper limit on the neutron EDM (nEDM) was
+given in 1957 [7] as ∼ 10−20 e·cm. During the past 60
+years of experiments, this upper limit has been improved
+by 6 orders of magnitude. The most recent experimen-
+tal result of the nEDM is 0.0(1.1)(0.2) × 10−26 e·cm [8],
+which is still around 5 orders of magnitude larger than
+the contribution that can be offered by the weak ��
+CP
+∗ jianliang@scnu.edu.cn
+phase. Currently, several experiments are aiming at im-
+proving the limit down to 10−28 e·cm in the next ∼10
+years. This still leaves plenty of room for the study of
+��
+CP from BSM interactions and the QCD θ term.
+As a reliable nonperturbative method for solving the
+strong interaction, lattice QCD provides us the possi-
+bility of studying the nucleon EDM (NEDM) from first
+principles and with both the statistical and systematic
+uncertainties under control. To be specific, lattice QCD
+can be used to calculate the ratio between the neutron
+and proton EDM induced by strong ��
+CP and the param-
+eter ¯θ, which is the most crucial theoretical input to de-
+termine ¯θ from experiments.
+Many lattice calculations have been carried out on this
+topic. However, there was a watershed in 2017 when it
+was pointed out [9] that all the previous lattice calcu-
+lations, e.g. [10–14], used a wrongly defined ��
+CP form
+factor such that all of those old results need a correc-
+tion. Although the fixing is numerically straight forward,
+none of the previous lattice calculations gives statisti-
+cally significant results after the fixing, leaving a great
+challenge to the lattice community. Since then, several
+attempts [15–18] have been made to tackle the problem,
+but the signal-to-noise ratios of the new results are still
+not satisfying, and no calculation performed directly at
+the physical point gives nonzero results.
+A possibility to bypass this difficulty is to perform the
+computations with several heavier pion masses and ex-
+arXiv:2301.04331v1 [hep-lat] 11 Jan 2023
+
+2
+trapolate to the physical point. However, only with chiral
+fermions can a correct chiral limit be reached at finite lat-
+tice spacings. Otherwise, extrapolating to the continuum
+limit for each pion mass becomes an inevitable prior step
+before a reliable chiral extrapolation, which complicates
+the calculation and potentially leads to hard-to-control
+systematic uncertainties. The best result, so far, of this
+approach, using clover fermions, obtained a 2-sigma sig-
+nal [16].
+In this article, we demonstrate that using chiral
+fermions to extrapolate to the physical point from heavier
+pion masses is the most efficient choice to study NEDM
+on the lattice at the current stage. We employ 3 gauge
+ensembles with different sea pion masses ranging from
+∼300 to ∼600 MeV and we use 3 to 4 valence pion masses
+on each lattice. Therefore, we can study both the valence
+and sea pion mass dependence of the NEDM and better
+control the chiral extrapolation. The results we obtain at
+the physical pion mass are dn = −0.00148 (14) (31) ¯θ e·fm
+and dp = 0.0038 (11) (8) ¯θ e·fm for neutron and proton,
+respectively.
+Nucleon EDM and the θ term: The QCD Lagrangian
+in Euclidean space with the θ term reads (detailed
+conventions can be found in the Supplemental Materi-
+als [19]):
+LE = ¯ψ
+�
+D/E + mq
+�
+ψ+1
+2Tr[F E
+µνF E,µν−i¯θ g2
+8π2 F E
+µν ˜F E,µν],
+(1)
+where ˜F E,µν = ϵµνρσF E
+ρσ. The effective parameter ¯θ =
+θ +
+1
+Nf ArgDet [M] where θ is the original coefficient of
+the θ term and M is the quark mass matrix generated
+by the spontaneous breaking of SU(2)×U(1) in the elec-
+troweak sector. For simplicity, we will not distinguish θ
+and ¯θ in the following content. A crucial point is that, if
+Det [M] = 0, phase of the UA(1) transformation is arbi-
+trary, which means one can always find a chiral rotation
+that lets ¯θ = 0, leaving no net effect of ��
+CP. This indi-
+cates a zero NEDM in the chiral limit [20], which poses a
+very strong constraint in the chiral extrapolation numer-
+ically. However, as mentioned before, for lattice fermions
+which violate the chiral symmetry this constraint cannot
+be used at finite lattice spacing.
+Given that θ is small, one can expand the theta term
+in the action in the path integral and obtain the corre-
+lation functions and matrix elements to the leading or-
+der in θ as θ⟨...⟩θ = ⟨...⟩ + iθ⟨...Qt⟩, where |0⟩θ denotes
+the vacuum with the θ term (namely, the θ vacuum),
+and Qt =
+�
+d4xqt(x) ≡
+g2
+16π2
+�
+d4xTr
+�
+F E
+µν(x) ˜F E,µν(x)
+�
+is the topological charge of the gauge field geometrically.
+Based on this expansion, the ��
+CP electromagnetic (EM)
+form factor F3(q2) can be extracted from normal and Qt
+weighted nucleon matrix elements with initial momentum
+Table I. Parameters of the RBC/UKQCD ensembles: label,
+sea and valence pion masses, and the number of configura-
+tions.
+label
+mπ,s (MeV)
+mπ,v (MeV)
+Ncfg
+24I005
+339
+282 321 348 389 805
+24I010
+432
+426 519 600 508
+24I020
+560
+432 525 606 552
+pi = (m,⃗0) and final momentum pf = (Ef, ⃗q) as [19]
+F3(q2) =
+2m
+Ef + m
+�
+�
+�
+2Ef
+qi
+Tr
+�
+ΓiM (3)Q
+4
+�
+Tr
+�
+ΓeM (2)� − α1GE(q2)
+�
+�
+� ,
+GE(q2) =
+2Ef
+Ef + m
+Tr
+�
+ΓeM (3)
+4
+�
+Tr
+�
+ΓeM (2)� , α1 = Tr
+�
+γ5M (2)Q�
+2Tr
+�
+ΓeM (2)�,
+(2)
+where the matrix elements are
+M (2) = ⟨N(pf)|N(pi)⟩,
+M (3)
+µ
+= ⟨N(pf)|Vµ(0)|N(pi)⟩,
+M (2)Q = ⟨N(pf)|Qt|N(pi)⟩,
+M (3)Q
+µ
+= ⟨N(pf)|QtVµ(0)|N(pi)⟩,
+(3)
+with Vµ being the EM current operator, Γe = 1+γ4
+2
+is the
+unpolarized spin projector, Γi = −iγ5γiΓe the polarized
+projector along the i’th direction, q2 = (pf −pi)2 = −Q2
+the momentum transfer, and qi the nonzero component
+of the momentum transfer. The above formalism is the
+same for both neutron and proton. In the end, the nu-
+cleon EDM can be extracted from the ��
+CP form factor
+F3(q2) in the forward limit for neutron and proton re-
+spectively using
+dn/p = F3,n/p
+�
+q2 → 0
+�
+2m
+θ.
+(4)
+An interesting fact, as seen in Eq. (2), is that the
+neutron ��
+CP form factor at the zero momentum trans-
+fer limit, F3,n(0) has no ��
+CP angle α1 dependence since
+GE,n(0) = 0, and thus one actually needs no information
+about M (2)Q in the neutron case.
+Numerical setups: This study is carried out on three
+2 + 1-flavor RBC/UKQCD gauge ensembles of domain
+wall fermions [21] with the same lattice spacing 0.1105(3)
+fm and lattice volume 243 × 64 but different sea quark
+masses. Using the overlap fermion action [22] on the HYP
+(hyper-cubic) smeared [23] gauge links, multiple partially
+quenched valence quark masses (as listed in Table I with
+other parameters) are calculated utilizing the multi-mass
+inversion algorithm; thus both the sea and valence pion
+mass dependencies of NEDM can be studied and the chi-
+ral extrapolation can be more reliable.
+
+3
+Figure 1.
+Illustration of the CDER technique used when
+computing the correlation functions with the local topolog-
+ical charge summed inside the sphere with radius R.
+Generally, using overlap fermions can be O(100) times
+more costly compared to the traditional Wilson-like dis-
+cretized fermion actions. To improve the computational
+efficiency, 12-12-12 grid sources with Z3-noise and Gaus-
+sian smearing are placed at tsrc = 0 and tsrc = 32
+in one inversion with randomly chosen spatial positions
+on different configurations, and low-mode substitution
+(LMS) [24] is applied to suppress the statistical contam-
+ination between different source positions. We also use
+the stochastic sandwich method (SSM) [25] with LMS to
+make the cost of using multiple nucleon sinks be additive
+instead of multiplicative. We use 8 sets of source noises
+and 16 sets of sink noises (for each of the source-sink sep-
+arations 6a, 7a, and 8a) to improve the statistics. Five
+nonzero momentum transfers are calculated such that we
+can reliably do the q2 extrapolation to get F3 (0); the de-
+tails of the q2 extrapolation are given in the Supplemental
+Materials [19].
+CDER improvement and results: To further suppress
+the statistical uncertainty of M (2)Q and M (3)Q, we take
+advantage of a technique called cluster decomposition er-
+ror reduction (CDER) for the disconnected insertion [26].
+As illustrated in Fig. 1, we write the total topologi-
+cal charge as the summation of the local charge den-
+sity qt(x) derived from the overlap operator [27, 28] as
+qt(x) =
+1
+2Tr [γ5Dov(x, x)], where the trace is over the
+color-spin indices, and convert the two-point function
+weighted with the total topological charge Qt into a sum-
+mation of the three-point functions involving qt(x)
+G(2)Q =
+�
+⃗x
+��
+r
+qt (x + r) χ (x) ¯χ (t0, G)
+�
+,
+(5)
+where χ is the nucleon interpolating operator, G denotes
+the source grid, and x = (tf, ⃗x). We then use the clus-
+ter decomposition property to limit the sum to a range
+commensurate with the correlation length
+G(2)Q ∼
+�
+⃗x
+�|r|R0 ⟨q(0 + r)Jµ(0)⟩ to estimate the contribution
+from the truncated tail. In this way, with the correlation data such as that in the right panel, the corresponding
+systematic uncertainty is estimated to be ∼10%. So the two methods give consistent systematic uncertainties and we
+choose ∼12% to be our final estimation.
+4) Chiral extrapolation: For the systematic uncertainty from the chiral extrapolation, we take the difference of the
+extrapolations with and without partially quenched data points to be our estimation. As shown in Fig. 9 and Fig. 10
+(the chiral fits for proton), the difference is around 3%. The small systematic uncertainty of chiral interpolation is
+understandable since the chiral limit provides a very strong constraint to the interpolation.
+The total systematic uncertainty is found to be 21%, which is simply calculated by quadrature from all the
+systematic uncertainties.
+
diff --git a/g9E3T4oBgHgl3EQfIQkV/content/tmp_files/load_file.txt b/g9E3T4oBgHgl3EQfIQkV/content/tmp_files/load_file.txt
new file mode 100644
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+page_content='Nucleon Electric Dipole Moment from the θ Term with Lattice Chiral Fermions Jian Liang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
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+page_content=' ∗ Andrei Alexandru,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='3 Terrence Draper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='4 Keh-Fei Liu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='4 Bigeng Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='4 Gen Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='5 and Yi-Bo Yang6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' 9 1Guangdong Provincial Key Laboratory of Nuclear Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Institute of Quantum Matter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' South China Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Guangzhou 51006,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' China 2Guangdong-Hong Kong Joint Laboratory of Quantum Matter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Southern Nuclear Science Computing Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' South China Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Guangzhou 51006,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' China 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' The George Washington University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' DC 20052,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' USA 4Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' University of Kentucky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Lexington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' KY 40506,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' USA 5Aix-Marseille Université,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Université de Toulon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' CPT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Marseille,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' France 6CAS Key Laboratory of Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Institute of Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Beijing 100190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' China 7School of Fundamental Physics and Mathematical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Hangzhou Institute for Advanced Study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' UCAS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Hangzhou 310024,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' China 8International Centre for Theoretical Physics Asia-Pacific,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Beijing/Hangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' China 9University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' School of Physical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' China We calculate the nucleon electric dipole moment (EDM) from the θ term with overlap fermions on three domain wall lattices with different sea pion masses at lattice spacing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='11 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Due to the chiral symmetry conserved by the overlap fermions, we have well defined topological charge and chiral limit for the EDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Thus, the chiral extrapolation can be carried out reliably at nonzero lattice spacings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' We use three to four different partially quenched valence pion masses for each sea pion mass and find that the EDM dependence on the valence and sea pion masses behaves oppositely, which can be described by partially quenched chiral perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' With the help of the cluster decomposition error reduction (CDER) technique, we determine the neutron and proton EDM at the physical pion mass to be dn = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='00148 (14) (31) ¯θ e·fm and dp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='0038 (11) (8) ¯θ e·fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' This work is a clear demonstration of the advantages of using chiral fermions in the nucleon EDM calculation and paves the road to future precise studies of the strong CP violation effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Introduction: Symmetries and their breaking are es- sential topics in modern physics, among which the dis- crete symmetries C (charge conjugation), P (parity), and T (time reversal) are of special importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' This is par- tially because the violation of the combined C and P symmetries is one of the three Sakharov conditions [1] that are necessary to give rise to the baryon asymmetry of the universe (BAU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' However, despite the great suc- cess of the standard model (SM), the weak baryogenesis mechanism from the CP violation (�� CP) within the SM contributes negligibly (∼ 16 orders of magnitude smaller than the observed BAU [2–6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' This poses a hint that, besides the possible θ term in QCD, there could exist beyond-standard-model (BSM) sources of �� CP and thus the study of �� CP plays an important role in the efforts of searching for BSM physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' The electric dipole moment of nucleons (NEDM) serves as an important observable to study�� CP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' The first exper- imental upper limit on the neutron EDM (nEDM) was given in 1957 [7] as ∼ 10−20 e·cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' During the past 60 years of experiments, this upper limit has been improved by 6 orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' The most recent experimen- tal result of the nEDM is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='0(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='1)(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='2) × 10−26 e·cm [8], which is still around 5 orders of magnitude larger than the contribution that can be offered by the weak �� CP ∗ jianliang@scnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='cn phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Currently, several experiments are aiming at im- proving the limit down to 10−28 e·cm in the next ∼10 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' This still leaves plenty of room for the study of �� CP from BSM interactions and the QCD θ term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' As a reliable nonperturbative method for solving the strong interaction, lattice QCD provides us the possi- bility of studying the nucleon EDM (NEDM) from first principles and with both the statistical and systematic uncertainties under control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' To be specific, lattice QCD can be used to calculate the ratio between the neutron and proton EDM induced by strong �� CP and the param- eter ¯θ, which is the most crucial theoretical input to de- termine ¯θ from experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Many lattice calculations have been carried out on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' However, there was a watershed in 2017 when it was pointed out [9] that all the previous lattice calcu- lations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' [10–14], used a wrongly defined �� CP form factor such that all of those old results need a correc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Although the fixing is numerically straight forward, none of the previous lattice calculations gives statisti- cally significant results after the fixing, leaving a great challenge to the lattice community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Since then, several attempts [15–18] have been made to tackle the problem, but the signal-to-noise ratios of the new results are still not satisfying, and no calculation performed directly at the physical point gives nonzero results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' A possibility to bypass this difficulty is to perform the computations with several heavier pion masses and ex- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='04331v1 [hep-lat] 11 Jan 2023 2 trapolate to the physical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' However, only with chiral fermions can a correct chiral limit be reached at finite lat- tice spacings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Otherwise, extrapolating to the continuum limit for each pion mass becomes an inevitable prior step before a reliable chiral extrapolation, which complicates the calculation and potentially leads to hard-to-control systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' The best result, so far, of this approach, using clover fermions, obtained a 2-sigma sig- nal [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' In this article, we demonstrate that using chiral fermions to extrapolate to the physical point from heavier pion masses is the most efficient choice to study NEDM on the lattice at the current stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' We employ 3 gauge ensembles with different sea pion masses ranging from ∼300 to ∼600 MeV and we use 3 to 4 valence pion masses on each lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Therefore, we can study both the valence and sea pion mass dependence of the NEDM and better control the chiral extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' The results we obtain at the physical pion mass are dn = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='00148 (14) (31) ¯θ e·fm and dp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='0038 (11) (8) ¯θ e·fm for neutron and proton, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Nucleon EDM and the θ term: The QCD Lagrangian in Euclidean space with the θ term reads (detailed conventions can be found in the Supplemental Materi- als [19]): LE = ¯ψ � D/E + mq � ψ+1 2Tr[F E µνF E,µν−i¯θ g2 8π2 F E µν ˜F E,µν], (1) where ˜F E,µν = ϵµνρσF E ρσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' The effective parameter ¯θ = θ + 1 Nf ArgDet [M] where θ is the original coefficient of the θ term and M is the quark mass matrix generated by the spontaneous breaking of SU(2)×U(1) in the elec- troweak sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' For simplicity, we will not distinguish θ and ¯θ in the following content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' A crucial point is that, if Det [M] = 0, phase of the UA(1) transformation is arbi- trary, which means one can always find a chiral rotation that lets ¯θ = 0, leaving no net effect of �� CP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' This indi- cates a zero NEDM in the chiral limit [20], which poses a very strong constraint in the chiral extrapolation numer- ically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' However, as mentioned before, for lattice fermions which violate the chiral symmetry this constraint cannot be used at finite lattice spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Given that θ is small, one can expand the theta term in the action in the path integral and obtain the corre- lation functions and matrix elements to the leading or- der in θ as θ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='⟩θ = ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='⟩ + iθ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='Qt⟩, where |0⟩θ denotes the vacuum with the θ term (namely, the θ vacuum), and Qt = � d4xqt(x) ≡ g2 16π2 � d4xTr � F E µν(x) ˜F E,µν(x) � is the topological charge of the gauge field geometrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Based on this expansion, the �� CP electromagnetic (EM) form factor F3(q2) can be extracted from normal and Qt weighted nucleon matrix elements with initial momentum Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Parameters of the RBC/UKQCD ensembles: label, sea and valence pion masses, and the number of configura- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' label mπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='s (MeV) mπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='v (MeV) Ncfg 24I005 339 282 321 348 389 805 24I010 432 426 519 600 508 24I020 560 432 525 606 552 pi = (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='⃗0) and final momentum pf = (Ef,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' ⃗q) as [19] F3(q2) = 2m Ef + m � � � 2Ef qi Tr � ΓiM (3)Q 4 � Tr � ΓeM (2)� − α1GE(q2) � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' GE(q2) = 2Ef Ef + m Tr � ΓeM (3) 4 � Tr � ΓeM (2)� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' α1 = Tr � γ5M (2)Q� 2Tr � ΓeM (2)�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' (2) where the matrix elements are M (2) = ⟨N(pf)|N(pi)⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' M (3) µ = ⟨N(pf)|Vµ(0)|N(pi)⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' M (2)Q = ⟨N(pf)|Qt|N(pi)⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' M (3)Q µ = ⟨N(pf)|QtVµ(0)|N(pi)⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' (3) with Vµ being the EM current operator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Γe = 1+γ4 2 is the unpolarized spin projector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Γi = −iγ5γiΓe the polarized projector along the i’th direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' q2 = (pf −pi)2 = −Q2 the momentum transfer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' and qi the nonzero component of the momentum transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' The above formalism is the same for both neutron and proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' In the end, the nu- cleon EDM can be extracted from the �� CP form factor F3(q2) in the forward limit for neutron and proton re- spectively using dn/p = F3,n/p � q2 → 0 � 2m θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' (4) An interesting fact, as seen in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' (2), is that the neutron �� CP form factor at the zero momentum trans- fer limit, F3,n(0) has no �� CP angle α1 dependence since GE,n(0) = 0, and thus one actually needs no information about M (2)Q in the neutron case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Numerical setups: This study is carried out on three 2 + 1-flavor RBC/UKQCD gauge ensembles of domain wall fermions [21] with the same lattice spacing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content='1105(3) fm and lattice volume 243 × 64 but different sea quark masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Using the overlap fermion action [22] on the HYP (hyper-cubic) smeared [23] gauge links, multiple partially quenched valence quark masses (as listed in Table I with other parameters) are calculated utilizing the multi-mass inversion algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' thus both the sea and valence pion mass dependencies of NEDM can be studied and the chi- ral extrapolation can be more reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Illustration of the CDER technique used when computing the correlation functions with the local topolog- ical charge summed inside the sphere with radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Generally, using overlap fermions can be O(100) times more costly compared to the traditional Wilson-like dis- cretized fermion actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' To improve the computational efficiency, 12-12-12 grid sources with Z3-noise and Gaus- sian smearing are placed at tsrc = 0 and tsrc = 32 in one inversion with randomly chosen spatial positions on different configurations, and low-mode substitution (LMS) [24] is applied to suppress the statistical contam- ination between different source positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' We also use the stochastic sandwich method (SSM) [25] with LMS to make the cost of using multiple nucleon sinks be additive instead of multiplicative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' We use 8 sets of source noises and 16 sets of sink noises (for each of the source-sink sep- arations 6a, 7a, and 8a) to improve the statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' Five nonzero momentum transfers are calculated such that we can reliably do the q2 extrapolation to get F3 (0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' the de- tails of the q2 extrapolation are given in the Supplemental Materials [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' CDER improvement and results: To further suppress the statistical uncertainty of M (2)Q and M (3)Q, we take advantage of a technique called cluster decomposition er- ror reduction (CDER) for the disconnected insertion [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' we write the total topologi- cal charge as the summation of the local charge den- sity qt(x) derived from the overlap operator [27,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' 28] as qt(x) = 1 2Tr [γ5Dov(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' x)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' where the trace is over the color-spin indices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' and convert the two-point function weighted with the total topological charge Qt into a sum- mation of the three-point functions involving qt(x) G(2)Q = � ⃗x �� r qt (x + r) χ (x) ¯χ (t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' G) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' (5) where χ is the nucleon interpolating operator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' G denotes the source grid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' and x = (tf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' ⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQfIQkV/content/2301.04331v1.pdf'}
+page_content=' We then use the clus- ter decomposition property to limit the sum to a range commensurate with the correlation length G(2)Q ∼ � ⃗x �|r| Q, we can still use the solution of the
+proposition, however, it is approximate. With the solution
+completed, we next evaluate our results.
+VI. RESULTS
+In this section, we evaluate the performance of the proposed
+beamforming solutions for cell-free ISAC MIMO systems. In
+particular, we consider a scenario where Mt = Mr with two
+APs placed at (25, 0) and (75, 0) in the Cartesian coordinates,
+as shown in Fig. 2. Each AP is equipped with a uniform
+
+6
+0
+20
+40
+60
+80
+100
+x (m)
+-10
+0
+10
+20
+30
+40
+50
+60
+y (m)
+AP
+UE
+Target
+UE 3
+UE 2
+UE 5
+UE 1
+AP 1
+AP 2
+UE 4
+The range the UEs and target are randomly placed
+Fig. 2.
+The simulation placement is illustrated. The UEs and target are
+randomly placed over the y-axis.
+linear array (ULA) along the x axis of Nt = 16 antennas.
+At y = 50m, we randomly place one sensing target and the
+U = 5 communications users along the x-axis. Specifically,
+the x coordinates of these locations are drawn from a uniform
+distribution in [0, 100]. For the communication channels, we
+adopt a LOS channel model and take σ2
+u = 1. For the sensing
+channels, we adopt the parameters ς2
+nr = 1 and ζmtmr = 0.1.
+The transmit power of the APs is Pm = 0dBW and the number
+of sensing streams Q = 1. In the following, we average the
+results over 1000 realizations. For this setup, we compare the
+following solutions:
+(i) NS Sensing - RZF Comm which designs the sensing
+beam as conjugate beamforming projected on the null
+space of the communication channels as in (15) and
+implements the communications beams according to the
+RZF design in (16).
+(ii) NS Sensing - OPT Comm which has the same sensing
+beam design as in (i) but designs the communication
+beam based on the max-min optimization in (18).
+(iii) CB Sensing - OPT Comm which first designs the
+sensing beam as the conjugate beamforming in (14) and
+then designs the communication beams to solve the max-
+min optimization in (18).
+(iv) JSC Beam Optimization which implements the com-
+munication and sensing beams based on Proposition 1
+that jointly optimizes the beamforming vectors based on
+the communication and sensing functions.
+Sensing and Communication Power Allocation: We first
+investigate the sensing and communication performance for
+different power allocation ratios. Specifically, in Fig. 3, we
+show the sensing SNR and communication SINR achieved
+by the different beamforming solutions for different values of
+ρ ∈ (0, 1). It is important to note here that for the beamforming
+solutions (i)-(iii), the communication and sensing beams are
+separately designed, and we directly allocate the communica-
+tion and sensing powers based on the ratio ρ. For the JSC beam
+optimization solution (iv), it implements the beamforming
+design in Proposition 1, which optimizes both the structure
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+0
+2
+4
+6
+8
+Comm SINR
+Communication
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Communication Power Ratio ( )
+0
+20
+40
+60
+80
+100
+Target SNR
+Sensing
+NS Sensing - RZF Comm
+NS Sensing - OPT Comm
+CB Sensing - OPT Comm
+JSC Beam Optimization
+Similar sensing performance
+Fig. 3. Performance of the solutions for different power allocation ratios for
+the communications and sensing. The proposed JSC optimization provides a
+significant gain for sensing while satisfying the best communication SINR.
+of the beams and the power allocation. Therefore, and for the
+sake of comparing with the other approaches, we plot the JSC
+optimization curve in Fig. 3 by setting the communication
+SINR threshold to be equal to the achieved SINR by solution
+(ii). This still respects the total power constraint, which is
+taken care of by (26). As seen in the figure, the first two
+solutions, (i) and (ii), achieve better communication SINR and
+less sensing SNR compared to solution (iii). This is expected
+as solution (iii) aims to maximize the sensing performance,
+irrespective of the communication, and hence, it causes some
+interference to the communication users. Interestingly, while
+achieving the best communication performance of the separate
+solutions, the joint solution provides very similar sensing
+performance to the MF sensing. This highlights the gain of
+the developed JSC beamforming design.
+Target distance to closes UE: To further investigate how
+the different beamforming approaches impact the trade-off
+between the sensing and communication performance, we
+evaluate this performance versus the distance between the
+sensing target and closest communication UE in Fig. 4. Note
+that, intuitively, as the sensing target gets closer to the com-
+munication users, the overlap between the communication and
+sensing channels’ subspaces increases, which can benefit or
+penalize the communication and sensing performance depend-
+ing on the beamforming design. In Fig. 4, we set the power
+ratio as 0.5 for the communication and sensing operation. This
+figure shows that for the smaller distances/separation between
+the sensing target and communication users, the conjugate
+beamforming sensing solution (solution (iii)) optimizes the
+sensing performance but causes non-negligible interference to
+the communication, which significantly degrades its perfor-
+mance. On the other side, solutions (i) and (ii), which prioritize
+the communication and keep the sensing beamforming in
+the null-space of the communication channels, optimize the
+communication SINR and degrade the sensing SNR. For the
+SINR constraint of the JSC optimization, we again adopt the
+SINR obtained by solution (ii), which achieves the best com-
+munication performance. Hence, the achieved communication
+
+7
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+1
+2
+3
+4
+Comm SINR
+Communication
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+Target-Closest UE Distance (m)
+20
+40
+60
+80
+Target SNR
+Sensing
+NS Sensing - RZF Comm
+NS Sensing - OPT Comm
+CB Sensing - OPT Comm
+JSC Beam Optimization
+MF sensing exceeds JSC
+with the comm. interference
+Joint opt.
+ gain
+Best comm. performance with joint opt.
+Fig. 4. Performance of the solutions versus the distance between the target and
+closest AP. The proposed JSC optimization provides almost a constant sensing
+SNR for different distances, with a significant gain over the NS solutions.
+SINR of this solution and JSC beam optimization are the same.
+The sensing SNR, however, enjoys the advantage of the joint
+beam optimization. Specifically, it provides almost a constant
+sensing performance for different target-closest UE distances:
+Achieving a close sensing performance to solution (i) when
+the separation between the sensing target and communication
+users is small and exceeds the performance of all the other
+three solutions when this separation is large, which highlights
+the potential of the joint beamforming design.
+APPENDIX
+A. Derivation of Sensing SNR
+The expectation of the nominator can be simplified as
+E
+� �
+mr∈Mr
+��GmrX
+��2
+�
+(32)
+=
+�
+mr∈Mr
+Tr
+�
+E
+�
+X X
+HG
+H
+mrGmr
+��
+(33)
+=
+�
+mr∈Mr
+Tr
+�
+E
+�
+X X
+H�
+�
+��
+�
+=I
+E
+�
+G
+H
+mrGmr
+� �
+(34)
+=
+�
+mr∈Mr
+�
+mt∈Mt
+�
+Tr E
+�
+Gmt,mrG
+H
+mt,mr
+�
++
+�
+m′
+t∈Mt\{mt}
+Tr E
+�
+Gmt,mrG
+H
+m′
+t,mr
+� �
+(35)
+which is obtained by applying the expansion of the Frobenius
+norm, interchanging expectation and trace, and permutating
+the inner terms of the trace operation several times. Here, due
+to the expectation over {αmt,mr} and independence of these
+random variables, we have E
+�
+Gmt,mrG
+H
+m′
+t,mr
+�
+= 0, which
+makes the latter line of (35) zero. Further, we also have
+Tr E
+�
+Gmt,mrG
+H
+mt,mr
+�
+(36)
+= Tr E
+�
+|αmt,mr|2 a(θmr)aH(θmt)FmtF
+H
+mta(θmt)aH(θmr)
+�
+(37)
+= σ2
+mt,mrTr E
+�
+�aH(θmt)FmtF
+H
+mta(θmt) aH(θmr)a(θmr)
+�
+��
+�
+=1
+�
+�
+(38)
+= σ2
+mt,mr
+��aH(θmt)Fmt
+��2 .
+(39)
+B. Proof of Proposition 1
+This extends the proof in [5]. For this purpose, we first note
+that in the problem formulation in (28), the sensing variables,
+Fq, are utilized together as a summation of all of the streams,
+both in the objective and constraints. Hence, if we define ¯F =
+�
+s∈S Fs, we can apply the optimization in terms of the user
+streams and this variable. After the solution, the individual
+sensing streams can be determined to enforce these constraints.
+To that end, we re-formulate the problem (28) as
+max
+{Fu},¯F
+�
+mr∈Mr
+mt∈Mt
+ζ2
+mt,mrTr
+�
+DmtADmt ¯F
+�
+(40a)
+�
+1 + γ−1�
+Tr (QuFu) − Tr
+�
+Qu ¯F
+�
+≥ σ2
+u,
+(40b)
+Tr
+�
+Dm ¯F
+�
+= Pm,
+∀m ∈ Mt
+(40c)
+Fu ∈ S+,
+∀u ∈ U,
+(40d)
+¯F −
+�
+u∈U
+Fu ∈ S+,
+(40e)
+¯F ∈ S+.
+(40f)
+Let us denote the variables obtained by the solution of this
+problem by {F′
+u} and ¯F′. Using this solution, we aim to con-
+struct an alternative optimal solution of rank-1. Specifically,
+we consider the following set of solutions
+¯F′′ = ¯F′,
+F′′
+u = f ′′
+u(f ′′
+u)H,
+f ′′
+u = (hH
+u F′
+uhu)− 1
+2 F′
+uhu.
+(41)
+Now, we want to show that the constructed set is also optimal.
+For this, we need to check if (i) the value of the objective is the
+same and (ii) the constraints are satisfied. First, the objective
+only contains the summation variable and provides the optimal
+value by definition. For (40b), we define vu = (hH
+u F′
+uhu)− 1
+2 ,
+and write
+Tr (QuF′′
+u) = Tr
+�
+huv2
+uhH
+u F′
+uhuhH
+u F′H
+u
+�
+= Tr (QuF′
+u) ,
+(42)
+where we used the cyclic permutation property of the trace and
+F′H
+u
+= f ′
+uf ′H
+u
+= F′
+u. With the addition of ¯F′′ = ¯F′, (40b) is
+satisfied. Similarly, the constraints (40c) and (40f) are already
+satisfied by ¯F′′ = ¯F′. Further, (40d) and the solution being
+rank-1 are also satisfied by the definition of F′′
+u in (41). For
+(40e), we have
+vH(F′
+u − F′′
+u)v = vHF′
+uv − (hH
+u F′
+uhu)−1 ��vHF′
+uhu
+��2 .
+(43)
+
+8
+From
+the
+Cauchy-Schwarz
+inequality,
+we
+have
+(vHF′
+uv)(hH
+u F′
+uhu)
+≥
+��vHF′
+uhu
+��2. Inserting this into
+(43), we obtain vH(F′
+u − F′′
+u)v
+≥
+0, which leads to
+vHF′′
+uv ≥ 0 since it is the summation of two semidefinite
+matrices, F′
+u − F′′
+u and F′
+u. Finally, (40e) can be shown via
+¯F′′ −
+�
+u∈U
+F′′
+u = ¯F′ −
+�
+u∈U
+F′
+u +
+�
+u∈U
+(F′
+u − F′′
+u)
+(44)
+which again leads to the summation of semidefinite matrices.
+Finally, for constructing the sensing matrices of the solution,
+we want to find Q rank-1 matrices whose summation is
+�
+q∈Q F′′
+q. For this purpose, we can utilize the eigendecompo-
+sition, i.e., �
+Q F′′
+q = UΛUH = �Q′
+q′=1 λq′uq′uH
+q′, and take
+the largest Q eigenvectors as the beams via f ′′
+q = √λu′ uq′.
+Here, it is important to note that it is only possible if the rank
+of the summation, Q′ = rank(�
+q∈Q F′′
+q), is smaller than or
+equal to the number of the sensing streams, Q. Otherwise, the
+summation cannot be constructed by Q rank-1 matrices, and
+the solution is approximate.
+REFERENCES
+[1] F. Liu, Y. Cui, C. Masouros, J. Xu, T. X. Han, Y. C. Eldar, and S. Buzzi,
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+[2] U. Demirhan and A. Alkhateeb, “Integrated sensing and communication
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+
diff --git a/j9FIT4oBgHgl3EQfqivf/content/tmp_files/load_file.txt b/j9FIT4oBgHgl3EQfqivf/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,468 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf,len=467
+page_content='1 Cell-Free ISAC MIMO Systems: Joint Sensing and Communication Beamforming Umut Demirhan and Ahmed Alkhateeb Abstract—This paper considers a cell-free integrated sensing and communication (ISAC) MIMO system, where distributed MIMO access points are jointly serving the communication users and sensing the targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' For this setup, we first develop two baseline approaches that separately design the sensing and communication beamforming vectors, namely communication- prioritized sensing beamforming and sensing-prioritized com- munication beamforming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Then, we consider the joint sensing and communication (JSC) beamforming design and derive the optimal structure of these JSC beamforming vectors based on a max-min fairness formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' The results show that the developed JSC beamforming is capable of achieving nearly the same communication signal-to-interference-plus-noise ratio (SINR) that of the communication-prioritized sensing beamform- ing solutions with almost the same sensing SNR of the sensing- prioritized communication beamforming approaches, yielding a promising strategy for cell-free ISAC MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' INTRODUCTION The integration of sensing functions into the communication systems is envisioned to be an integral part of the 6G and future communication systems [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' If the hardware and wireless resources are efficiently shared, this will enable the communication infrastructure to have sensing capabilities at minimal cost and open the sensing frequency bands for wireless communication operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Achieving that, however, requires the careful design of the various aspects of the inte- grated sensing and communication (ISAC) systems, including the transmission waveform, the post-processing of the received signals, and the MIMO beamforming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' While these problems have recently attracted increasing research interest, the prior work has mainly focused on the single ISAC basestation case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' In practice, however, multiple ISAC basestations will operate in the same geographical region, frequency band, and time, causing interference on each other for both the sensing and communication functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' This motivates the coordination be- tween these distributed nodes to improve both communication and sensing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' This ultimately leads to cell-free ISAC MIMO systems, where distributed ISAC basestations jointly serve the same set of communication users and sense the same targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' With this motivation, this paper investigates the joint sensing and communication beamforming design of these cell-free ISAC MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Prior work has mainly focused on the single-node case and investigated the design of the joint-sensing and communication (JSC) waveform [3], post-processing [4], and beamforming [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' For example, in [3], the authors investigated the JSC waveform design for correlated and uncorrelated waveforms The authors are with the School of Electrical, Computer and Energy Engi- neering, Arizona State University, (Email: udemirhan, alkhateeb@asu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' and the trade-offs between communication and sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' For beamforming, the work in [5] investigated the JSC beamform- ing design of co-located MIMO system with monostatic radar that serves multiple users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' For distributed nodes, but assuming that each basestation serves only one user, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=', not in a cell-free MIMO setup, [6]–[8] studied the power allocation and beam- forming design problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' More relevantly, the optimization of the JSC power allocation has been investigated for distributed multi-antenna systems that consider a single user served by basestations or in a cell-free setup [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' The authors in this work, however, adopted fixed beam designs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=', regularized zero beamforming for the communication with the sensing beamforming in the nullspace of the the communication chan- nels without further optimization, and focused on optimizing the power allocated to these beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Since these cell-free ISAC MIMO systems rely mainly on beamforming in their dual- function operation, it is very important to optimize the design of these JSC beams, which to the best of our knowledge, has not been previously investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' With this motivation, we propose and compare various beamforming strategies for the cell-free ISAC MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' To investigate the JSC transmit beamforming in cell-free massive MIMO systems, in this paper, we consider a system model with many APs and UEs, where the APs jointly serve the UEs and sense the targets in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' For beamforming, we first consider two baseline strategies that we call communication-prioritized sensing and sensing-prioritized communication beamforming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' In these strategies, either the sensing or the communication beamforming is given the prior- ity to be design first without accounting for the other function, and then the beamforming of the other function is designed in a way that does not affect the performance of the higher-priority function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' After that, we consider the case when the sensing and communication beamforming is jointly designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' For this, we formulate a JSC beamforming problem, that aims to maximize the sensing SNR while satisfying the communication SINR constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' We then re-formulate this problem as a non-convex semidefinite problem (SDP) and apply semidefinite relaxation (SDR) to find the optimal beamforming structure for a large set of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Using numerical simulations, We then evaluate the proposed approaches and show that the JSC design provides near-optimal performance for both sensing and communication thanks to the co-design for the two functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Notation: We use the following notation throughout this paper: A is a matrix, a is a vector, a is a scalar, A is a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' AT , AH, A∗, A−1, A† are transpose, Hermitian (conjugate transpose), conjugate, inverse, and pseudo-inverse of A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' ∥a∥ is the l2-norm of a and ∥A∥F is the Frobenius norms of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' CN(µ, Σ) is a complex Gaussian arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='11328v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='IT] 26 Jan 2023 2 random vector with mean µ and covariance Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' E [·] and ⊗ denote expectation and Kronecker product, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' S+ is the set of hermitian positive semidefinite matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' SYSTEM MODEL We consider a cell-free massive MIMO ISAC system with M access points (APs) and U communication users, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' In the downlink, and without loss of generality, we assume that a subset Mt (out of the M APs) are transmitting communication and sensing waveforms to jointly serve the U users, where |Mt| = Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Simultaneously, a subset Mr (out of the M APs) is receiving the possible reflections/scattering of the transmitted waveforms on the various targets/objects in the environment, with |Mr| = Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' It is important to note here that the subsets Mt and Mr may generally have no, partial, or full overlap, which means that none, some, or all the APs could be part of Mt and Mr and are simultaneously transmitting and receiving signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' The transmitting and receiving APs are equipped with Nt and Nr antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Further, for simplicity, all the APs are assumed to have digital beamforming capabilities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=', each antenna element has a dedicated radio frequency (RF) chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' The UEs are equipped with single antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' The APs are connected to a central processing unit that allows joint design and processing, and they are assumed to be fully synchronized for both sensing and communication purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Signal Model In this subsection, we define the joint sensing and commu- nication signal model for the downlink transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' The APs jointly transmit U communication streams, {xu[ℓ]}u∈U, and Q sensing streams, {xq[ℓ]}q∈Q, where Q = {U +1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' , U +Q} and with ℓ denoting the ℓ’s symbol in these communica- tion/sensing streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' For ease of exposition, we also define the overall set of streams as S = U ∪ Q = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' , S} with S = U +Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' If xm[ℓ] ∈ CNt denotes the transmit signal at the transmitting AP m due to the ℓ-th symbol, we can then write xm[ℓ] = � u∈ U fmuxu[ℓ] � �� � Communication + � q∈Q fmqxq[ℓ] � �� � Sensing = � s∈S fmsxs[ℓ], (1) where xs[ℓ] ∈ C is the ℓ-th symbol of the s-th stream, fms ∈ CNt is the beamforming vector for this stream applied by AP m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' The symbols are assumed to be of unit average energy, E[|xs|2] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' The beamforming vectors are subject to the total power constraint, Pm, given as E[∥xm[ℓ]∥2] = � s∈S ∥fms∥2 ≤ Pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (2) Further, by stacking the beamforming vectors of stream s of all the APs, we define the beamforming vector fs: fs = �f T 1s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' f T Mts �T ∈ CMtNt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (3) For each stream s, we denote the sequence of L transmit symbols as xs = �xs[1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' , xs[L]�T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Given this notation, we make the following assumption, which is commonly adopted Sensing Target User 1 User u User U Communication and Sensing Beams Target Reflection Central Processing Unit AP 1 AP 2 AP m AP M-1 AP M Transmitting AP Receiving AP Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' The system model with the joint sensing and communication transmissions is illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' The APs serve multiple users while aiming to sense the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' in the literature [5]: The messages of the radar and communi- cation signals are statistically independent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=', E[xsxH s ] = I and E[xsxH s′ ] = 0 for s, s′ ∈ S with s ̸= s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Note that the radar signal generation with these properties may be achieved through pseudo-random coding [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Communication Model We denote the communication channel between UE u and AP m as hmu ∈ CNt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Further, by stacking the channels between user u and all the APs, we get hu ∈ CMtNt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' considering a block fading channel model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' where the channel remains constant over the transmission of the L symbols,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' we can write the received signal at UE u as y(c) u [ℓ] = � m∈Mt hH muxm[ℓ] + nu = � m∈Mt hH mufmuxu[ℓ] � �� � Desired Signal (DS) + � u′∈U\\{u} � m∈Mt hH mufmu′xu′[ℓ] � �� � Multi-user Interference (MUI) + � q∈Q � m∈Mt hH mufmqxq[ℓ] � �� � Sensing Interference (SI) + nu[ℓ] ���� Noise ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (4) where nu[ℓ] ∼ CN(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' σ2 u) is the receiver noise of UE u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Then, the communication SINR of UE u can be obtained as SINR(c) u = E[|DS|2] E[|MUI|2] + E[|SI|2] + E[|Noise|2] , (5) where the detailed expression is given in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Sensing Model For the sensing channel model, we consider a single- point reflector, as commonly adopted in the literature [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Specifically, the transmit signal is scattered from the single- point reflector and received by the receiving APs in Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' With a single path model, the channel between the transmitting AP 3 SINR(c) u = ��� � m∈Mt hH mufmu ��� 2 � u′∈U\\{u} ��� � m∈Mt hH mufmu′ ��� 2 + � q∈Q ��� � m∈Mt hH mufmq ��� 2 + σ2u = ��hH u fu ��2 � u′∈U\\{u} |hH u fu′|2 + � q∈Q |hH u fq|2 + σ2u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (6) mt and the receiving AP mr through the reflector is defined as Gmtmr = αmtmra(θmr)aH(θmt), (7) where αmtmr ∼ CN(0, ζ2 mtmr) is the combined sensing channel gain, which includes the effects due to the path- loss and radar cross section (RCS) of the target, and a(θ) is the array response vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' The angles of departure/arrival of the transmitting AP mt and receiving AP mr from the point reflector are respectively denoted by θmt and θmr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' We consider the Swerling-I model for the sensing channel [10], which assumes that the fluctuations of RCS are slow and the sensing channel does not change within the transmission of the L sensing and communication symbols in xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' With this model, the signal received at AP mr at instant ℓ can be written as y(s) mr[ℓ] = � mt∈Mt Gmtmr xmt[ℓ] + nm[ℓ] = � mt∈Mt αmtmra(θmr)aH(θmt) xmt[ℓ] + nmr[ℓ], (8) where nmr[ℓ] ∈ CNr is the receiver noise at AP mr and has the distribution CN(0, ς2 mrI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' To write the received radar signal due to the L symbols in a compact form, we introduce Fm = [fm1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' , fmS] ∈ CNt×S, (9) X = [x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' , xS]T ∈ CS×L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (10) Then, as an equivalent of (1), we can write the transmit signal of the L symbols from each AP mt as Xmt = FmtX ∈ CNt×L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (11) With that, we can re-write the sensing signal in (8) at each receiving AP mr, due to the L symbols, in a compact form as Y(s) mr = � mt∈M αmtmra(θmr)aH(θmt)Fmt � �� � ≜Gmtmr � �� � ≜Gmr X + Nmr, (12) with Gmr denoting the beam-space sensing channel of the receiving AP mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' With the purpose of having a general sensing objective that is correlated with the performance of various sensing tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=', detection [8], range/Doppler/angle estimation and tracking), we adopt the joint SNR of the received signals as the sensing objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Note that utilization of the joint SNR requires a joint processing of the radar signal at the Mr sensing receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' The sensing SNR can be written as in equation (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' where the derivation is provided in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Here we note that the sensing SNR is scaled with the contribution of all the (communication and sensing) streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Our objective is then to design the cell-free communication beamforming {fu}u∈U and the sensing beamforming {fq}q∈Q to optimize the communication SINR and the sensing SNR defined in (6) and (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' It is important to note here that, in this paper, we focus on the beamforming design problem assuming that the communication channel and the sensing target angles are known to the transmit APs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' In the next three sections, we present the proposed beamforming strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' COMMUNICATION-PRIORITIZED SENSING BEAMFORMING DESIGN In this section, we investigate the scenario where the communication has a higher priority, and where the commu- nication beams are already designed a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' In this case, the objective is to design the sensing beams to optimize the sensing performance while not affecting the communication performance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=', not causing any interference to the U communication users).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Note that in this section and the next section, Section IV, we assume that Q = 1 since we have one sensing target and that the total power is divided with a fixed ratio ρ, leading to P c m = ρPm for the communication power and P s m = Pm − P c m for the sensing power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' This makes it interesting for the future work to explore the joint optimization of the beamforming and power allocation in cell-free ISAC MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Next, we present two sensing beamforming design solutions for the cases (i) when the communication users are not present and when (ii) they are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Conjugate Sensing Beamforming: When the communica- tion users are not present (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=', U = 0), for example during downtimes, the system can completely focus on the sensing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' In this case, and given the single target sensing model, the conjugate sensing beamforming solution becomes optimal, as it directly maximizes the sensing SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' With this solution, the sensing beamforming vectors can be written as f CB mq = �pmq Nt a(θm), (14) where pmq = P s m is the power allocated for the sensing beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Communication-Prioritized Optimal Sensing Solution: When the communication users exist (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=', U ≥ 1), and since the communication has a higher priority, a straight- forward optimal sensing beamforming approach is to project the optimal sensing beams (constructed through conjugate beamforming) to the null-space of the communication chan- nels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' This way, the interference contribution of the sensing beam to the communication channels is eliminated while the sensing SNR is maximized within the communication null space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Let Hm = [hm1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' hmU] ∈ CNt×U denote the full 4 SNR(s) = E �� mr∈Mr ��GmrX ��2 F � E �� mr∈Mr ∥Nmr∥2 F � = � mr∈Mr � mt∈Mt ζ2 mtmr ��aH(θmt)Fmt ��2 � mr∈Mr LNrς2nr ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (13) channel matrix from the transmit AP m to all the UEs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' then the NS sensing beamforming can be constructed as f NS mq = √pmq � I − Hm � HH mHm �† HH m � a(θm) ��� � I − Hm (HH mHm)† HH m � a(θm) ��� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (15) where we again set the allocated power pmq = P s m as we consider a single sensing beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' SENSING-PRIORITIZED COMMUNICATION BEAMFORMING DESIGN In this section, we consider the scenario where the sensing has a higher priority, and where the sensing beams are already designed a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' In this case, the objective is to design the communication beams to optimize the communication perfor- mance while minimizing the impact of the sensing interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' It is important to note here that an interesting difference be- tween the communication and sensing optimization problems is that while the sensing signals could cause interference that degrades the communication performance, the communication signals could generally be leveraged to further enhance the sensing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Next, we present two communication beamforming design solutions for the cases when (i) the sensing target is not present and when (ii) it is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Regularized Zero-forcing Beamforming: When the sens- ing target is not present, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=', Q = 0, a near-optimal commu- nication beamforming design is the regularized zero-forcing (RZF) [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' This solution allows a trade-off between the multi- user interference and noise terms of the SINR through a regu- larization parameter λ, that is added to the ZF beamforming: ˜f RZF u = � λI + � u′∈U hu′hH u′ �−1 hu, (16) which then can be normalized to satisfy the power constraints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=', f RZF mu = √pmu(˜f RZF mu /|˜f RZF mu |).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' We here again adopt pmu = Pm/U with an equal power between the beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' For the RZF, it is preferable to have a higher regularization parameter in the scenarios with a higher noise and smaller regularization parameter in scenarios with more interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' For further details, we refer to [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Sensing-Prioritized Optimal Communication Solution: For the case when the sensing beam is designed a priori, we derive a max-min fair rate optimal communication beamform- ing solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' First, this max-min problem can be written as max {fmu} min u SINR(c) u (17a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' � u∈U ∥fmu∥2 ≤ P c m, ∀m ∈ Mt, (17b) where the objective is quasiconvex and shows a similar structure to the optimal beamforming formulation for the cell- free massive MIMO networks with only the communication objective [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' For a given minimum SINR constraint γ, the problem can be written as the following feasibility problem find {fmu} (18a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' SINR(c) u ≥ γ, ∀u ∈ U, (18b) � u∈U ∥fmu∥2 ≤ P c m, ∀m ∈ Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (18c) Here, we note that the SINR constraint (18b) is in a fractional form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' This, however, can be converted to a second- order cone constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' For this purpose, we can re-write the constraint as � 1 + 1 γ � ����� � m∈Mt hH mufmu ����� 2 ≥ � u′∈U ����� � m∈Mt hH mufmu′ ����� 2 + � q∈Q ����� � m∈Mt hH mufmq ����� 2 + σ2 u (19) Now, taking the square root of both sides, we can convert the given form to a second-order cone constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' The square root, however, leaves an absolute on the left-hand side, which is a non-linear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' This can be simplified as the real part of the variable [11], since any angular rotation (e−jψ) to the expression inside the absolute does not change the value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=', ����� � m∈Mt hH mufmu ����� = ����� � m∈Mt hH mufmue−jψ ����� = Re � � m∈Mt hH mufmu � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (20) This can be seen as selecting the optimal solution with a specific angular rotation from the set of infinite rotations ψ ∈ [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Finally, we can write the constraint (18b) as a second- order cone as follows �� 1 + 1 γ � Re � � m∈Mt hH mufmu � ≥ ��������� � m∈Mt hH mufm1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' � m∈Mt hH mufmS σu ��������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (21) When (18b) is replaced with (21), it results in a second-order cone problem and can be solved by the convex solvers [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Using the bisection algorithm, the maximum SINR value, γ⋆, can be obtained by solving the convex feasibility problem (18) for different values of γ within a predetermined range [γmin, γmax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' This computes the optimal solution to (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' 5 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' JOINT SENSING AND COMMUNICATION BEAMFORMING OPTIMIZATION A more desirable approach for cell-free joint sensing and communication MIMO systems is to jointly optimize the beamforming vectors for the sensing and communication func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Specifically, our objective is to maximize the sensing SNR together with the communication SINR of the UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Towards this objective, we reformulate (18) as a sensing SNR maximization problem with constraints on the minimum communication SINRs: max {fms} SNR(s) (22a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' SINR(c) u ≥ γ, ∀u ∈ U, (22b) � s∈S ∥fms∥2 ≤ Pm, ∀m ∈ Mt, (22c) where the objective, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=', the maximization of the convex SNR expression SNR(s), is non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Hence, a similar approach to the beamforming optimization in the previous section can not be adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' The problem in (22), however, can be cast as a semidefinite program, which allows applying a semidef- inite relaxation for the non-convex objective [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' With the relaxation, the problem becomes convex, and it can be solved with the convex solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' The result obtained from the relaxed problem can then be cast to the original problem’s space with a method designed specifically for the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' In the following, we present the details of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' To reformulate (22) as an SDP, we first re-define the beamforming optimization variables as matrices: Fs = fsf H s , ∀s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Writing (22) in terms of Fs instead of fs eliminates the quadratic terms in the sensing SNR and communica- tion SINR expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' This SDP formulation, however, by construction introduces two new constraints: (i) The convex hermitian positive semi-definiteness constraint Fs ∈ S+, where S+ is the set of hermitian positive semidefinite matrices, and (ii) the non-convex rank-1 constraint rank(Fs) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Further, we need to write the problem in terms of these newly introduced variables, {Fs}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' For this purpose, we define the selection matrix Dmt = diag(dmt) ⊗ INt×Nt, where dm = [dm1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' , dmMt] is an indicator vector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=', dmm = 1 and dmm′ = 0 ∀m, m′ ∈ Mt with m ̸= m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' In addition, we define A = ˜a˜aH with ˜a = [a(θ1)T , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' , a(θMt)T ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Now, we can write the objective of (22) in terms of these variables as SNR(s) = � mr∈Mr � mt∈Mt ζ2 mt,mrTr � DmtADmt � s∈S Fs � � mr∈Mr LNrς2nr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (23) For the constraints of the problem in (22), we define Qu = huhH u and re-write the SINR in (6) in terms of the new variables as SINR(c) u = Tr (QuFu) � u′∈U\\{u} Tr (QuFu′) + � q∈Q Tr (QuFq) + σ2u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (24) With this, we can write the constraint in (22b) and the power constraint in (22c) as � 1 + γ−1� Tr (QuFu) − Tr � Qu � s∈S Fs � ≥ σ2 u, (25) � s∈S Tr (DmFs) ≤ Pm, ∀m ∈ Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (26) By collecting these expressions together, we can write the SDP form of our problem in (22) as max {Fs} SNR(s) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (25) and (26), rank(Fs) = 1, ∀s ∈ S, Fs ∈ S+, ∀s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (27) which can be relaxed by removing the rank-1 constraint as follows max {Fs} SNR(s) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (25) and (26), Fs ∈ S+, ∀s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (28) This relaxed problem in (28) can be solved via CVX and convex SDP solvers [13], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Then, if the matrices obtained by this solution, denoted by {F′ s}, are rank-1, then they are optimal for (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' The optimal beamforming vectors, fs, in this case, can be obtained as the eigenvector of F′ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' For the case the matrices are not rank-1, we make the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' There exists a solution to the problem (27), denoted by {F′′ s}, that satisfies rank(F′′ u) = 1, ∀u ∈ U and � q∈Q F′′ q = � s∈S F′ s − � U∈U F′′ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (29) The communication beamforming vectors of this solution can be given as f ′′ u = (hH u F′ uhu)− 1 2 F′ uhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (30) Further, if rank(� q∈Q F′′ q) ≤ Q, the sensing beamforming vectors of this solution can be constructed by f ′′ q = � λq−U uq−U, (31) with λi and ui being the i-th largest eigenvalue of � q∈Q F′′ q and the corresponding eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' The proof slightly extends the solution in [5, Theo- rem 1], which we provide in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' In the case rank(� q∈Q F′′ q) > Q, we can still use the solution of the proposition, however, it is approximate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' With the solution completed, we next evaluate our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' RESULTS In this section, we evaluate the performance of the proposed beamforming solutions for cell-free ISAC MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' In particular, we consider a scenario where Mt = Mr with two APs placed at (25, 0) and (75, 0) in the Cartesian coordinates, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Each AP is equipped with a uniform 6 0 20 40 60 80 100 x (m) 10 0 10 20 30 40 50 60 y (m) AP UE Target UE 3 UE 2 UE 5 UE 1 AP 1 AP 2 UE 4 The range the UEs and target are randomly placed Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' The simulation placement is illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' The UEs and target are randomly placed over the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' linear array (ULA) along the x axis of Nt = 16 antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' At y = 50m, we randomly place one sensing target and the U = 5 communications users along the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Specifically, the x coordinates of these locations are drawn from a uniform distribution in [0, 100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' For the communication channels, we adopt a LOS channel model and take σ2 u = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' For the sensing channels, we adopt the parameters ς2 nr = 1 and ζmtmr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' The transmit power of the APs is Pm = 0dBW and the number of sensing streams Q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' In the following, we average the results over 1000 realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' For this setup, we compare the following solutions: (i) NS Sensing - RZF Comm which designs the sensing beam as conjugate beamforming projected on the null space of the communication channels as in (15) and implements the communications beams according to the RZF design in (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (ii) NS Sensing - OPT Comm which has the same sensing beam design as in (i) but designs the communication beam based on the max-min optimization in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (iii) CB Sensing - OPT Comm which first designs the sensing beam as the conjugate beamforming in (14) and then designs the communication beams to solve the max- min optimization in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (iv) JSC Beam Optimization which implements the com- munication and sensing beams based on Proposition 1 that jointly optimizes the beamforming vectors based on the communication and sensing functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Sensing and Communication Power Allocation: We first investigate the sensing and communication performance for different power allocation ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Specifically, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' 3, we show the sensing SNR and communication SINR achieved by the different beamforming solutions for different values of ρ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' It is important to note here that for the beamforming solutions (i)-(iii), the communication and sensing beams are separately designed, and we directly allocate the communica- tion and sensing powers based on the ratio ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' For the JSC beam optimization solution (iv), it implements the beamforming design in Proposition 1, which optimizes both the structure 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='9 1 0 2 4 6 8 Comm SINR Communication 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='9 1 Communication Power Ratio ( ) 0 20 40 60 80 100 Target SNR Sensing NS Sensing - RZF Comm NS Sensing - OPT Comm CB Sensing - OPT Comm JSC Beam Optimization Similar sensing performance Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Performance of the solutions for different power allocation ratios for the communications and sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' The proposed JSC optimization provides a significant gain for sensing while satisfying the best communication SINR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' of the beams and the power allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Therefore, and for the sake of comparing with the other approaches, we plot the JSC optimization curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' 3 by setting the communication SINR threshold to be equal to the achieved SINR by solution (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' This still respects the total power constraint, which is taken care of by (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' As seen in the figure, the first two solutions, (i) and (ii), achieve better communication SINR and less sensing SNR compared to solution (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' This is expected as solution (iii) aims to maximize the sensing performance, irrespective of the communication, and hence, it causes some interference to the communication users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Interestingly, while achieving the best communication performance of the separate solutions, the joint solution provides very similar sensing performance to the MF sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' This highlights the gain of the developed JSC beamforming design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Target distance to closes UE: To further investigate how the different beamforming approaches impact the trade-off between the sensing and communication performance, we evaluate this performance versus the distance between the sensing target and closest communication UE in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Note that, intuitively, as the sensing target gets closer to the com- munication users, the overlap between the communication and sensing channels’ subspaces increases, which can benefit or penalize the communication and sensing performance depend- ing on the beamforming design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' 4, we set the power ratio as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='5 for the communication and sensing operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' This figure shows that for the smaller distances/separation between the sensing target and communication users, the conjugate beamforming sensing solution (solution (iii)) optimizes the sensing performance but causes non-negligible interference to the communication, which significantly degrades its perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' On the other side, solutions (i) and (ii), which prioritize the communication and keep the sensing beamforming in the null-space of the communication channels, optimize the communication SINR and degrade the sensing SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' For the SINR constraint of the JSC optimization, we again adopt the SINR obtained by solution (ii), which achieves the best com- munication performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Hence, the achieved communication 7 0 5 10 15 20 25 30 35 40 45 1 2 3 4 Comm SINR Communication 0 5 10 15 20 25 30 35 40 45 Target-Closest UE Distance (m) 20 40 60 80 Target SNR Sensing NS Sensing - RZF Comm NS Sensing - OPT Comm CB Sensing - OPT Comm JSC Beam Optimization MF sensing exceeds JSC with the comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' interference Joint opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' gain Best comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' performance with joint opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Performance of the solutions versus the distance between the target and closest AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' The proposed JSC optimization provides almost a constant sensing SNR for different distances, with a significant gain over the NS solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' SINR of this solution and JSC beam optimization are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' The sensing SNR, however, enjoys the advantage of the joint beam optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Specifically, it provides almost a constant sensing performance for different target-closest UE distances: Achieving a close sensing performance to solution (i) when the separation between the sensing target and communication users is small and exceeds the performance of all the other three solutions when this separation is large, which highlights the potential of the joint beamforming design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Derivation of Sensing SNR The expectation of the nominator can be simplified as E � � mr∈Mr ��GmrX ��2 � (32) = � mr∈Mr Tr � E � X X HG H mrGmr �� (33) = � mr∈Mr Tr � E � X X H� � �� � =I E � G H mrGmr � � (34) = � mr∈Mr � mt∈Mt � Tr E � Gmt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='mrG H mt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='mr � + � m′ t∈Mt\\{mt} Tr E � Gmt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='mrG H m′ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='mr � � (35) which is obtained by applying the expansion of the Frobenius norm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' interchanging expectation and trace,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' and permutating the inner terms of the trace operation several times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Here, due to the expectation over {αmt,mr} and independence of these random variables, we have E � Gmt,mrG H m′ t,mr � = 0, which makes the latter line of (35) zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Further, we also have Tr E � Gmt,mrG H mt,mr � (36) = Tr E � |αmt,mr|2 a(θmr)aH(θmt)FmtF H mta(θmt)aH(θmr) � (37) = σ2 mt,mrTr E � �aH(θmt)FmtF H mta(θmt) aH(θmr)a(θmr) � �� � =1 � � (38) = σ2 mt,mr ��aH(θmt)Fmt ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (39) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Proof of Proposition 1 This extends the proof in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' For this purpose, we first note that in the problem formulation in (28), the sensing variables, Fq, are utilized together as a summation of all of the streams, both in the objective and constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Hence, if we define ¯F = � s∈S Fs, we can apply the optimization in terms of the user streams and this variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' After the solution, the individual sensing streams can be determined to enforce these constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' To that end, we re-formulate the problem (28) as max {Fu},¯F � mr∈Mr mt∈Mt ζ2 mt,mrTr � DmtADmt ¯F � (40a) � 1 + γ−1� Tr (QuFu) − Tr � Qu ¯F � ≥ σ2 u, (40b) Tr � Dm ¯F � = Pm, ∀m ∈ Mt (40c) Fu ∈ S+, ∀u ∈ U, (40d) ¯F − � u∈U Fu ∈ S+, (40e) ¯F ∈ S+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (40f) Let us denote the variables obtained by the solution of this problem by {F′ u} and ¯F′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Using this solution, we aim to con- struct an alternative optimal solution of rank-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Specifically, we consider the following set of solutions ¯F′′ = ¯F′, F′′ u = f ′′ u(f ′′ u)H, f ′′ u = (hH u F′ uhu)− 1 2 F′ uhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (41) Now, we want to show that the constructed set is also optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' For this, we need to check if (i) the value of the objective is the same and (ii) the constraints are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' First, the objective only contains the summation variable and provides the optimal value by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' For (40b), we define vu = (hH u F′ uhu)− 1 2 , and write Tr (QuF′′ u) = Tr � huv2 uhH u F′ uhuhH u F′H u � = Tr (QuF′ u) , (42) where we used the cyclic permutation property of the trace and F′H u = f ′ uf ′H u = F′ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' With the addition of ¯F′′ = ¯F′, (40b) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Similarly, the constraints (40c) and (40f) are already satisfied by ¯F′′ = ¯F′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Further, (40d) and the solution being rank-1 are also satisfied by the definition of F′′ u in (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' For (40e), we have vH(F′ u − F′′ u)v = vHF′ uv − (hH u F′ uhu)−1 ��vHF′ uhu ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' (43) 8 From the Cauchy-Schwarz inequality, we have (vHF′ uv)(hH u F′ uhu) ≥ ��vHF′ uhu ��2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Inserting this into (43), we obtain vH(F′ u − F′′ u)v ≥ 0, which leads to vHF′′ uv ≥ 0 since it is the summation of two semidefinite matrices, F′ u − F′′ u and F′ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Finally, (40e) can be shown via ¯F′′ − � u∈U F′′ u = ¯F′ − � u∈U F′ u + � u∈U (F′ u − F′′ u) (44) which again leads to the summation of semidefinite matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Finally, for constructing the sensing matrices of the solution, we want to find Q rank-1 matrices whose summation is � q∈Q F′′ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' For this purpose, we can utilize the eigendecompo- sition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=', � Q F′′ q = UΛUH = �Q′ q′=1 λq′uq′uH q′, and take the largest Q eigenvectors as the beams via f ′′ q = √λu′ uq′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Here, it is important to note that it is only possible if the rank of the summation, Q′ = rank(� q∈Q F′′ q), is smaller than or equal to the number of the sensing streams, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
+page_content=' Otherwise, the summation cannot be constructed by Q rank-1 matrices, and the solution is approximate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FIT4oBgHgl3EQfqivf/content/2301.11328v1.pdf'}
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+arXiv:2301.03452v1 [math.AP] 9 Jan 2023
+QUANTITATIVE COMPACTNESS ESTIMATES FOR
+STOCHASTIC CONSERVATION LAWS
+K. H. KARLSEN
+Abstract. We present a quantitative compensated compactness estimate for
+stochastic conservation laws, which generalises a previous result of Golse &
+Perthame [8] for deterministic equations. With a stochastic modification of
+Kruˇzkov’s interpolation lemma, this estimate provides bounds on the rate at
+which a sequence of vanishing viscosity solutions becomes compact.
+1. Introduction
+During the last decade many authors investigated the well-posedness of hyper-
+bolic conservation laws perturbed by stochastic source terms, see [5, 6] and the list
+of references in [7]. The initial-value problem for these Itˆo-type SPDEs take the
+form
+∂tu + divf(x, u) = σ(x, u) ˙W (t) + R(x, u),
+(t, x) ∈ (0, T ) × Rd,
+u(0, x) = u0(x),
+x ∈ Rd,
+(1.1)
+where f = (f1, . . . , fd) is the flux vector, R is the “deterministic” source term,
+u0 ∈ L∞(Rd) is the initial function, and T > 0 is a final time. The term σ ˙W(t) is
+a stochastic forcing term, where W is a cylindrical Wiener process [4] with noise
+amplitude σ. We will refer to the SPDEs (1.1) as stochastic conservation laws.
+Stochastic conservation laws are used to model a wide variety of physical systems
+that are subject to random fluctuations and have wave-propagating behavior.
+We fix a stochastic basic S consisting of a complete probability space (Ω, F, P),
+and a complete right-continuous filtration {Ft}t∈[0,T ]. The solution u, the Wiener
+process W, and all other relevant processes, are always understood as defined on
+S and to be appropriately measurable with respect to the filtration {Ft}t∈[0,T ].
+We refer to [7, Pages 38, 40] for precise regularity and growth assumptions on f,
+σ, R.
+For a precise definition of entropy/kinetic solutions and a corresponding
+well-posedness theorem, see [7, Section 5]. Under the assumptions that R ≡ 0 and
+f = f(u), we refer to the original works [6] (on Rd) and [5] (on Td).
+In this paper, we are interested in deriving quantitative estimates that can be
+used to prove the convergence in L1
+loc of sequences {un}n∈N of approximate solutions
+to (1.1). As a concrete example, consider the parabolic SPDE
+∂tun + divf(x, un) − εn∆un = σ(x, un) ˙W(t) + R(x, un),
+(1.2)
+where εn
+n↑∞
+−−−→ 0. For the well-posedness of classical solutions to (1.2), see [6] under
+the assumptions that R ≡ 0 and f = f(u) does not depend on x. For the general
+context provided by (1.2), see [9] and [7, Theorem 5.1].
+Date: January 10, 2023.
+1
+
+2
+KARLSEN
+In the study of SPDEs on Rd, weight functions are sometimes used. These weight
+functions are used to control the growth of solutions as they approach infinity,
+which in turn allows for the derivation of optimal conditions on the coefficients of
+the equations. The use of weighted Lp spaces facilitates the analysis of stochastic
+conservation laws on Rd (see, e.g., [9]). Denote by Lp(χdx) the weighted Lp space
+of functions for which
+�
+Rd |u(x)|p χ(x) dx < ∞,
+where χ is a weight function. The collection of relevant weights, denoted by W,
+consists of χ ∈ C1(Rd) ∩ L1(Rd) for which χ(x) > 0 and |∇χ(x)| ≤ Cχχ(x), for all
+x ∈ Rd, where Cχ > 0 is a constant depending only on χ. A simple example of a
+(smooth) weight function includes χ(x) = χN(x) = (1 + |x|2)−N, N > d/2. Any
+weight function χ ∈ W satisfies the properties
+(1.3)
+|χ(x + z) − χ(x)| ≲ χ(x) |z| ,
+sup
+|x−y|≤R
+χ(x)
+χ(y) ≲R 1,
+which are used repeatedly in this paper. Clearly, Lp(Rd) ⊂ Lp(χdx), p ∈ [1, ∞).
+Moreover, χ−1 ∈ L∞
+loc(Rd) implies that Lp(χdx) ⊂ Lp
+loc(Rd). Since χ ∈ L1(Rd), we
+also have Lq(χdx) ⊂ Lp(χdx) for all q, p such that 1 ≤ p < q < ∞.
+Now regarding a priori estimates for un, one can prove that there is an n-
+independent constant C = C(χ, p, r) such that
+(1.4)
+E ∥un∥r
+L∞(0,T ;Lp(χdx)) ≤ C,
+E
+�����
+�
+R+
+�
+R
+εn |∇un|2 χ(x) dx dt
+�����
+r
+≤ C,
+∀p, r ∈ [2, ∞) and for any χ ∈ W, see [6], [9], and [7, Remark 5.9]. In the general
+case, un does not exhibit n-independent L∞ and BV estimates [2, p. 711].
+In [2] (see also [5]), the authors derived some basic quantitative compactness
+estimates. These n-uniform estimates, which were used and further refined in [9, 10]
+and [3], take the form
+(1.5)
+E
+�
+Rd
+�
+Rd Jδ(z) |un(t, x + z) − un(t, x − z)| χ(x) dx dz ≲T δµx,
+for any t ∈ (0, T ) and some µx ∈ (0, 1), where {Jδ}δ>0 is a mollifier sequence. One
+can prove that (1.5) implies a “fractional BV ” estimate of the form
+(1.6)
+E sup
+|z|<δ
+� T
+0
+∥un(t, · + z) − un(t, ·)∥L1(χdx) dt ≲T δµx,
+for any δ > 0, see, e.g., [2, Lemma 2] and Proposition 2.5 herein. Estimates like
+(1.5) are often linked to the L1 stability `ala Kruˇzkov of the solution operator.
+Using the approximating SPDE and a modification [2, 10] of an interpolation
+technique due to Kruˇzkov, one can use the spatial estimate (1.5) to establish that
+there exists µt ∈ (0, 1) such that
+(1.7)
+E sup
+τ∈(0,δ)
+� T −δ
+0
+∥un(t + τ, ·) − un(t, ·)∥L1(χdx) dt ≲T δµt,
+see Proposition 2.6 for a general estimate of this type.
+We refer to translation estimates like (1.6) and (1.7) as quantitative compactness
+estimates, see Section 2 for further discussion and refinements. They can be used to
+derive convergence results (via Cauchy sequence arguments) and error estimates for
+
+STOCHASTIC CONSERVATION LAWS
+3
+approximate solutions. Moreover, as part of the stochastic compactness method,
+one can use them to show that the laws L(un) of un form a tight sequence of
+probability measures on L1(χdx), which allows for the application of Skorokhod’s
+representation theorem.
+In [6], the authors establish convergence of the viscosity approximations (1.2)
+using compensated compactness, assuming d = 1, R ≡ 0, and the genuine nonlin-
+earity of f = f(u) (f ′′ ̸= 0 a.e.). The main result of our paper is a refinement of
+the compensated compactness approach—in the spirit of [8]—that leads to a spatial
+compactness estimate like (1.6) for the viscosity approximation, under a strength-
+ened nonlinearity condition (|f ′′| ≥ c > 0). A temporal estimate (1.7) follows from
+this estimate via Proposition 2.6. Roughly speaking, in Section 3, we prove (1.6)
+with δx = 1
+4 −
+1
+4p, for any finite p > 2, assuming that the viscosity approximation
+un is uniformly bounded in Lp
+ω,t,x for any finite p, see (1.4).
+In the case that un is bounded in L∞
+ω,t,x, we recover δx = 1
+4, which coincides with
+the known Besov regularity exponent (in x) of entropy solutions to conservation
+laws with one convex entropy and an entropy production that is a signed Radon
+measure [8, Theorem 5]. The quantitative version of compensated compactness
+allows for non-homogenous/discontinuous flux functions f = f(x, u), in which case
+a signed measure arises naturally. The details will be presented elsewhere.
+For simplicity of presentation, we will in what follows assume that W is a real-
+valued Wiener process and that σ(x, u) is a real-valued function. The extension to
+a cylindrical Wiener process with corresponding operator-valued noise amplitude is
+standard, as discussed in [4] and the references cited earlier.
+2. Quantitative compactness estimates
+A subset K of a metric space (X, d) is precompact if its closure K is compact. A
+subset K of a complete metric space (X, d) is precompact if and only if it is totally
+bounded, meaning that for every ε > 0 there exists a finite cover of K of open balls of
+radius ε. We will use the well-known Kolmogorov–Riesz–Fr´echet characterization of
+precompact subsets of L1
+loc(Rd) in terms of the uniform continuity of the translation
+in L1(Rd), see, e.g., [1, Theorem 4.26].
+Using the fact that translation is continuous in L1, we have the following simple
+but useful lemma.
+Lemma 2.1. Let U ⊂b L1(χdx), χ ∈ W. Fix any J ∈ L1(Rd) with supp (J) ⊂
+B(0, R), R > 0. Then K := J ⋆ U = {J ⋆ u : u ∈ U} is precompact in L1
+loc(Rd).
+Proof. Clearly, as χ > 0 on any set D ⊂⊂ Rd, if J ⋆ Uχ := {(J ⋆ u)χ : u ∈ U} is
+precompact in L1
+loc(Rd), then K = J ⋆ U is precompact in L1
+loc(Rd) as well.
+Let us verify the precompactness of J ⋆ Uχ using the Kolmogorov–Riesz–Fr´echet
+theorem. First, we claim that the set J ⋆ Uχ is bounded in L1(Rd). Indeed,
+∥J ⋆ uχ∥L1(Rd) ≤
+�
+Rd
+�
+Rd |J(y)| |u(x − y)| χ(x) dx dy
+(2.1)
+=
+�
+Rd
+�
+Rd |J(y)| |u(x − y)| χ(x − y)
+χ(x)
+χ(x − y) dx dy
+(1.3)
+≲R ∥J∥L1(Rd) ∥u∥L1(χdx) .
+
+4
+KARLSEN
+Next, we verify the translation condition. For any translation z ∈ Rd,
+∥J ⋆ uχ)(· + z) − J ⋆ uχ∥L1(Rd) ≤
+�
+Rd |J ⋆ u(x + z) − J ⋆ u(x)| χ(x) dx dy
++
+�
+Rd J ⋆ u(x + z) |χ(x + z) − χ(x)| dx =: I1 + I2,
+where I1
+(1.3)
+≲R ∥J(· + z) − J∥L1(Rd) ∥u∥L1(χdx) and
+I2
+(1.3)
+≤ |z|
+�
+Rd |J ⋆ u(x + z)χ(x + z)|
+χ(x)
+χ(x + z) dx
+(1.3)
+≲ |z| ∥J ⋆ uχ∥L1(Rd)
+(2.1)
+≲R |z| ∥J∥L1(Rd) ∥u∥L1(χdx) .
+Consequently, as |z| → 0, ∥(J ⋆ uχ)(· + z) − J ⋆ uχ∥L1(Rd) → 0, uniformly in J ⋆uχ
+with u ∈ U. An application of [1, Theorem 4.26] concludes the proof.
+□
+Definition 2.2 (approximate identity). An approximate identity (or a convolution
+kernel) is a family {Jδ}δ>0 of L1(Rd) functions Jδ : Rd → R such that
+(i) Jδ is nonnegative and
+�
+Rd Jδ(x) dx = 1, for each δ > 0,
+(ii) for each h > 0,
+lim
+δ→0
+�
+|x|≥h
+Jδ(x) dx = 0.
+Remark 2.3. One simple (compactly supported) example is the standard (Friedrichs)
+mollifier. More generally, given a positive function J with
+�
+J = 1, the rescaled
+family Jδ(x) =
+1
+δd J(x/δ) supplies an approximate identify.
+Fix an approximate identity {Jδ}δ>0. Consider a sequence {un}n≥1 for which
+∥un∥L1(χdx) ≲ 1 and
+(2.2)
+lim
+δ→0 lim sup
+n→∞ ∥un − Jδ ⋆ un∥L1(χdx) = 0.
+Then {un}n≥1 is precompact in L1
+loc(Rd).
+Indeed, as {un}n≥1 is bounded in
+L1(χdx), Lemma 2.1 supplies the precompactnes in L1(D) of the sequence {Jδ ⋆ un}n≥1,
+for each fixed δ > 0, and for any D ⊂⊂ Rd. Therefore, it is totally bounded in
+L1(D). In view of (2.2) and since χ > 0 on D, the sequence {un}n≥1 is also totally
+bounded—and thus precompact—in L1(D).
+Lemma 2.4. Let {Jδ}δ>0 be an approximate identity, and consider a sequence
+{un}n∈N of functions on Rd. Suppose ∥un∥L1(χdx) ≲ 1 and, for any δ > 0,
+(2.3)
+�
+Rd
+�
+Rd Jδ(z) |un(x + z) − un(x − z)| χ(x) dz dx ≤ ρ(δ),
+where ρ : [0, ∞) → [0, ∞) is an increasing function that is continuous at 0 with
+ρ(0) = 0 (ρ is independent of n). Then {u}n∈N is precompact in L1
+loc(Rd).
+
+STOCHASTIC CONSERVATION LAWS
+5
+Proof. Since
+�
+Rd |un(x) − Jδ ⋆ un(x)| χ(x) dx
+≤
+�
+Rd
+�
+Rd Jδ(x − y) |un(x) − un(y)| χ(x) dy dx
+= 2d
+�
+Rd
+�
+Rd J δ
+2 (z) |un(˜x + z) − un(˜x − z)| χ(˜x + z) dz d˜x
+(1.3)
+≲
+�
+Rd
+�
+Rd J δ
+2 (z) |un(˜x + z) − un(˜x − z)| χ(˜x) dz d˜x
+(2.3)
+≤ ρ
+�
+δ/2
+�
+≤ ρ(δ)
+δ↓0
+−−→ 0,
+uniformly in n, the compactness condition (2.2) follows.
+□
+In what follows, we return to functions un = un(ω, t, x) : Ω × (0, T ) × Rd, n ∈ N,
+depending also on the probability (ω) and temporal (t) variables. The estimate
+(2.3), for suitable choices of the “modulus of continuity” ρ(·) and the approximate
+identity {Jδ}δ>0, can be turned into a fractional BV estimate like (1.6). This fact
+is related to known links between Sobolev, Besov, and Nikolskii fractional spaces
+(see, for example, [13]). More generally, we have
+Proposition 2.5 (quantitative compactness estimate in space). Fix a weight χ ∈
+W and a standard Friedrichs mollifier {Jδ}δ>0. Consider a sequence {un}n∈N of
+functions satisfying E
+� T
+0
+�
+Rd |un(t, x)| χ(x) dx dt ≲ 1 and
+(2.4)
+E
+� T
+0
+�
+Rd
+�
+Rd Jδ(z) |un(t, x + z) − un(t, x − z)| χ(x) dz dx dt ≤ ρx(δ),
+where ρx : [0, ∞) → [0, ∞) is an increasing function that is continuous at 0 with
+ρx(0) = 0 (ρx is independent of n but may depend on χ, T ).
+Then un satisfies the quantitative compactness (spatial translation) estimate
+(2.5)
+E sup
+|z|<δ
+� T
+0
+�
+Rd |un(t, x + z) − un(t, x)| χ(x) dx dt ≤ Cρx(δ),
+δ > 0,
+where the constant C = C(χ, T ) is independent of n.
+Proof. For |z| > 0 and δ > 0,
+� T
+0
+�
+Rd |un(t, x + z) − un(t, x)| χ(x) dx dt
+≤
+� T
+0
+�
+Rd |Jδ ⋆ un(t, x + z) − Jδ ⋆ un(t, x)| χ(x) dx dt
++ 2
+� T
+0
+�
+Rd |Jδ ⋆ un(t, x) − un(t, x)| χ(x) dx dt =: A(z) + B,
+(2.6)
+where, by assumption,
+B ≤ 2
+� T
+0
+�
+Rd
+�
+Rd Jδ(x − y) |un(x) − un(y)| χ(x) dx dy dt
+= 2−d+1 E
+� T
+0
+�
+Rd
+�
+Rd J δ
+2 (z) |un(˜x + z) − un(˜x − z)| χ(˜x + z) d˜x dz dt
+(1.3)
+≲ E
+� T
+0
+�
+Rd
+�
+Rd J δ
+2 (z) |un(˜x + z) − un(˜x − z)| χ(˜x) d˜x dz dt
+(2.4)
+≲ ρx(δ/2),
+
+6
+KARLSEN
+so B ≲ ρx(δ). For any δ > 0, we introduce the random variable
+σu(δ) := sup
+|z|<δ
+� T
+0
+�
+Rd |un(t, x + z) − un(t, x)| χ(x) dx dt,
+which is a sub-additive/increasing modulus of continuity (for each fixed ω ∈ Ω).
+We obtain from (2.6) that
+(2.7)
+E σu(δ) ≲ E sup
+|z|<δ
+A(z) + ρx(δ).
+Note that
+Jδ ⋆ un(t, x + z) − Jδ ⋆ un(t, x)
+=
+�
+Rd κz,δ(y)un(t, x − y) dy =
+�
+Rd κz,δ(y)
+�
+un(t, x − y) − un(t, x)
+�
+dy,
+with κz,δ(y) := Jδ(y + z) − Jδ(y) satisfying �
+Rd κz,δ(y) dy = 0 and supp (κz,δ) ⊂
+B(0, |z| + δ). For |z| ≤ rδ, with r ∈ (0, 1) to be fixed later,
+A(z) ≤
+� T
+0
+�
+Rd
+�
+Rd |κz,δ(y)| |un(t, x) − un(t, x − y)| dyχ(x) dx
+≤
+�
+Rd |κz,δ(y)| dy σu
+�
+|z| + δ
+�
+≤ C1 |z| ∥∇Jδ∥L∞(Rd) |B(0, |z| + δ)| σu
+�
+|z| + δ
+�
+≤ C2
+|z| (|z| + δ)d
+δd+1
+σu(|z| + δ) ≤ C3rσu(rδ + δ) ≤ C4rσu(rδ),
+using also the sub-additivity of σu(·).
+Consequently, going back to (2.7), E σu(rδ) ≤ C5r E σu(rδ) + C6ρx(rδ). Fixing
+r ∈ (0, 1) such that C5r = 1
+2, we arrive at E σu(rδ) ≤ 1
+2 E σu(rδ) + C6ρx(δ) and
+thus E σu(δ) ≤ C7ρx(δ). This concludes the proof of (2.5).
+□
+One can use the spatial estimate (2.4) to derive a quantitative compactness
+estimate in time. To do that we need a version of a celebrated interpolation result
+due to Kruˇzkov [11], which trades spatial regularity, here quantified in terms of
+(2.4), for temporal L1(χdx) continuity.
+Proposition 2.6 (quantitative compactness estimate in time). Fix m ∈ N and
+a weight χ ∈ W ∩ W m,∞(Rd) such that |Dαχ(x)| ≲ χ(x) for any multi-index α
+with |α| ≤ m. Let {Jδ}δ>0 be an approximate identity such that the support of Jδ
+is bounded independently of δ > 0 and ∥DαJδ∥L1(Rd) ≲ δ−|α|, for |α| ≤ m, which
+includes, e.g., a standard Friedrichs mollifier.
+Consider a sequence {un}n∈N of
+
+STOCHASTIC CONSERVATION LAWS
+7
+functions on Ω × (0, T ) × Rd, with T > 0 fixed, satisfying
+E
+� T
+0
+�
+Rd |un(t, x)| χ(x) dx dt ≲ 1,
+(2.8)
+dun =
+�
+|α|≤mF
+DαF (α)
+n
+dt +
+�
+|α|≤mG
+DαG(α)
+n
+dW
+in D′(Rd), a.s.,
+(2.9)
+E
+� T
+0
+�
+Rd
+���F (α)
+n
+(t, x)
+��� χ(x) dx dt ≲ 1,
+∀ |α| ≤ mF , mF ≤ m.
+(2.10)
+E
+� T
+0
+�
+Rd
+���G(α)
+n (t, x)
+���
+2
+χ(x) dx dt ≲ 1,
+∀ |α| ≤ mG, mG ≤ m.
+(2.11)
+Suppose the spatial compactness condition (2.4) holds. Then, for any δ ∈ (0, T ),
+(2.12)
+E sup
+τ∈(0,δ)
+� T −δ
+0
+�
+Rd |un(t + τ, x) − un(t, x)| χ(x) dx dt ≤ ρt(δ),
+where ρt : [0, ∞) → [0, ∞) is an increasing function that is continuous at 0 with
+ρt(0) = 0 (ρt is independent of n but may depend on T, χ), see also (2.18).
+Proof. Recall that un satisfies the L1 bound (2.8). For ν > 0, set
+(2.13)
+vn,ν(t, x) =
+�
+Rd
+1
+2d Jν
+�x − y
+2
+�
+sign
+�
+dn(t, y)
+�
+dy,
+where dn(t, x) := un(t + τ, x) − un(t, x), for τ ∈ (0, δ), δ ∈ (0, T ). We have
+(2.14)
+∥Dαvn,ν(t, ·)∥L∞(Rd) ≲ 1/ν|α|,
+|α| ≤ m,
+where the right-hand side is independent of n, t. In other words, we have a.s. that
+vn,ν ∈ L∞(0, T ; W m,∞(Rd)).
+Note that, by
+��|a| − a sgn(b)
+�� ≤ 2 |a − b| ∀a, b ∈ R,
+��|dn(t, x)| − dn(t, x)vn,ν(t, x)
+�� ≤
+1
+2d−1
+�
+Rd Jν
+�x − y
+2
+�
+|dn(t, x) − dn(t, y)| dy,
+As a result,
+� T −τ
+0
+�
+Rd
+��|dn(t, x)| − dn(t, x)vn,ν(t, x)
+��χ(x) dx dt
+≤
+1
+2d−1
+� T −τ
+0
+�
+Rd
+�
+Rd Jν
+�x − y
+2
+�
+|dn(t, x) − dn(t, y)| χ(x) dx dy dt
+= 2
+� T −τ
+0
+�
+Rd
+�
+Rd Jν(z) |dn(t, ˜x + z) − dn(t, ˜x − z)| χ(˜x + z) d˜x dz dt.
+Utilising (1.3) (and the bounded support of Jν), Jν(z)χ(˜x+z) ≲ Jν(z)χ(˜x). Hence
+(2.15)
+E
+� T −τ
+0
+�
+Rd
+��|dn(t, x)| − dn(t, x)vn,ν(t, x)
+��χ(x) dx dt
+(2.4)
+≲ ρx(ν).
+
+8
+KARLSEN
+The weak form of the SPDE (2.9) implies
+����
+�
+Rd dn(t, x)v(t, x) χ(x) dx
+���� ≤
+�
+|α|≤mF
+����
+� t+τ
+t
+�
+Rd F (α)
+n
+(s, x) · Dα�
+v(t, x)χ(x)
+�
+dx ds
+����
++
+�
+|α|≤mG
+����
+�
+Rd
+�� t+τ
+t
+G(α)
+n (s, x) dW(s)
+�
+Dα�
+v(t, x) χ(x)
+�
+dx
+���� ,
+(2.16)
+for all v ∈ L∞(0, T ; W m,∞(Rd)). Combining (2.16) and (2.15) yields
+I := E
+� T −δ
+0
+sup
+τ∈(0,δ)
+�
+Rd |dn(t, x)| χ(x) dx dt ≤ C1ρx(ν) + A(δ, ν) + B(δ, ν),
+where
+A(δ, ν) =
+�
+|α|≤mF
+� T −δ
+0
+E
+� t+δ
+t
+�
+Rd
+���F (α)
+n
+(s, x)
+���
+��Dα�
+vn,ν(t, x)χ(x)
+��� dx ds dt,
+B(δ, ν) =
+�
+|α|≤mG
+� T −δ
+0
+E sup
+τ∈(0,δ)
+���M (α)(τ; t)
+��� dt,
+M (α)(τ; t) =
+�
+Rd
+�� t+τ
+t
+G(α)
+n (s, x) dW(s)
+�
+Dα�
+vn,ν(t, x) χ(x)
+�
+dx,
+and vn,ν is defined in (2.13). By assumption, the weight function χ belongs to
+W m,∞ and satisfies |Dαχ(x)| ≲ χ(x) for |α| ≤ m. Thus, by (2.14),
+(2.17)
+��Dα�
+vn,ν(t, x)χ(x)
+��� ≲ χ(x)
+ν|α| ,
+and so
+I ≤ C1ρx(ν) +
+δ
+νmF
+�
+|α|≤mF
+E
+� T
+0
+�
+Rd
+���F (α)
+n
+(t, x)
+��� χ(x) dx dt + B(δ, ν)
+(2.10)
+≤
+C1ρx(ν) + C2
+δ
+νmF + B(δ, ν),
+for some (n, δ, ν)-independent constants C1, C2.
+Next, using first (2.17) and then the Cauchy–Schwarz inequality, it follows that
+��M (α)(τ; t)
+�� ≲χ
+1
+νmG
+���
+� t+τ
+t
+G(α)
+n (s) dW(s)
+���
+L2(χdx). By the (Hilbert-space valued)
+BDG inequality [4, page 174], and again the Cauchy–Schwarz inequality,
+B(δ, ν) ≲χ,T
+1
+νmG
+�
+|α|≤mG
+�
+E
+� T
+0
+� t+δ
+t
+�
+Rd
+���G(α)
+n (s, x)
+���
+2
+χ(x) dx ds dt
+�1/2 (2.11)
+≲
+δ1/2
+νmG ,
+so that I ≤ C1ρx(ν) + C2
+δ
+νmF + C3 δ1/2
+νmG . The claim (2.12) follows by setting
+(2.18)
+ρt(δ) = inf
+ν>0
+�
+C1ρx(ν) + C2
+δ
+νmF + C3
+δ1/2
+νmG
+�
+.
+□
+
+STOCHASTIC CONSERVATION LAWS
+9
+3. Quantitative compensated compactness
+Consider the viscosity approximation un of (1.1) with d = 1, R ≡ 0, and f =
+f(u).
+According to [6], see also [9] and [7, Remark 5.9], there exists a unique
+solution uε—continuous in t and smooth in x—of the SPDEs
+duε + ∂xf(uε) dt = ε∂2
+xxuε dt + σ(x, uε) dW,
+(3.1)
+dη(uε) + ∂xq(uε) dt = ε∂2
+xxη(uε) dt − µε
+(3.2)
++ 1
+2η′′(uε)σ2(x, uε) dt + η′(uε)σ(x, uε) dW,
+where η ∈ C2(R), q′ = η′f ′, and µε := η′′(uε)ε |∂xuε|2. Moreover, ∀p, r ∈ [2, ∞)
+and for any weight χ ∈ W,
+E ∥uε∥r
+L∞(0,T ;Lp(χdx)) ≤ Cχ,
+E
+�����
+� T
+0
+�
+R
+ε |∂xuε|2 χ(x) dx dt
+�����
+r
+≤ Cχ,
+and
+E
+�����
+� T
+0
+�
+R
+χ d |µε|
+�����
+r
+≤ Cχ.
+(3.3)
+The entropy balance (3.2) follows from the viscous SPDE (3.1) and the spatial and
+temporal (Itˆo) chain rules.
+The flux f and entropy η are assumed to satisfy the nonlinearity assumptions
+of the next lemma, which is a global version of [8, Lemma 5.2] that will be utilised
+later. We refer to [8] for the proof.
+Lemma 3.1. Suppose f, η are C1 functions such that
+f ′(v) − f ′(w) ≥ Cf (v − w)pf ,
+−∞ ≤ w < v ≤ ∞, pf ≥ 1,
+(3.4)
+η′(v) − η′(w) ≥ Cη (v − w)pη ,
+−∞ ≤ w < v ≤ ∞, pη ≥ 1,
+(3.5)
+for some constants Cf, Cη > 0. Then, for all v, w ∈ R,
+(w − v)
+�
+q(w) − q(v)
+�
+−
+�
+η(w) − η(v)
+��
+f(w) − f(v)
+�
+≥ Cf,η |w − v|pf +pη+2 ,
+where Cf,η =
+CfCη
+(1+pf +pη)(2+pf +pη).
+Remark 3.2. Suppose f ∈ W 2,∞(R) is such that |f ′′(ξ)| ≥ c > 0 for a.e. ξ ∈ R.
+This is a “quantitive” version of the classical nonlinearity condition f ′′(ξ) ̸= 0
+for a.e. ξ ∈ R [12, 14]. Then, choosing η = f as an entropy flux, it follows that
+(w − v)
+�
+q(w) − q(v)
+�
+−
+�
+f(w) − f(v)
+�2 ≥ c2
+8 |w − v|4.
+Remark 3.3. Compared to [8, Lemma 5.2], the assumptions in Lemma 3.1 are
+global, since we do not know that uε is bounded in L∞
+ω,t,x.
+In what follows, we
+always assume η ∈ W 2,∞
+loc (R) is such that η, η′, η′′ are at most polynominally growing
+(including η = f): for some p0 ≥ 2,
+(3.6)
+|η(u)| ≲ 1 + |u|p0 ,
+|η′(u)| ≲ 1 + |u|p0−1 ,
+|η′′(u)| ≲ 1 + |u|p0−2 .
+To ensure the validity of the a priori estimates in (3.3), we assume that
+(3.7)
+|σ(x, u)| ≲ 1 + |u| .
+The main result is the following spatial compensated compactness estimate:
+
+10
+KARLSEN
+Theorem 3.4 (quantitative compactness estimate, viscosity approximation). Fix
+a weight χ ∈ W. Let un be a classical solution to the viscous SPDEs (3.1), (3.2)
+with ε = εn → 0 as n → ∞.
+Suppose (3.4), (3.5), (3.6), and (3.7) hold.
+Set
+µ := 1 − 1
+p, p ∈ [2, ∞). Then, for any z ∈ R with |z| < 1,
+(3.8)
+E
+� T
+0
+�
+R
+|un(t, x + z) − un(t, x)|pf +pη+2 χ(x) dx dt ≲χ,p |z|µ .
+Remark 3.5. By H¨older’s inequality, it follows from (3.8) that
+(3.9)
+E
+� T
+0
+�
+R
+|un(t, x + z) − un(t, x)| χ(x) dx dt ≤ C |z|µx ,
+where µx :=
+µ
+pf +pη+2 and C = C(T, χ, p). Note carefully that this estimate appears
+to be slightly weaker than the spatial compactness estimate (1.6) as there is a
+missing sup|z|<δ inside the expectation operator. It is not immediately clear how to
+recover this supremum; the available maximal (martingale) inequalities apply only
+to the temporal variable t and not an arbitrary parameter z.
+However, we can
+use Proposition 2.5 to recover the supremum by considering a standard Friedrichs
+mollifier {Jδ}δ>0. Observe that (3.9) implies
+E
+� T
+0
+�
+R
+�
+R
+Jδ(z) |un(t, x + z) − un(t, x − z)| χ(x) dz dx dt
+≤ 2C
+�
+R
+Jδ(z) |z|µx dz ≤ 2Cδµx.
+(3.10)
+Given (3.9), Proposition 2.5—via the estimate (3.10)—supplies the spatial estimate
+(1.6) with sup|z|<δ inside the expectation operator.
+Besides, by Proposition 2.6,
+(3.10) also implies the temporal compactness estimate (1.7).
+It is worth noting that these spatial and temporal estimates do not require the
+measure µε in the entropy balance equation (3.2) to be positive. This means that
+they can be used to analyze stochastic conservation laws with discontinuous (BV )
+flux, which will be discussed further in future work.
+Remark 3.6. Under the assumption of Remark 3.2, we have pf + pη + 2 = 4; thus
+the L1
+x compactness estimate (1.6) holds with ρx(δ) = δµx and µx := 1
+4 −
+1
+4p, for
+any p ∈ [2, ∞). If ∥uε∥L∞
+ω,t,x ≲ 1, then we can improve this to µx = 1/4, which is
+consistent with [8].
+3.1. Stochastic interaction lemma. Consider the two Itˆo SPDEs
+dA + ∂xB dt = CA dt + σA dW,
+dD + ∂xE dt = CD dt + σD dW,
+(3.11)
+which hold weakly in x, almost surely. Here A = A(ω, t, x), D = D(ω, t, x) belong to
+C([0, T ]; L1(Rd)) a.s., satisfy E
+���
+A, D
+�
+(t)
+��2
+L1(R) < ∞ ∀t ∈ [0, T ], and A, D → 0 as
+|x| → 0, for each fixed (ω, t). Besides,
+�
+B, CA, E, CD
+�
+=
+�
+B, CA, E, CD
+�
+(ω, t, x) sat-
+isfy E
+� T
+0
+���
+B, CA, E, CD
+�
+(t)
+��
+L1(R) dx dt < ∞. The noise amplitudes
+�
+σA, σD
+�
+=
+�
+σA, σD
+�
+(ω, t, x) satisfy E
+� T
+0
+���
+σA(t), σD(t)
+���2
+Lp(R) dt < ∞ for p = 1, 2.
+The processes A, D, σA, σD and W live on the stochastic basis introduced in
+Section 1, and they are assumed to be appropriately measurable with respect to
+the filtration, so all relevant stochastic integrals are well-defined in the sense of Itˆo.
+
+STOCHASTIC CONSERVATION LAWS
+11
+In what follows, we need the spatial anti-derivatives of A and D,
+A(t, y) :=
+� y
+−∞
+A(t, y) dx,
+D(t, x) :=
+� ∞
+x
+D(t, y) dy,
+as well the spatial anti-derivatives of σA and σD,
+ΣA(t, y) :=
+� y
+−∞
+σA(t, x) dx,
+ΣD(t, x) :=
+� ∞
+x
+σD(t, y) dy.
+The following lemma is a stochastic adaptation of a result from Golse and
+Perthame [8, Section 2].
+Lemma 3.7 (stochastic interaction identity). For t ∈ [0, T ], define
+I(t) =
+��
+x)3 >
+< (TOT(x) · (x− < x >)2)3/2 >,
+(2)
+skewness y =
+< TOT(y) · (y− < y >)3 >
+< (TOT(y) · (y− < y >)2)3/2 >.
+(3)
+Here TOT(x) is the TOT observed on strip x, and <> represents the means value. The
+ability to determine the head-tail, called the head-tail power Pht, is defined as
+Pht = Ntrue
+N
+,
+(4)
+where N is the total number of events, and Ntrue is the number of events that were correctly
+determined by the skewness. Determinations of Ntrue are discussed in the followings.
+In our previous work, we selected events with small θele and large skewness to increase the
+head-tail power at a cost of lowering the selection efficiency to less than one half [4]. The
+analysis was updated so that the the selection efficiency was recovered while the Pht was
+retained; the use of skewness x and skewness y were determined according to the azimuth
+direction of the tracks. For the tracks along the X-coordinate direction (0 ◦ ≤ |φazi| < 45 ◦),
+skewness x was used, and skewness y was used for the tracks with 45◦ ≤ |φazi| < 90◦). In
+addition, number of hit strips were increased by the operation at a high gas gains.
+The raw values of skewness were found to be correlated with θele as shown in the upper
+panels of Figure 8. The skewness were corrected according to sin θele with cubic functions
+and the corrected skewness values shown in the lower panels of Figure 8 were used for further
+discussions.
+8/21
+
+13.32
+13.28
+13.24
+13.20
+13.16
+13.12
+13.08
+13.04
+13.00
+12.96
+12.92
+12.88
+12.84
+12.80
+12.76
+12.72
+12.68
+12.64
+12.60
+12.56
+x (cm)
+0
+2
+4
+6
+8
+10
+12
+14
+TOT(x)
+skewness x = 0.0071
+5.72
+5.68
+5.64
+5.60
+5.56
+5.52
+5.48
+5.44
+5.40
+5.36
+5.32
+5.28
+5.24
+5.20
+5.16
+5.12
+5.08
+5.04
+5.00
+4.96
+4.92
+4.88
+y (cm)
+0
+2
+4
+6
+8
+10
+12
+14
+TOT(y)
+skewness y = -0.0021
+Fig. 7: TOT values of an event along each X (left panel) and Y (right panel) strip.
+0.2
+0.0
+0.2
+skewness y (raw)
+1.0
+0.5
+0.0
+0.5
+1.0
+sin
+ele
+(
+90
+azi <
+45 )
+0.2
+0.0
+0.2
+skewness x (raw)
+1.0
+0.5
+0.0
+0.5
+1.0
+sin
+ele
+(
+45
+azi < 45 )
+0.2
+0.0
+0.2
+skewness y (raw)
+1.0
+0.5
+0.0
+0.5
+1.0
+sin
+ele
+(45
+azi < 90 )
+0.2
+0.0
+0.2
+skewness y
+1.0
+0.5
+0.0
+0.5
+1.0
+sin
+ele
+Corrected(
+90
+azi <
+45 )
+0.2
+0.0
+0.2
+skewness x
+1.0
+0.5
+0.0
+0.5
+1.0
+sin
+ele
+Corrected (
+45
+azi < 45 )
+0.2
+0.0
+0.2
+skewness y
+1.0
+0.5
+0.0
+0.5
+1.0
+sin
+ele
+Corrected(45
+azi < 90 )
+Fig. 8: Correlation between skewnesses and sin θele. The distributios before and after the
+correction are shown in the upper and lower figures, respectively.
+Figures. 9 and 10 show skewness distributions of a 252Cf source data after all cuts for
+three energy ranges. Neutron irradiation data from +X and −X directions are shown with
+red and blue histograms in the upper panels of Figure 9. They show different skewness x
+distributions as expected while the skewness x distributions for the ±Y direction irradiation
+data (lower panels of Figure 9) did not show significant difference. Same trend was confirmed
+for skewness y as shown in Figure 10. Ntrue was defined by discriminating at skewness = 0
+and Pht values were calculated. Averaged Pht values for 50–100 keV, 100–200 keV, and 200–
+400 keV energy ranges were (52.4 ± 1.1)%, (52.9 ± 1.4)%, and (53.6 ± 2.0)%, respectively.
+9/21
+
+Details of Pht are summarized in Table 1. The error of Pht in each irradiation direction is
+the standard deviation of head-tail power determined for each period. The overall head-tail
+power error is the standard deviation of the Phts in each irradiation direction. Head-tail
+powers equivalent to those of Ref. [4] were achieved without any specific selection for the
+head-tail determination.
+0.2
+0.0
+0.2
+skewness x
+0.00
+0.05
+0.10
+0.15
+0.20
+Normalized counts
+50-100 keV
+50-100 keV
+0.2
+0.0
+0.2
+skewness x
+100-200 keV
+100-200 keV
+252Cf (25.5,0,0) cm
+252Cf (-25.5,0,0) cm
+0.2
+0.0
+0.2
+skewness x
+200-400 keV
+200-400 keV
+0.2
+0.0
+0.2
+skewness x
+0.00
+0.05
+0.10
+0.15
+0.20
+Normalized counts
+50-100 keV
+50-100 keV
+0.2
+0.0
+0.2
+skewness x
+100-200 keV
+100-200 keV
+252Cf (0,25.5,0) cm
+252Cf (0,-25.5,0) cm
+0.2
+0.0
+0.2
+skewness x
+200-400 keV
+200-400 keV
+Fig. 9: Distribution of skewness x at each energy. Events are normalized to unity.
+Energy range
+Pht (+x) (%)
+Pht (-x) (%)
+Pht (+y) (%)
+Pht (-y) (%)
+Pht (average) (%)
+50–100 keV
+52.2±0.9
+53.3 ±1.2
+52.2 ±1.1
+51.9 ±0.9
+52.4 ±1.1
+100–200 keV
+52.6 ±1.4
+53.2 ±1.2
+53.5 ±1.2
+52.5 ±1.0
+52.9 ±1.2
+200–400 keV
+53.3 ±1.6
+52.4 ±1.0
+54.9 ±2.8
+53.8 ±1.6
+53.6 ±2.0
+Table 1: Head-tail powers in unit of % for each direction and energy range.
+2.6.
+Efficiencies
+There are two types of efficiencies regarding this study; the detection-selection and the
+directional efficiencies. The former, or the “absolute” efficiency, determines the number of
+detected-and-selected events while the latter, or the “relative” one, determines the directional
+distribution of these events without changing the total number. A data-set of recoil events
+isotropic in terms of the position and the direction was used to measure the efficiencies.
+10/21
+
+0.2
+0.0
+0.2
+skewness y
+0.00
+0.05
+0.10
+0.15
+0.20
+Normalized counts
+50-100 keV
+50-100 keV
+0.2
+0.0
+0.2
+skewness y
+100-200 keV
+100-200 keV
+252Cf (25.5,0,0) cm
+252Cf (-25.5,0,0) cm
+0.2
+0.0
+0.2
+skewness y
+200-400 keV
+200-400 keV
+0.2
+0.0
+0.2
+skewness y
+0.00
+0.05
+0.10
+0.15
+0.20
+Normalized counts
+50-100 keV
+50-100 keV
+0.2
+0.0
+0.2
+skewness y
+100-200 keV
+100-200 keV
+252Cf (0,25.5,0) cm
+252Cf (0,-25.5,0) cm
+0.2
+0.0
+0.2
+skewness y
+200-400 keV
+200-400 keV
+Fig. 10: Distribution of skewnes y at each energy. Events are normalized to unity.
+The isotropic data-set was made by summing-up the time-normalized data obtained by
+irradiating the detector with neutrons from a 252Cf source placed at six positions in ±X,
+±Y , and ±Z directions.
+The detection-selection efficiency is defined as the number of nuclear recoil events after all
+selections divided by the expected number of nuclear recoils in the fiducial volume. Here,
+the expected number of nuclear recoils is estimated by the Geant4 simulation. Results are
+shown in Figure 11. It should be noted that the increase of the detection efficiency seen
+below 100 keV is due to the contamination of the gamma-ray events and is not real. The
+contamination is removed with the selections to a negligible level. The detection efficiency
+is about 60% above 200 keV. The main reason of not reaching at 100% is that the gas
+gain being not high enough to trigger all the nuclear recoil events. The detection-selection
+efficiency above 200 keV is half of the detection efficiency because of the mean value for
+the TOTsum-Length selection. A 20%-reduction of the detection-selection efficiency from
+NEWAGE2021 should also attribute to the additional cut, which still gives a large advantage
+in the signal-to-noise ratio if we consider the gain on the rejection shown in Figure 6. The
+detection-selection efficiency shown in Figure 11, or the ”absolute” efficiency, can be used
+to calculate the expected number of events for a given WIMP or background model. It can
+also be used to unfold the measured energy spectrum and obtain an ”effective” spectrum
+for the comparison of the background rates.
+11/21
+
+The directional efficiency is expressed as a sky map, or the relative response in the elevation
+(θele) - azimuth (φazi) plane, for isotropic recoils. The possible non-homogeneity of the direc-
+tional efficiency mainly originates from the reconstruction algorithm. The 3D recoil direction,
+including the sense (head-tail) of the track, is reconstructed from the TOT-distributions of
+X and Y strips. Figure 12 shows the obtained θele-φazi distributions of an isotropic recoil
+calibration data. Since this map is to know the ”relative” or reconstruction efficiency of
+the directions, the color map is a relative one to be used with the total number of events
+being conserved. It is seen that the tracks tend to be reconstructed to align with the strips,
+i.e. φazi = 0◦, ±90◦, 180◦ for the tracks parallel to the detection plane, or the tracks with
+θele ∼ 0. The directional efficiencies shown in Figure 12, or the relative efficiency, can be used
+to make an expected recoil distribution for a given number of expected events calculated by
+the detection-selection efficiency.
+50
+100
+150
+200
+250
+300
+350
+400
+Energy (keV)
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+Efficiency
+THIS WORK (detection)
+THIS WORK (detection+selection)
+NEWAGE2021 (detection+selection)
+Fig. 11: Nuclear recoil efficiencies as a function of the energy. The cyan and the blue his-
+tograms are the detection and detection-selection efficiencies of nuclear recoil of this study,
+respectively. The gray histograms is the result of NEWAGE2021 [6].
+2.7.
+Angular resolution
+The angular resolution was evaluated by comparing the distribution of the recoil angle γ of
+neutron irradiation data with the simulated ones smeared by various angular resolutions.
+Here γ is the angle between the incoming neutron direction and the reconstructed nuclear-
+recoil direction. Since the head-tails of the tracks are determined and considered in the
+analysis independent from the effetct of the angular resolution, the angular resolution was
+evaluated with the distribution of absolute value of cos γ. χ2
+ang value defined by Eq. (5) was
+calculated for a given angular resolution σang.
+χ2
+ang =
+Nbin
+�
+i
+(Ndata
+i
+− NMC
+i
+(σang))2
+Ndata
+i
+,
+(5)
+12/21
+
+-150°-120° -90° -60° -30°
+0°
+30°
+60°
+90° 120° 150°
+-75°
+-60°
+-45°
+-30°
+-15°
+0°
+15°
+30°
+45°
+60°
+75°
+50-100 keV
+5
+10
+15
+20
+25
+Direction efficiency (%)
+Fig. 12: Directional efficiency in the detector coordinate system.
+where Ndata
+i
+is the number of events in the i-th bin of the histogram of measured | cos γ|,
+and NMC
+i
+is the number of events in the i-th bin of the histogram of the | cos γ| distribution
+simulated by Geant4 smeared with the angular resolution, and Nbin is the number of bins
+in that histogram. The angular resolution at the minimum χ2
+ang value was adopted. The
+angular resolution was 58.1+5.8
+−2.8 degree in the energy range of 50–100 keV.
+3.
+Experiment
+A direction-sensitive dark matter search was performed in Laboratory B, Kamioka Observa-
+tory (36.25’N, 137.18’E), located 2700 m water equivalent underground. The measurement
+was carried out from December 12th, 2017 to March 26th, 2020, subdivided into eight peri-
+ods. The period was renewed when the detector was evacuated and filled with new CF4
+gas. The period information is summarized in Table 2. The Z-axis of the NEWAGE-0.3b”
+detector was aligned to the direction of S30◦E. The target gas was CF4 at 76 Torr (0.1 atm)
+with a mass of 10 g in an effective volume of 28 × 24 × 41 cm3 (27.6 L). The total live time
+is 318 days corresponding to an exposure of 3.18 kg·days.
+Various environmental parameters were monitored during the measurement to confirm the
+stability of the detector. Figure 13 shows the time dependences of the integrated exposure,
+the gas gain and the energy resolution. The energy calibrations and the efficiency measure-
+ments were performed approximately every two weeks. The energy scale was corrected by
+the monitored gas gain. The mean value of the energy resolution was 12.4% with a standard
+deviation of 3.0% during the measurement. No variation of the energy resolution beyond
+errors was observed.
+The event selections described in subsection 2.3 and 2.4 were applied to the data. Figure 14
+shows the energy spectrum after each event selection. The statistic errors are shown for
+the spectrum after all selections. For a comparison with NEWAGE2021 result, an energy
+spectrum divided by the detection-selection efficiency is shown in Figure 15 as ”This work”
+13/21
+
+Period
+Date
+Gas gain
+Live time (days)
+Exposure (kg·days)
+RUN20-1
+2017/12/12 – 2018/01/18
+2000
+13.5
+0.135
+RUN20-2
+2018/01/23 – 2018/02/23
+1750
+20.0
+0.200
+RUN21
+2018/02/28 – 2018/06/01
+1550
+58.6
+0.586
+RUN22-1
+2018/06/06 – 2018/08/24
+1110
+52.5
+0.525
+RUN22-2
+2018/09/20 – 2018/11/29
+1200
+60.5
+0.605
+RUN23
+2018/12/05 – 2019/04/12
+1750
+45.9
+0.459
+RUN24
+2019/04/26 – 2019/06/27
+1800
+49.4
+0.494
+RUN25
+2020/03/04 – 2020/03/26
+1950
+17.6
+0.176
+Total
+2017/12/12 – 2020/03/26
+318.0
+3.180
+Table 2: Summary of the measurement periods with gas gains (at the start of each RUN), live
+times, and exposures. RUN22-1 and RUN22-2 are the data analyzed in NEWAGE2021 [6].
+0
+2
+Exporsure (kg days)
+0
+2000
+Gas gain
+0
+100
+200
+300
+400
+500
+600
+700
+800
+Day from December 12th, 2017
+0
+20
+Energy resolution (%)
+Fig. 13: Cumulative exposure, gas gains, and energy resolutions during the measurement.
+RUN20–25. The rate of this work is comparable to the that of NEWAGE2021. This is
+reasonable because there is no change in terms of the hardware-level radioactive background.
+We have achieved the same count rate as that of NEWAGE2021 The energy spectrum of
+this work has smaller statistical errors due to the increase of the statistics by a factor of 2.4.
+Figure 16 shows the directions of measured nuclear recoil events in the detector coordinate
+(a) and the galactic coordinate (b), respectively. The cos θCYGNUS was calculated for each
+event in Figure 16 (b) and distributions are shown in Figure 17. The cos θCYGNUS is binned
+by four and the energy is binned every 10 keV.
+14/21
+
+100
+200
+300
+400
+Energy (keV)
+10
+1
+101
+103
+105
+107
+Counts
+No cut
+Fiducial cut
+Length-Energy cut
+TOTsum/Energy cut
+TOTsum-Length cut
+Expected Rn BG
+Expected gamma-ray BG
+BG upper error (1 )
+Roundness cut
+Fig. 14: Energy spectra after each selection step. The grey, orange, blue, magenta, and
+green lines are the energy spectra after no cut, Fiducial volume cut, Length-Energy cut,
+TOTsum/Energy cut, and TOTsum-Length cut, respectively. The black dots with error bars
+are the final data sample after the Roundness cut. The fill stacked green and red spectra are
+the expected gamma-ray and radon background ones estimated by the simulation. The gray
+shaded area is a 1σ error in the background.
+4.
+Results
+A directional WIMP search analysis was performed with an assumption of the standard
+halo model. Here the Maxwell distribution with a velocity dispersion of 220 km/sec, and an
+escape velocity of 650 km/sec were assumed [12]. The local density of 0.3 GeV/c2/cm3 was
+assumed. The spin parameter λ2J(J + 1) for the 19F of 0.647 was used in this analysis [13].
+The spectra of cos θCYGNUS for each energy bin as shown in Figure 17 were simultaneously
+compared with sum distributions of WIMP signal and isotropic background using the binned
+likelihood ratio method.
+A statistic value χ2 was defined as Eq. (6).
+χ2 = 2
+n
+�
+i=0
+m
+�
+j=0
+�
+(NMC
+i,j
+− Ndata
+i,j
+) + Ndata
+i,j
+ln
+�Ndata
+i,j
+NMC
+i,j
+��
++ α2
+E + α2
+BG,
+(6)
+where,
+NMC
+i,j
+= NDM
+i,j (σχ−p, mχ, ξE) + NBG
+i,j (ξE, ξBG),
+(7)
+αE = ξE
+σE
+,
+(8)
+αBG = ξBG
+σBG
+.
+(9)
+15/21
+
+0
+100
+200
+300
+400
+Energy (keV)
+10
+3
+10
+2
+10
+1
+Rate (counts/keV/kg/days)
+THIS WORK
+NEWAGE2021
+Fig. 15: Energy spectra divided by the detection-selection efficiency. Red histogram is the
+energy spectrum of this work. Black histogram is the energy spectrum of NEWAGE2021.
+Subscripts i and j are the bin-number of the cos θCYGNUS and the energy, respectively.
+The expected and measured number of events in bin i, j are described as NMC
+i,j
+and Ndata
+i,j
+,
+respectively. NMC
+i,j
+is written as Eq. 7, where NDM
+i,j
+is the expected number of the WIMP-
+nucleus scatterings, and NBG
+i,j
+is the expected number of background events. σχ−p is the
+WIMP-proton cross section. NBG
+i,j
+was estimated using the Geant4 simulation based on
+the flux measurements of the ambient gamma-rays, the ambient neutrons, the alpha rays
+from the radon, and the alpha rays from the LAµ-PIC surface. The dominant background
+components in the energy range of 50–100 keV were the ambient gamma-rays and the alpha
+rays from the radon (see Ref. [6] for details). Expected background spectra are shown in
+Figure 14 for reference. The largest systematic uncertainty of the expected rate arise from the
+energy scale uncertainty. This uncertainty was estimated from the discrepancy of the energy
+calibration between 10B, 220Rn, and 222Rn measurements discussed in subsection 2.2. The
+uncertainty was evaluated in each run. The weighted average of the energy scale uncertainty
+was +13.2% and -2.3%. The uncertainties of the background rate are the measurement errors
+of radioactivities for the ambient gamma-rays and the radons. Here the ambient gamma-
+ray flux was measured with a CsI scintillator[14] and the radon background was estimated
+with the high energy spectrum of this work. Nuisance parameters αE and αBG considering
+the systematic uncertainty from the energy scale σE and the background estimation σBG
+are defined as Equations (8) and (9). Possible shifts of the energy scale and the number of
+expected backgrounds are expressed as ξE and ξBG.
+χ2 was minimized for a given WIMP mass with σχ−p, pull-terms αE and αBG as fitting
+parameters. We first explain the procedure for the WIMP mass of 150 GeV/c2 case. A
+minimum χ2/NDF of 20.4/17 was obtained for σχ−p=14.6 pb. The left panel in Figure 17
+16/21
+
+(a) Nuclear-recoil directions in the detector coordinate
+(b) Nuclear-recoil directions in the galaxy coordinate
+Fig. 16: (a) Nuclear recoil directions of final data sample in the detector coordinate. The X-
+axis and Y-axis are φazi and θele in the detector coordinate system, respectively. (b) Nuclear
+recoil directions of final data sample in the galactic coordinate. The X-axis and Y-axis
+are the longitude and latitude of the galactic coordinate, respectively. The direction of the
+galactic center is (0,0) and that of Cygnus is (-90,0). The orange, red, pink, purple, and blue
+points indicate the energy ranges of 50–60 keV, 60–70 keV, 70–80 keV, 80–90 keV, and 90–
+100 keV, respectively. The color contours in the background are the directional efficiencies
+in each coordinate system.
+shows the cos θCYGNUS distributions of the best-fit case. A chi-square distribution was created
+from a dummy sample of isotropic background model using Monte Carlo simulations. This
+test gave the p-value of 60% for the measured result. Observed distribution was thus found
+to be consistent with the background-only model. Since no significant WIMP excess was
+17/21
+
+obtained, an upper limit at 90% confidence level (C.L.) was set for the spin-dependent
+WIMP-proton scattering cross section. The likelihood ratio L is defined as,
+L = exp
+�
+−χ2(σχ−p) − χ2
+min
+2
+�
+.
+(10)
+Here, χ2(σχ−p) and χ2
+min are the value of χ2 and the minimum value of χ2 calculated by
+varying σχ−p, respectively. The 90% C.L. upper limit of the WIMP-proton cross section,
+σlimit
+χ−p , is determined as follows,
+� σlimit
+χ−p
+0
+Ldσχ−p
+� ∞
+0 Ldσχ−p
+= 0.9.
+(11)
+Using the above equation, the 90% C.L. upper limit of the spin-dependent cross section was
+found to be 25.7 pb for a WIMP mass of 150 GeV/c2. The cos θCYGNUS distributions with
+the upper limit of 90% C.L. are shown in the right panels of Figure 17.
+Upper limits of the cross sections were obtained for other WIMP masses by the same
+procedure. Figure 18 shows the upper limits at 90% C.L. of the spin-dependent WIMP-
+proton cross sections as a function of the WIMP mass. Compared to the NEWAGE2020
+results, which was analyzed by the 3D-vector method using the standard µ-PIC, this upper
+limit updates by about one order of magnitude. This is due to the reduction of surface
+background events with the LAµ-PIC. Furthermore, compared to the NEWAGE2021 result,
+the statistics of the 2.4 factor and an updated analysis including the background estimation,
+improved the limits by a factor of about two for WIMPs heavier than 100 GeV/c2.
+5.
+Discussions
+A new limit by a directional dark matter search with a 3D-vector analysis was obtained
+by this work. Although we started to search the region of one of the interpretations of the
+DAMA/LIBRA’s annual modulation signal [18], a significant improvement of the sensitivity
+is needed for the search of the region of more interest. The improvements can be realized
+mainly in three aspects: the detection-selection efficiency, the energy threshold, and the
+backgrounds.
+The detection-selection efficiency at 50–60 keV is 12.5%, which indicates the statistics
+can be increased by a factor of eight at most for a same exposure by an improvement of
+the detection-selection efficiency. A measurement with a higher gas gain will increase the
+trigger efficiency. A better gamma-ray rejection analysis, e.g. introducing machine-learning
+methods, would compensate the expected increase of the gamma-ray background rate and
+allow us to operate the detector at a higher gas gain. Shielding the detector is an independent
+hardware approach to reduce the gamma-ray background events.
+The current energy threshold (50 keV) is mainly limited by the track length of the recoil
+events. Typical length of the track of fluorine nuclear recoil below 50 keV in CF4 gas at
+76 Torr (0.1 atm) is less than 1 mm. This is comparable to the strip pitch of 0.4 mm and one
+can deduce that the angular resolution and gamma-ray rejection both get worth below this
+point. One solution is to operate the CF4 gas at a lower pressure than 76 Torr to allow the
+nuclei and electrons run longer and improve the angular resolution and gamma-ray rejection
+below 50 keV.
+The remaining background sources are the ambient gamma-rays and internal radons as
+shown in Figure 14. We have already discussed the gamma-ray reduction above so we discuss
+18/21
+
+0
+5
+50-60 keV
+Best fit
+0
+5
+50-60 keV
+90 % C.L.
+0
+5
+60-70 keV
+0
+5
+60-70 keV
+0
+5
+Counts
+70-80 keV
+0
+5
+Counts
+70-80 keV
+0
+5
+80-90 keV
+0
+5
+80-90 keV
+1.0
+0.5
+0.0
+0.5
+1.0
+cos
+CYGNUS
+0
+5
+90-100 keV
+data
+NDM
+NBG
+1.0
+0.5
+0.0
+0.5
+1.0
+cos
+CYGNUS
+0
+5
+90-100 keV
+data
+NDM
+NBG
+Fig. 17: cos θCYGNUS distributions (identical black histograms in both panels) for the final
+date sample in the 50–100 keV energy ranges. The best fit and 90% upper limit distributions
+for the WIMP mass of 150 GeV/c2 are shown with color histograms in the left and right
+panels, respectively.
+the reduction of radon background here. The LAµ-PIC, significantly reduced the surface
+alpha rays in NEWAGE2021, still contains some material which emanates the radon gas [5].
+A new version of the µ-PIC series, LBGµ-PIC currently being developed. The material used
+for the LBGµ-PIC is carefully selected so that the total radon emanation is less than 1/10
+of the LAµ-PIC.
+With the improvements described above, we aim to explore the region claimed by
+DAMA/LIBRA [18] and to improve the sensitivity to reach limits by other direct search
+experiments.
+6.
+Conclusion
+A direction-sensitive direct dark matter search was carried out at Kamioka Observatory
+with a total live time of 318.0 days corresponding to an exposure of 3.18 kg·days. A new
+gamma-ray rejection cut, which improved the gamma-ray rejection power to 8.8 × 10−7
+while maintaining the detection-selection efficiency of the nuclear recoil at about 20% was
+introduced. This enabled us to use the high gas gain data, which was not used in the
+previous study due to the deterioration of the gamma-ray rejection power. The exposure
+was increased by a factor of 2.4. A 3D-vector reconstruction with a head-tail determination
+power of 52.4% in the energy range of 50–100 keV was also used for this study. As a result of
+the directional WIMP-search analysis, an upper limit of the spin-dependent WIMP-proton
+cross section of 25.7 pb for a WIMP mass of 150 GeV/c2 was derived. This limit marked the
+best direction-sensitive limit.
+19/21
+
+101
+102
+103
+WIMP mass (GeV/c2)
+100
+101
+102
+103
+104
+SD WIMP-proton
+p (pb)
+DMTPC 2012
+NEWAGE2015
+NEWAGE2020 3D-vector
+NEWAGE2021
+THIS WORK 3D-vector
+DRIFT 2017
+DAMA/LIBRA
+Fig. 18: 90% C.L. upper limits of the spin-dependent WIMP-proton scattering cross section
+as a function of the WIMP mass. The red line is the result of this work. The green line
+is the result of our previous work (NEWAGE2021 [6]) and the purple line is the result
+with the 3D-vector directional analysis for NEWAGE2020 [4]. The gray line is the result
+of NEWAGE2015 [15]. The solid light-blue shows the results from the directional analysis
+of DMTPC [16]. The blue line is the limit curve for DRIFT [17], which is a gas detector
+but non-directional analysis. The gray area is an interpretation of the allowed region of
+DAMA/LIBRA [18].
+Acknowledgment
+This work was partially supported by KAKENHI Grant-in-Aids (19H05806, 19684005,
+23684014, 26104005, and 21H04471).
+References
+[1] A. Arbey and F. Mahmoudi, Progress in Particle and Nuclear Physics, 119, 103865 (2021).
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+[3] D. N. Spergel, Phys. Rev. D, 37, 1353–1355 (Mar 1988).
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+[5] T. Hashimoto, K. Miuchi, et al., Nucl. Inst. Meth. A, 977, 164285 (2020).
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+page_content=' Atsushi Takada2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
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+page_content=' and Kentaro Miuchi1 1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Graduate School of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Kobe University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Rokkodai-cho,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
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+page_content=' Kobe-shi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Hyogo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
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+page_content=' Japan ∗E-mail: kasshiy21@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='com 2Division of Physics and Astronomy Graduate School of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Kyoto University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Kitashirakawaoiwake-cho,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
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+page_content=' Kyoto-shi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Kyoto,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
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+page_content=' Japan 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Tohoku University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Aramakiazaaoba 6-3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Aoba-ku,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Sendai-shi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Miyagi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
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+page_content=' Japan 4Kamioka Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Institute for Cosmic Ray Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' the University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Higashi-Mozumi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
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+page_content=' Japan 5Kavli Institute for the Physics and Mathematics of the Universe (WPI),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' the University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 5-1-5 Kashiwanoha,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
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+page_content=' Subject Index Dark matter, WIMP, µTPC, NEWAGE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Introduction Existence of the dark matter in the universe is nowadays widely believed because the dark matter naturally explains observational results in various scales of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Weakly Interactive Massive Particles (WIMPs), which are promising candidates of the dark matter, © The Author(s) 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Published by Oxford University Press on behalf of the Physical Society of Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='org/licenses/by-nc/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='04779v1 [hep-ex] 12 Jan 2023 have been searched for by a number of direct search experiments pursuing for the nuclear recoil by WIMPs [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' However, no conclusive evidence of the direct detection of WIMPs was obtained yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' There are two possible characteristic signatures for the direct detection of the dark mat- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' One is the annual modulation in the energy spectrum caused by the Earth’s motion around the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The modulation amplitude is expected to be a few percent [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The other is the directional non-isotropy of the nuclei recoils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Since the Solar System is orbiting in the Milkyway Galaxy, the incoming direction of the dark matter is biased to the direction of the Solar System’s motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The directional distribution of the nuclear recoil also has an asymmetry and this asymmetric ratio can be as large as tenfold in some cases [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Thus, the observation of the non-isotropic signal for the nuclear recoil direction distribution is expected to be a strong evidence for the dark matter detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' NEWAGE (NEw generation WIMP search with an Advanced Gaseous tracker Experiment) is a direction-sensitive direct WIMP search experiment using a low-pressure gaseous micro Time Projection Chamber (µ-TPC) for the detection of three-dimensional (3D) tracks of recoil nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' NEWAGE started direction-sensitive direct WIMP searches in an underground laboratory in 2007 and has updated the results since then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' In 2020, head-tail determinations of the nuclear tracks were implemented and a limit by a vector-like tracking analysis was obtained (NEWAGE2020 results [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' In 2021, the limit was updated by installing a low alpha ray emission rate detector called LAµ-PIC [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Here the limit was obtained without the vector-like analysis (NEWAGE2021 results) because of the limited statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' In this paper, we report a result of a direction-sensitive dark matter search with a new gamma-ray rejection cut and a vector analysis for 3D-tracks (3D-vector analysis) for a data 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='4 times larger than NEWAGE2021 results in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Detector A gaseous time projection chamber, NEWAGE-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='3b”, was used for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The detector overview is described in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Energy calibration using alpha rays are discussed in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Event selections already implemented in our previous analysis are summarized in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' An event selection newly-added for this work utilizing the track information for a better gamma-ray rejection is described in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The reconstruction method of the 3D-vector tracks is explained in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='5 as the head-tail analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Finally, the detector performances on the efficiencies and the angular resolution of the nuclear recoil are shown in subsections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' NEWAGE-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='3b” NEWAGE-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='3b”, refurbished in 2018 by replacing the readout device (micro pixel chamber, µ-PIC) with a low alpha-emission rate one (LAµ-PIC [5]), was used for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Figure 1 shows schematic drawings of the NEWAGE-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='3b” detector and its detection scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The detection volume was 31 × 31× 41 cm3 in size and was filled with low-pressure gas of CF4 at 76 Torr (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='1 atm) for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The LAµ-PIC has a pixel structure of 768 × 768 with a pitch of 400 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Amplified charge at each pixel is read through 768 anode (hereafter X- axis) and 768 cathode(hereafter Y-axis) strips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Signals read through the strips are processed by Amplifier-Shaper-Discriminator chips (SONY CXA3653Q [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The processed signals are 2/21 then divided into two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' One is compared with a threshold voltage in the chips and the time- over-thresholds (TOTs) of 768 + 768 strips are recorded with a 100 MHz frequency clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The other 768 cathode strips were grouped into four channels and their waveformes were recorded with a 100 MHz flash analog-to-digital converters (FADCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' A detected track is parameterized with its energy, length, elevation angle θele, azimuth angle Φazi (see Figure 1) and some other parameters defined in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' LAμ-PIC GEM Drift Plane 4 mm (GAP reagin) 10B glass plate (-5, -12, 0) cm CF4 gas 76 Torr Copper wire 400 μm Electric field 41 cm θele φazi Y X Field cage X Y Z 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='72 cm Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 1: Schematic drawings of the NEWAGE-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='3b” detector and its detection scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' A recoil nucleus shown with red markers passes through the gas volume and ionizes the gas molecules (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The ionized electrons are drifted toward the readout plane by the electric field, amplified by the GEM [8], and further amplified by the LAµ-PIC before being detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The image on the left is a magnified view of the LAµ-PIC with an electrode structure of a 400 µm pitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Energy calibration The energy calibration was performed with alpha rays produced by 10B(n, α)7Li reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' A glass plate coated with a 10B layer was set in the TPC volume as illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Thermal neutrons were irradiated from the outside of the chamber, captured in the 10B layer, and then produced continuous spectrum up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='5 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The obtained spectrum is a sum of the thermal neutron capture events and elastic scattering events by fast neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' By comparing these spectra with the simulation results by Geant4 [9], the gas gain and the energy resolution were determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Figure 2 shows one of the calibration results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='5 MeV edge of the thermal neutrons was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The detector gas contains rare gas radon isotopes, 220Rn and 222Rn, emitted from the detector materials as natural contaminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The high-energy calibration was performed by the alpha rays from radon isotopes and their progenies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 220Rn produces alpha rays with 3/21 energies of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='05 MeV, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='29 MeV, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='78 MeV, and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='79 MeV, 222Rn produces alpha rays with energies of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='49 MeV, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='00 MeV, and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='69 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Because the ratio of 220Rn to 222Rn were not known, the measured spectra were fit with the simulated spectrum of 220Rn and 222Rn separately and the difference was treated as the systematic error of the energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 500 1000 1500 2000 2500 Energy (keV) 0 50 100 150 200 250 Counts Gas gain = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='75×103 MC Measured Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 2: Energy spectrum of alpha rays from a 10B glass plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The black and blue histograms are the measured data and the simulated results, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Standard event selections Several event selections had been established as standard event selections by NEWAGE2021 analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' These selections aim to cut non-physical electronics noise events and electron track events mainly originating from ambient gamma-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The standard event selections are briefly explained here, while details can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Fiducial volume cut A fiducial volume of 28 × 24 × 41 cm3 was defined in the detection volume of 31 × 31 × 41 cm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Any events were required to be fully contained in the fiducial volume so as to discriminate the events from the walls of the TPC field cage and the 10B glass plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Length-Energy cut The amount of energy loss by a charged particle per a unit length depends on the particle type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Electron events were discriminated by setting a maximum track length for a given energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' TOTsum/Energy cut The energy deposition on each strip was recorded as TOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' A total TOTs of all strips were defined as TOTsum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Since the nuclear recoil events have larger TOTsum than those of the electron recoil events for a given energy, electron events were discriminated by setting a minimum TOTsum/energy value for a given energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' (See the left panel of Figure 3, for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=') 4/21 Roundness cut “Roundness” was defined as the root-mean-square deviation of a track from the best-fit straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Nuclear recoil events with a short drift distance have small roundnesses because they are less affected by the gas diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Background events in the gas region between the LAµ-PIC and the GEM were discriminated by setting a minimum roundness value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' TOTsum-Length cut The detector was operated at a higher gas gain (typically 1800) than that of NEWAGE2021 (1200) aiming for a better detection efficiency of nuclear recoil events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' One of the expected drawbacks of the high-gain operation was the increase of the background gamma-ray events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Figure 3 shows the TOTsum/Energy distributions as functions of the energy after the fiducial volume cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The gas gains of the left and right panels are 1200 and 1800, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' It should be noted that each calibration run with the source had been conducted at a common live time of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='18 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' It is therefore clearly seen that the detection efficiency of electron events (137Cs data) are significantly larger in a measurement at a high gas gain because the number of shown events are increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' It is also seen that the TOTsum/Energy of electron events in the high-gain data have a large component which excess the selection line of TOTsum/Energy selection shown with a red line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' This result indicated that the standard event selections were not sufficient for the high-gain operation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' A new cut, “TOTsum-Length cut”, was implemented in order to improve the discrimina- tion power against the gamma-ray events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Nuclear recoil events have large TOTsums and short track lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' On the other hand, the electron recoil events have smaller TOTsums and longer tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Figure 4 shows the track length distributions as a function of TOTsum for the irradiation with a 252Cf source and a 137Cs source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Here the gas gain is 1800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Since our energy threshold is set to be 50 keV, the data in an energy range of 50–60 keV are selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' We confirmed a good separation of the electron (seen in both plots) and nuclear distributions (seen only in the 252Cf plot) in this parameter space even for a high-gain operation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' In order to discriminate electron events, an empirical function written by L = (S/β)α, (1) was introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Here L is the track length, S is the TOTsum, and α and β are parameters for the cut definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Here α was fixed within a run while β was an energy-dependent parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' We first determined α and β values in the 50–60 keV energy range for each period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The period is a set of data taken under a same detector condition and will be summarized in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The parameters were determined so that they would give the best rejection of gamma-ray events while retaining the selection efficiency of nuclear recoil events to be greater than 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Here, the selection efficiency for a specific selection is defined as the ratio of the remaining number of events to that before the selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' We then fixed α and determined β for a given energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Figure 5 shows the energy dependence of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The black and blue dots represent the data with a 252Cf and a 137Cs sources, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The distribution of β values of the nuclear recoils events was fit with Gaussian in every 10 keV energy bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The region between the mean and upper 3σ of the Gaussian indicated with red lines in Figure 5 was set as the nuclear recoil region and the rest was rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Gamma-ray rejection powers with 5/21 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 3: TOTsum/Energy distributions as functions of energy (after the fiducial volume cut).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The left and right panels show the distributions corresponding to the gas gain of 1200 and 1800, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The black gradation distribution is obtained with a 252Cf neutron source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The blue point distribution is obtained with a 137Cs gamma-ray source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The red-dashed lines indicate the cut lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Each calibration run with the source had been conducted at a common live time of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='18 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 0 100 200 300 TOTsum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 Length (cm) 252Cf = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='3 = 250 0 100 200 300 TOTsum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 Length (cm) 137Cs = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='3 = 250 100 101 Counts 100 101 102 Counts Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 4: Distributions of track length as a function of TOTsum in the energy range of 50– 60 keV after the fiducial volume cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The left (black gradient) is the data with a 252Cf neutron source and the right (blue gradient) is the data with a 137Cs gamma-ray source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The red line in the figure is L = (S/β)α (α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='3 and β = 250).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Since the 252Cf source emits not only neutrons but also gamma-rays, the distribution has two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' and without this cut are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' A gamma-ray rejection power of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='8 × 10−7 was achieved, which is about two orders of magnitude better than that in NEWAGE2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 6/21 5 5 252 Cf 252Cf 102 102 137 CS 137 CS 4 4 Gas gain ～ 1200 Gas gain _ 1800 Counts Counts 101 101 2 100 100 0 200 300 400 100 300 400 100 200 Energy (keV) Energy (keV)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 5: Energy dependence of β at α=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='3 after the TOTsum cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The black gradient is for the 252Cf neutron source calibration data and the blue dots are for the 137Cs gamma-ray source calibration data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The dashed red lines indicate the mean value and the 3σ cut line by Gaussian fit, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The events between the cut lines are selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Head-tail analysis Importance of the track sense recognition, or the head-tail determination, has been stressed for years [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' We started to use the head-tail determination for the direction-sensitive dark matter search analysis with a limited efficiency in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' An analysis update improved the efficiency and head-tail determinations for 3D tracks, or 3D-vector analysis, were used for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The first step in reconstructing the direction of a track is to obtain the relative arrival times of ionized electrons in the readout strips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' These relative arrival times on X or Y strips are converted into relative Z positions taking account the drift velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The charge detected on the strip, or a hit, is thus assigned a (X, Z) or (Y, Z) hit-position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Angles of a track in the X-Z and Y-Z planes are known by fitting the hit-positions with straight lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 3D-axial directions of the tracks in the detector coordinate system are determined from these two angles in the X-Z and Y-Z planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' These reconstructed tracks are not 3D-vector ones at this stage because the head-tail of the track is not determined yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The head-tail of a track can be determined by observing the asymmetry of the energy deposition along its trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The fluorine-nuclear track with an energy of our interest (less than 400 keV) is known to deposit its energy large at the starting point and small around its end point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' This phenomena can be observed as large TOTs at the starting point and small TOTs around its end point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Figure 7 shows observed TOT distributions of an event along X and Y strips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' This event was obtained with a 252Cf source placed at (25 cm, 0 cm, 0 cm) so that we expect to observe fluorine nucleus tracks running from +X to -X directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' An asymmetry of the TOT distribution along the X-axis is seen while that along the Y-axis is more symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 7/21 800 102 700 600 500 Counts 400 101 300 200 137 CS B 100 100 50 100 150 200 250 300 350 400 Energy (keV)50 100 150 200 250 300 350 400 Energy (keV) 10 7 10 6 10 5 10 4 Gamma-ray rejection power Previous cut After TOT-sum-Length cut Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 6: Gamma-ray rejection powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The magenda dots are the result using the TOTsum- Length cut and green is the one without the TOTsum-Length cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The TOTsum-Length cut introduced in this study improved the results by two orders of magnitude in the energy range of 50–70 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' This asymmetry is quantified by parameters skewneesses defined as following equations, skewness x = < TOT(x) · (x− < x >)3 > < (TOT(x) · (x− < x >)2)3/2 >, (2) skewness y = < TOT(y) · (y− < y >)3 > < (TOT(y) · (y− < y >)2)3/2 >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' (3) Here TOT(x) is the TOT observed on strip x, and <> represents the means value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The ability to determine the head-tail, called the head-tail power Pht, is defined as Pht = Ntrue N , (4) where N is the total number of events, and Ntrue is the number of events that were correctly determined by the skewness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Determinations of Ntrue are discussed in the followings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' In our previous work, we selected events with small θele and large skewness to increase the head-tail power at a cost of lowering the selection efficiency to less than one half [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The analysis was updated so that the the selection efficiency was recovered while the Pht was retained;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' the use of skewness x and skewness y were determined according to the azimuth direction of the tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' For the tracks along the X-coordinate direction (0 ◦ ≤ |φazi| < 45 ◦), skewness x was used, and skewness y was used for the tracks with 45◦ ≤ |φazi| < 90◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' In addition, number of hit strips were increased by the operation at a high gas gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The raw values of skewness were found to be correlated with θele as shown in the upper panels of Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The skewness were corrected according to sin θele with cubic functions and the corrected skewness values shown in the lower panels of Figure 8 were used for further discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 8/21 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='32 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
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+page_content='56 x (cm) 0 2 4 6 8 10 12 14 TOT(x) skewness x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
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+page_content='04 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
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+page_content='88 y (cm) 0 2 4 6 8 10 12 14 TOT(y) skewness y = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0021 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 7: TOT values of an event along each X (left panel) and Y (right panel) strip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
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+page_content='0 sin ele Corrected( 90 azi < 45 ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
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+page_content='0 sin ele Corrected ( 45 azi < 45 ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
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+page_content='2 skewness y 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 sin ele Corrected(45 azi < 90 ) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 8: Correlation between skewnesses and sin θele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The distributios before and after the correction are shown in the upper and lower figures, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 9 and 10 show skewness distributions of a 252Cf source data after all cuts for three energy ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Neutron irradiation data from +X and −X directions are shown with red and blue histograms in the upper panels of Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' They show different skewness x distributions as expected while the skewness x distributions for the ±Y direction irradiation data (lower panels of Figure 9) did not show significant difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Same trend was confirmed for skewness y as shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Ntrue was defined by discriminating at skewness = 0 and Pht values were calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Averaged Pht values for 50–100 keV, 100–200 keV, and 200– 400 keV energy ranges were (52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='1)%, (52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='4)%, and (53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='6 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0)%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 9/21 Details of Pht are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The error of Pht in each irradiation direction is the standard deviation of head-tail power determined for each period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The overall head-tail power error is the standard deviation of the Phts in each irradiation direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Head-tail powers equivalent to those of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' [4] were achieved without any specific selection for the head-tail determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 skewness x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='20 Normalized counts 50-100 keV 50-100 keV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 skewness x 100-200 keV 100-200 keV 252Cf (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='5,0,0) cm 252Cf (-25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='5,0,0) cm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 skewness x 200-400 keV 200-400 keV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 skewness x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='20 Normalized counts 50-100 keV 50-100 keV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 skewness x 100-200 keV 100-200 keV 252Cf (0,25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='5,0) cm 252Cf (0,-25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='5,0) cm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 skewness x 200-400 keV 200-400 keV Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 9: Distribution of skewness x at each energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Events are normalized to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Energy range Pht (+x) (%) Pht (-x) (%) Pht (+y) (%) Pht (-y) (%) Pht (average) (%) 50–100 keV 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='9 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='3 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='1 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='9 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='9 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='4 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='1 100–200 keV 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='6 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='4 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='5 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='5 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='9 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 200–400 keV 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='3 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='6 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='4 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='9 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='8 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='8 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='6 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='6 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 Table 1: Head-tail powers in unit of % for each direction and energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Efficiencies There are two types of efficiencies regarding this study;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' the detection-selection and the directional efficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The former, or the “absolute” efficiency, determines the number of detected-and-selected events while the latter, or the “relative” one, determines the directional distribution of these events without changing the total number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' A data-set of recoil events isotropic in terms of the position and the direction was used to measure the efficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 10/21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 skewness y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='20 Normalized counts 50-100 keV 50-100 keV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 skewness y 100-200 keV 100-200 keV 252Cf (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='5,0,0) cm 252Cf (-25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='5,0,0) cm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 skewness y 200-400 keV 200-400 keV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 skewness y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='20 Normalized counts 50-100 keV 50-100 keV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 skewness y 100-200 keV 100-200 keV 252Cf (0,25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='5,0) cm 252Cf (0,-25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='5,0) cm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 skewness y 200-400 keV 200-400 keV Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 10: Distribution of skewnes y at each energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Events are normalized to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The isotropic data-set was made by summing-up the time-normalized data obtained by irradiating the detector with neutrons from a 252Cf source placed at six positions in ±X, ±Y , and ±Z directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The detection-selection efficiency is defined as the number of nuclear recoil events after all selections divided by the expected number of nuclear recoils in the fiducial volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Here, the expected number of nuclear recoils is estimated by the Geant4 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Results are shown in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' It should be noted that the increase of the detection efficiency seen below 100 keV is due to the contamination of the gamma-ray events and is not real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The contamination is removed with the selections to a negligible level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The detection efficiency is about 60% above 200 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The main reason of not reaching at 100% is that the gas gain being not high enough to trigger all the nuclear recoil events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The detection-selection efficiency above 200 keV is half of the detection efficiency because of the mean value for the TOTsum-Length selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' A 20%-reduction of the detection-selection efficiency from NEWAGE2021 should also attribute to the additional cut, which still gives a large advantage in the signal-to-noise ratio if we consider the gain on the rejection shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The detection-selection efficiency shown in Figure 11, or the ”absolute” efficiency, can be used to calculate the expected number of events for a given WIMP or background model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' It can also be used to unfold the measured energy spectrum and obtain an ”effective” spectrum for the comparison of the background rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 11/21 The directional efficiency is expressed as a sky map, or the relative response in the elevation (θele) - azimuth (φazi) plane, for isotropic recoils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The possible non-homogeneity of the direc- tional efficiency mainly originates from the reconstruction algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The 3D recoil direction, including the sense (head-tail) of the track, is reconstructed from the TOT-distributions of X and Y strips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Figure 12 shows the obtained θele-φazi distributions of an isotropic recoil calibration data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Since this map is to know the ”relative” or reconstruction efficiency of the directions, the color map is a relative one to be used with the total number of events being conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' It is seen that the tracks tend to be reconstructed to align with the strips, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' φazi = 0◦, ±90◦, 180◦ for the tracks parallel to the detection plane, or the tracks with θele ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The directional efficiencies shown in Figure 12, or the relative efficiency, can be used to make an expected recoil distribution for a given number of expected events calculated by the detection-selection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 50 100 150 200 250 300 350 400 Energy (keV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='8 Efficiency THIS WORK (detection) THIS WORK (detection+selection) NEWAGE2021 (detection+selection) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 11: Nuclear recoil efficiencies as a function of the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The cyan and the blue his- tograms are the detection and detection-selection efficiencies of nuclear recoil of this study, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The gray histograms is the result of NEWAGE2021 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Angular resolution The angular resolution was evaluated by comparing the distribution of the recoil angle γ of neutron irradiation data with the simulated ones smeared by various angular resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Here γ is the angle between the incoming neutron direction and the reconstructed nuclear- recoil direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Since the head-tails of the tracks are determined and considered in the analysis independent from the effetct of the angular resolution, the angular resolution was evaluated with the distribution of absolute value of cos γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' χ2 ang value defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' (5) was calculated for a given angular resolution σang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' χ2 ang = Nbin � i (Ndata i − NMC i (σang))2 Ndata i , (5) 12/21 150°-120° -90° -60° -30° 0° 30° 60° 90° 120° 150° 75° 60° 45° 30° 15° 0° 15° 30° 45° 60° 75° 50-100 keV 5 10 15 20 25 Direction efficiency (%) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 12: Directional efficiency in the detector coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' where Ndata i is the number of events in the i-th bin of the histogram of measured | cos γ|, and NMC i is the number of events in the i-th bin of the histogram of the | cos γ| distribution simulated by Geant4 smeared with the angular resolution, and Nbin is the number of bins in that histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The angular resolution at the minimum χ2 ang value was adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The angular resolution was 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='1+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='8 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='8 degree in the energy range of 50–100 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Experiment A direction-sensitive dark matter search was performed in Laboratory B, Kamioka Observa- tory (36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='25’N, 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='18’E), located 2700 m water equivalent underground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The measurement was carried out from December 12th, 2017 to March 26th, 2020, subdivided into eight peri- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The period was renewed when the detector was evacuated and filled with new CF4 gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The period information is summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The Z-axis of the NEWAGE-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='3b” detector was aligned to the direction of S30◦E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The target gas was CF4 at 76 Torr (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='1 atm) with a mass of 10 g in an effective volume of 28 × 24 × 41 cm3 (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='6 L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The total live time is 318 days corresponding to an exposure of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='18 kg·days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Various environmental parameters were monitored during the measurement to confirm the stability of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Figure 13 shows the time dependences of the integrated exposure, the gas gain and the energy resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The energy calibrations and the efficiency measure- ments were performed approximately every two weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The energy scale was corrected by the monitored gas gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The mean value of the energy resolution was 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='4% with a standard deviation of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0% during the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' No variation of the energy resolution beyond errors was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The event selections described in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='4 were applied to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Figure 14 shows the energy spectrum after each event selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The statistic errors are shown for the spectrum after all selections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' For a comparison with NEWAGE2021 result, an energy spectrum divided by the detection-selection efficiency is shown in Figure 15 as ”This work” 13/21 Period Date Gas gain Live time (days) Exposure (kg·days) RUN20-1 2017/12/12 – 2018/01/18 2000 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='135 RUN20-2 2018/01/23 – 2018/02/23 1750 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='200 RUN21 2018/02/28 – 2018/06/01 1550 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='586 RUN22-1 2018/06/06 – 2018/08/24 1110 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='525 RUN22-2 2018/09/20 – 2018/11/29 1200 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='605 RUN23 2018/12/05 – 2019/04/12 1750 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='459 RUN24 2019/04/26 – 2019/06/27 1800 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='494 RUN25 2020/03/04 – 2020/03/26 1950 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='176 Total 2017/12/12 – 2020/03/26 318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='180 Table 2: Summary of the measurement periods with gas gains (at the start of each RUN), live times, and exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' RUN22-1 and RUN22-2 are the data analyzed in NEWAGE2021 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 0 2 Exporsure (kg days) 0 2000 Gas gain 0 100 200 300 400 500 600 700 800 Day from December 12th, 2017 0 20 Energy resolution (%) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 13: Cumulative exposure, gas gains, and energy resolutions during the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' RUN20–25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The rate of this work is comparable to the that of NEWAGE2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' This is reasonable because there is no change in terms of the hardware-level radioactive background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' We have achieved the same count rate as that of NEWAGE2021 The energy spectrum of this work has smaller statistical errors due to the increase of the statistics by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Figure 16 shows the directions of measured nuclear recoil events in the detector coordinate (a) and the galactic coordinate (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The cos θCYGNUS was calculated for each event in Figure 16 (b) and distributions are shown in Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The cos θCYGNUS is binned by four and the energy is binned every 10 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 14/21 100 200 300 400 Energy (keV) 10 1 101 103 105 107 Counts No cut Fiducial cut Length-Energy cut TOTsum/Energy cut TOTsum-Length cut Expected Rn BG Expected gamma-ray BG BG upper error (1 ) Roundness cut Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 14: Energy spectra after each selection step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The grey, orange, blue, magenta, and green lines are the energy spectra after no cut, Fiducial volume cut, Length-Energy cut, TOTsum/Energy cut, and TOTsum-Length cut, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The black dots with error bars are the final data sample after the Roundness cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The fill stacked green and red spectra are the expected gamma-ray and radon background ones estimated by the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The gray shaded area is a 1σ error in the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Results A directional WIMP search analysis was performed with an assumption of the standard halo model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Here the Maxwell distribution with a velocity dispersion of 220 km/sec, and an escape velocity of 650 km/sec were assumed [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The local density of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='3 GeV/c2/cm3 was assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The spin parameter λ2J(J + 1) for the 19F of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='647 was used in this analysis [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The spectra of cos θCYGNUS for each energy bin as shown in Figure 17 were simultaneously compared with sum distributions of WIMP signal and isotropic background using the binned likelihood ratio method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' A statistic value χ2 was defined as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' χ2 = 2 n � i=0 m � j=0 � (NMC i,j − Ndata i,j ) + Ndata i,j ln �Ndata i,j NMC i,j �� + α2 E + α2 BG, (6) where, NMC i,j = NDM i,j (σχ−p, mχ, ξE) + NBG i,j (ξE, ξBG), (7) αE = ξE σE , (8) αBG = ξBG σBG .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' (9) 15/21 0 100 200 300 400 Energy (keV) 10 3 10 2 10 1 Rate (counts/keV/kg/days) THIS WORK NEWAGE2021 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 15: Energy spectra divided by the detection-selection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Red histogram is the energy spectrum of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Black histogram is the energy spectrum of NEWAGE2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Subscripts i and j are the bin-number of the cos θCYGNUS and the energy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The expected and measured number of events in bin i, j are described as NMC i,j and Ndata i,j , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' NMC i,j is written as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 7, where NDM i,j is the expected number of the WIMP- nucleus scatterings, and NBG i,j is the expected number of background events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' σχ−p is the WIMP-proton cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' NBG i,j was estimated using the Geant4 simulation based on the flux measurements of the ambient gamma-rays, the ambient neutrons, the alpha rays from the radon, and the alpha rays from the LAµ-PIC surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The dominant background components in the energy range of 50–100 keV were the ambient gamma-rays and the alpha rays from the radon (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' [6] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Expected background spectra are shown in Figure 14 for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The largest systematic uncertainty of the expected rate arise from the energy scale uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' This uncertainty was estimated from the discrepancy of the energy calibration between 10B, 220Rn, and 222Rn measurements discussed in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The uncertainty was evaluated in each run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The weighted average of the energy scale uncertainty was +13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='2% and -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The uncertainties of the background rate are the measurement errors of radioactivities for the ambient gamma-rays and the radons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Here the ambient gamma- ray flux was measured with a CsI scintillator[14] and the radon background was estimated with the high energy spectrum of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Nuisance parameters αE and αBG considering the systematic uncertainty from the energy scale σE and the background estimation σBG are defined as Equations (8) and (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Possible shifts of the energy scale and the number of expected backgrounds are expressed as ξE and ξBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' χ2 was minimized for a given WIMP mass with σχ−p, pull-terms αE and αBG as fitting parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' We first explain the procedure for the WIMP mass of 150 GeV/c2 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' A minimum χ2/NDF of 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='4/17 was obtained for σχ−p=14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='6 pb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The left panel in Figure 17 16/21 (a) Nuclear-recoil directions in the detector coordinate (b) Nuclear-recoil directions in the galaxy coordinate Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 16: (a) Nuclear recoil directions of final data sample in the detector coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The X- axis and Y-axis are φazi and θele in the detector coordinate system, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' (b) Nuclear recoil directions of final data sample in the galactic coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The X-axis and Y-axis are the longitude and latitude of the galactic coordinate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The direction of the galactic center is (0,0) and that of Cygnus is (-90,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The orange, red, pink, purple, and blue points indicate the energy ranges of 50–60 keV, 60–70 keV, 70–80 keV, 80–90 keV, and 90– 100 keV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The color contours in the background are the directional efficiencies in each coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' shows the cos θCYGNUS distributions of the best-fit case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' A chi-square distribution was created from a dummy sample of isotropic background model using Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' This test gave the p-value of 60% for the measured result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Observed distribution was thus found to be consistent with the background-only model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Since no significant WIMP excess was 17/21 obtained, an upper limit at 90% confidence level (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=') was set for the spin-dependent WIMP-proton scattering cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The likelihood ratio L is defined as, L = exp � −χ2(σχ−p) − χ2 min 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' (10) Here, χ2(σχ−p) and χ2 min are the value of χ2 and the minimum value of χ2 calculated by varying σχ−p, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' upper limit of the WIMP-proton cross section, σlimit χ−p , is determined as follows, � σlimit χ−p 0 Ldσχ−p � ∞ 0 Ldσχ−p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' (11) Using the above equation, the 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' upper limit of the spin-dependent cross section was found to be 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='7 pb for a WIMP mass of 150 GeV/c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The cos θCYGNUS distributions with the upper limit of 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' are shown in the right panels of Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Upper limits of the cross sections were obtained for other WIMP masses by the same procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Figure 18 shows the upper limits at 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' of the spin-dependent WIMP- proton cross sections as a function of the WIMP mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Compared to the NEWAGE2020 results, which was analyzed by the 3D-vector method using the standard µ-PIC, this upper limit updates by about one order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' This is due to the reduction of surface background events with the LAµ-PIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Furthermore, compared to the NEWAGE2021 result, the statistics of the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='4 factor and an updated analysis including the background estimation, improved the limits by a factor of about two for WIMPs heavier than 100 GeV/c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Discussions A new limit by a directional dark matter search with a 3D-vector analysis was obtained by this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Although we started to search the region of one of the interpretations of the DAMA/LIBRA’s annual modulation signal [18], a significant improvement of the sensitivity is needed for the search of the region of more interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The improvements can be realized mainly in three aspects: the detection-selection efficiency, the energy threshold, and the backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The detection-selection efficiency at 50–60 keV is 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='5%, which indicates the statistics can be increased by a factor of eight at most for a same exposure by an improvement of the detection-selection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' A measurement with a higher gas gain will increase the trigger efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' A better gamma-ray rejection analysis, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' introducing machine-learning methods, would compensate the expected increase of the gamma-ray background rate and allow us to operate the detector at a higher gas gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Shielding the detector is an independent hardware approach to reduce the gamma-ray background events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The current energy threshold (50 keV) is mainly limited by the track length of the recoil events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Typical length of the track of fluorine nuclear recoil below 50 keV in CF4 gas at 76 Torr (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='1 atm) is less than 1 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' This is comparable to the strip pitch of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='4 mm and one can deduce that the angular resolution and gamma-ray rejection both get worth below this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' One solution is to operate the CF4 gas at a lower pressure than 76 Torr to allow the nuclei and electrons run longer and improve the angular resolution and gamma-ray rejection below 50 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The remaining background sources are the ambient gamma-rays and internal radons as shown in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' We have already discussed the gamma-ray reduction above so we discuss 18/21 0 5 50-60 keV Best fit 0 5 50-60 keV 90 % C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 0 5 60-70 keV 0 5 60-70 keV 0 5 Counts 70-80 keV 0 5 Counts 70-80 keV 0 5 80-90 keV 0 5 80-90 keV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 cos CYGNUS 0 5 90-100 keV data NDM NBG 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 cos CYGNUS 0 5 90-100 keV data NDM NBG Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 17: cos θCYGNUS distributions (identical black histograms in both panels) for the final date sample in the 50–100 keV energy ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The best fit and 90% upper limit distributions for the WIMP mass of 150 GeV/c2 are shown with color histograms in the left and right panels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' the reduction of radon background here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The LAµ-PIC, significantly reduced the surface alpha rays in NEWAGE2021, still contains some material which emanates the radon gas [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' A new version of the µ-PIC series, LBGµ-PIC currently being developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The material used for the LBGµ-PIC is carefully selected so that the total radon emanation is less than 1/10 of the LAµ-PIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' With the improvements described above, we aim to explore the region claimed by DAMA/LIBRA [18] and to improve the sensitivity to reach limits by other direct search experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Conclusion A direction-sensitive direct dark matter search was carried out at Kamioka Observatory with a total live time of 318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='0 days corresponding to an exposure of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='18 kg·days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' A new gamma-ray rejection cut, which improved the gamma-ray rejection power to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='8 × 10−7 while maintaining the detection-selection efficiency of the nuclear recoil at about 20% was introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' This enabled us to use the high gas gain data, which was not used in the previous study due to the deterioration of the gamma-ray rejection power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The exposure was increased by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' A 3D-vector reconstruction with a head-tail determination power of 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='4% in the energy range of 50–100 keV was also used for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' As a result of the directional WIMP-search analysis, an upper limit of the spin-dependent WIMP-proton cross section of 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='7 pb for a WIMP mass of 150 GeV/c2 was derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' This limit marked the best direction-sensitive limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 19/21 101 102 103 WIMP mass (GeV/c2) 100 101 102 103 104 SD WIMP-proton p (pb) DMTPC 2012 NEWAGE2015 NEWAGE2020 3D-vector NEWAGE2021 THIS WORK 3D-vector DRIFT 2017 DAMA/LIBRA Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' 18: 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' upper limits of the spin-dependent WIMP-proton scattering cross section as a function of the WIMP mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The red line is the result of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The green line is the result of our previous work (NEWAGE2021 [6]) and the purple line is the result with the 3D-vector directional analysis for NEWAGE2020 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The gray line is the result of NEWAGE2015 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The solid light-blue shows the results from the directional analysis of DMTPC [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The blue line is the limit curve for DRIFT [17], which is a gas detector but non-directional analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' The gray area is an interpretation of the allowed region of DAMA/LIBRA [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Acknowledgment This work was partially supported by KAKENHI Grant-in-Aids (19H05806, 19684005, 23684014, 26104005, and 21H04471).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Arbey and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Mahmoudi, Progress in Particle and Nuclear Physics, 119, 103865 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Baum, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Freese, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Kelso, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
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+page_content=' 21/21' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE3T4oBgHgl3EQf5Qsz/content/2301.04779v1.pdf'}
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+Causal Inference in Recommender Systems: A
+Survey of Strategies for Bias Mitigation,
+Explanation, and Generalization
+Yaochen Zhu, Jing Ma, and Jundong Li
+Abstract In the era of information overload, recommender systems (RSs) have be-
+come an indispensable part of online service platforms. Traditional RSs estimate
+user interests and predict their future behaviors by utilizing correlations in the obser-
+vational user historical activities, user profiles, and the content of interacted items.
+However, since the inherent causal reasons that lead to the observed user behaviors
+are not considered, multiple types of biases could exist in the generated recommen-
+dations. In addition, the causal motives that drive user activities are usually entangled
+in these RSs, where the explainability and generalization abilities of recommenda-
+tions cannot be guaranteed. To address these drawbacks, recent years have witnessed
+an upsurge of interest in enhancing traditional RSs with causal inference techniques.
+In this survey, we provide a systematic overview of causal RSs and help readers
+gain a comprehensive understanding of this promising area. We start with the basic
+concepts of traditional RSs and their limitations due to the lack of causal reasoning
+ability. We then discuss how different causal inference techniques can be introduced
+to address these challenges, with an emphasis on debiasing, explainability promo-
+tion, and generalization improvement. Furthermore, we thoroughly analyze various
+evaluation strategies for causal RSs, focusing especially on how to reliably estimate
+their performance with biased data if the causal effects of interests are unavailable.
+Finally, we provide insights into potential directions for future causal RS research.
+Yaochen Zhu
+Department of Electrical and Computer Engineering, University of Virginia, e-mail: uqp4qh@
+virginia.edu
+Jing Ma
+Department of Computer Science, University of Virginia, e-mail: jm3mr@virginia.edu
+Jundong Li
+Department of Electrical and Computer Engineering, Department of Computer Science, and School
+of Data Science, University of Virginia, e-mail: jl6qk@virginia.edu
+1
+arXiv:2301.00910v1 [cs.IR] 3 Jan 2023
+
+2
+Yaochen Zhu, Jing Ma, and Jundong Li
+1 Introduction
+With information growing exponentially on the web, recommender systems (RSs)
+are playing an increasingly pivotal role in modern online services, due to their
+ability to automatically deliver items1 to users based on their personalized interests.
+Traditional RSs can be mainly categorized into three classes [9]: Collaborative
+filtering-based methods [10], content-based methods [11], and hybrid methods [12].
+Collaborative filtering-based RSs estimate user interests and predict their future
+behaviors by exploiting their past activities, such as browsing, clicking, purchases,
+etc. Content-based methods, on the other hand, predict new recommendations by
+matching user interests with item content. Hybrid methods combine the advantages
+of both worlds, where collaborative information and user/item feature information
+are comprehensively considered to generate more accurate recommendations.
+Although recent years have witnessed substantial achievements for all three
+classes of RSs introduced above, a great limitation of these methods is that they
+can only estimate user interests and predict future recommendations based on cor-
+relations in the observational user historical behaviors and user/item features, which
+guarantee no causal implications [13, 14]. For example, a collaborative filtering-
+based RS may discover that several drama shows from a certain genre tend to have
+high ratings from a group of users, and conclude that we should keep recommending
+drama shows from the same genre to these users. But there is an important question:
+Are the high ratings caused by the fact that the users indeed like drama shows from
+this genre, or they were limitedly exposed to drama shows from the same genre (i.e.,
+exposure bias), and if given a chance, they would prefer something new to watch? In
+addition, a content-based RS may observe that micro-videos with certain features are
+associated with more clicks and conclude that these features may reflect the current
+trend of user interests. But are the clicks because these micro-videos tend to have
+sensational titles as clickbait where users could be easily deceived? Moreover, if the
+titles of these micro-videos are changed to the ones that reflect their true content,
+would users still click them? The above questions are causal in nature because they
+either ask about the effects of an intervention (e.g., what the rating would be if a new
+drama show is made exposed to the user) or a counterfactual outcome (e.g., would
+the user still click a micro-video if its title had been changed to faithfully reflect the
+content), rather than mere associations in the observational data. According to Pearl
+[15], these questions lie on Rungs 2 and 3 of the Ladder of Causality, i.e., interven-
+tional and counterfactual reasoning, and they cannot be answered by traditional RSs
+that reason only with associations, which lie on Rung 1 of the ladder.
+Why are these causal questions important for RSs? The first reason is that failing
+to address them may easily incur bias in recommendations, which can get unnoticed
+for a long time. If the collaborative filtering-based RSs mentioned above mistake
+exposure bias for user interests, they would amplify the bias by continuously recom-
+mending users with similar items; eventually, recommendations will lose serendipity,
+1 We use the term item in a broad sense to refer to anything recommendable to users, such as news
+[1], jobs [2], articles [3], music [4], movies [5], micro-videos [6], PoIs [7], hashtags [8], etc.
+
+Causal Inference for Recommender Systems: A survey
+3
+and users’ online experience can be severely degraded. Similarly, for the content-
+based micro-video RSs, if they cannot distinguish clicks due to user interests from
+the ones deceived by clickbait, they may over-recommend micro-videos with sen-
+sational titles, which is unfair to the uploaders of high-quality micro-videos who
+put much effort into designing the content. In addition, understanding the cause of
+user activities can help improve the explainability of recommendations. Consider the
+causal question of whether a user purchases an item due to its quality or its low price.
+Pursuing the causal explanations behind user behaviors can help service providers to
+enhance the RS algorithm based on users’ personalized preferences. Finally, causal
+inference allows us to identify and base recommendations on causal relations that
+are stable and invariant, while discarding other correlations that are undesirable or
+susceptible to change. Take restaurant recommendations as an example. Users can
+choose a restaurant because of its convenience (e.g., going to a nearby fast food shop
+to quickly grab a bite, but they do not necessarily like it, a non-stable correlation)
+or due to their personal interests (e.g., traveling far away for a hot-pot restaurant, a
+stable causal relation). If an RS can properly disentangle users’ intent that causally
+affects their previous restaurant visits, even if the convenience levels of different
+restaurants may change due to various internal or external reasons such as users’
+moving to a new place, the system can still adapt well to the new situation. From this
+aspect, the generalization ability of the causal RSs can be substantially improved.
+This survey provides a systematic overview of recent advances in causal RS
+research. The organization is illustrated in Fig. 1. We start with the fundamental con-
+cepts of traditional RSs and their limitation of correlational reasoning in Section 2.
+Then Section 3 recaps two important causal inference paradigms in machine learning
+and statistics, and shows their connections with the recommendation task. Section 4
+thoroughly discusses how different causal inference techniques can be introduced to
+address the limitations of traditional RSs, with an emphasis on debiasing, explainabil-
+ity promotion, and generalization improvement. Section 5 summarizes the general
+evaluation strategies for causal RSs. Finally, Sections 6 and 7 discuss prospective
+open questions and future directions for causal RSs and conclude this survey.
+2 Recommender System Basics
+To keep this survey compact, we confine our discussions to simple RSs with 𝐼
+users and 𝐽 items. The main data for the RSs, i.e., users’ historical behaviors, are
+represented by a user-item rating matrix R ∈ R𝐼×𝐽, where a non-zero element 𝑟𝑖 𝑗
+denotes user 𝑖’s rating to item 𝑗, and a zero element 𝑟0
+𝑖𝑘 indicates the rating is
+missing 2. To make the discussions of RSs compatible with causal inference, we take
+a probabilistic view of R [17], where 𝑟𝑖 𝑗 is assumed to be the realized value of the
+2 We use rating to refer to any user-item interaction that can be represented by a numerical value.
+This includes both explicit feedback such as likes/dislikes, and implicit feedback such as views and
+clicks. When 𝑟𝑖 𝑗 represents implicit feedback, the missing elements 𝑟0
+𝑖𝑘 in R may be used as weak
+negative feedback in the training phase [16]. This may complicate the causal problems. Therefore,
+we assume RSs are trained on observed ratings to simplify the discussion unless specified otherwise.
+
+4
+Yaochen Zhu, Jing Ma, and Jundong Li
+Causal Inference in
+Recommendations
+Causal
+Debiasing
+Causal
+Explanation
+RCM-based
+promotes
+Exposure Bias
+Unfairness
+Section 3
+Causal
+Generalization
+Popularity Bias
+Intervention-based
+Causal Inference
+Recommender Systems
+SCM-based
+CF-based
+Content-based
+Hybrid
+promotes
+Disentangle-based
+Causal Embeddings
+Colliding Effects
+Clickbait
+Section 2
+Section 4.1
+Future Directions
+Section 4.2
+Sections 6,7
+Section 4.3
+Evaluation Strategies
+Section 5
+Fig. 1: An overview of the structure of this survey and connections between different sections.
+random variable 𝑅 dependent on user 𝑖 and item 𝑗3. In addition to R, an RS usually
+has access to side information like user features f𝑢
+𝑖 ∈ R𝐾 𝑢
+𝐹 , such as her age, gender,
+location, etc., or item features f𝑣
+𝑗 ∈ R𝐾 𝑣
+𝐹 , such as its content and textual description.
+𝐾𝑢
+𝐹 and 𝐾𝑣
+𝐹 are the dimensions of user and item features, respectively. The main
+purpose of an RS is to predict users’ ratings for previously uninteracted items (i.e.,
+the missing values 𝑟0
+𝑖𝑘 in R) based on the observed ratings 𝑟𝑖 𝑗 in R and the available
+user and item side information such as f𝑢
+𝑖 and f𝑣
+𝑗 , such that new relevant items can
+be properly recommended based on users’ personalized interests.
+2.1 Collaborative Filtering
+Collaborative filtering-based RSs recommend new items by leveraging user ratings in
+the past. They generally consider the ratings 𝑟𝑖 𝑗 as being generated from a user latent
+variable u𝑖 ∈ R𝐾 that represents user interests and an item latent variable v𝑗 ∈ R𝐾
+that encodes the item attributes (i.e., item latent semantic information), where 𝐾
+is the dimension of the latent space. Here we list three widely-used collaborative
+filtering-based RSs, which will be frequently used as examples in this survey:
+• Matrix Factorization (MF) [18]. MF models 𝑟𝑖 𝑗 with the inner product between
+u𝑖 and v𝑗, where 𝑟𝑖 𝑗 ∼ N (u𝑇
+𝑖 · v𝑗, 𝜎2
+𝑖 𝑗) and 𝜎2
+𝑖 𝑗 is the predetermined variance4.
+3 However, we do not distinguish random variables and their specific realizations if there is no risk
+of confusion. For simplicity, we assume 𝑅 to be Gaussian unless specified otherwise.
+4 For works that do not explicitly treat 𝑟𝑖 𝑗 as a random variable, we assume it follows a Gaussian
+distribution with zero variance. The generative process then becomes as 𝑟𝑖 𝑗 = u𝑇
+𝑖 · v𝑗.
+
+Causal Inference for Recommender Systems: A survey
+5
+• Deep Matrix Factorization (DMF) [19]. DMF extends MF by applying deep
+neural networks (DNNs) [20], i.e., 𝑓 𝑢
+𝑛𝑛, 𝑓 𝑣
+𝑛𝑛 : R𝐾 → R𝐾 ′, to u𝑖 and v𝑗, where
+the ratings are assumed to be generated as 𝑟𝑖 𝑗 ∼ N ( 𝑓 𝑢
+𝑛𝑛(u𝑖)𝑇 · 𝑓 𝑣
+𝑛𝑛(v𝑗), 𝜎2
+𝑖 𝑗).
+• Auto-encoder (AE)-based RSs [21, 22] model user 𝑖’s ratings to all 𝐽 items
+as r𝑖 ∼ N ( 𝑓 𝑢
+𝑛𝑛(u𝑖), 𝝈2
+𝑖 · I𝐾), where 𝑓 𝑢
+𝑛𝑛 : R𝐾 → R𝐽 is a DNN and item latent
+variables v𝑗 for all items are implicit in last layer weights of the decoder [23].
+In the training phase, the models learn the latent variables u𝑖, v𝑗 and the associated
+function 𝑓𝑛𝑛 by fitting on the observed ratings 𝑟𝑖 𝑗 (e.g., via maximum likelihood
+estimation, which essentially estimates the conditional distribution 𝑝(𝑟𝑖 𝑗|u𝑖, v𝑗) from
+the observational data [24]). Afterward, we can use them to predict new ratings for
+previously uninteracted items 𝑘, e.g., ˆ𝑟MF
+𝑖𝑘 = u𝑇
+𝑖 ·v𝑘 for MF, ˆ𝑟DMF
+𝑖𝑘
+= 𝑓 𝑢
+𝑛𝑛(u𝑖)𝑇 · 𝑓 𝑣
+𝑛𝑛(v𝑘)
+for DMF, and ˆ𝑟AE
+𝑖𝑘 = 𝑓 𝑢
+𝑛𝑛(u𝑖)𝑘 for AE-based RSs, where the top ones that best match
+users’ interests can be selected as recommendations.
+Traditional collaborative filtering-based RSs reasons with correlations.
+Ideally, we would expect u𝑖, v𝑗 and 𝑓𝑛𝑛 to capture the causal influence of user
+interests and item attributes on ratings, i.e., what the rating would be if item
+𝑗 is made exposed to user 𝑖 [24]. However, since the collected rating data are
+observational rather than experimental, what actually learned by u𝑖, v𝑗, and
+𝑓 are the co-occurrence patterns in users’ past behaviors, which guarantee
+no causal implications. Consequently, spurious correlations and biases can be
+captured by the model, which will be amplified in future recommendations
+[25]. Furthermore, the learned user latent variable u𝑖 generally entangles dif-
+ferent factors that causally determine user interests. From this perspective, the
+explainability and generalization of these methods cannot be guaranteed.
+2.2 Content-Based Recommender Systems
+Personalized content-based RSs (CBRSs) estimate user interests based on the fea-
+tures of the items they have interacted with. These models typically encode user
+interests into user latent variables u𝑖 ∈ R𝐾 and assume that the ratings are generated
+by matching user interests with item content, i.e., 𝑟𝑖 𝑗 ∼ N ( 𝑓 (u𝑖, f𝑣
+𝑗 ), 𝜎𝑖 𝑗), where 𝑓
+is a matching function. The training of personalized CBRSs follow similar steps as
+collaborative filtering, where u𝑖 and 𝑓 are learned by fitting on the observed ratings
+(which essentially estimates the conditional distribution 𝑝(𝑟𝑖 𝑗|u𝑖, f𝑣
+𝑗 ) from the obser-
+vational data), and new ratings can be predicted by ˆ𝑟𝑖𝑘 = 𝑓 (u𝑖, f𝑣
+𝑘). The key step of
+building a CBRS is to create item features f𝑣
+𝑗 that can best reflect user interests, which
+crucially depends on the item being recommended. For example, for micro-videos,
+the visual, audio, and textual modalities are comprehensively considered such that
+users’ interest in different aspects of a micro-video can be well captured [26].
+
+6
+Yaochen Zhu, Jing Ma, and Jundong Li
+Traditional content-based RSs cannot model the causal influence of user
+interests u𝑖 and item content f𝑣
+𝑗 on user rating 𝑟𝑖 𝑗. The reason is that, factors
+other than users’ interests in the item content, such as users’ being deceived
+by clickbaits (e.g., sensational titles of micro-videos) [27], etc., can create an
+undesirable association between item content f𝑣
+𝑗 and user ratings 𝑟𝑖 𝑗 in the
+observed dataset, where the bias can be captured by the user latent variables
+u𝑖 and the matching function 𝑓 , and perpetuates into future recommendations.
+2.3 Hybrid Recommendation
+Hybrid RSs combine user/item side information with collaborative filtering to en-
+hance the recommendations. A commonly-used hybrid strategy is to augment user
+and item latent variables u𝑖 and v𝑗 with user/item side information f𝑣
+𝑖 and f𝑣
+𝑗 in
+existing collaborative filtering methods by replacing u𝑖 and v 𝑗 with u+
+𝑖 = [u𝑖||f𝑢
+𝑖 ]
+and v+
+𝑗 = [v 𝑗||f𝑣
+𝑗 ] in MF, DMF, and AE-based RSs, where [·||·] represents vector
+concatenation [28, 29]. The dimensions of u𝑖 and v𝑗 that encode the collaborative
+information are adjusted accordingly to make u+
+𝑖 and v+
+𝑗 compatible in the model.
+Another important class of hybrid RS is the factorization machine (FM) [30] and its
+extensions like [31, 32], which can be viewed as learning a bi-linear function 𝑓 𝑓 𝑚
+where the ratings are generated by 𝑟𝑖 𝑗 ∼ N ( 𝑓 𝑓 𝑚(u𝑖, v𝑗, f𝑢
+𝑖 , f𝑣
+𝑗 ), 𝜎2
+𝑖 𝑗).
+Simple hybrid strategies cannot break the correlational reasoning lim-
+itation of collaborative filtering and content-based RSs, because the ob-
+jective of the hybridization is still to improve the models’ fitting on the ob-
+servational user historical behaviors (i.e., estimating conditional distribution
+𝑝(𝑟𝑖 𝑗|u𝑖, v𝑗, f𝑢
+𝑖 , f𝑣
+𝑗 ) from the data), where the causal reasons that lead to the
+observed user behaviors are not considered. However, the idea of introducing
+extra user/item side information is important for building causal RSs. The rea-
+son is that, combined with the domain knowledge of human experts, the side
+information can help form more comprehensive causal relations among the
+variables of interests, such as user interests, item attributes, historical ratings,
+and other important covariates that may lead to spurious correlations and bi-
+ases, which is usually a crucial step for causal reasoning in recommendations.
+3 Causal Recommender Systems: Preliminaries
+In the previous section, we discussed the recommendation strategies of the tradi-
+tional RSs and their limitations due to correlational reasoning on observational user
+
+Causal Inference for Recommender Systems: A survey
+7
+behaviors. In this section, we introduce two causal inference frameworks, i.e., Ru-
+bin’s potential outcome framework (also known as the Rubin causal model, RCM)
+[33] and Pearl’s structural causal model (SCM) [34], in the context of RSs, aiming
+to provide a theoretically rigorous basis for reasoning with correlation and causation
+in recommendations. We show that both RCM and SCM are powerful frameworks to
+build RSs with causal reasoning ability (i.e., causal RSs), but they are best suited for
+different tasks and questions. The discussions in this section serve as the foundation
+for more in-depth discussions of the state-of-the-art causal RS models in Section 4.
+3.1 Rubin’s Potential Outcome Framework
+3.1.1 Motivation of Applications in RSs
+To understand the correlational reasoning nature of traditional RSs, we note that
+naively fitting models on the observed ratings can only answer the question “what
+the rating would be if we observe an item was exposed to the user". Since item
+exposure is not randomized in the collected dataset 5, the predicate “the item was
+exposed to the user" per se contains extra information about the user-item pair (e.g.,
+the item could be more popular than other non-exposed items), which cannot be
+generalized to the rating predictions of arbitrary user-item pairs. Therefore, what
+RS asks is essentially an interventional question (and therefore a causal inference
+question), i.e., what the rating would be if an item is made exposed to the user. To
+address this question, RCM-based RSs draw inspiration from clinical trials, where
+exposing a user to an item is compared to subjecting a patient to a treatment, and the
+user ratings are analogous to the outcomes of the patients after the treatment [39, 40].
+Accordingly, RCM-based RSs aim to estimate the causal effects of the treatments
+(exposing a user to an item) on the outcomes (user ratings), despite the possible
+correlations between the treatment assignment and the outcome observations [39].
+3.1.2 Definitions and Objectives
+We first introduce necessary symbols and definitions to connect RCM with RSs. We
+consider the unit as the user-item pair (𝑖, 𝑗) that can receive the treatment “exposing
+user 𝑖 to item 𝑗”, and the population as all user-item pairs PO = {(𝑖, 𝑗), 1 ≤ 𝑖, 𝑗 ≤
+𝐼, 𝐽} [41]. We start by using a binary scalar 𝑎𝑖 𝑗 to denote the exposure status of
+item 𝑗 for user 𝑖, i.e., the assigned treatment. We further define the rating potential
+outcome 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) as user 𝑖’s rating to item 𝑗 if the item is made exposed to
+the user and 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 0) as the rating if the item is not exposed [42]. For user 𝑖, if
+she rated item 𝑗, we observe 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) = 𝑟𝑖 𝑗. Otherwise, we observe the baseline
+potential outcome 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 0) = 0, which is usually ignored in debias-oriented
+5 which can be attributed to multiple reasons such as users’ self-search [35], the recommendations
+of previous models [36], the position where the items are displayed [37], item popularity [38], etc.
+Generally, RCM-based causal RSs are agnostic to the specific reason that causes the exposure bias.
+
+8
+Yaochen Zhu, Jing Ma, and Jundong Li
+5
+5
+5
+5
+5
+5
+5
+5
+5
+5
+5
+5
+5
+5
+1
+1
+1
+1
+1
+1
+1
+1
+Horror
+Lover
+Romance
+Lover
+Horror
+Romance
+5
+5
+5
+5
+5
+5
+5
+5
+5
+5
+5
+5
+5
+5
+5
+5
+Horror
+Lover
+Romance
+Lover
+Horror
+Lover
+Romance
+Lover
+Horror
+Romance
+Horror
+Romance
+(a) Observed Ratings
+(b) Rating Potential Outcomes
+(c) Predicted Ratings
+Fig. 2: A classical example of exposure bias in RSs [43]. The example is composed of two horror
+lovers who always rate horror movies with five while hating romance movies, and two romance
+lovers would who do exactly the opposite. (a) shows the observed ratings 𝑟𝑖 𝑗. (b) shows the rating
+potential outcomes 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1). (c) shows the rating predictions of an RS that maximizes the
+likelihood of the observed ratings in (a), but the RS is bad because it predicts all ratings to five.
+causal RS research [39, 43]6. Similar to clinical trials, we can define the treatment
+group T = {(𝑖, 𝑗) : 𝑎𝑖 𝑗 = 1} as the set of user-item pairs where user 𝑖 is exposed
+to item 𝑗, and define the non-treatment group NT = {(𝑖, 𝑘) : 𝑎𝑖𝑘 = 0} accordingly.
+The purpose of RSs, under the RCM framework, can be framed as utilizing the
+observed ratings from units in the treatment group T to unbiasedly estimate the
+rating potential outcomes for units from the population PO, despite the possible
+correlations between item exposures 𝑎𝑖 𝑗 and user ratings 𝑟𝑖 𝑗 in the collected data.
+3.1.3 Causal Analysis of Traditional RSs
+Traditional RSs naively train a rating prediction model that best fits the ratings in
+the treatment group T (e.g., via maximum likelihood introduced in Section 2) to
+estimate the unobserved rating potential outcomes 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) for user-item pairs in
+NT [46], which neglect the fact that exposure bias can lead to a systematic difference
+in the distribution of 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) between T and NT. For example, users tend to
+rate items they like in reality, which could lead to the following spurious correlation
+between item exposure 𝑎𝑖 𝑗 and rating potential outcome 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1):
+𝑝(𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) is high|𝑎𝑖 𝑗 = 1) > 𝑝(𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) is high|𝑎𝑖 𝑗 = 0),
+(1)
+i.e., users who have rated an item 𝑗 may have systematically higher ratings than
+users who haven’t rated it yet. In this case, traditional RSs may have a tendency to
+overestimate the ratings for units in NT (see Fig. 2 for an intuitive example). The-
+oretically, RCM attributes the exposure bias in the collected dataset to the violation
+of the unconfoundedness assumption [33] defined as follows:
+𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) ⊥ 𝑎𝑖 𝑗.
+(2)
+6 In the uplift evaluation of RSs that aims to estimate how recommendations change user behaviors
+[44], 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 0) may be used to represent user 𝑖’s rating to item 𝑗 through self-searching [45].
+
+Causal Inference for Recommender Systems: A survey
+9
+The rationale is that, if Eq. (2) holds, the exposure of user 𝑖 to item 𝑗 (i.e., 𝑎𝑖 𝑗)
+is independent of the rating potential outcome 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1), which implies that
+𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) in T and NT follows the same distribution. Therefore, the exposure of
+the items is randomized, and exposure bias such as Eq. (1) will not exist [42].
+3.1.4 Potential Outcome Estimation with the RCM Framework
+One classic solution from the RCM-based framework to address the exposure bias
+is that we find user and item covariates 𝐶, such that in each data stratum specified by
+𝐶 = c, users’ exposure to items are randomized [33]. The property of the covariates
+𝐶 can be formulated as the conditional unconfoundedness assumption as follows:
+𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) ⊥ 𝑎𝑖 𝑗 | c.
+(3)
+𝐶 is sometimes non-rigorously referred to as confounder in the literature, but we will
+see its formal definition in the next subsection. If Eq. (3) holds, the item exposures are
+independent of the rating potential outcomes in each data stratum specified by 𝐶 = c,
+and the exposure bias can be attributed solely to the discrepancy in the distribution
+of the covariates 𝐶 = c between the treatment group T and the population PO,
+i.e., 𝑝(c|𝑎𝑖 𝑗 = 1) and 𝑝(c)7 Therefore, we can reweight the observed ratings in
+T based on the covariates 𝐶 to address the bias, such that they can be viewed
+as pseudo randomized samples. This leads to inverse propensity weighting (IPW),
+which eliminates the exposure bias from the data’s perspective [39]. In addition,
+we can also adjust the influence of 𝐶 in the RS model, where the exposure bias is
+addressed from the model side [42]. Both methods will be discussed in Section 4.1.1.
+•! Attention: Extra Assumptions Required by Most RCM-based RSs
+In addition to unconfoundedness, most RCM-based RS need two extra assumptions
+to identify the causal effects of item exposures on ratings: (1) The stable unit
+treatment assumption (SUTVA), which states that items exposed to one user cannot
+affect ratings of another user. (2) The positivity assumption, which states that every
+user has a positive chance of being exposed to every item [33]. For RCM-based
+causal RSs introduced in this survey, these two assumptions are tacitly accepted.
+7 ! We can gain an intuition of this claim from Fig. 2. Suppose covariates 𝐶 represent the two-
+dimensional features (user type, movie type). Given 𝐶 = c, 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) ⊥ 𝑎𝑖 𝑗 | c described in
+Eq. (3) is satisfied because in each data stratum specified by 𝐶 = c (i.e., the four 2×2 blocks in Fig.
+2-(b)), 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) is constant. Fig. 2-(a) shows that for the treatment group T, 𝑝(c|𝑎𝑖 𝑗 = 1) = 1/2
+for c ∈ C1 = {(horror fan, horror movie), (romance fan, romance movie) } and 𝑝(c|𝑎𝑖 𝑗 = 1) = 0
+for c ∈ C2 = {(horror fan, romance movie), (romance fan, horror movie) }. In contrast, for the
+population PO, 𝑝(c) = 1/4 for c ∈ C1 ∪ C2. Therefore, in the treatment group T, user-item pairs
+with covariates in C1 are over-represented, while those with covariates in C2 are under-represented.
+However, we also note that this case is too extreme to be addressed by RCM, as 𝑝(c|𝑎𝑖 𝑗 = 1) = 0
+for 𝐶 ∈ C2 violates the positivity assumption mentioned in the above attention box.
+
+10
+Yaochen Zhu, Jing Ma, and Jundong Li
+3.2 Pearl’s Structural Causal Model
+3.2.1 Motivation of Applications in RS
+Different from RCM that uses rating potential outcomes to reason with causality and
+attributes the biases in observed user behaviors to non-randomized item exposures,
+Pearl’s structural causal model (SCM) delves deep into the causal mechanism that
+generates the observed outcomes (and biases) and represents it with a causal graph
+𝐺 = (N, E). The nodes N specify the variables of interests, which in the context
+of RS could be user interests 𝑈, item attributes 𝑉, observed ratings 𝑅, and other
+important covariates 𝐶, such as item popularity, user features, etc8. The directed
+edges E between nodes represent their causal relations determined by researchers’
+domain knowledge. Each node 𝑋 ∈ N is associated with a structural equation
+𝑝𝐺(𝑋|𝑃𝑎(𝑋))9, which describes how the parent nodes 𝑃𝑎(𝑋) causally influence 𝑋
+(i.e., the response of 𝑋 when setting nodes in 𝑃𝑎(𝑋) to specific values)
+Although RCM and SCM are generally believed to be fundamentally equivalent
+[34], both have their unique advantages. Compared to RCM, the key advantage of
+SCM is that causal graph offers an intuitive and straightforward way to encode and
+communicate domain knowledge and substantive assumptions of researchers, which
+is beneficial even for the RCM-based RSs [42]. Furthermore, SCM is more flexible
+as it can represent and reason with the causal effects between any subset of nodes in
+the causal graph (e.g., between two causes 𝑈,𝑉 and one outcome 𝑅), as well as the
+causal effects along specific paths (e.g., 𝑈 → 𝑅 and 𝑈𝑐 → 𝑅). Therefore, SCMs are
+broadly applicable to multiple problems in RSs (not limited to exposure bias), such
+as clickbait bias, unfairness, entanglement, domain adaptation, etc [14].
+•! Attention: Two Caveats of SCM-based Causal RSs.
+There are two caveats of SCM-based causal RSs. (1) Causal graphs for RSs often
+involve user, item latent variables𝑈,𝑉 that encode user interests and item attributes.
+Most works infer them alongside the estimation of structural equations and treat them
+as if they were observed when analyzing the causal relations. Alternatively, this can
+be viewed as representing users and items with their IDs (i.e., 𝑖 and 𝑗) in the causal
+graph and subsuming the embedding process into the structural equations [47]. (2)
+Generally, the causal graph should describe the causal mechanism that generates the
+observed data, because it allows us to distinguish invariant, causal relations from
+undesirable correlations. For example, we may argue that item popularity 𝐶 should
+be determined by item attributes 𝑉, i.e., 𝑉 → 𝐶. But to describe the generation
+of the observed ratings, causal relation 𝐶 → 𝑉 is usually assumed instead as item
+popularity causally influences the exposure probability of each item [48].
+8 In causal graphs, the subscripts 𝑖, 𝑗 for each node are omitted for simplicity.
+9 We also omit the mutually independent exogenous variables for each node and summarize their
+randomness into the structural equations with probability distributions [14]. Subscript 𝐺 is used to
+distinguish structural equations from other conditional relationships that can be inferred from 𝐺.
+
+Causal Inference for Recommender Systems: A survey
+11
+R
+U
+Uc
+Cu
+Cv
+V
+Cu
+U
+R
+Cu
+U
+R
+U
+R
+Uc
+(a) A generic causal graph for RS
+(b) The chain structure
+(c) The fork structure
+(d) The V-structure
+Fig. 3: (a): A generic causal graph for RS that depicts the causal influence of user interests 𝑈, user
+conformity to the popularity trend𝑈𝑐, and item attributes 𝑉 on the observed ratings 𝑅. Specifically,
+the causal paths 𝑈 → 𝑅 and 𝑉 → 𝑅 are confounded by 𝐶𝑢 and 𝐶𝑣, which represent user features
+and item popularity, respectively. (b)(c)(d): Three atomic structures identified from (a).
+3.2.2 Atomic Structures of Causal Graphs
+The structure of causal graphs represents researchers’ domain knowledge regard-
+ing the causal generation process of the observational data, which is the key to
+distinguishing stable, causal relations from other undesirable correlations between
+variables of interest. Here, we use a generic causal graph applicable to RSs in Fig.
+3-(a) as a running example to illustrate three atomic graph structures:
+• Chain, e.g., 𝐶𝑢 → 𝑈 → 𝑅. In a chain, the successor node is assumed to be
+causally influenced by the ancestor nodes. In the example, 𝑈 is a direct cause of
+𝑅, whereas 𝐶𝑢 indirectly influences 𝑅 via 𝑈 as a mediator.
+• Fork, e.g., 𝑈 ← 𝐶𝑢 → 𝑅. In the fork, 𝐶𝑢 is called a confounder as it causally
+influences two children 𝑈 and 𝑅. From a probabilistic perspective, 𝑈 and 𝑅
+are not independent unless conditioned on the confounder 𝐶𝑢 [49]. This leads
+to the tricky part of a fork structure, i.e., confounding effect [34], where an
+unobserved 𝐶𝑢 can lead to spurious correlations between 𝑈 and 𝑅.
+• V-structure, e.g., 𝑈 → 𝑅 ← 𝑈𝑐. In the V-structure, 𝑅 is called a collider
+because it is under the causal influence of two parents, i.e., 𝑈 and 𝑈𝑐. An inter-
+esting property of the V-structure is the colliding effects [34], where observing
+𝑅 creates a dependence on 𝑈 and 𝑈𝑐, even if they are marginally independent.
+Confounders can lead to non-causal dependencies among variables in the obser-
+vational dataset. This could introduce bias in traditional RSs, where the confounding
+effects are mistaken as causal relations. Confounding bias is a generic problem in
+RSs [24], which will be further analyzed in the following subsections. In addition,
+abstracted V-structure usually leads to the entanglement of causes, which could jeop-
+ardize the explainability of RSs. For example, a user’s purchase of an item may be due
+to her interest, i.e., 𝑈, or her conformity to the popularity trend, i.e., 𝑈𝑐. Since most
+RSs summarize both into a user latent variable 𝑈, the V-structure 𝑈 → 𝑅 ← 𝑈𝑐 is
+abstracted away, where the two causes of the purchase cannot be distinguished.
+
+12
+Yaochen Zhu, Jing Ma, and Jundong Li
+U
+R
+V
+(b) Confounded true SCM
+(a) SCM assumed by non-causal RS
+Cv
+(c) SCM under intervention
+U
+R
+V
+U
+R
+do(V)
+Cv
+Fig. 4: (a): SCM assumed by non-causal collaborative filtering-based RS. (b): The confounded
+SCM that depicts the true data generation process. (c): SCM under intervention 𝑑𝑜(𝑉 ).
+3.2.3 Causal Analysis of Traditional RSs
+In this section, we investigate the susceptibility of traditional collaborative filtering-
+based RSs to the confounding bias. As discussed in Section 2.1, a commonality
+of these models is that they estimate conditional distribution 𝑝(𝑟𝑖 𝑗|u𝑖, v𝑗) from
+observed ratings and use it to predict new ratings. For 𝑝(𝑟𝑖 𝑗|u𝑖, v𝑗) to represent the
+causal influence of user interests u𝑖 and item attributes v 𝑗 on ratings 𝑟𝑖 𝑗 (which, in
+the context of collaborative filtering, means the rating of any arbitrary item 𝑗 that is
+made exposed to user 𝑖 [24]), the causal graph 𝐺1 of Fig. 4-(a) is tacitly assumed,
+i.e., no unobserved confounders for causal paths 𝑈 → 𝑅 and 𝑉 → 𝑅10.
+However, in reality, both 𝑈 → 𝑅 [25, 51] and 𝑉 → 𝑅 [52, 53] can be confounded,
+where the confounding effects can be implicitly captured by 𝑝(𝑟𝑖 𝑗|u𝑖, v𝑗) that bias
+future recommendations. To reveal the bias, we consider the scenario where the
+causal path 𝑉 → 𝑅 is confounded by 𝐶𝑣 (e.g., item popularity). We assume the
+causal path 𝐶𝑣 → 𝑉 denotes the causal influence of 𝐶𝑣 on the exposure probability
+of item 𝑉 [48]. In this case, the observed ratings are generated according to the
+causal graph 𝐺2 in Fig. 4-(b). Utilizing the law of total probability, the conditional
+distribution 𝑝(𝑟𝑖 𝑗|u𝑖, v𝑗) estimated from the confounded data can be calculated as:
+𝑝(𝑟𝑖 𝑗|u𝑖, v𝑗) =
+∑︁
+c
+𝑝(c|v𝑗) · 𝑝𝐺2(𝑟𝑖 𝑗|u𝑖, v𝑗, c) = E𝑝(𝐶𝑣 |v 𝑗) [𝑝𝐺2(𝑟𝑖 𝑗|u𝑖, v𝑗, 𝐶𝑣)].
+(4)
+The issue of Eq. (4) is that, the 𝑝(c|v𝑗) term is not causal (as we only have an edge
+𝐶𝑣 → 𝑉 in the causal graph but not 𝑉 → 𝐶𝑣). In fact, 𝑝(c|v𝑗) represents abductive
+reasoning because it infers the cause c (e.g., item popularity) from the effect v𝑗 (i.e.,
+item 𝑗 is exposed to user 𝑖) and uses the inferred c to support the prediction of the
+rating 𝑟𝑖 𝑗. However, such reasoning cannot be generalized to the rating prediction of
+an arbitrary item v 𝑗, i.e., an item that is made exposed to the user. In other words,
+uncontrolled confounder 𝐶𝑣 leaves open a backdoor path (i.e., non-causal path)
+between 𝑉 and 𝑅, such that non-causal dependence of 𝑅 on 𝑉 exists in the data,
+which can be captured by traditional RSs and bias future recommendations. 11
+10 This corresponds to the case where item exposures are randomized (see the discussions in Section
+3.1.3), as the user-item pair (𝑈, 𝑉 ) is not determined by other factors associated with 𝑅 [50].
+11 The similarity between this section and Section 3.1.1 shows us the connection between RCM-
+based and SCM-based causal RSs, where the claim that when item exposure is not randomized,
+
+Causal Inference for Recommender Systems: A survey
+13
+3.2.4 Causal Reasoning with SCM
+To calculate the causal effect of u𝑖 and v𝑗 on 𝑟𝑖 𝑗, we should conduct intervention
+on 𝑈 and 𝑉. This means that we set 𝑈, 𝑉 to u𝑖, v𝑗 regardless of the values of their
+parent nodes in the causal graph, including the confounder 𝐶𝑣 (because these nodes
+determine the exposure of item 𝑗 to user 𝑖 in the observed data). SCM denotes the
+intervention with do-operator as 𝑝(𝑟𝑖 𝑗|𝑑𝑜(u𝑖, v𝑗)) to distinguish it from the con-
+ditional distribution 𝑝(𝑟𝑖 𝑗|u𝑖, v 𝑗) that reasons with correlations in the observational
+data. Consider again the causal graph 𝐺2 illustrated in Fig. 4-(b). The intervention
+on node 𝑉 can be realized by removing all the incoming edges for node 𝑉 and setting
+the structural equation 𝑝𝐺2(𝑉|𝐶𝑣) deterministically as 𝑉 = v𝑗, while other struc-
+tural equations remain intact (Fig. 4-(c)). If the confounder 𝐶𝑣 can be determined
+and measured for each item, the interventional distribution 𝑝(𝑟𝑖 𝑗|𝑑𝑜(u𝑖, v𝑗)) can be
+directly calculated from the confounded data via backdoor adjustment [34] as:
+𝑝(𝑟𝑖 𝑗|𝑑𝑜(u𝑖, v𝑗)) =
+∑︁
+c
+𝑝𝐺2(c) · 𝑝𝐺2(𝑟𝑖 𝑗|u𝑖, v𝑗, c) = E𝑝𝐺2 (𝐶𝑣) [𝑝𝐺2(𝑟𝑖 𝑗|u𝑖, v𝑗, 𝐶𝑣)],
+(5)
+which, compared with Eq. (4), blocks the abductive inference of c from v𝑗, such that
+the causal influence of u𝑖, v𝑗 on 𝑟𝑖 𝑗 can be properly identified.
+Backdoor adjustment requires all confounders to be determined and measured in
+advance, but there are other SCM-based causal inference methods that can estimate
+causal effects with unknown confounders, and we refer readers to [54, 55] for details.
+Moreover, causal graphs allow us to conduct other types of causal reasoning based
+on the encoded causal knowledge, such as debiasing for non-confounder-induced
+biases (e.g., clickbait bias and unfairness), causal disentanglement, and causal gen-
+eralization [56]. These will be thoroughly discussed in the next section.
+4 Causal Recommender Systems: The State-of-the-Art
+Based on the preliminary knowledge of RSs and causal inference discussed in previ-
+ous sections, we are ready to introduce the state-of-the-art causal RSs. Specifically,
+we focus on three important topics, i.e., bias mitigation, explainability promotion,
+and generalization improvement, as well as their inter-connections, where various
+limitations of traditional RSs due to correlational reasoning can be well addressed.
+4.1 Causal Debiasing for Recommendations
+The correlational reasoning of traditional RSs can inherit multiple types of biases in
+the observational user behaviors and amplify them in future recommendations [46].
+“observing that an item was exposed to the user per se contains extra information about the user-item
+pair" is mathematically transformed into the abductive inference of c from v𝑗 by 𝑝(c|v𝑗).
+
+14
+Yaochen Zhu, Jing Ma, and Jundong Li
+The biases may result in various consequences, such as the discrepancy between
+offline evaluation and online metrics, loss of diversity, reduced recommendation
+quality, offensive recommendations, etc. Causal inference can distinguish stable
+causal relations from spurious correlations and biases that could negatively influence
+the recommendations, such that the robustness of recommendations can be improved.
+4.1.1 Exposure Bias
+Exposure bias in RSs broadly refers to the bias in observed ratings due to non-
+randomized item exposures. From the RCM’s perspective, exposure bias can be
+defined as the bias where users are favorably exposed to items depending on their
+expected ratings for them (i.e., rating potential outcomes) [43]. Exposure bias occurs
+due to various reasons, such as users’ self-search or the recommendation of the
+previous RSs [36], which leads to the down-weighting of items less likely to be
+exposed to users. Since item exposures can be naturally compared with treatments
+in clinical trials, we discuss the debiasing strategies with the RCM framework.
+Inverse Propensity Weighting (IPW). IPW-based causal RSs aim to reweight the
+biased observed ratings 𝑟𝑖 𝑗 for user-item pairs in the treatment group, i.e., T =
+{(𝑖, 𝑗) : 𝑎𝑖 𝑗 = 1}, to create pseudo randomized samples [57] for unbiased training
+of RS models that aim to predict the rating potential outcomes 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) for the
+population PO = {(𝑖, 𝑗), 1 ≤ 𝑖, 𝑗 ≤ 𝐼, 𝐽}. Intuitively, we can set the weight of 𝑟𝑖 𝑗
+for units in T to be the inverse of item 𝑗’s exposure probability to user 𝑖, such that
+under-exposed items can be up-weighted and vice versa. If for each user-item pair,
+the covariates c that satisfy the conditional unconfoundedness assumption in Eq. (3)
+are available, the exposure probability 𝑒𝑖 𝑗 can be unbiasedly estimated from c via
+𝑒𝑖 𝑗 = 𝑝(𝑎𝑖 𝑗 = 1|c) = E[𝑎𝑖 𝑗|c],
+(6)
+which is formally known as propensity score in causal inference literature [58].
+•> Background: The Balancing Property of Propensity Scores.
+Propensity scores have the following property called balancing [33, 59], which is the
+key to proving the unbiasedness of IPW-based RSs:
+E
+� 𝑟𝑖 𝑗
+𝑒𝑖 𝑗
+���𝑎𝑖 𝑗 = 1
+�
+= E
+�𝑟𝑖 𝑗 · 𝑎𝑖 𝑗
+𝑒𝑖 𝑗
+�
+= E
+�
+E
+�𝑟𝑖 𝑗 · 𝑎𝑖 𝑗
+𝑒𝑖 𝑗
+���c
+��
+= E
+�
+E
+�𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) · 𝑎𝑖 𝑗
+𝑒𝑖 𝑗
+���c
+�� (𝑎)= E
+�E[𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) | c] · E[𝑎𝑖 𝑗 | c]
+𝑒𝑖 𝑗
+�
+= E
+�E[𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) | c] · 𝑒𝑖 𝑗
+𝑒𝑖 𝑗
+�
+= E[𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1)],
+(7)
+where the step (𝑎) follows the conditional unconfoundedness assumption in Eq. (3).
+
+Causal Inference for Recommender Systems: A survey
+15
+We first discuss the implementation of IPW-based RS and its unbiasedness if
+user and item covariates c that satisfy Eq. (3) are available and the propensity scores
+𝑒𝑖 𝑗 can be calculated exactly as Eq. (6). We denote the rating predictor of an RS
+that aims to predict the rating potential outcome 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) as ˆ𝑟𝑖 𝑗 and assume
+𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) follows the unit-variance Gaussian distribution. Ideally, we would like
+ˆ𝑟𝑖 𝑗 to maximize the log-likelihood on the rating potential outcomes 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1)
+for all user-item pairs in PO, which is equivalent to the minimization of the mean
+squared error (MSE) loss between ˆ𝑟𝑖 𝑗 and 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) as follows:
+LTrue =
+1
+𝐼 × 𝐽
+∑︁
+𝑖, 𝑗
+(ˆ𝑟𝑖 𝑗 − 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1))2.
+(8)
+However, since 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) is unobservable for user-item pairs in the non-treatment
+group NT, LTrue is impossible to calculate. Therefore, traditional RSs only maxi-
+mize the log-likelihood of the observed ratings for user-item pairs in the treatment
+group T, which leads to the empirical MSE loss as follows:
+LObs =
+1
+|(𝑖, 𝑗) : 𝑎𝑖 𝑗 = 1|
+∑︁
+(𝑖, 𝑗):𝑎𝑖 𝑗=1
+(ˆ𝑟𝑖 𝑗 − 𝑟𝑖 𝑗)2,
+(9)
+where |(𝑖, 𝑗) : 𝑎𝑖 𝑗 = 1| is the number of observed ratings. When exposure bias exists,
+item exposure 𝑎𝑖 𝑗 depends on the rating potential outcome 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1). Therefore,
+LObs is a biased estimator for LTrue, because the observed ratings for user-item pairs
+in the treatment group T are biased samples from the rating potential outcomes of
+the population PO (see Fig. 5-(a) and Fig. 5-(b) for an example). To remedy the
+bias, IPW-based causal RSs reweight the observed ratings 𝑟𝑖 𝑗 in T by the inverse of
+the propensity scores, i.e.,
+1
+𝑒𝑖 𝑗 , which leads to the following new training objective:
+LIPW =
+1
+𝐼 × 𝐽
+∑︁
+(𝑖, 𝑗):𝑎𝑖 𝑗=1
+1
+𝑒𝑖 𝑗
+· (ˆ𝑟𝑖 𝑗 − 𝑟𝑖 𝑗)2.
+(10)
+The proof for the unbiasedness of LIPW for LTrue can be achieved by utilizing the
+balancing property of propensity scores in Eq. (7), where we substitute (ˆ𝑟𝑖 𝑗 − 𝑟𝑖 𝑗)2
+for 𝑟𝑖 𝑗 in the LHS of Eq. (7) and treat the rating predictor ˆ𝑟𝑖 𝑗 as constant [39]. We
+also provide a toy example in Fig. 5 to intuitively show the calculation of 𝑒𝑖 𝑗, the
+biasedness of LObs and the unbiasedness of LIPW. The objective for IPW-based RSs
+defined in Eq. (10) is model-agnostic. Therefore, it is applicable to all traditional
+RSs we introduced in Section 2. For example, for MF-based RSs, we can plug in
+ˆ𝑟MF
+𝑖 𝑗
+= u𝑇
+𝑖 · v𝑗, for DMF-based RSs, we plug in ˆ𝑟DMF
+𝑖 𝑗
+= 𝑓 𝑢
+𝑛𝑛(u𝑖)𝑇 · 𝑓 𝑣
+𝑛𝑛(v𝑗), etc.
+In practice, since the conditional unconfoundedness assumption in Eq. (3) is
+untestable, it is usually infeasible to calculate the exact value of 𝑒𝑖 𝑗 based on user/item
+covariates that satisfy Eq. (3). Nevertheless, we can still calculate approximate
+propensity scores ˜𝑒𝑖 𝑗 and reweight the observed ratings by 1/ ˜𝑒𝑖 𝑗, but the unbiasedness
+of Eq. (10) after the reweighting cannot be guaranteed. Here we introduce two
+strategies for the approximate estimation. If user/item features f𝑢
+𝑖 and f𝑣
+𝑗 are available,
+
+16
+Yaochen Zhu, Jing Ma, and Jundong Li
+5
+5
+5
+5
+5
+5
+5
+5
+5
+5
+5
+5
+5
+5
+1
+1
+1
+1
+1
+1
+1
+1
+Horror
+Lover
+Romance
+Lover
+Horror
+Romance
+5
+5
+5
+5
+5
+5
+5
+5
+5
+5
+5
+5
+5
+5
+5
+5
+Horror
+Lover
+Romance
+Lover
+Horror
+Lover
+Romance
+Lover
+Horror
+Romance
+Horror
+Romance
+(a) Observed Ratings
+(b) Rating Potential Outcomes
+(d) Predicted Ratings
+1
+1
+Horror
+Lover
+Romance
+Lover
+Horror
+Romance
+(c) Propensity Scores
+3
+4
+1
+4
+1
+4
+3
+4
+Fig. 5: An example adapted from Fig. (2) where the positivity assumption holds. Suppose again
+covariates 𝐶 represent the two-dimensional features (user type, movie type). (a) shows the observed
+ratings; (b) shows rating potential outcomes; (d) shows the predicted rating potential outcome of
+an RS model. The propensity scores 𝑒𝑖 𝑗 = 𝑝(𝑎𝑖 𝑗 |c) = E[𝑎𝑖 𝑗 |c] are shown in (c). Based on (a)(d)
+and Eq. (9), LObs = (5 − 1)2 × 2/8 = 4. Based on (b)(d) and Eq. (8), 𝐿True = (5 − 1)2 × 8/16 = 8.
+Based on (a)(c)(d) and Eq. (10), LIPW =
+1
+1/4 (5 − 1)2 × 2/16 = 8, which is unbiased for 𝐿True.
+˜𝑒𝑖 𝑗 can be estimated with logistic regression [39] as follows:
+˜𝑒LR
+𝑖 𝑗 = Sigmoid
+�� ∑︁
+𝑘
+𝑤𝑢
+𝑘 𝑓 𝑢
+𝑖𝑘
+�
++
+� ∑︁
+𝑘
+𝑤𝑣
+𝑘 𝑓 𝑣
+𝑗𝑘
+�
++ 𝑏𝑖 + 𝑏 𝑗
+�
+,
+(11)
+where Sigmoid(𝑥) = (1 + exp(−𝑥))−1, 𝑤𝑢
+𝑘 and 𝑤𝑣
+𝑘 are the regression coefficients,
+and 𝑏𝑖, 𝑏 𝑗 are the user and item-specific offsets, respectively. If user/item features
+f𝑢
+𝑖 and f𝑣
+𝑗 are not available, we can crudely approximate 𝑒𝑖 𝑗 based on the exposure
+data alone. For example, we can estimate ˜𝑒𝑖 𝑗 with Poisson factorization [60] as:
+˜𝑒PF
+𝑖 𝑗 ≈ 1 − exp
+�
+−𝝅𝑇
+𝑖 · 𝜸 𝑗
+�
+,
+(12)
+where 𝝅𝒊 and 𝜸𝒋 are trainable user and item embeddings with Gamma prior, and they
+can be inferred from the exposure data as discussed in [61]. Additional strategies to
+calculate the propensity scores can be found in [62, 63, 64, 65].
+The advantage of IPW is that the unbiasedness of Eq. (10) for rating potential
+outcome estimation can be guaranteed if the propensity scores 𝑒𝑖 𝑗 are correctly
+estimated. However, the accuracy of the propensity score estimation models relies
+heavily on the domain knowledge and expertise of human experts, which is untestable
+by experiments. In addition, IPW suffers from a large variance and numerical in-
+stability issues, especially when the estimated propensity scores 𝑒𝑖 𝑗 are very small.
+Therefore, variance reduction techniques such as clipping and multi-task learning
+are usually applied to improve the stability of the training dynamics [66, 67, 68].
+Substitute Confounder Adjustment. IPW-based RSs address exposure bias from
+the data’s perspective: They reweight the biased observational dataset to create
+a pseudo randomized dataset that allows unbiased training of RSs. Confounder
+adjustment-based methods, in contrast, estimate confounders 𝐶 that cause the expo-
+sure bias and adjust their effects in the rating prediction model (A simple adjustment
+
+Causal Inference for Recommender Systems: A survey
+17
+strategy is to control 𝐶 as extra covariates12). For the adjustment to be unbiased,
+classical causal inference requires the conditional unconfoundedess assumption in
+Eq. (3) hold, i.e., no unobserved confounders [33], which is generally infeasible
+in practice. Fortunately, recent advances in multi-cause causal inference [69] have
+shown that we can control substitute confounders estimated from item co-exposure
+data instead, where exposure bias can be mitigated with weaker assumptions.
+We use a𝑖 = [𝑎𝑖1, · · · , 𝑎𝑖𝐽] to denote the exposure status of all 𝐽 items to
+user 𝑖, which can be viewed as a bundle treatment in clinical trials [70]. Wang
+et al. [42] showed that if we can estimate user-specific latent variables 𝝅𝑖, such
+that conditional on 𝝅𝑖, the exposures of different items to the user are mutually
+independent, controlling 𝝅𝑖 can eliminate the influence of multi-cause confounders
+c𝑚
+𝑖 (i.e., confounders that simultaneously affect the exposure of multiple items and
+ratings). A simple proof of the claim is that, if c𝑚
+𝑖 can still influence a𝑖 and r𝑖 after
+conditioning on 𝝅𝑖, since c𝑚
+𝑖 is an unobserved common cause for the exposure of
+different items, 𝑎𝑖 𝑗 cannot be conditionally independent (see the discussion of the
+fork structure in section 3.2.2), which renders a contradiction. The rigorous proof
+can be found in [69]. Wang et al. further assumed that 𝑝(a𝑖|𝝅𝑖) = Π𝑗 𝑝(𝑎𝑖 𝑗|𝝅𝑖) =
+Π 𝑗Poission(𝝅𝑇
+𝑖 ·𝜸𝒋) and used the Poisson factorization to infer 𝝅𝑖 and 𝜸𝒋. Afterward,
+exposure bias can be mitigated by controlling 𝝅𝑖 as extra covariates in the RS model
+[33]. For example, controlling 𝝅𝑖 in MF-based RSs leads to the following adjustment:
+𝑟adj
+𝑖 𝑗 (𝑎𝑖 𝑗 = 1) ∼ N
+�
+u𝑇
+𝑖 · v 𝑗
+������
+user interests
++
+∑︁
+𝑘
+𝑤𝑘𝜋𝑖𝑘
+��������������
+adj. for expo. bias
+, 𝜎2
+𝑖 𝑗
+�
+.
+(13)
+The property of propensity scores can be utilized to further simplify Eq. (13): If un-
+confoundedness in Eq. (3) holds for 𝐶 = 𝝅𝑖, it will also hold for 𝐶 = ˜𝑒𝑖 𝑗 = 𝑝(𝑎𝑖 𝑗|𝝅𝑖)
+[58]. Therefore, we can control the approximate propensity scores estimated by 𝝅𝑖,
+i.e., ˜𝑒𝑖 𝑗 = 𝝅𝑇
+𝑖 · 𝜸𝒋, which leads to the simplified adjustment formula:
+𝑟adj
+𝑖 𝑗 (𝑎𝑖 𝑗 = 1) ∼ N
+�
+u𝑇
+𝑖 · v𝑗 + 𝑤𝑖 · ˜𝑒𝑖 𝑗, 𝜎2
+𝑖 𝑗
+�
+,
+(14)
+where 𝑤𝑖 is a user-specific coefficient that captures the influence of ˜𝑒𝑖 𝑗 on ratings.
+Despite the success in addressing exposure bias with weaker assumptions, one
+limitation of the above method is that, since Poisson factorization is a shallow model,
+it may fail to capture the complex influences of multi-cause confounders on item
+co-exposures. To address this problem, recent works have introduced deep neural
+networks (DNNs) to infer the user-specific substitute confounders 𝝅𝑖 from bundle
+treatment a𝑖 [71, 72]. These methods generally assume that a𝑖 are generated from 𝝅𝑖
+via 𝑝(a𝑖|𝝅𝑖) parameterized by a deep generative network 𝑓 exp
+𝑛𝑛 as:
+𝑝(a𝑖|𝝅𝑖) = Π𝑗Bernoulli(Sigmoid( 𝑓 exp
+𝑛𝑛 (𝝅𝑖) 𝑗)),
+(15)
+12 Consider again the toy example in Fig. 5. If we know exactly the user type and item type c for
+each user-item pair, the predictions can be unbiased even if the item exposures are non-randomized.
+
+18
+Yaochen Zhu, Jing Ma, and Jundong Li
+(a) SCM that considers item popularity
+M
+(b) SCM under intervention
+U
+R
+V
+M
+U
+R
+do(V)
+Fig. 6: (a): SCM that explicitly models item popularity. (b): SCM under intervention 𝑑𝑜(𝑉 ).
+where the intractable posterior of 𝝅𝑖 is then approximated with a Gaussian distribu-
+tion parameterized by DNNs via the variational auto-encoding Bayes algorithm [73],
+i.e., 𝑞(𝝅𝒊|a𝑖) = N ( 𝑓 𝝁
+𝑛𝑛(a𝑖), diag( 𝑓 𝜎2
+𝑛𝑛 (a𝑖))), where 𝑓 𝝁
+𝑛𝑛 and 𝑓 𝝈2
+𝑛𝑛 are two DNNs that
+calculate the posterior mean and variance (before diagonalization) of 𝝅𝑖. With deep
+generative models introduced to estimate the substitute confounders 𝝅𝑖, non-linear
+influences of multi-cause confounders on item exposures can be adjusted in the RS
+models, where exposure bias can be further mitigated in recommendations.
+The key advantage of substitute confounder estimation-based causal RSs is that
+controlling confounders in the potential outcome prediction model generally leads
+to lower variance than IPW-based methods [42]. However, these models need to
+estimate substitute confounders 𝝅𝑖 from the item co-exposures and introduce extra
+parameters in the RS models to adjust their influences, which may incur extra bias if
+the confounders and the parameters are not correctly estimated. In addition, exposure
+bias due to single-cause confounders cannot be addressed by these methods.
+4.1.2 Popularity Bias
+Popularity bias can be viewed as a special kind of exposure bias where users are
+overly exposed to popular items [74, 75]. Therefore, it can be addressed with tech-
+niques introduced in the previous section, especially the IPW-based methods [76].
+The reason is that, if we define the popularity of an item as its exposure rate:
+𝑚 𝑗 =
+�
+𝑖 𝑎𝑖 𝑗
+�
+𝑗
+�
+𝑖 𝑎𝑖 𝑗
+,
+(16)
+we can view 𝑚 𝑗 as pseudo propensity scores and use IPW to reweight the observed
+ratings. Alternatively, we can also analyze and address popularity bias with the
+structural causal model (SCM), where the causal mechanism that generates the
+observed ratings under the influence of item popularity is deeply investigated.
+The discussion is mainly based on the popularity-bias deconfounding (PD) al-
+gorithm proposed in [48]. PD assumes that the relations among user interests u𝑖,
+item latent attributes v 𝑗, item popularity 𝑚 𝑗, and observed ratings 𝑟𝑖 𝑗 can be repre-
+sented by the causal graph illustrated in Fig. 6, where item popularity can be clearly
+identified as a confounder that spuriously correlates the item attributes and the user
+
+Causal Inference for Recommender Systems: A survey
+19
+ratings. PD aims to eliminate such spurious correlations with backdoor adjustment,
+such that the causal influences of u𝑖 and v𝑗 on 𝑟𝑖 𝑗 (which represents users’ interests
+on intrinsic item properties) can be properly identified. Recall that backdoor ad-
+justment with SCM involves two stages: (1) During the training phase, the relevant
+structural equations in the causal graph are estimated from the collected dataset. (2)
+Afterward, we adjust the influence of confounders according to Eq. (5) to remove
+the spurious correlations. Therefore, we need to estimate 𝑝𝐺(𝑟𝑖 𝑗|u𝑖, v𝑗, 𝑚 𝑗) with the
+observed ratings 𝑟𝑖 𝑗 and item popularty 𝑚 𝑗 and infer the latent variables u𝑖 and v𝑗.
+In PD, 𝑝𝐺(𝑟𝑖 𝑗|u𝑖, v𝑗, 𝑚 𝑗) is modeled as a variant of MF as follows:
+𝑝𝐺(𝑟𝑖 𝑗|u𝑖, v𝑗, 𝑚 𝑗) ∝ Elu(u𝑇
+𝑖 · v𝑗)
+����������������������
+user interests
+×
+𝑚𝜆
+𝑗
+����
+pop. bias
+,
+(17)
+where 𝜆 is a hyper-parameter that denotes our belief toward the strength of influence
+of item popularity on ratings, and the function Elu (defined as Elu(𝑥) = 𝑒(𝑥) if 𝑥 < 0
+else 𝑥 + 1) makes the RHS of Eq. (17) a proper unnormalized probability density
+function. After u𝑖, v 𝑗 are estimated from the datasets with Eq. (17), we conduct an
+intervention on the item node 𝑉 in the causal graph (see Eq. (5)), where the spurious
+correlation due to item popularity can be eliminated with backdoor adjustment:
+𝑝(𝑟𝑖 𝑗|𝑑𝑜(u𝑖, v𝑗)) ∝ E𝑝(𝑚𝑗) [Elu(u𝑇
+𝑖 ·v𝑗) ×𝑚𝜆
+𝑗] = Elu(u𝑇
+𝑖 ·v𝑗) ×E𝑝(𝑚𝑗) [𝑚𝜆
+𝑗]. (18)
+Since the second term E𝑝(𝑚𝑗) [𝑚𝜆
+𝑗] in Eq. (18) is a constant and Elu is a monotonically
+increasing function, they have no influence on the ranking of the uninteracted items
+in the prediction phase. Therefore, we can drop them and use ˆ𝑟𝑖 𝑗 = u𝑇
+𝑖 · v𝑗 as the
+unbiased rating predictor to generate future recommendations.
+Generally, the debiasing mechanism of PD is very intuitive and universal among
+backdoor adjustment-based causal RSs [25, 24]: When fitting the RS model on
+the biased training set, we explicitly introduce the item popularity 𝑚 𝑗 (i.e., the
+confounder) in Eq. (17) to explain away the spurious correlation between item
+attributes and the observed user ratings. Therefore, the user/item latent variables
+u𝑖 and v𝑗 used to generate future recommendations, i.e., ˆ𝑟𝑖 𝑗 = u𝑇
+𝑖 · v𝑗, can focus
+exclusively on estimating users’ true interests on intrinsic item properties.
+Is popularity bias always bad? Recently, more researchers have begun to
+believe that popularity bias is not necessarily bad for RSs, because some items
+are popular because they per se have better quality than other items or they
+catch the current trends of user interests, where more recommendations for
+these items can be well-justified [77, 78]. Therefore, rather than setting the
+interventional distribution of item popularity to 𝑝(𝑚 𝑗), PD introduced above
+as well as some other methods [48] further propose to make it correspond to
+item qualities or reflect the future popularity predictions. We will introduce
+these strategies in Section 4.3 regarding causal generalizations of RSs.
+
+20
+Yaochen Zhu, Jing Ma, and Jundong Li
+(a) SCM that considers item content feature
+Fc and item exposure feature Fb
+Fb
+(b) SCM that models the undesirable direct
+effect of item exposure feature Fb
+U
+R
+V
+Fc
+Fb
+U
+R
+V*
+Fb*
+Fc*
+Fig. 7: (a): The SCM that considers both the causal influences of item content feature 𝐹 𝑐 and
+item exposure feature 𝐹 𝑏 on item latent variable 𝑉 . (b): The counterfactual SCM where 𝑉 ∗ is
+determined by baseline value 𝐹 𝑏∗ and 𝐹 𝑐∗ to model the undesirable direct effects of 𝐹 𝑏.
+4.1.3 Clickbait Bias
+Different from previous subsections that mainly focus on causal debiasing strategies
+for collaborative filtering-based RSs, this section discusses content-based recom-
+mendations. Specifically, we discuss the clickbait bias, which is defined as the bias
+of overly recommending items with attractive exposure features such as sensational
+titles but with low content qualities. The discussion is mainly based on [27]. We
+assume that item features f𝑣
+𝑗 can be further decomposed into the item content feature
+f𝑐
+𝑗 that captures item content information and the item exposure feature f𝑏
+𝑗 whose
+main purpose is to attract users’ attention. Taking micro-video as an example, item
+content feature f𝑐
+𝑗 can be the audiovisual content of the video, whereas item exposure
+feature f𝑏
+𝑗 can be its title, which is not obliged to describe its content faithfully.
+The relations among user interests u𝑖, item exposure feature f𝑏
+𝑗 , item content
+feature f𝑐
+𝑗 , item fused features v𝑗, and the observed ratings 𝑟𝑖 𝑗 are depicted in the
+causal graph in Fig. 7-(a). We note that clickbait bias occurs when a user’s recorded
+click on an item because she was cheated by the item exposure feature f𝑏
+𝑗 before
+viewing the item content f𝑐
+𝑗 . Therefore, the bias can be defined as the direct influence
+of f𝑏
+𝑗 on ratings 𝑟𝑖 𝑗 represented by the causal path 𝐹𝑏 → 𝑅. To eliminate the clickbait
+bias, we need to block the direct influence of 𝐹𝑏 on rating predictions, such that the
+item content quality can be comprehensively considered in recommendations.
+As with SCM-based causal RSs, we first estimate structural equations of interest
+in the causal graph, i.e., 𝑝𝐺(v𝑗|f𝑏
+𝑗 , f𝑐
+𝑗 ) and 𝑝𝐺(𝑟𝑖 𝑗|u𝑖, v𝑗, f𝑏
+𝑗 ). Since distributions
+in [27] are reasoned in a deterministic manner (i.e., Gaussian distributions with
+infinite precision), we keep the discussion consistent with them. Specifically, we
+use v 𝑗 (f𝑏
+𝑗 , f𝑐
+𝑗 ) = 𝑓 𝑓 𝑓 (f𝑏
+𝑗 , f𝑐
+𝑗 ) to represent the structural equation 𝑝𝐺(v𝑗|f𝑏
+𝑗 , f𝑐
+𝑗 ),
+where 𝑓 𝑓 𝑓 is the feature fusion function that aggregates f𝑏
+𝑗 , f𝑐
+𝑗 into v 𝑗, and use
+𝑟𝑖 𝑗 (u𝑖, v𝑗, f𝑏
+𝑗 ) to represent the structural equation 𝑝𝐺(𝑟𝑖 𝑗|u𝑖, v𝑗, f𝑏
+𝑗 ), respectively.
+To explicitly disentangle the influence of item exposure feature f𝑏
+𝑗 and item latent
+variable v𝑗 on the observed ratings, 𝑟𝑖 𝑗 (u𝑖, v𝑗, f𝑏
+𝑗 ) is assumed to factorize as follows:
+
+Causal Inference for Recommender Systems: A survey
+21
+𝑟𝑖 𝑗 (u𝑖, v𝑗, f𝑏
+𝑗 ) = 𝑓 𝑢𝑣
+𝑛𝑛 (u𝑖, v𝑗)
+������������������
+user interests
+· Sigmoid
+�
+𝑓 𝑢 𝑓
+𝑛𝑛 (u𝑖, f𝑏
+𝑗 )
+�
+����������������������������������������������������
+potential clickbait bias
+,
+(19)
+where the Sigmoid function provides necessary non-linearity in the fusion process.
+Essentially, Eq. (19) represents the causal mechanism that generates the observed
+ratings, which entangles both user interests in item content and clickbait bias.
+However, after learning the latent variables u𝑖, v𝑗 and functions 𝑓 𝑢 𝑓
+𝑛𝑛 , 𝑓 𝑢𝑣
+𝑛𝑛 via Eq.
+(19), removing clickbait bias from the rating predictions is not as straightforward
+as the PD algorithm, because we should eliminate only the direct influence of item
+exposure feature f𝑏
+𝑗 on ratings 𝑟𝑖 𝑗, while preserving its indirect influence mediated by
+item latent variable v𝑗, such that all available item features can be comprehensively
+considered in recommendations. To achieve this purpose, we first calculate the natural
+direct effect (NDE) [79] of item exposure feature f𝑏
+𝑗 on ratings 𝑟𝑖 𝑗 as follows:
+NDE(u𝑖, v∗
+𝑗, f𝑏
+𝑗 ) = 𝑟𝑖 𝑗 (u𝑖, v∗
+𝑗, f𝑏
+𝑗 ) − 𝑟𝑖 𝑗 (u𝑖, v∗
+𝑗, f𝑏∗
+𝑗 ),
+(20)
+where v∗
+𝑗 = 𝑓 𝑓 𝑓
+𝑛𝑛 (f𝑏∗
+𝑗 , f𝑐∗
+𝑗 ), and the baseline values f𝑏∗
+𝑗 , f𝑐∗
+𝑗
+are treated as if the
+corresponding features are missing from the item [27]. Since the second term
+𝑟𝑖 𝑗 (u𝑖, v∗
+𝑗, f𝑏∗
+𝑗 ) in Eq. (20) denotes the user’s rating to a “void” item and can be
+viewed as a constant, it will not affect the rank of the items. So we only adjust the
+first term of Eq. (20), which reasons with user 𝑖’s rating to item 𝑗 in a counterfactual
+world where item 𝑗 has only the exposure feature f𝑏
+𝑗 but no content and fused features
+f𝑐∗
+𝑗 and v∗
+𝑗, in Eq. (19) (Fig. 7-(b)). The adjustment leads to the following estimator,
+ˆ𝑟𝑖 𝑗 = 𝑟𝑖 𝑗 (u𝑖, v𝑗, f𝑏
+𝑗 ) − 𝑟𝑖 𝑗 (u𝑖, v∗
+𝑗, f𝑏
+𝑗 ) ≜
+𝑟𝑖 𝑗 (u𝑖, v𝑗, f𝑏
+𝑗 )
+��������������������������
+user interests + clickbait
+− 𝑟𝑖 𝑗 (u𝑖, v∗
+𝑗, f𝑏
+𝑗 )
+��������������������������
+clickbait bias
+.
+(21)
+Eq. (21) removes the direct influence of f𝑏
+𝑗 on rating predictions, such that item
+content quality can be comprehensively considered in future recommendations.
+4.1.4 Unfairness
+Recently, with the growing concern of algorithmic fairness, RSs are expected to
+show no discrimination against users from certain demographic groups [80, 81, 82].
+However, traditional RSs may capture the undesirable associations between users’
+sensitive information and their historical activities, which leads to potentially offen-
+sive recommendations to the users. Causal inference can help identify and address
+such unfair associations, where fairness can be promoted in future recommendations.
+This section focuses on the user-oriented fairness discussed in [83], which is defined
+as the bias where RS discriminately treats users with certain sensitive attributes.
+When considering the user-oriented fairness for RSs, a subset of user features f𝑖,
+which we denote as s𝑖, is assumed to contain the sensitive information of users, such
+
+22
+Yaochen Zhu, Jing Ma, and Jundong Li
+R
+S
+F
+U
+V
+R
+(a) Causal Generation Process of
+the Observational Dataset
+(b) Causal Decision Process of
+the Traditional RSs
+(b)
+(a)
+Fig. 8: The SCM that reasons with the causal decision mechanism of traditional RSs. Observed
+user ratings 𝑅 can be causally driven by user features 𝐹, including sensitive features 𝑆, which can
+then unfairly influence the inference of user latent variables 𝑈 and new rating predictions ˆ𝑅.
+as gender, race, and age. Features s𝑖 are sensitive because recommendations that
+improperly rely on these features may be offensive to users, which degrade both their
+online experiences and their trust in the system. The causal graph that depicts the
+causal decision mechanism of most traditional RSs is illustrated in Fig. 8 [83]. From
+Fig. 8 we can find that the user historical behaviors, i.e., the observed ratings 𝑟𝑖 𝑗, are
+causally driven by user features f𝑖, including user sensitive features s𝑖. Therefore, the
+user latent variables u𝑖 inferred from 𝑟𝑖 𝑗 could capture sensitive user information in
+s𝑖, which unfairly influences the rating predictions ˆ𝑟𝑖 𝑗 in the future.
+To address this problem, Li et al. [83] proposed to disentangle the user sensitive
+features s𝑖 from the user latent variable u𝑖, such that the unfair influence of s𝑖 on
+u𝑖 represented by the causal chain 𝑆 → 𝑅 → 𝑈 can be maximally suppressed in
+the future recommendations. A common strategy to achieve the disentanglement is
+adversarial training [84], where we train a discriminator 𝑓 cls
+𝑛𝑛 (u𝑖) → s𝑖 that predicts
+the sensitive features s𝑖 from user latent variables u𝑖 alongside the RS. While fitting
+the RS on the observe ratings 𝑟𝑖 𝑗, we constrain the inferred u𝑖 to fool the discriminator
+𝑓 cls
+𝑛𝑛 by making wrong predictions about s𝑖, which discourages u𝑖 from capturing
+sensitive information in 𝑟𝑖 𝑗 due to its unfair correlations with s𝑖. Here we take the
+MF-based RS as an example to show the details. We use LRec to denote the original
+training objective of the MF-based RS that maximizes the log-likelihood on observed
+ratings 𝑟𝑖 𝑗 and use Lcls to denote the loss function of the discriminator 𝑓 cls
+𝑛𝑛 . The
+adjusted training objective LFair with fairness constraint becomes the following:
+LFair = LRec(u𝑇
+𝑖 · v𝑗, 𝑟𝑖 𝑗)
+������������������������������������
+user interests
+−𝜆 · Lcls( 𝑓 cls
+𝑛𝑛 (u𝑖), s𝑖)
+������������������������������������
+fairness constraint
+,
+(22)
+where 𝜆 is a hyper-parameter that balances the recommendation performance and
+the fairness objective. Generally, a higher 𝜆 leads to better fairness, but it also
+restricts the capacity of the user latent variables u𝑖, which could negatively impact
+the recommendation performance. Although here we use the MF-based RS as an
+example, it is straightforward to generalize Eq. (22) to DMF or AE-based RS by
+replacing the u𝑇
+𝑖 · v𝑗 term with the corresponding rating estimators.
+
+Causal Inference for Recommender Systems: A survey
+23
+U
+R
+Uc
+(b) Generalization to PoI recommendation
+(a) Causal graph for DICE
+U
+R
+Uc
+user interests
+user conformity
+purchases
+visits
+user interests
+Geo. influence
+Fig. 9: Causal Graphs for DICE (a) and its generalization to PoI recommendations (b).
+4.2 Causal Explanation in Recommendations
+In previous sections, we have introduced causality to address various types of bias
+and spurious correlation issues for traditional RSs. In this section, we use causality
+to explain the user decision process. Specifically, we discuss an interesting question
+aiming to disentangle users’ intent that causally explains their past behaviors, i.e.,
+did a user purchase an item because she conformed to the current trend or because
+she really liked it? The tricky part of this question is that: in reality, we only observe
+the effects, i.e., the purchases, which can be explained by both causes.
+4.2.1 Disentangling Interest and Conformity with Causal Embedding
+The discussion is based on DICE proposed in [56]. To simplify the discussion, we
+consider 𝑟𝑖 𝑗 as implicit feedback and define the set of user, positive item (𝑗 : 𝑟𝑖 𝑗 =
+1), negative item (𝑘 : 𝑟𝑖𝑘 = 0) triplets as R 𝑝𝑛 = {(𝑖, 𝑗, 𝑘)|𝑟𝑖 𝑗 = 1 ∧ 𝑟𝑖𝑘 = 0}.
+The popularity of each item 𝑗, i.e., 𝑚 𝑗, which reflects the current trend, can be
+calculated with Eq. (16). Observing that the causal relation between user interests
+𝑈, user conformity 𝑈𝑐 and observed ratings 𝑅 can be represented as a V-structure
+in Fig. 9-(a), DICE exploits the colliding effect to achieve the disentanglement, i.e.,
+outcomes that cannot be explained by one cause are more likely caused by another
+(see discussions in Section 3.2.2). Therefore, although users’ interests cannot be
+directly estimated from their ratings 𝑟𝑖 𝑗 due to entanglement, their conformity to the
+trend can be estimated by the popularity level of item 𝑗, and positive feedback not
+likely caused by conformity has a higher chance of reflecting users’ true interests.
+In implementation, DICE assumes that the observed ratings 𝑟𝑖 𝑗 can be decom-
+posed into the sum of a conformity part 𝑟𝑐
+𝑖 𝑗 = 𝑓 𝑐(u𝑐
+𝑖 , v𝑐
+𝑗) and a user interests part
+𝑟𝑖
+𝑖 𝑗 = 𝑓 𝑖(u𝑖
+𝑖, v𝑖
+𝑗), where u𝑐,𝑖
+𝑖 , v𝑐,𝑖
+𝑗
+are learnable user, item embeddings that reflect
+user 𝑖’s interests in (i.e., superscript 𝑖) and conformity to (i.e., superscript 𝑐) item
+𝑗, respectively. According to the colliding effect of causal graphs, we can split the
+triplets in R 𝑝𝑛 into two parts: In the first part R (1)
+𝑝𝑛, positive item 𝑎 in the triplet has
+a higher popularity level than the negative item 𝑏, i.e., 𝑚𝑎 > 𝑚𝑏. In this case, we
+can draw two general conclusions from this triplet: (1) Overall, the user prefers item
+𝑎 over 𝑏; (2) She is more likely to conform to item 𝑎 than item 𝑏 due to 𝑎’s higher
+
+24
+Yaochen Zhu, Jing Ma, and Jundong Li
+popularity. These conclusions lead to the two inequalities as follows:
+∀(𝑖, 𝑎, 𝑏) ∈ R (1)
+𝑝𝑛, we have
+�
+𝑟𝑐
+𝑖𝑎 > 𝑟𝑐
+𝑖𝑏 (conformity)
+𝑟𝑖
+𝑖𝑎 + 𝑟𝑐
+𝑖𝑎 > 𝑟𝑖
+𝑖𝑏 + 𝑟𝑐
+𝑖𝑏 (overall preference),
+(23)
+where the dependency of 𝑟𝑐,𝑖
+𝑖{𝑎,𝑏} on latent variables u𝑐,𝑖
+𝑖 , v𝑐,𝑖
+{𝑎,𝑏} are omitted for
+simplicity. The second part, i.e., R (2)
+𝑝𝑛, is the key to achieving disentanglement,
+because for every triplet (𝑖, 𝑐, 𝑑) in R (2)
+𝑝𝑛, the negative item 𝑑 is more popular than
+the positive item 𝑐. In this case, user 𝑖 could have simply conformed to the trend
+and chosen item 𝑑 to consume, but instead, she actively chose the less popular
+item 𝑐. Therefore, we can draw one more specific conclusion that leads to the
+disentanglement between user interests and conformity: The choice of item 𝑐 over 𝑑
+is more likely due to user interests. Therefore, we can form three inequalities as:
+∀(𝑖, 𝑐, 𝑑) ∈ R (2)
+𝑝𝑛, we have
+�
+𝑟𝑖
+𝑖𝑐 > 𝑟𝑖
+𝑖𝑑 (interests), 𝑟𝑐
+𝑖𝑐 < 𝑟𝑐
+𝑖𝑑 (conformity),
+𝑟𝑖
+𝑖𝑐 + 𝑟𝑐
+𝑖𝑐 > 𝑟𝑖
+𝑖𝑑 + 𝑟𝑐
+𝑖𝑑 (overall preference).
+(24)
+The inequalities in Eqs. (23) and (24) can be solved by ranking-based loss in RSs, such
+as Bayesian personalized ranking (BPR) [85], where the disentangled embeddings
+u𝑐,𝑖
+𝑖 , v𝑐,𝑖
+𝑗
+and the match functions 𝑓 𝑐,𝑖(·, ·) can be learned from R (1)
+𝑝𝑛 and R (2)
+𝑝𝑛. Finally,
+we form a rating predictor ˆ𝑟𝑖 𝑗 = 𝑓 𝑖(u𝑖
+𝑖, v𝑖
+𝑗) + 𝑓 𝑐(u𝑐
+𝑖 , v𝑐
+𝑗) for future recommendations.
+4.2.2 Generalizations of DICE
+DICE disentangles the user intent and promotes the explainability of RSs from
+the data’s perspective: It partitions the triplets (𝑖, 𝑗, 𝑘) in R 𝑝𝑛 into two disjoint
+subsets R (1)
+𝑝𝑛 and R (2)
+𝑝𝑛 based on the relative popularity of the positive and negative
+items, and shows that the triplets in R (2)
+𝑝𝑛 are informative to distinguish the user
+interests from their conformity to the popularity trend. The basic idea of DICE is
+generalizable to promote explainability for other types of recommendation tasks, if
+we can find alternative causal explanations to challenge the assumption that the
+observed positive feedback in these tasks can be attributed solely to user interests.
+For example, in point-of-interests (PoI) recommendations, the target items are
+specific point locations that users may find useful or interesting to visit, such as
+restaurants, grocery stores, and malls [7]. In this task, the location of a PoI is
+an important alternative explanation for users’ visits to the PoI other than user
+interests, because nearby POIs are more convenient to visit than the remote ones
+[86]. Therefore, to disentangle user interests from potential geographical factors
+that could causally influence users’ choices, we can take a similar strategy as DICE
+and partition the user historical visit records according to the distance of positive
+and negative PoIs to users. Then, the disentangled user interest embeddings can be
+estimated based on the partitioned dataset with the same ranking-based approach.
+
+Causal Inference for Recommender Systems: A survey
+25
+4.2.3 Other Works on Explainable RSs
+Explanable recommendation is a broad topic [87], where disentangling user’s intent
+based on data partitioning is a small part. There are also plenty of works that
+focus on improving the explainability of RSs from the model’s side, where specific
+disentanglement modules, such as prototype learning [88], context modeling [89],
+and aspect modeling [90], are designed and integrated with traditional RS models to
+further enhance their transparency and explainability. We refer interested readers to
+the corresponding papers as well as [91, 92] for further investigation.
+4.3 Causal Generalization of Recommendations
+After estimating the causal relations from potentially biased and entangled obser-
+vational datasets, the generalization ability of RSs can be substantially enhanced,
+because even if the context (or environment) in which we make recommendations
+changes (e.g., item popularity, user conformity, etc.), we can still basing the rec-
+ommendations on causal relations that are stable and invariant, while discarding or
+correcting other undesirable correlations that are transient and susceptible to change
+[56, 93]. In this section, we use the PD algorithm for popularity bias and the DICE
+algorithm for causal explainability as two examples to show how the generalization
+of RSs can be improved with causal intervention and disentanglement.
+4.3.1 Generalization Based on Intervention
+First, we take the PD algorithm as an example to show how causal intervention
+can improve the generalization of RSs within a dynamic environment. In RS, it is
+generally assumed that user interests can remain unchanged for a certain period of
+time, i.e., the causal structure 𝑈 → 𝑅 ← 𝑉 in Fig. (6) represents the stable user
+interests on intrinsic item properties. However, the popularity of different items, i.e.,
+the context in which we make recommendations, can shift rapidly during the same
+period [78]. Recall that PD disentangles the causal influences of user interests and
+item popularity on ratings via the product of two terms, i.e., Elu(u𝑇
+𝑖 · v𝑗) and 𝑚𝜆
+𝑗, as
+Eq. (17). Suppose 𝑚 𝑗 represents the current popularity level of item 𝑗. If we predict
+that the popularity of item 𝑗 will change to 𝑚′
+𝑗 in the future [6], we can conduct
+an intervention that sets 𝑀 to the predicted value 𝑚′
+𝑗 in the structural equation
+𝑝𝐺(𝑅|𝑈,𝑉, 𝑀) and predict future ratings 𝑟′
+𝑖 𝑗 via the following formula:
+𝑝𝐺(𝑟′
+𝑖 𝑗|u𝑖, v 𝑗, 𝑑𝑜(𝑚′
+𝑗)) ∝ Elu(u𝑇
+𝑖 · v𝑗)
+����������������������
+stable user interests
+×
+(𝑚′
+𝑗)𝜆
+����
+future popularity
+,
+(25)
+
+26
+Yaochen Zhu, Jing Ma, and Jundong Li
+where the user, item latent variables u𝑖 and v𝑗 learned from the current time step
+remain unaltered. With the influence of future changes in item popularity on ratings
+considered in the predictions, service providers can make strategic decisions to allo-
+cate resources for items with different popularity potentials. In contrast, traditional
+RSs could mistakenly capture the influence of the current popularity level of items
+on ratings as user interests. Therefore, they will not generalize well when the item
+popularity 𝑚 𝑗 changes to a different level 𝑚′
+𝑗 due to time evolution.
+4.3.2 Generalization Based on Disentanglement
+In addition, causal disentanglement can promote the generalization of RSs by iden-
+tifying and basing recommendations on causes that are more robust to potential
+changes in the environments [94, 95]. For example, if users’ conformity and interest
+are disentangled based on their historical behaviors, if a user’s conformity reduces to
+a low level due to certain reasons, since user interests are assumed to be stable within
+a certain period of time, we can still use the learned user/item interest variables u𝑖
+𝑖,
+v𝑖
+𝑗 to make recommendations based on their interests, where the previously esti-
+mated unreliable user conformity information can be discarded or down-weighted.
+In contrast, for traditional RSs, different factors that causally influence their histor-
+ical behaviors are entangled as a single user latent variable u𝑖. Therefore, even if
+some less stable causes of user behaviors are known to change (e.g., in the PoI RS
+introduced above, a user could move to a new place where the convenience levels of
+different PoIs change for the user), these models will still utilize the outdated causes
+to make recommendations, which could fail to generalize to the new environment.
+5 Evaluation Strategies for Causal RSs
+In the previous sections, we have discussed various causal inference techniques that
+are promising to address multiple types of biases, entanglement, and generalization
+problems in traditional RSs. However, without a well-designed model evaluation
+strategy, it is difficult to tell whether the proposed causal RS model is indeed effective,
+nor can we guarantee that the model will perform reliably after deploying in a real-
+world environment. The evaluation of causal models is particularly difficult, because
+the groundtruths, i.e., the causal effects of interest, are usually infeasible. Despite
+the challenges, there are several strategies that can reliably evaluate causal RSs
+with biased real-world data, and we will thoroughly discuss them in this section. In
+addition, we also compile the available real-world datasets that conduct randomized
+experiments to eliminate exposure bias to facilitate future causal RS research.
+
+Causal Inference for Recommender Systems: A survey
+27
+5.1 Evaluation Strategies for Traditional RSs
+The assessment of traditional RSs generally follows three steps: First, the observed
+ratings 𝑟𝑖 𝑗 in the rating matrix R are split into the non-overlapping training set R𝑡𝑟
+and test set R𝑡𝑒, usually by randomly holding out a certain percentage of the observed
+ratings from each user. Then, the proposed RS is trained on ratings in R𝑡𝑟 to learn
+the latent variables and the associated functional models (see Section 2). Finally,
+the trained RS predicts the missing ratings in R𝑡𝑟 for each user, where the results
+are compared with the held-out ratings in R𝑡𝑒 to evaluate the model performance.
+The quality of rating predictions can be measured by accuracy-based metrics such
+as mean squared error (MSE) and mean absolute error (MAE), and ranking-based
+metrics such as recall, precision, normalized discounted cumulative gain (NDCG),
+etc. More information on these evaluation metrics can be found in [96].
+5.2 Challenges for the Evaluation of Causal RSs
+The above evaluation strategy, however, is not directly applicable to causal RSs,
+because ratings in R𝑡𝑒 may have the same spurious correlation and bias as ratings
+in R𝑡𝑟, which makes the evaluation on R𝑡𝑒 a biased measure of the true model
+performance. Therefore, to unbiasedly evaluate the effectiveness of causal RSs, it
+is ideal that we have a biased/entangled training set R𝑏
+𝑡𝑟 to learn the model, and an
+unbiased/disentangled test set R𝑢𝑏
+𝑡𝑒 for model evaluation, such that the effectiveness
+of the causal debiasing/disentangling algorithm can be directly verified from exper-
+iments. However, such unbiased/disentangled test set R𝑢𝑏
+𝑡𝑒 can be difficult to acquire
+and expansive to establish. Therefore, we first introduce common data simulation
+strategies for causal RS evaluation. We then discuss how real-world datasets can be
+directly utilized to further promote the credibility of causal RS research.
+5.3 Evaluation Based on Simulated Datasets
+A good dataset simulation strategy to evaluate causal RSs should have the following
+properties: (1) The generation mechanisms of the bias and entanglement to be studied
+are clearly identified, credibly designed, and can be adjusted in a flexible manner;
+(2) The available real-world information is utilized as much as possible.
+5.3.1 Simulation Based on Generative Models
+One promising dataset simulation strategy that satisfies the above criteria is to use
+deep generative models. Here we take exposure bias as an example to demonstrate
+how it can be simulated from real-world datasets [71]. The simulation is composed of
+
+28
+Yaochen Zhu, Jing Ma, and Jundong Li
+two phases. In the training phase, two variational auto-encoders (VAEs) [22, 73] are
+trained on the exposure and rating data in a real-world dataset (e.g., the MovieLens
+dataset [5]), which results in two decoder networks 𝑓 𝑎
+𝑛𝑛 and 𝑓 𝑟
+𝑛𝑛 that generate item
+exposures a𝑖 ∈ {0, 1}𝐽 and user ratings r𝑖 ∈ R𝐽 from 𝐾-dimensional Gaussian
+user latent variables u𝑎
+𝑖 ∼ N (0, I𝐾) and u𝑟
+𝑖 ∼ N (0, I𝐾), respectively. The decoders
+capture the generative distributions of item exposures and user ratings based on
+the data of real users, where the available real-world information is effectively
+utilized. In the generation phase, for each hypothetical user 𝑖′, we draw a confounder
+c𝑖′ ∼ N (0, I𝐾) that simultaneously affects u𝑎
+𝑖′ and u𝑟
+𝑖′. Then, to simulate the exposure
+bias, we set u𝑎
+𝑖′ = c𝑖′ and u𝑟
+𝑖′ = 𝜆 · c𝑖′ + (1 − 𝜆)𝝐𝑖′ and use 𝑓 𝑎
+𝑛𝑛, 𝑓 𝑟
+𝑛𝑛 to generate the
+simulated item exposures a𝑖′ and ratings r𝑖′, where 𝝐𝑖′ ∼ N (0, I𝐾) and hyper-
+parameter 𝜆 controls the strength of the confounding bias. Finally, we mask r𝑖′ with
+a𝑖′ to form the biased training set R𝑏
+𝑡𝑟, and keep the generated ratings r𝑖′ unmasked
+in the test set R𝑢𝑏
+𝑡𝑒 for an unbiased evaluation of model performance.
+The advantage of dataset simulation strategies based on generative models is
+that the true causal mechanisms of interest, such as the rating potential outcomes,
+are available in the evaluation stage, which is generally impossible for real-world
+datasets. Therefore, the effectiveness of causal RSs can be easily verified based on
+the simulated groundtruths. In addition, the simulations are flexible as the strength
+of biases and entanglements can be set into different levels (e.g., 𝜆 in the example),
+where the sensitivity and robustness of causal RSs can be thoroughly investigated.
+5.3.2 Test Set Intervention
+Another reliable dataset simulation strategy is test set intervention, where an in-
+tervened test set is created from the original test set, such that it has a different
+bias/entanglement distribution from the training set [56, 60, 97]. For example, to
+study the popularity bias, we can first select observed ratings from R such that 90%
+of the interacted items are popular and 10% are unpopular to form the training set
+R𝑡𝑟 [98]. We then select from the remaining ratings, i.e., the original test set R𝑡𝑒, a
+subset with a different ratio of popular and unpopular items (e.g., 10% popular and
+90% unpopular) to form the intervened test set R𝑖𝑛𝑡
+𝑡𝑒 . If the causal RSs trained on
+R𝑡𝑟 can still perform well on the intervened test set R𝑖𝑛𝑡
+𝑡𝑒 , the model’s invariance to
+the popularity bias can be supported. A similar test set intervention strategies can be
+used to evaluate the disentanglement of user interests and conformity for DICE [56].
+The advantage of the test set intervention-based causal RS evaluation strategy is
+that extra assumptions that cannot be justified by real-world information are mini-
+mally introduced, because the intervention is usually achieved by selecting samples
+from the original test set to change the data distribution, which does not introduce
+extra assumptions of the generative mechanisms or hypothetical users, items, and
+ratings. From this perspective, the evaluation results based on test set intervention
+may be more credible compared with the generative model-based strategies.
+
+Causal Inference for Recommender Systems: A survey
+29
+5.4 Evaluation Based on Real-world Datasets
+5.4.1 Randomized Experiments
+For the study of exposure bias, it is feasible to establish-bias free real-world datasets,
+where ratings for either every item or randomly selected items are collected from a
+subset of users. This can be extremely expansive and user-unfriendly, but recent years
+have witnessed a growing interest in causal RS research from the industry, where
+more such randomized datasets are established and released to facilitate causal RS
+research. The available real-world datasets are compiled as follows:
+• Coat datasets13 [39] (2016). The Coat dataset is a small-scale dataset crowd-
+sourced from the Amazon Mechanical Turkers platform with 300 users and 290
+items. Specifically, each Turker is first asked to self-select 24 coats to rate, where
+the ratings form the biased training set R𝑏
+𝑡𝑟. Then each Turker is asked to rate 16
+random coats, and these ratings form the unbiased test set R𝑢𝑏
+𝑡𝑒 .
+• Yahoo! R3 dataset14 [99, 100] (2009). The Yahoo! R3 dataset is collected from
+the Yahoo! Music platform. The biased training set R𝑏
+𝑡𝑟 is composed of 300,000
+self-supplied ratings from 15,400 users to 1,000 items. In addition, a subset of
+5,400 users is presented with ten randomly selected items to rate, and the ratings
+are used to create the unbiased test set R𝑢𝑏
+𝑡𝑒 .
+• KuaiRec dataset15 [101] (2022). The KuaiRec dataset is established based on
+a popular micro-video sharing platform, KuaiShou, in China (known as Kwai
+internationally). The dataset records self-supplied ratings from 7,176 users to
+10,728 items as the biased training set R𝑏
+𝑡𝑟. The unbiased test set R𝑢𝑏
+𝑡𝑒 is composed
+of a subset of 1,411 users and 3,327 items, where the ratings between these users
+and items are almost fully observed (with 99.6% density).
+The statistics of the datasets are summarized in Table 1 for reference. There are also
+randomized datasets for some related topics such as click-through rate prediction
+[102], i.e., Criteo Ads datasets16 [103], and bandit-based RS [104], i.e., Open Bandit
+dataset17 [105], where the sources are also provided in case the readers are interested.
+From Table 1 we can find that, the Coat dataset is small in scale. While for the
+Yahoo! R3 dataset, the training set is comparatively large (15,400 users and 1,000
+items), the randomized experiment conducted to establish the unbiased test set is
+small-scale in comparison (16 and 10 randomly exposed items per user, respectively).
+Therefore, although these ratings are unbiased due to randomization, they may not
+capture well-rounded user interests and therefore induce a high evaluation variance.
+13 https://www.cs.cornell.edu/~schnabts/mnar/
+14 https://webscope.sandbox.yahoo.com/catalog.php?datatype=r&did=3
+15 https://github.com/chongminggao/KuaiRec
+16 http://cail.criteo.com/criteo-uplift-prediction-dataset/
+17 https://research.zozo.com/data.html
+
+30
+Yaochen Zhu, Jing Ma, and Jundong Li
+Dataset
+# Users # Items Item Type
+Training Sets
+Test Sets
+Coat
+300
+290
+Coat
+24 i/u (self-supplied)
+16 i/u (random)
+Yahoo! R3 15,400
+1,000
+Music
+300,000 r (self-supplied)
+10 i/u (random) for 5,400 u
+KuaiRec
+7,176
+10,728
+Video
+16.3% r (self-supplied) 99.6% r for 1,411 u and 3,327 i
+Table 1: Characteristics of the currently available real-world causal recommendation datasets, where
+the test sets are devoid of exposure bias either due to randomized item exposures or fully observed
+ratings. In the table, terms like 24 i/u mean that every user rates 24 items, the term 300,000 r denotes
+the number of observed ratings, and terms like 16.3% r represent the density of interactions.
+For the recently released KuaiRec datasets, large-scale experiments are conducted
+on users to establish the bias-free test set, where the 1,411 users’ ratings for 3,327
+items are almost fully collected. Therefore, it may be a promising new benchmark
+that allows the evaluation of more complex causal RS models with a lower variance.
+5.4.2 Qualitative Evaluation and Case Study
+For other types of biases in RSs that cannot be attributed to non-randomized item
+exposures (e.g., clickbait bias and unfairness), the establishment of bias-free test sets
+is more challenging. For example, when studying the clickbait bias, it is difficult to
+determine whether a user clicked an item due to interests or clickbait. Similarly, when
+examining the user-oriented fairness of RSs, we cannot know if the generated items
+are offensive to the users. Under such circumstances, we can still conduct case studies
+for qualitative model evaluations, where we manually select some representative
+samples from the original test set and observe whether the trained causal RS model
+would respond as expected to these samples [106].
+Consider the evaluation of the robustness of a causal RS to clickbait bias. We
+can select some representative items with low-quality content but highly-attractive
+exposure features and other items with high-quality content but normal exposure
+features from the original test set. Then, we obtain rating predictions for items from
+these two groups and draw comparisons. If the studied causal RS indeed ranks items
+in the second group higher than those in the first group, we can likely conclude that
+the model is robust to clickbait bias because the quality of the item content, not
+its exposure features, is prioritized in recommendations. In addition, to evaluate the
+user-oriented fairness of a causal RS, we can analyze the generated recommendation
+for users from certain demographic groups. If the recommended items tend to capture
+the social stereotypes that are negatively associated with user sensitive features, we
+can conclude that the model is still discriminatory against users.
+
+Causal Inference for Recommender Systems: A survey
+31
+6 Future Directions
+Despite the recent achievements in marrying causal inference with traditional RSs
+to address their various limitations of correlational reasoning on observational user
+data, causal RS research is still in its emerging stage. Several promising directions
+could be pursued to further advance this field. In this section, we identify four
+interesting and important open problems worthy of exploration in the future.
+First, the assumptions required by existing causal RSs could be too strong, which
+may not hold in reality. For example, most RCM-based causal RSs rely on SUTVA
+to exclude the interference of item exposures for different users. However, if users
+are connected by a social network, they may interact closely with each other or be
+heavily affected by the influencers in the network [107]. Consequently, SUTVA can
+be violated because the recommendations made to one user may causally affect the
+ratings of others (i.e., the spill-over effects [108, 109]). In addition, the positivity
+assumptions may also be violated if some users never click certain types of items
+(i.e., non-compliance and defiers [33]). Therefore, it is crucial to further weaken the
+assumptions of causal RSs to make them more practical for real-world applications.
+In addition, there currently lacks a universal causal model for RSs that can be
+applied for different causal reasoning purposes. Most SCM-based causal RSs are
+designed to address one specific type of bias or entanglement problem, where other
+issues are tacitly assumed to be absent and omitted from the causal graph. Moreover,
+even for causal RSs that address the same problem, several varieties of causal graphs
+that include different sets of variables and relationships can be assumed, which
+leads to inconsistency between different works. Therefore, it would be promising
+and beneficial to have a generic and widely-accepted causal model that is able to
+comprehensively address multiple types of causal problems in recommendations.
+Furthermore, certain types of biases in RSs are double-blade swords, where the
+positive side is seldom investigated. Consider the item exposure bias discussed in
+Section 4.1.1. We should note that some items are more likely to be exposed because
+they have higher quality than other items. Therefore, the higher exposure rate of these
+items can be well justified and may be utilized to further enhance the recommendation
+performance. In addition, recent research also found that confounders that spuriously
+correlate item exposures and user ratings may also help explain the co-occurrence
+patterns of different items [71]. Therefore, how to properly identify and utilize the
+positive side of biases while maximally suppressing their negative effects is of great
+importance and deserves more in-depth investigations in the future.
+Finally, although recent years have witnessed the establishment and release of
+more real-world datasets for causal RS research from the industry, many causal RS
+models still rely heavily on simulated datasets for evaluation. The simulation can lead
+to the over-simplification of the problem and is often designed to correspond exactly
+with the debiasing/disentanglement mechanism of the proposed model. Therefore,
+the effectiveness of these methods in more complicated real-world scenarios is still
+uncertain due to the lack of model deployment and online tests. As such, to more
+convincingly demonstrate the practical utility of causal RSs, more collaborations
+with the industry are highly expected.
+
+32
+Yaochen Zhu, Jing Ma, and Jundong Li
+7 Summary
+In this survey, we provide a comprehensive overview of recent advances in causal
+inference for RSs. We start by pointing out issues of traditional RSs that rely on
+correlations in observed user behaviors and user/item features. We then introduce two
+mainstream causal inference frameworks, i.e., Rubin’s RCM and Pearl’s SCM, which
+provide deeper insights into these issues and the foundation for moving traditional
+RSs to the upper rungs of the Ladder of Causality. Specifically, we thoroughly
+discuss several state-of-the-art causal RS models that lead to enhanced robustness
+to various biases and improved explainability. In addition, since causal RSs can
+base recommendations on causal relationships that are stable and invariant, we also
+demonstrate that their generalization abilities can be significantly improved. Finally,
+we introduce evaluation strategies for causal RSs, with an emphasis on how to
+reliably estimate the model performance based on biased real-world data. We further
+compile real-world datasets where expensive randomized experiments are conducted
+on users, which reflects growing attention to causal RSs from the industry.
+Overall, causal RS is still a relatively new and under-explored research topic. More
+efforts are urgently demanded to systematize the existing works and conduct deeper
+investigations for further improvements. Accordingly, we point out four interesting
+and practically important open problems in causal RSs. We hope that this survey
+can help readers gain a comprehensive understanding of the main idea of applying
+causality in RSs and encourage further progress in this promising area.
+Acknowledgements. This work is supported by the National Science Foundation
+under grants IIS-2006844, IIS-2144209, IIS-2223769, CNS-2154962, and BCS-
+2228534, the JP Morgan Chase Faculty Research Award, and the Cisco Faculty
+Research Award.
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf,len=1310
+page_content='Causal Inference in Recommender Systems: A Survey of Strategies for Bias Mitigation, Explanation, and Generalization Yaochen Zhu, Jing Ma, and Jundong Li Abstract In the era of information overload, recommender systems (RSs) have be- come an indispensable part of online service platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Traditional RSs estimate user interests and predict their future behaviors by utilizing correlations in the obser- vational user historical activities, user profiles, and the content of interacted items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' However, since the inherent causal reasons that lead to the observed user behaviors are not considered, multiple types of biases could exist in the generated recommen- dations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In addition, the causal motives that drive user activities are usually entangled in these RSs, where the explainability and generalization abilities of recommenda- tions cannot be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' To address these drawbacks, recent years have witnessed an upsurge of interest in enhancing traditional RSs with causal inference techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In this survey, we provide a systematic overview of causal RSs and help readers gain a comprehensive understanding of this promising area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' We start with the basic concepts of traditional RSs and their limitations due to the lack of causal reasoning ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' We then discuss how different causal inference techniques can be introduced to address these challenges, with an emphasis on debiasing, explainability promo- tion, and generalization improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Furthermore, we thoroughly analyze various evaluation strategies for causal RSs, focusing especially on how to reliably estimate their performance with biased data if the causal effects of interests are unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Finally, we provide insights into potential directions for future causal RS research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Yaochen Zhu Department of Electrical and Computer Engineering, University of Virginia, e-mail: uqp4qh@ virginia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='edu Jing Ma Department of Computer Science, University of Virginia, e-mail: jm3mr@virginia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='edu Jundong Li Department of Electrical and Computer Engineering, Department of Computer Science, and School of Data Science, University of Virginia, e-mail: jl6qk@virginia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='edu 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='00910v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='IR] 3 Jan 2023 2 Yaochen Zhu, Jing Ma, and Jundong Li 1 Introduction With information growing exponentially on the web, recommender systems (RSs) are playing an increasingly pivotal role in modern online services, due to their ability to automatically deliver items1 to users based on their personalized interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Traditional RSs can be mainly categorized into three classes [9]: Collaborative filtering-based methods [10], content-based methods [11], and hybrid methods [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Collaborative filtering-based RSs estimate user interests and predict their future behaviors by exploiting their past activities, such as browsing, clicking, purchases, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Content-based methods, on the other hand, predict new recommendations by matching user interests with item content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Hybrid methods combine the advantages of both worlds, where collaborative information and user/item feature information are comprehensively considered to generate more accurate recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Although recent years have witnessed substantial achievements for all three classes of RSs introduced above, a great limitation of these methods is that they can only estimate user interests and predict future recommendations based on cor- relations in the observational user historical behaviors and user/item features, which guarantee no causal implications [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' For example, a collaborative filtering- based RS may discover that several drama shows from a certain genre tend to have high ratings from a group of users, and conclude that we should keep recommending drama shows from the same genre to these users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' But there is an important question: Are the high ratings caused by the fact that the users indeed like drama shows from this genre, or they were limitedly exposed to drama shows from the same genre (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', exposure bias), and if given a chance, they would prefer something new to watch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In addition, a content-based RS may observe that micro-videos with certain features are associated with more clicks and conclude that these features may reflect the current trend of user interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' But are the clicks because these micro-videos tend to have sensational titles as clickbait where users could be easily deceived?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Moreover, if the titles of these micro-videos are changed to the ones that reflect their true content, would users still click them?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The above questions are causal in nature because they either ask about the effects of an intervention (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', what the rating would be if a new drama show is made exposed to the user) or a counterfactual outcome (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', would the user still click a micro-video if its title had been changed to faithfully reflect the content), rather than mere associations in the observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' According to Pearl [15], these questions lie on Rungs 2 and 3 of the Ladder of Causality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', interven- tional and counterfactual reasoning, and they cannot be answered by traditional RSs that reason only with associations, which lie on Rung 1 of the ladder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Why are these causal questions important for RSs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The first reason is that failing to address them may easily incur bias in recommendations, which can get unnoticed for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' If the collaborative filtering-based RSs mentioned above mistake exposure bias for user interests, they would amplify the bias by continuously recom- mending users with similar items;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' eventually, recommendations will lose serendipity, 1 We use the term item in a broad sense to refer to anything recommendable to users, such as news [1], jobs [2], articles [3], music [4], movies [5], micro-videos [6], PoIs [7], hashtags [8], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Causal Inference for Recommender Systems: A survey 3 and users’ online experience can be severely degraded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Similarly, for the content- based micro-video RSs, if they cannot distinguish clicks due to user interests from the ones deceived by clickbait, they may over-recommend micro-videos with sen- sational titles, which is unfair to the uploaders of high-quality micro-videos who put much effort into designing the content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In addition, understanding the cause of user activities can help improve the explainability of recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Consider the causal question of whether a user purchases an item due to its quality or its low price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Pursuing the causal explanations behind user behaviors can help service providers to enhance the RS algorithm based on users’ personalized preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Finally, causal inference allows us to identify and base recommendations on causal relations that are stable and invariant, while discarding other correlations that are undesirable or susceptible to change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Take restaurant recommendations as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Users can choose a restaurant because of its convenience (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', going to a nearby fast food shop to quickly grab a bite, but they do not necessarily like it, a non-stable correlation) or due to their personal interests (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', traveling far away for a hot-pot restaurant, a stable causal relation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' If an RS can properly disentangle users’ intent that causally affects their previous restaurant visits, even if the convenience levels of different restaurants may change due to various internal or external reasons such as users’ moving to a new place, the system can still adapt well to the new situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' From this aspect, the generalization ability of the causal RSs can be substantially improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' This survey provides a systematic overview of recent advances in causal RS research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The organization is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' We start with the fundamental con- cepts of traditional RSs and their limitation of correlational reasoning in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Then Section 3 recaps two important causal inference paradigms in machine learning and statistics, and shows their connections with the recommendation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Section 4 thoroughly discusses how different causal inference techniques can be introduced to address the limitations of traditional RSs, with an emphasis on debiasing, explainabil- ity promotion, and generalization improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Section 5 summarizes the general evaluation strategies for causal RSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Finally, Sections 6 and 7 discuss prospective open questions and future directions for causal RSs and conclude this survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 2 Recommender System Basics To keep this survey compact, we confine our discussions to simple RSs with 𝐼 users and 𝐽 items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The main data for the RSs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', users’ historical behaviors, are represented by a user-item rating matrix R ∈ R𝐼×𝐽, where a non-zero element 𝑟𝑖 𝑗 denotes user 𝑖’s rating to item 𝑗, and a zero element 𝑟0 𝑖𝑘 indicates the rating is missing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' To make the discussions of RSs compatible with causal inference, we take a probabilistic view of R [17], where 𝑟𝑖 𝑗 is assumed to be the realized value of the 2 We use rating to refer to any user-item interaction that can be represented by a numerical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' This includes both explicit feedback such as likes/dislikes, and implicit feedback such as views and clicks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' When 𝑟𝑖 𝑗 represents implicit feedback, the missing elements 𝑟0 𝑖𝑘 in R may be used as weak negative feedback in the training phase [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' This may complicate the causal problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, we assume RSs are trained on observed ratings to simplify the discussion unless specified otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 4 Yaochen Zhu, Jing Ma, and Jundong Li Causal Inference in Recommendations Causal Debiasing Causal Explanation RCM-based promotes Exposure Bias Unfairness Section 3 Causal Generalization Popularity Bias Intervention-based Causal Inference Recommender Systems SCM-based CF-based Content-based Hybrid promotes Disentangle-based Causal Embeddings Colliding Effects Clickbait Section 2 Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1 Future Directions Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='2 Sections 6,7 Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='3 Evaluation Strategies Section 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 1: An overview of the structure of this survey and connections between different sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' random variable 𝑅 dependent on user 𝑖 and item 𝑗3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In addition to R, an RS usually has access to side information like user features f𝑢 𝑖 ∈ R𝐾 𝑢 𝐹 , such as her age, gender, location, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', or item features f𝑣 𝑗 ∈ R𝐾 𝑣 𝐹 , such as its content and textual description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 𝐾𝑢 𝐹 and 𝐾𝑣 𝐹 are the dimensions of user and item features, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The main purpose of an RS is to predict users’ ratings for previously uninteracted items (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', the missing values 𝑟0 𝑖𝑘 in R) based on the observed ratings 𝑟𝑖 𝑗 in R and the available user and item side information such as f𝑢 𝑖 and f𝑣 𝑗 , such that new relevant items can be properly recommended based on users’ personalized interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1 Collaborative Filtering Collaborative filtering-based RSs recommend new items by leveraging user ratings in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' They generally consider the ratings 𝑟𝑖 𝑗 as being generated from a user latent variable u𝑖 ∈ R𝐾 that represents user interests and an item latent variable v𝑗 ∈ R𝐾 that encodes the item attributes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', item latent semantic information), where 𝐾 is the dimension of the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Here we list three widely-used collaborative filtering-based RSs, which will be frequently used as examples in this survey: Matrix Factorization (MF) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' MF models 𝑟𝑖 𝑗 with the inner product between u𝑖 and v𝑗, where 𝑟𝑖 𝑗 ∼ N (u𝑇 𝑖 · v𝑗, 𝜎2 𝑖 𝑗) and 𝜎2 𝑖 𝑗 is the predetermined variance4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 3 However, we do not distinguish random variables and their specific realizations if there is no risk of confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' For simplicity, we assume 𝑅 to be Gaussian unless specified otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 4 For works that do not explicitly treat 𝑟𝑖 𝑗 as a random variable, we assume it follows a Gaussian distribution with zero variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The generative process then becomes as 𝑟𝑖 𝑗 = u𝑇 𝑖 · v𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Causal Inference for Recommender Systems: A survey 5 Deep Matrix Factorization (DMF) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' DMF extends MF by applying deep neural networks (DNNs) [20], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', 𝑓 𝑢 𝑛𝑛, 𝑓 𝑣 𝑛𝑛 : R𝐾 → R𝐾 ′, to u𝑖 and v𝑗, where the ratings are assumed to be generated as 𝑟𝑖 𝑗 ∼ N ( 𝑓 𝑢 𝑛𝑛(u𝑖)𝑇 · 𝑓 𝑣 𝑛𝑛(v𝑗), 𝜎2 𝑖 𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Auto-encoder (AE)-based RSs [21, 22] model user 𝑖’s ratings to all 𝐽 items as r𝑖 ∼ N ( 𝑓 𝑢 𝑛𝑛(u𝑖), 𝝈2 𝑖 · I𝐾), where 𝑓 𝑢 𝑛𝑛 : R𝐾 → R𝐽 is a DNN and item latent variables v𝑗 for all items are implicit in last layer weights of the decoder [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In the training phase, the models learn the latent variables u𝑖, v𝑗 and the associated function 𝑓𝑛𝑛 by fitting on the observed ratings 𝑟𝑖 𝑗 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', via maximum likelihood estimation, which essentially estimates the conditional distribution 𝑝(𝑟𝑖 𝑗|u𝑖, v𝑗) from the observational data [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Afterward, we can use them to predict new ratings for previously uninteracted items 𝑘, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', ˆ𝑟MF 𝑖𝑘 = u𝑇 𝑖 ·v𝑘 for MF, ˆ𝑟DMF 𝑖𝑘 = 𝑓 𝑢 𝑛𝑛(u𝑖)𝑇 · 𝑓 𝑣 𝑛𝑛(v𝑘) for DMF, and ˆ𝑟AE 𝑖𝑘 = 𝑓 𝑢 𝑛𝑛(u𝑖)𝑘 for AE-based RSs, where the top ones that best match users’ interests can be selected as recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Traditional collaborative filtering-based RSs reasons with correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Ideally, we would expect u𝑖, v𝑗 and 𝑓𝑛𝑛 to capture the causal influence of user interests and item attributes on ratings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', what the rating would be if item 𝑗 is made exposed to user 𝑖 [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' However, since the collected rating data are observational rather than experimental, what actually learned by u𝑖, v𝑗, and 𝑓 are the co-occurrence patterns in users’ past behaviors, which guarantee no causal implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Consequently, spurious correlations and biases can be captured by the model, which will be amplified in future recommendations [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Furthermore, the learned user latent variable u𝑖 generally entangles dif- ferent factors that causally determine user interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' From this perspective, the explainability and generalization of these methods cannot be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='2 Content-Based Recommender Systems Personalized content-based RSs (CBRSs) estimate user interests based on the fea- tures of the items they have interacted with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' These models typically encode user interests into user latent variables u𝑖 ∈ R𝐾 and assume that the ratings are generated by matching user interests with item content, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', 𝑟𝑖 𝑗 ∼ N ( 𝑓 (u𝑖, f𝑣 𝑗 ), 𝜎𝑖 𝑗), where 𝑓 is a matching function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The training of personalized CBRSs follow similar steps as collaborative filtering, where u𝑖 and 𝑓 are learned by fitting on the observed ratings (which essentially estimates the conditional distribution 𝑝(𝑟𝑖 𝑗|u𝑖, f𝑣 𝑗 ) from the obser- vational data), and new ratings can be predicted by ˆ𝑟𝑖𝑘 = 𝑓 (u𝑖, f𝑣 𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The key step of building a CBRS is to create item features f𝑣 𝑗 that can best reflect user interests, which crucially depends on the item being recommended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' For example, for micro-videos, the visual, audio, and textual modalities are comprehensively considered such that users’ interest in different aspects of a micro-video can be well captured [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 6 Yaochen Zhu, Jing Ma, and Jundong Li Traditional content-based RSs cannot model the causal influence of user interests u𝑖 and item content f𝑣 𝑗 on user rating 𝑟𝑖 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The reason is that, factors other than users’ interests in the item content, such as users’ being deceived by clickbaits (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', sensational titles of micro-videos) [27], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', can create an undesirable association between item content f𝑣 𝑗 and user ratings 𝑟𝑖 𝑗 in the observed dataset, where the bias can be captured by the user latent variables u𝑖 and the matching function 𝑓 , and perpetuates into future recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='3 Hybrid Recommendation Hybrid RSs combine user/item side information with collaborative filtering to en- hance the recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' A commonly-used hybrid strategy is to augment user and item latent variables u𝑖 and v𝑗 with user/item side information f𝑣 𝑖 and f𝑣 𝑗 in existing collaborative filtering methods by replacing u𝑖 and v 𝑗 with u+ 𝑖 = [u𝑖||f𝑢 𝑖 ] and v+ 𝑗 = [v 𝑗||f𝑣 𝑗 ] in MF, DMF, and AE-based RSs, where [·||·] represents vector concatenation [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The dimensions of u𝑖 and v𝑗 that encode the collaborative information are adjusted accordingly to make u+ 𝑖 and v+ 𝑗 compatible in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Another important class of hybrid RS is the factorization machine (FM) [30] and its extensions like [31, 32], which can be viewed as learning a bi-linear function 𝑓 𝑓 𝑚 where the ratings are generated by 𝑟𝑖 𝑗 ∼ N ( 𝑓 𝑓 𝑚(u𝑖, v𝑗, f𝑢 𝑖 , f𝑣 𝑗 ), 𝜎2 𝑖 𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Simple hybrid strategies cannot break the correlational reasoning lim- itation of collaborative filtering and content-based RSs, because the ob- jective of the hybridization is still to improve the models’ fitting on the ob- servational user historical behaviors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', estimating conditional distribution 𝑝(𝑟𝑖 𝑗|u𝑖, v𝑗, f𝑢 𝑖 , f𝑣 𝑗 ) from the data), where the causal reasons that lead to the observed user behaviors are not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' However, the idea of introducing extra user/item side information is important for building causal RSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The rea- son is that, combined with the domain knowledge of human experts, the side information can help form more comprehensive causal relations among the variables of interests, such as user interests, item attributes, historical ratings, and other important covariates that may lead to spurious correlations and bi- ases, which is usually a crucial step for causal reasoning in recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 3 Causal Recommender Systems: Preliminaries In the previous section, we discussed the recommendation strategies of the tradi- tional RSs and their limitations due to correlational reasoning on observational user Causal Inference for Recommender Systems: A survey 7 behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In this section, we introduce two causal inference frameworks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', Ru- bin’s potential outcome framework (also known as the Rubin causal model, RCM) [33] and Pearl’s structural causal model (SCM) [34], in the context of RSs, aiming to provide a theoretically rigorous basis for reasoning with correlation and causation in recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' We show that both RCM and SCM are powerful frameworks to build RSs with causal reasoning ability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', causal RSs), but they are best suited for different tasks and questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The discussions in this section serve as the foundation for more in-depth discussions of the state-of-the-art causal RS models in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1 Rubin’s Potential Outcome Framework 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1 Motivation of Applications in RSs To understand the correlational reasoning nature of traditional RSs, we note that naively fitting models on the observed ratings can only answer the question “what the rating would be if we observe an item was exposed to the user".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Since item exposure is not randomized in the collected dataset 5, the predicate “the item was exposed to the user" per se contains extra information about the user-item pair (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', the item could be more popular than other non-exposed items), which cannot be generalized to the rating predictions of arbitrary user-item pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, what RS asks is essentially an interventional question (and therefore a causal inference question), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', what the rating would be if an item is made exposed to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' To address this question, RCM-based RSs draw inspiration from clinical trials, where exposing a user to an item is compared to subjecting a patient to a treatment, and the user ratings are analogous to the outcomes of the patients after the treatment [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Accordingly, RCM-based RSs aim to estimate the causal effects of the treatments (exposing a user to an item) on the outcomes (user ratings), despite the possible correlations between the treatment assignment and the outcome observations [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='2 Definitions and Objectives We first introduce necessary symbols and definitions to connect RCM with RSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' We consider the unit as the user-item pair (𝑖, 𝑗) that can receive the treatment “exposing user 𝑖 to item 𝑗”, and the population as all user-item pairs PO = {(𝑖, 𝑗), 1 ≤ 𝑖, 𝑗 ≤ 𝐼, 𝐽} [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' We start by using a binary scalar 𝑎𝑖 𝑗 to denote the exposure status of item 𝑗 for user 𝑖, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', the assigned treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' We further define the rating potential outcome 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) as user 𝑖’s rating to item 𝑗 if the item is made exposed to the user and 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 0) as the rating if the item is not exposed [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' For user 𝑖, if she rated item 𝑗, we observe 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) = 𝑟𝑖 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Otherwise, we observe the baseline potential outcome 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 0) = 0, which is usually ignored in debias-oriented 5 which can be attributed to multiple reasons such as users’ self-search [35], the recommendations of previous models [36], the position where the items are displayed [37], item popularity [38], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Generally, RCM-based causal RSs are agnostic to the specific reason that causes the exposure bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 8 Yaochen Zhu, Jing Ma, and Jundong Li 5 5 5 5 5 5 5 5 5 5 5 5 5 5 1 1 1 1 1 1 1 1 Horror Lover Romance Lover Horror Romance 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 Horror Lover Romance Lover Horror Lover Romance Lover Horror Romance Horror Romance (a) Observed Ratings (b) Rating Potential Outcomes (c) Predicted Ratings Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 2: A classical example of exposure bias in RSs [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The example is composed of two horror lovers who always rate horror movies with five while hating romance movies, and two romance lovers would who do exactly the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (a) shows the observed ratings 𝑟𝑖 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (b) shows the rating potential outcomes 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (c) shows the rating predictions of an RS that maximizes the likelihood of the observed ratings in (a), but the RS is bad because it predicts all ratings to five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' causal RS research [39, 43]6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Similar to clinical trials, we can define the treatment group T = {(𝑖, 𝑗) : 𝑎𝑖 𝑗 = 1} as the set of user-item pairs where user 𝑖 is exposed to item 𝑗, and define the non-treatment group NT = {(𝑖, 𝑘) : 𝑎𝑖𝑘 = 0} accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The purpose of RSs, under the RCM framework, can be framed as utilizing the observed ratings from units in the treatment group T to unbiasedly estimate the rating potential outcomes for units from the population PO, despite the possible correlations between item exposures 𝑎𝑖 𝑗 and user ratings 𝑟𝑖 𝑗 in the collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='3 Causal Analysis of Traditional RSs Traditional RSs naively train a rating prediction model that best fits the ratings in the treatment group T (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', via maximum likelihood introduced in Section 2) to estimate the unobserved rating potential outcomes 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) for user-item pairs in NT [46], which neglect the fact that exposure bias can lead to a systematic difference in the distribution of 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) between T and NT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' For example, users tend to rate items they like in reality, which could lead to the following spurious correlation between item exposure 𝑎𝑖 𝑗 and rating potential outcome 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1): 𝑝(𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) is high|𝑎𝑖 𝑗 = 1) > 𝑝(𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) is high|𝑎𝑖 𝑗 = 0), (1) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', users who have rated an item 𝑗 may have systematically higher ratings than users who haven’t rated it yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In this case, traditional RSs may have a tendency to overestimate the ratings for units in NT (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 2 for an intuitive example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The- oretically, RCM attributes the exposure bias in the collected dataset to the violation of the unconfoundedness assumption [33] defined as follows: 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) ⊥ 𝑎𝑖 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (2) 6 In the uplift evaluation of RSs that aims to estimate how recommendations change user behaviors [44], 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 0) may be used to represent user 𝑖’s rating to item 𝑗 through self-searching [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Causal Inference for Recommender Systems: A survey 9 The rationale is that, if Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (2) holds, the exposure of user 𝑖 to item 𝑗 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', 𝑎𝑖 𝑗) is independent of the rating potential outcome 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1), which implies that 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) in T and NT follows the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, the exposure of the items is randomized, and exposure bias such as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (1) will not exist [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='4 Potential Outcome Estimation with the RCM Framework One classic solution from the RCM-based framework to address the exposure bias is that we find user and item covariates 𝐶, such that in each data stratum specified by 𝐶 = c, users’ exposure to items are randomized [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The property of the covariates 𝐶 can be formulated as the conditional unconfoundedness assumption as follows: 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) ⊥ 𝑎𝑖 𝑗 | c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (3) 𝐶 is sometimes non-rigorously referred to as confounder in the literature, but we will see its formal definition in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' If Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (3) holds, the item exposures are independent of the rating potential outcomes in each data stratum specified by 𝐶 = c, and the exposure bias can be attributed solely to the discrepancy in the distribution of the covariates 𝐶 = c between the treatment group T and the population PO, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', 𝑝(c|𝑎𝑖 𝑗 = 1) and 𝑝(c)7 Therefore, we can reweight the observed ratings in T based on the covariates 𝐶 to address the bias, such that they can be viewed as pseudo randomized samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' This leads to inverse propensity weighting (IPW), which eliminates the exposure bias from the data’s perspective [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In addition, we can also adjust the influence of 𝐶 in the RS model, where the exposure bias is addressed from the model side [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Both methods will be discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Attention: Extra Assumptions Required by Most RCM-based RSs In addition to unconfoundedness, most RCM-based RS need two extra assumptions to identify the causal effects of item exposures on ratings: (1) The stable unit treatment assumption (SUTVA), which states that items exposed to one user cannot affect ratings of another user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (2) The positivity assumption, which states that every user has a positive chance of being exposed to every item [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' For RCM-based causal RSs introduced in this survey, these two assumptions are tacitly accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 7 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' We can gain an intuition of this claim from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Suppose covariates 𝐶 represent the two- dimensional features (user type, movie type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Given 𝐶 = c, 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) ⊥ 𝑎𝑖 𝑗 | c described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (3) is satisfied because in each data stratum specified by 𝐶 = c (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', the four 2×2 blocks in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 2-(b)), 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 2-(a) shows that for the treatment group T, 𝑝(c|𝑎𝑖 𝑗 = 1) = 1/2 for c ∈ C1 = {(horror fan, horror movie), (romance fan, romance movie) } and 𝑝(c|𝑎𝑖 𝑗 = 1) = 0 for c ∈ C2 = {(horror fan, romance movie), (romance fan, horror movie) }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In contrast, for the population PO, 𝑝(c) = 1/4 for c ∈ C1 ∪ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, in the treatment group T, user-item pairs with covariates in C1 are over-represented, while those with covariates in C2 are under-represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' However, we also note that this case is too extreme to be addressed by RCM, as 𝑝(c|𝑎𝑖 𝑗 = 1) = 0 for 𝐶 ∈ C2 violates the positivity assumption mentioned in the above attention box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 10 Yaochen Zhu, Jing Ma, and Jundong Li 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='2 Pearl’s Structural Causal Model 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1 Motivation of Applications in RS Different from RCM that uses rating potential outcomes to reason with causality and attributes the biases in observed user behaviors to non-randomized item exposures, Pearl’s structural causal model (SCM) delves deep into the causal mechanism that generates the observed outcomes (and biases) and represents it with a causal graph 𝐺 = (N, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The nodes N specify the variables of interests, which in the context of RS could be user interests 𝑈, item attributes 𝑉, observed ratings 𝑅, and other important covariates 𝐶, such as item popularity, user features, etc8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The directed edges E between nodes represent their causal relations determined by researchers’ domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Each node 𝑋 ∈ N is associated with a structural equation 𝑝𝐺(𝑋|𝑃𝑎(𝑋))9, which describes how the parent nodes 𝑃𝑎(𝑋) causally influence 𝑋 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', the response of 𝑋 when setting nodes in 𝑃𝑎(𝑋) to specific values) Although RCM and SCM are generally believed to be fundamentally equivalent [34], both have their unique advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Compared to RCM, the key advantage of SCM is that causal graph offers an intuitive and straightforward way to encode and communicate domain knowledge and substantive assumptions of researchers, which is beneficial even for the RCM-based RSs [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Furthermore, SCM is more flexible as it can represent and reason with the causal effects between any subset of nodes in the causal graph (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', between two causes 𝑈,𝑉 and one outcome 𝑅), as well as the causal effects along specific paths (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', 𝑈 → 𝑅 and 𝑈𝑐 → 𝑅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, SCMs are broadly applicable to multiple problems in RSs (not limited to exposure bias), such as clickbait bias, unfairness, entanglement, domain adaptation, etc [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Attention: Two Caveats of SCM-based Causal RSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' There are two caveats of SCM-based causal RSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (1) Causal graphs for RSs often involve user, item latent variables𝑈,𝑉 that encode user interests and item attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Most works infer them alongside the estimation of structural equations and treat them as if they were observed when analyzing the causal relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Alternatively, this can be viewed as representing users and items with their IDs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', 𝑖 and 𝑗) in the causal graph and subsuming the embedding process into the structural equations [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (2) Generally, the causal graph should describe the causal mechanism that generates the observed data, because it allows us to distinguish invariant, causal relations from undesirable correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' For example, we may argue that item popularity 𝐶 should be determined by item attributes 𝑉, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', 𝑉 → 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' But to describe the generation of the observed ratings, causal relation 𝐶 → 𝑉 is usually assumed instead as item popularity causally influences the exposure probability of each item [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 8 In causal graphs, the subscripts 𝑖, 𝑗 for each node are omitted for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 9 We also omit the mutually independent exogenous variables for each node and summarize their randomness into the structural equations with probability distributions [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Subscript 𝐺 is used to distinguish structural equations from other conditional relationships that can be inferred from 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Causal Inference for Recommender Systems: A survey 11 R U Uc Cu Cv V Cu U R Cu U R U R Uc (a) A generic causal graph for RS (b) The chain structure (c) The fork structure (d) The V-structure Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 3: (a): A generic causal graph for RS that depicts the causal influence of user interests 𝑈, user conformity to the popularity trend𝑈𝑐, and item attributes 𝑉 on the observed ratings 𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Specifically, the causal paths 𝑈 → 𝑅 and 𝑉 → 𝑅 are confounded by 𝐶𝑢 and 𝐶𝑣, which represent user features and item popularity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (b)(c)(d): Three atomic structures identified from (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='2 Atomic Structures of Causal Graphs The structure of causal graphs represents researchers’ domain knowledge regard- ing the causal generation process of the observational data, which is the key to distinguishing stable, causal relations from other undesirable correlations between variables of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Here, we use a generic causal graph applicable to RSs in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 3-(a) as a running example to illustrate three atomic graph structures: Chain, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', 𝐶𝑢 → 𝑈 → 𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In a chain, the successor node is assumed to be causally influenced by the ancestor nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In the example, 𝑈 is a direct cause of 𝑅, whereas 𝐶𝑢 indirectly influences 𝑅 via 𝑈 as a mediator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Fork, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', 𝑈 ← 𝐶𝑢 → 𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In the fork, 𝐶𝑢 is called a confounder as it causally influences two children 𝑈 and 𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' From a probabilistic perspective, 𝑈 and 𝑅 are not independent unless conditioned on the confounder 𝐶𝑢 [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' This leads to the tricky part of a fork structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', confounding effect [34], where an unobserved 𝐶𝑢 can lead to spurious correlations between 𝑈 and 𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' V-structure, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', 𝑈 → 𝑅 ← 𝑈𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In the V-structure, 𝑅 is called a collider because it is under the causal influence of two parents, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', 𝑈 and 𝑈𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' An inter- esting property of the V-structure is the colliding effects [34], where observing 𝑅 creates a dependence on 𝑈 and 𝑈𝑐, even if they are marginally independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Confounders can lead to non-causal dependencies among variables in the obser- vational dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' This could introduce bias in traditional RSs, where the confounding effects are mistaken as causal relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Confounding bias is a generic problem in RSs [24], which will be further analyzed in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In addition, abstracted V-structure usually leads to the entanglement of causes, which could jeop- ardize the explainability of RSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' For example, a user’s purchase of an item may be due to her interest, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', 𝑈, or her conformity to the popularity trend, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', 𝑈𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Since most RSs summarize both into a user latent variable 𝑈, the V-structure 𝑈 → 𝑅 ← 𝑈𝑐 is abstracted away, where the two causes of the purchase cannot be distinguished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 12 Yaochen Zhu, Jing Ma, and Jundong Li U R V (b) Confounded true SCM (a) SCM assumed by non-causal RS Cv (c) SCM under intervention U R V U R do(V) Cv Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 4: (a): SCM assumed by non-causal collaborative filtering-based RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (b): The confounded SCM that depicts the true data generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (c): SCM under intervention 𝑑𝑜(𝑉 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='3 Causal Analysis of Traditional RSs In this section, we investigate the susceptibility of traditional collaborative filtering- based RSs to the confounding bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' As discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1, a commonality of these models is that they estimate conditional distribution 𝑝(𝑟𝑖 𝑗|u𝑖, v𝑗) from observed ratings and use it to predict new ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' For 𝑝(𝑟𝑖 𝑗|u𝑖, v𝑗) to represent the causal influence of user interests u𝑖 and item attributes v 𝑗 on ratings 𝑟𝑖 𝑗 (which, in the context of collaborative filtering, means the rating of any arbitrary item 𝑗 that is made exposed to user 𝑖 [24]), the causal graph 𝐺1 of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 4-(a) is tacitly assumed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', no unobserved confounders for causal paths 𝑈 → 𝑅 and 𝑉 → 𝑅10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' However, in reality, both 𝑈 → 𝑅 [25, 51] and 𝑉 → 𝑅 [52, 53] can be confounded, where the confounding effects can be implicitly captured by 𝑝(𝑟𝑖 𝑗|u𝑖, v𝑗) that bias future recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' To reveal the bias, we consider the scenario where the causal path 𝑉 → 𝑅 is confounded by 𝐶𝑣 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', item popularity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' We assume the causal path 𝐶𝑣 → 𝑉 denotes the causal influence of 𝐶𝑣 on the exposure probability of item 𝑉 [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In this case, the observed ratings are generated according to the causal graph 𝐺2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 4-(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Utilizing the law of total probability, the conditional distribution 𝑝(𝑟𝑖 𝑗|u𝑖, v𝑗) estimated from the confounded data can be calculated as: 𝑝(𝑟𝑖 𝑗|u𝑖, v𝑗) = ∑︁ c 𝑝(c|v𝑗) · 𝑝𝐺2(𝑟𝑖 𝑗|u𝑖, v𝑗, c) = E𝑝(𝐶𝑣 |v 𝑗) [𝑝𝐺2(𝑟𝑖 𝑗|u𝑖, v𝑗, 𝐶𝑣)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (4) The issue of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (4) is that, the 𝑝(c|v𝑗) term is not causal (as we only have an edge 𝐶𝑣 → 𝑉 in the causal graph but not 𝑉 → 𝐶𝑣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In fact, 𝑝(c|v𝑗) represents abductive reasoning because it infers the cause c (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', item popularity) from the effect v𝑗 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', item 𝑗 is exposed to user 𝑖) and uses the inferred c to support the prediction of the rating 𝑟𝑖 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' However, such reasoning cannot be generalized to the rating prediction of an arbitrary item v 𝑗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', an item that is made exposed to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In other words, uncontrolled confounder 𝐶𝑣 leaves open a backdoor path (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', non-causal path) between 𝑉 and 𝑅, such that non-causal dependence of 𝑅 on 𝑉 exists in the data, which can be captured by traditional RSs and bias future recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 11 10 This corresponds to the case where item exposures are randomized (see the discussions in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='3), as the user-item pair (𝑈, 𝑉 ) is not determined by other factors associated with 𝑅 [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 11 The similarity between this section and Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1 shows us the connection between RCM- based and SCM-based causal RSs, where the claim that when item exposure is not randomized, Causal Inference for Recommender Systems: A survey 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='4 Causal Reasoning with SCM To calculate the causal effect of u𝑖 and v𝑗 on 𝑟𝑖 𝑗, we should conduct intervention on 𝑈 and 𝑉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' This means that we set 𝑈, 𝑉 to u𝑖, v𝑗 regardless of the values of their parent nodes in the causal graph, including the confounder 𝐶𝑣 (because these nodes determine the exposure of item 𝑗 to user 𝑖 in the observed data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' SCM denotes the intervention with do-operator as 𝑝(𝑟𝑖 𝑗|𝑑𝑜(u𝑖, v𝑗)) to distinguish it from the con- ditional distribution 𝑝(𝑟𝑖 𝑗|u𝑖, v 𝑗) that reasons with correlations in the observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Consider again the causal graph 𝐺2 illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 4-(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The intervention on node 𝑉 can be realized by removing all the incoming edges for node 𝑉 and setting the structural equation 𝑝𝐺2(𝑉|𝐶𝑣) deterministically as 𝑉 = v𝑗, while other struc- tural equations remain intact (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 4-(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' If the confounder 𝐶𝑣 can be determined and measured for each item, the interventional distribution 𝑝(𝑟𝑖 𝑗|𝑑𝑜(u𝑖, v𝑗)) can be directly calculated from the confounded data via backdoor adjustment [34] as: 𝑝(𝑟𝑖 𝑗|𝑑𝑜(u𝑖, v𝑗)) = ∑︁ c 𝑝𝐺2(c) · 𝑝𝐺2(𝑟𝑖 𝑗|u𝑖, v𝑗, c) = E𝑝𝐺2 (𝐶𝑣) [𝑝𝐺2(𝑟𝑖 𝑗|u𝑖, v𝑗, 𝐶𝑣)], (5) which, compared with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (4), blocks the abductive inference of c from v𝑗, such that the causal influence of u𝑖, v𝑗 on 𝑟𝑖 𝑗 can be properly identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Backdoor adjustment requires all confounders to be determined and measured in advance, but there are other SCM-based causal inference methods that can estimate causal effects with unknown confounders, and we refer readers to [54, 55] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Moreover, causal graphs allow us to conduct other types of causal reasoning based on the encoded causal knowledge, such as debiasing for non-confounder-induced biases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', clickbait bias and unfairness), causal disentanglement, and causal gen- eralization [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' These will be thoroughly discussed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 4 Causal Recommender Systems: The State-of-the-Art Based on the preliminary knowledge of RSs and causal inference discussed in previ- ous sections, we are ready to introduce the state-of-the-art causal RSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Specifically, we focus on three important topics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', bias mitigation, explainability promotion, and generalization improvement, as well as their inter-connections, where various limitations of traditional RSs due to correlational reasoning can be well addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1 Causal Debiasing for Recommendations The correlational reasoning of traditional RSs can inherit multiple types of biases in the observational user behaviors and amplify them in future recommendations [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' “observing that an item was exposed to the user per se contains extra information about the user-item pair" is mathematically transformed into the abductive inference of c from v𝑗 by 𝑝(c|v𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 14 Yaochen Zhu, Jing Ma, and Jundong Li The biases may result in various consequences, such as the discrepancy between offline evaluation and online metrics, loss of diversity, reduced recommendation quality, offensive recommendations, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Causal inference can distinguish stable causal relations from spurious correlations and biases that could negatively influence the recommendations, such that the robustness of recommendations can be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1 Exposure Bias Exposure bias in RSs broadly refers to the bias in observed ratings due to non- randomized item exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' From the RCM’s perspective, exposure bias can be defined as the bias where users are favorably exposed to items depending on their expected ratings for them (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', rating potential outcomes) [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Exposure bias occurs due to various reasons, such as users’ self-search or the recommendation of the previous RSs [36], which leads to the down-weighting of items less likely to be exposed to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Since item exposures can be naturally compared with treatments in clinical trials, we discuss the debiasing strategies with the RCM framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Inverse Propensity Weighting (IPW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' IPW-based causal RSs aim to reweight the biased observed ratings 𝑟𝑖 𝑗 for user-item pairs in the treatment group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', T = {(𝑖, 𝑗) : 𝑎𝑖 𝑗 = 1}, to create pseudo randomized samples [57] for unbiased training of RS models that aim to predict the rating potential outcomes 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) for the population PO = {(𝑖, 𝑗), 1 ≤ 𝑖, 𝑗 ≤ 𝐼, 𝐽}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Intuitively, we can set the weight of 𝑟𝑖 𝑗 for units in T to be the inverse of item 𝑗’s exposure probability to user 𝑖, such that under-exposed items can be up-weighted and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' If for each user-item pair, the covariates c that satisfy the conditional unconfoundedness assumption in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (3) are available, the exposure probability 𝑒𝑖 𝑗 can be unbiasedly estimated from c via 𝑒𝑖 𝑗 = 𝑝(𝑎𝑖 𝑗 = 1|c) = E[𝑎𝑖 𝑗|c], (6) which is formally known as propensity score in causal inference literature [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' > Background: The Balancing Property of Propensity Scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Propensity scores have the following property called balancing [33, 59], which is the key to proving the unbiasedness of IPW-based RSs: E � 𝑟𝑖 𝑗 𝑒𝑖 𝑗 ���𝑎𝑖 𝑗 = 1 � = E �𝑟𝑖 𝑗 · 𝑎𝑖 𝑗 𝑒𝑖 𝑗 � = E � E �𝑟𝑖 𝑗 · 𝑎𝑖 𝑗 𝑒𝑖 𝑗 ���c �� = E � E �𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) · 𝑎𝑖 𝑗 𝑒𝑖 𝑗 ���c �� (𝑎)= E �E[𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) | c] · E[𝑎𝑖 𝑗 | c] 𝑒𝑖 𝑗 � = E �E[𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) | c] · 𝑒𝑖 𝑗 𝑒𝑖 𝑗 � = E[𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1)], (7) where the step (𝑎) follows the conditional unconfoundedness assumption in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Causal Inference for Recommender Systems: A survey 15 We first discuss the implementation of IPW-based RS and its unbiasedness if user and item covariates c that satisfy Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (3) are available and the propensity scores 𝑒𝑖 𝑗 can be calculated exactly as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' We denote the rating predictor of an RS that aims to predict the rating potential outcome 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) as ˆ𝑟𝑖 𝑗 and assume 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) follows the unit-variance Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Ideally, we would like ˆ𝑟𝑖 𝑗 to maximize the log-likelihood on the rating potential outcomes 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) for all user-item pairs in PO, which is equivalent to the minimization of the mean squared error (MSE) loss between ˆ𝑟𝑖 𝑗 and 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) as follows: LTrue = 1 𝐼 × 𝐽 ∑︁ 𝑖, 𝑗 (ˆ𝑟𝑖 𝑗 − 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (8) However, since 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1) is unobservable for user-item pairs in the non-treatment group NT, LTrue is impossible to calculate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, traditional RSs only maxi- mize the log-likelihood of the observed ratings for user-item pairs in the treatment group T, which leads to the empirical MSE loss as follows: LObs = 1 |(𝑖, 𝑗) : 𝑎𝑖 𝑗 = 1| ∑︁ (𝑖, 𝑗):𝑎𝑖 𝑗=1 (ˆ𝑟𝑖 𝑗 − 𝑟𝑖 𝑗)2, (9) where |(𝑖, 𝑗) : 𝑎𝑖 𝑗 = 1| is the number of observed ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' When exposure bias exists, item exposure 𝑎𝑖 𝑗 depends on the rating potential outcome 𝑟𝑖 𝑗 (𝑎𝑖 𝑗 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, LObs is a biased estimator for LTrue, because the observed ratings for user-item pairs in the treatment group T are biased samples from the rating potential outcomes of the population PO (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 5-(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 5-(b) for an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' To remedy the bias, IPW-based causal RSs reweight the observed ratings 𝑟𝑖 𝑗 in T by the inverse of the propensity scores, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', 1 𝑒𝑖 𝑗 , which leads to the following new training objective: LIPW = 1 𝐼 × 𝐽 ∑︁ (𝑖, 𝑗):𝑎𝑖 𝑗=1 1 𝑒𝑖 𝑗 (ˆ𝑟𝑖 𝑗 − 𝑟𝑖 𝑗)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (10) The proof for the unbiasedness of LIPW for LTrue can be achieved by utilizing the balancing property of propensity scores in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (7), where we substitute (ˆ𝑟𝑖 𝑗 − 𝑟𝑖 𝑗)2 for 𝑟𝑖 𝑗 in the LHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (7) and treat the rating predictor ˆ𝑟𝑖 𝑗 as constant [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' We also provide a toy example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 5 to intuitively show the calculation of 𝑒𝑖 𝑗, the biasedness of LObs and the unbiasedness of LIPW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The objective for IPW-based RSs defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (10) is model-agnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, it is applicable to all traditional RSs we introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' For example, for MF-based RSs, we can plug in ˆ𝑟MF 𝑖 𝑗 = u𝑇 𝑖 · v𝑗, for DMF-based RSs, we plug in ˆ𝑟DMF 𝑖 𝑗 = 𝑓 𝑢 𝑛𝑛(u𝑖)𝑇 · 𝑓 𝑣 𝑛𝑛(v𝑗), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In practice, since the conditional unconfoundedness assumption in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (3) is untestable, it is usually infeasible to calculate the exact value of 𝑒𝑖 𝑗 based on user/item covariates that satisfy Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Nevertheless, we can still calculate approximate propensity scores ˜𝑒𝑖 𝑗 and reweight the observed ratings by 1/ ˜𝑒𝑖 𝑗, but the unbiasedness of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (10) after the reweighting cannot be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Here we introduce two strategies for the approximate estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' If user/item features f𝑢 𝑖 and f𝑣 𝑗 are available,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 16 Yaochen Zhu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Jing Ma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' and Jundong Li 5 5 5 5 5 5 5 5 5 5 5 5 5 5 1 1 1 1 1 1 1 1 Horror Lover Romance Lover Horror Romance 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 Horror Lover Romance Lover Horror Lover Romance Lover Horror Romance Horror Romance (a) Observed Ratings (b) Rating Potential Outcomes (d) Predicted Ratings 1 1 Horror Lover Romance Lover Horror Romance (c) Propensity Scores 3 4 1 4 1 4 3 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 5: An example adapted from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (2) where the positivity assumption holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Suppose again covariates 𝐶 represent the two-dimensional features (user type, movie type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (a) shows the observed ratings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (b) shows rating potential outcomes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (d) shows the predicted rating potential outcome of an RS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The propensity scores 𝑒𝑖 𝑗 = 𝑝(𝑎𝑖 𝑗 |c) = E[𝑎𝑖 𝑗 |c] are shown in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Based on (a)(d) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (9), LObs = (5 − 1)2 × 2/8 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Based on (b)(d) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (8), 𝐿True = (5 − 1)2 × 8/16 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Based on (a)(c)(d) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (10), LIPW = 1 1/4 (5 − 1)2 × 2/16 = 8, which is unbiased for 𝐿True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' ˜𝑒𝑖 𝑗 can be estimated with logistic regression [39] as follows: ˜𝑒LR 𝑖 𝑗 = Sigmoid �� ∑︁ 𝑘 𝑤𝑢 𝑘 𝑓 𝑢 𝑖𝑘 � + � ∑︁ 𝑘 𝑤𝑣 𝑘 𝑓 𝑣 𝑗𝑘 � + 𝑏𝑖 + 𝑏 𝑗 � , (11) where Sigmoid(𝑥) = (1 + exp(−𝑥))−1, 𝑤𝑢 𝑘 and 𝑤𝑣 𝑘 are the regression coefficients, and 𝑏𝑖, 𝑏 𝑗 are the user and item-specific offsets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' If user/item features f𝑢 𝑖 and f𝑣 𝑗 are not available, we can crudely approximate 𝑒𝑖 𝑗 based on the exposure data alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' For example, we can estimate ˜𝑒𝑖 𝑗 with Poisson factorization [60] as: ˜𝑒PF 𝑖 𝑗 ≈ 1 − exp � −𝝅𝑇 𝑖 · 𝜸 𝑗 � , (12) where 𝝅𝒊 and 𝜸𝒋 are trainable user and item embeddings with Gamma prior, and they can be inferred from the exposure data as discussed in [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Additional strategies to calculate the propensity scores can be found in [62, 63, 64, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The advantage of IPW is that the unbiasedness of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (10) for rating potential outcome estimation can be guaranteed if the propensity scores 𝑒𝑖 𝑗 are correctly estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' However, the accuracy of the propensity score estimation models relies heavily on the domain knowledge and expertise of human experts, which is untestable by experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In addition, IPW suffers from a large variance and numerical in- stability issues, especially when the estimated propensity scores 𝑒𝑖 𝑗 are very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, variance reduction techniques such as clipping and multi-task learning are usually applied to improve the stability of the training dynamics [66, 67, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Substitute Confounder Adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' IPW-based RSs address exposure bias from the data’s perspective: They reweight the biased observational dataset to create a pseudo randomized dataset that allows unbiased training of RSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Confounder adjustment-based methods, in contrast, estimate confounders 𝐶 that cause the expo- sure bias and adjust their effects in the rating prediction model (A simple adjustment Causal Inference for Recommender Systems: A survey 17 strategy is to control 𝐶 as extra covariates12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' For the adjustment to be unbiased, classical causal inference requires the conditional unconfoundedess assumption in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (3) hold, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', no unobserved confounders [33], which is generally infeasible in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Fortunately, recent advances in multi-cause causal inference [69] have shown that we can control substitute confounders estimated from item co-exposure data instead, where exposure bias can be mitigated with weaker assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' We use a𝑖 = [𝑎𝑖1, · · · , 𝑎𝑖𝐽] to denote the exposure status of all 𝐽 items to user 𝑖, which can be viewed as a bundle treatment in clinical trials [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' [42] showed that if we can estimate user-specific latent variables 𝝅𝑖, such that conditional on 𝝅𝑖, the exposures of different items to the user are mutually independent, controlling 𝝅𝑖 can eliminate the influence of multi-cause confounders c𝑚 𝑖 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', confounders that simultaneously affect the exposure of multiple items and ratings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' A simple proof of the claim is that, if c𝑚 𝑖 can still influence a𝑖 and r𝑖 after conditioning on 𝝅𝑖, since c𝑚 𝑖 is an unobserved common cause for the exposure of different items, 𝑎𝑖 𝑗 cannot be conditionally independent (see the discussion of the fork structure in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='2), which renders a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The rigorous proof can be found in [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' further assumed that 𝑝(a𝑖|𝝅𝑖) = Π𝑗 𝑝(𝑎𝑖 𝑗|𝝅𝑖) = Π 𝑗Poission(𝝅𝑇 𝑖 ·𝜸𝒋) and used the Poisson factorization to infer 𝝅𝑖 and 𝜸𝒋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Afterward, exposure bias can be mitigated by controlling 𝝅𝑖 as extra covariates in the RS model [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' For example, controlling 𝝅𝑖 in MF-based RSs leads to the following adjustment: 𝑟adj 𝑖 𝑗 (𝑎𝑖 𝑗 = 1) ∼ N � u𝑇 𝑖 · v 𝑗 ������ user interests + ∑︁ 𝑘 𝑤𝑘𝜋𝑖𝑘 �������������� adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' for expo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' bias , 𝜎2 𝑖 𝑗 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (13) The property of propensity scores can be utilized to further simplify Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (13): If un- confoundedness in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (3) holds for 𝐶 = 𝝅𝑖, it will also hold for 𝐶 = ˜𝑒𝑖 𝑗 = 𝑝(𝑎𝑖 𝑗|𝝅𝑖) [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, we can control the approximate propensity scores estimated by 𝝅𝑖, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', ˜𝑒𝑖 𝑗 = 𝝅𝑇 𝑖 · 𝜸𝒋, which leads to the simplified adjustment formula: 𝑟adj 𝑖 𝑗 (𝑎𝑖 𝑗 = 1) ∼ N � u𝑇 𝑖 · v𝑗 + 𝑤𝑖 · ˜𝑒𝑖 𝑗, 𝜎2 𝑖 𝑗 � , (14) where 𝑤𝑖 is a user-specific coefficient that captures the influence of ˜𝑒𝑖 𝑗 on ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Despite the success in addressing exposure bias with weaker assumptions, one limitation of the above method is that, since Poisson factorization is a shallow model, it may fail to capture the complex influences of multi-cause confounders on item co-exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' To address this problem, recent works have introduced deep neural networks (DNNs) to infer the user-specific substitute confounders 𝝅𝑖 from bundle treatment a𝑖 [71, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' These methods generally assume that a𝑖 are generated from 𝝅𝑖 via 𝑝(a𝑖|𝝅𝑖) parameterized by a deep generative network 𝑓 exp 𝑛𝑛 as: 𝑝(a𝑖|𝝅𝑖) = Π𝑗Bernoulli(Sigmoid( 𝑓 exp 𝑛𝑛 (𝝅𝑖) 𝑗)), (15) 12 Consider again the toy example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' If we know exactly the user type and item type c for each user-item pair, the predictions can be unbiased even if the item exposures are non-randomized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 18 Yaochen Zhu, Jing Ma, and Jundong Li (a) SCM that considers item popularity M (b) SCM under intervention U R V M U R do(V) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 6: (a): SCM that explicitly models item popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (b): SCM under intervention 𝑑𝑜(𝑉 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' where the intractable posterior of 𝝅𝑖 is then approximated with a Gaussian distribu- tion parameterized by DNNs via the variational auto-encoding Bayes algorithm [73], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', 𝑞(𝝅𝒊|a𝑖) = N ( 𝑓 𝝁 𝑛𝑛(a𝑖), diag( 𝑓 𝜎2 𝑛𝑛 (a𝑖))), where 𝑓 𝝁 𝑛𝑛 and 𝑓 𝝈2 𝑛𝑛 are two DNNs that calculate the posterior mean and variance (before diagonalization) of 𝝅𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' With deep generative models introduced to estimate the substitute confounders 𝝅𝑖, non-linear influences of multi-cause confounders on item exposures can be adjusted in the RS models, where exposure bias can be further mitigated in recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The key advantage of substitute confounder estimation-based causal RSs is that controlling confounders in the potential outcome prediction model generally leads to lower variance than IPW-based methods [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' However, these models need to estimate substitute confounders 𝝅𝑖 from the item co-exposures and introduce extra parameters in the RS models to adjust their influences, which may incur extra bias if the confounders and the parameters are not correctly estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In addition, exposure bias due to single-cause confounders cannot be addressed by these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='2 Popularity Bias Popularity bias can be viewed as a special kind of exposure bias where users are overly exposed to popular items [74, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, it can be addressed with tech- niques introduced in the previous section, especially the IPW-based methods [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The reason is that, if we define the popularity of an item as its exposure rate: 𝑚 𝑗 = � 𝑖 𝑎𝑖 𝑗 � 𝑗 � 𝑖 𝑎𝑖 𝑗 , (16) we can view 𝑚 𝑗 as pseudo propensity scores and use IPW to reweight the observed ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Alternatively, we can also analyze and address popularity bias with the structural causal model (SCM), where the causal mechanism that generates the observed ratings under the influence of item popularity is deeply investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The discussion is mainly based on the popularity-bias deconfounding (PD) al- gorithm proposed in [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' PD assumes that the relations among user interests u𝑖, item latent attributes v 𝑗, item popularity 𝑚 𝑗, and observed ratings 𝑟𝑖 𝑗 can be repre- sented by the causal graph illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 6, where item popularity can be clearly identified as a confounder that spuriously correlates the item attributes and the user Causal Inference for Recommender Systems: A survey 19 ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' PD aims to eliminate such spurious correlations with backdoor adjustment, such that the causal influences of u𝑖 and v𝑗 on 𝑟𝑖 𝑗 (which represents users’ interests on intrinsic item properties) can be properly identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Recall that backdoor ad- justment with SCM involves two stages: (1) During the training phase, the relevant structural equations in the causal graph are estimated from the collected dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (2) Afterward, we adjust the influence of confounders according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (5) to remove the spurious correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, we need to estimate 𝑝𝐺(𝑟𝑖 𝑗|u𝑖, v𝑗, 𝑚 𝑗) with the observed ratings 𝑟𝑖 𝑗 and item popularty 𝑚 𝑗 and infer the latent variables u𝑖 and v𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In PD, 𝑝𝐺(𝑟𝑖 𝑗|u𝑖, v𝑗, 𝑚 𝑗) is modeled as a variant of MF as follows: 𝑝𝐺(𝑟𝑖 𝑗|u𝑖, v𝑗, 𝑚 𝑗) ∝ Elu(u𝑇 𝑖 · v𝑗) ���������������������� user interests × 𝑚𝜆 𝑗 ���� pop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' bias , (17) where 𝜆 is a hyper-parameter that denotes our belief toward the strength of influence of item popularity on ratings, and the function Elu (defined as Elu(𝑥) = 𝑒(𝑥) if 𝑥 < 0 else 𝑥 + 1) makes the RHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (17) a proper unnormalized probability density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' After u𝑖, v 𝑗 are estimated from the datasets with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (17), we conduct an intervention on the item node 𝑉 in the causal graph (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (5)), where the spurious correlation due to item popularity can be eliminated with backdoor adjustment: 𝑝(𝑟𝑖 𝑗|𝑑𝑜(u𝑖, v𝑗)) ∝ E𝑝(𝑚𝑗) [Elu(u𝑇 𝑖 ·v𝑗) ×𝑚𝜆 𝑗] = Elu(u𝑇 𝑖 ·v𝑗) ×E𝑝(𝑚𝑗) [𝑚𝜆 𝑗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (18) Since the second term E𝑝(𝑚𝑗) [𝑚𝜆 𝑗] in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (18) is a constant and Elu is a monotonically increasing function, they have no influence on the ranking of the uninteracted items in the prediction phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, we can drop them and use ˆ𝑟𝑖 𝑗 = u𝑇 𝑖 · v𝑗 as the unbiased rating predictor to generate future recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Generally, the debiasing mechanism of PD is very intuitive and universal among backdoor adjustment-based causal RSs [25, 24]: When fitting the RS model on the biased training set, we explicitly introduce the item popularity 𝑚 𝑗 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', the confounder) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (17) to explain away the spurious correlation between item attributes and the observed user ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, the user/item latent variables u𝑖 and v𝑗 used to generate future recommendations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', ˆ𝑟𝑖 𝑗 = u𝑇 𝑖 · v𝑗, can focus exclusively on estimating users’ true interests on intrinsic item properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Is popularity bias always bad?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Recently, more researchers have begun to believe that popularity bias is not necessarily bad for RSs, because some items are popular because they per se have better quality than other items or they catch the current trends of user interests, where more recommendations for these items can be well-justified [77, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, rather than setting the interventional distribution of item popularity to 𝑝(𝑚 𝑗), PD introduced above as well as some other methods [48] further propose to make it correspond to item qualities or reflect the future popularity predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' We will introduce these strategies in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='3 regarding causal generalizations of RSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 20 Yaochen Zhu, Jing Ma, and Jundong Li (a) SCM that considers item content feature Fc and item exposure feature Fb Fb (b) SCM that models the undesirable direct effect of item exposure feature Fb U R V Fc Fb U R V* Fb* Fc* Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 7: (a): The SCM that considers both the causal influences of item content feature 𝐹 𝑐 and item exposure feature 𝐹 𝑏 on item latent variable 𝑉 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (b): The counterfactual SCM where 𝑉 ∗ is determined by baseline value 𝐹 𝑏∗ and 𝐹 𝑐∗ to model the undesirable direct effects of 𝐹 𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='3 Clickbait Bias Different from previous subsections that mainly focus on causal debiasing strategies for collaborative filtering-based RSs, this section discusses content-based recom- mendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Specifically, we discuss the clickbait bias, which is defined as the bias of overly recommending items with attractive exposure features such as sensational titles but with low content qualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The discussion is mainly based on [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' We assume that item features f𝑣 𝑗 can be further decomposed into the item content feature f𝑐 𝑗 that captures item content information and the item exposure feature f𝑏 𝑗 whose main purpose is to attract users’ attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Taking micro-video as an example, item content feature f𝑐 𝑗 can be the audiovisual content of the video, whereas item exposure feature f𝑏 𝑗 can be its title, which is not obliged to describe its content faithfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The relations among user interests u𝑖, item exposure feature f𝑏 𝑗 , item content feature f𝑐 𝑗 , item fused features v𝑗, and the observed ratings 𝑟𝑖 𝑗 are depicted in the causal graph in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 7-(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' We note that clickbait bias occurs when a user’s recorded click on an item because she was cheated by the item exposure feature f𝑏 𝑗 before viewing the item content f𝑐 𝑗 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, the bias can be defined as the direct influence of f𝑏 𝑗 on ratings 𝑟𝑖 𝑗 represented by the causal path 𝐹𝑏 → 𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' To eliminate the clickbait bias, we need to block the direct influence of 𝐹𝑏 on rating predictions, such that the item content quality can be comprehensively considered in recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' As with SCM-based causal RSs, we first estimate structural equations of interest in the causal graph, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', 𝑝𝐺(v𝑗|f𝑏 𝑗 , f𝑐 𝑗 ) and 𝑝𝐺(𝑟𝑖 𝑗|u𝑖, v𝑗, f𝑏 𝑗 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Since distributions in [27] are reasoned in a deterministic manner (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', Gaussian distributions with infinite precision), we keep the discussion consistent with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Specifically, we use v 𝑗 (f𝑏 𝑗 , f𝑐 𝑗 ) = 𝑓 𝑓 𝑓 (f𝑏 𝑗 , f𝑐 𝑗 ) to represent the structural equation 𝑝𝐺(v𝑗|f𝑏 𝑗 , f𝑐 𝑗 ), where 𝑓 𝑓 𝑓 is the feature fusion function that aggregates f𝑏 𝑗 , f𝑐 𝑗 into v 𝑗, and use 𝑟𝑖 𝑗 (u𝑖, v𝑗, f𝑏 𝑗 ) to represent the structural equation 𝑝𝐺(𝑟𝑖 𝑗|u𝑖, v𝑗, f𝑏 𝑗 ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' To explicitly disentangle the influence of item exposure feature f𝑏 𝑗 and item latent variable v𝑗 on the observed ratings,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 𝑟𝑖 𝑗 (u𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' v𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' f𝑏 𝑗 ) is assumed to factorize as follows: Causal Inference for Recommender Systems: A survey 21 𝑟𝑖 𝑗 (u𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' v𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' f𝑏 𝑗 ) = 𝑓 𝑢𝑣 𝑛𝑛 (u𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' v𝑗) ������������������ user interests Sigmoid � 𝑓 𝑢 𝑓 𝑛𝑛 (u𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' f𝑏 𝑗 ) � ���������������������������������������������������� potential clickbait bias ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (19) where the Sigmoid function provides necessary non-linearity in the fusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Essentially, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (19) represents the causal mechanism that generates the observed ratings, which entangles both user interests in item content and clickbait bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' However, after learning the latent variables u𝑖, v𝑗 and functions 𝑓 𝑢 𝑓 𝑛𝑛 , 𝑓 𝑢𝑣 𝑛𝑛 via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (19), removing clickbait bias from the rating predictions is not as straightforward as the PD algorithm, because we should eliminate only the direct influence of item exposure feature f𝑏 𝑗 on ratings 𝑟𝑖 𝑗, while preserving its indirect influence mediated by item latent variable v𝑗, such that all available item features can be comprehensively considered in recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' To achieve this purpose, we first calculate the natural direct effect (NDE) [79] of item exposure feature f𝑏 𝑗 on ratings 𝑟𝑖 𝑗 as follows: NDE(u𝑖, v∗ 𝑗, f𝑏 𝑗 ) = 𝑟𝑖 𝑗 (u𝑖, v∗ 𝑗, f𝑏 𝑗 ) − 𝑟𝑖 𝑗 (u𝑖, v∗ 𝑗, f𝑏∗ 𝑗 ), (20) where v∗ 𝑗 = 𝑓 𝑓 𝑓 𝑛𝑛 (f𝑏∗ 𝑗 , f𝑐∗ 𝑗 ), and the baseline values f𝑏∗ 𝑗 , f𝑐∗ 𝑗 are treated as if the corresponding features are missing from the item [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Since the second term 𝑟𝑖 𝑗 (u𝑖, v∗ 𝑗, f𝑏∗ 𝑗 ) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (20) denotes the user’s rating to a “void” item and can be viewed as a constant, it will not affect the rank of the items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' So we only adjust the first term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (20), which reasons with user 𝑖’s rating to item 𝑗 in a counterfactual world where item 𝑗 has only the exposure feature f𝑏 𝑗 but no content and fused features f𝑐∗ 𝑗 and v∗ 𝑗, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (19) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 7-(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The adjustment leads to the following estimator, ˆ𝑟𝑖 𝑗 = 𝑟𝑖 𝑗 (u𝑖, v𝑗, f𝑏 𝑗 ) − 𝑟𝑖 𝑗 (u𝑖, v∗ 𝑗, f𝑏 𝑗 ) ≜ 𝑟𝑖 𝑗 (u𝑖, v𝑗, f𝑏 𝑗 ) �������������������������� user interests + clickbait − 𝑟𝑖 𝑗 (u𝑖, v∗ 𝑗, f𝑏 𝑗 ) �������������������������� clickbait bias .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (21) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (21) removes the direct influence of f𝑏 𝑗 on rating predictions, such that item content quality can be comprehensively considered in future recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='4 Unfairness Recently, with the growing concern of algorithmic fairness, RSs are expected to show no discrimination against users from certain demographic groups [80, 81, 82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' However, traditional RSs may capture the undesirable associations between users’ sensitive information and their historical activities, which leads to potentially offen- sive recommendations to the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Causal inference can help identify and address such unfair associations, where fairness can be promoted in future recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' This section focuses on the user-oriented fairness discussed in [83], which is defined as the bias where RS discriminately treats users with certain sensitive attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' When considering the user-oriented fairness for RSs, a subset of user features f𝑖, which we denote as s𝑖, is assumed to contain the sensitive information of users, such 22 Yaochen Zhu, Jing Ma, and Jundong Li R S F U V R (a) Causal Generation Process of the Observational Dataset (b) Causal Decision Process of the Traditional RSs (b) (a) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 8: The SCM that reasons with the causal decision mechanism of traditional RSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Observed user ratings 𝑅 can be causally driven by user features 𝐹, including sensitive features 𝑆, which can then unfairly influence the inference of user latent variables 𝑈 and new rating predictions ˆ𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' as gender, race, and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Features s𝑖 are sensitive because recommendations that improperly rely on these features may be offensive to users, which degrade both their online experiences and their trust in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The causal graph that depicts the causal decision mechanism of most traditional RSs is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 8 [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 8 we can find that the user historical behaviors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', the observed ratings 𝑟𝑖 𝑗, are causally driven by user features f𝑖, including user sensitive features s𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, the user latent variables u𝑖 inferred from 𝑟𝑖 𝑗 could capture sensitive user information in s𝑖, which unfairly influences the rating predictions ˆ𝑟𝑖 𝑗 in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' To address this problem, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' [83] proposed to disentangle the user sensitive features s𝑖 from the user latent variable u𝑖, such that the unfair influence of s𝑖 on u𝑖 represented by the causal chain 𝑆 → 𝑅 → 𝑈 can be maximally suppressed in the future recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' A common strategy to achieve the disentanglement is adversarial training [84], where we train a discriminator 𝑓 cls 𝑛𝑛 (u𝑖) → s𝑖 that predicts the sensitive features s𝑖 from user latent variables u𝑖 alongside the RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' While fitting the RS on the observe ratings 𝑟𝑖 𝑗, we constrain the inferred u𝑖 to fool the discriminator 𝑓 cls 𝑛𝑛 by making wrong predictions about s𝑖, which discourages u𝑖 from capturing sensitive information in 𝑟𝑖 𝑗 due to its unfair correlations with s𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Here we take the MF-based RS as an example to show the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' We use LRec to denote the original training objective of the MF-based RS that maximizes the log-likelihood on observed ratings 𝑟𝑖 𝑗 and use Lcls to denote the loss function of the discriminator 𝑓 cls 𝑛𝑛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The adjusted training objective LFair with fairness constraint becomes the following: LFair = LRec(u𝑇 𝑖 · v𝑗, 𝑟𝑖 𝑗) ������������������������������������ user interests −𝜆 · Lcls( 𝑓 cls 𝑛𝑛 (u𝑖), s𝑖) ������������������������������������ fairness constraint , (22) where 𝜆 is a hyper-parameter that balances the recommendation performance and the fairness objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Generally, a higher 𝜆 leads to better fairness, but it also restricts the capacity of the user latent variables u𝑖, which could negatively impact the recommendation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Although here we use the MF-based RS as an example, it is straightforward to generalize Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (22) to DMF or AE-based RS by replacing the u𝑇 𝑖 · v𝑗 term with the corresponding rating estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Causal Inference for Recommender Systems: A survey 23 U R Uc (b) Generalization to PoI recommendation (a) Causal graph for DICE U R Uc user interests user conformity purchases visits user interests Geo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' influence Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 9: Causal Graphs for DICE (a) and its generalization to PoI recommendations (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='2 Causal Explanation in Recommendations In previous sections, we have introduced causality to address various types of bias and spurious correlation issues for traditional RSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In this section, we use causality to explain the user decision process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Specifically, we discuss an interesting question aiming to disentangle users’ intent that causally explains their past behaviors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', did a user purchase an item because she conformed to the current trend or because she really liked it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The tricky part of this question is that: in reality, we only observe the effects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', the purchases, which can be explained by both causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1 Disentangling Interest and Conformity with Causal Embedding The discussion is based on DICE proposed in [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' To simplify the discussion, we consider 𝑟𝑖 𝑗 as implicit feedback and define the set of user, positive item (𝑗 : 𝑟𝑖 𝑗 = 1), negative item (𝑘 : 𝑟𝑖𝑘 = 0) triplets as R 𝑝𝑛 = {(𝑖, 𝑗, 𝑘)|𝑟𝑖 𝑗 = 1 ∧ 𝑟𝑖𝑘 = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The popularity of each item 𝑗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', 𝑚 𝑗, which reflects the current trend, can be calculated with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Observing that the causal relation between user interests 𝑈, user conformity 𝑈𝑐 and observed ratings 𝑅 can be represented as a V-structure in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 9-(a), DICE exploits the colliding effect to achieve the disentanglement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', outcomes that cannot be explained by one cause are more likely caused by another (see discussions in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, although users’ interests cannot be directly estimated from their ratings 𝑟𝑖 𝑗 due to entanglement, their conformity to the trend can be estimated by the popularity level of item 𝑗, and positive feedback not likely caused by conformity has a higher chance of reflecting users’ true interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In implementation, DICE assumes that the observed ratings 𝑟𝑖 𝑗 can be decom- posed into the sum of a conformity part 𝑟𝑐 𝑖 𝑗 = 𝑓 𝑐(u𝑐 𝑖 , v𝑐 𝑗) and a user interests part 𝑟𝑖 𝑖 𝑗 = 𝑓 𝑖(u𝑖 𝑖, v𝑖 𝑗), where u𝑐,𝑖 𝑖 , v𝑐,𝑖 𝑗 are learnable user, item embeddings that reflect user 𝑖’s interests in (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', superscript 𝑖) and conformity to (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', superscript 𝑐) item 𝑗, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' According to the colliding effect of causal graphs, we can split the triplets in R 𝑝𝑛 into two parts: In the first part R (1) 𝑝𝑛, positive item 𝑎 in the triplet has a higher popularity level than the negative item 𝑏, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', 𝑚𝑎 > 𝑚𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In this case, we can draw two general conclusions from this triplet: (1) Overall, the user prefers item 𝑎 over 𝑏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (2) She is more likely to conform to item 𝑎 than item 𝑏 due to 𝑎’s higher 24 Yaochen Zhu, Jing Ma, and Jundong Li popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' These conclusions lead to the two inequalities as follows: ∀(𝑖, 𝑎, 𝑏) ∈ R (1) 𝑝𝑛, we have � 𝑟𝑐 𝑖𝑎 > 𝑟𝑐 𝑖𝑏 (conformity) 𝑟𝑖 𝑖𝑎 + 𝑟𝑐 𝑖𝑎 > 𝑟𝑖 𝑖𝑏 + 𝑟𝑐 𝑖𝑏 (overall preference), (23) where the dependency of 𝑟𝑐,𝑖 𝑖{𝑎,𝑏} on latent variables u𝑐,𝑖 𝑖 , v𝑐,𝑖 {𝑎,𝑏} are omitted for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The second part, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', R (2) 𝑝𝑛, is the key to achieving disentanglement, because for every triplet (𝑖, 𝑐, 𝑑) in R (2) 𝑝𝑛, the negative item 𝑑 is more popular than the positive item 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In this case, user 𝑖 could have simply conformed to the trend and chosen item 𝑑 to consume, but instead, she actively chose the less popular item 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, we can draw one more specific conclusion that leads to the disentanglement between user interests and conformity: The choice of item 𝑐 over 𝑑 is more likely due to user interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, we can form three inequalities as: ∀(𝑖, 𝑐, 𝑑) ∈ R (2) 𝑝𝑛, we have � 𝑟𝑖 𝑖𝑐 > 𝑟𝑖 𝑖𝑑 (interests), 𝑟𝑐 𝑖𝑐 < 𝑟𝑐 𝑖𝑑 (conformity), 𝑟𝑖 𝑖𝑐 + 𝑟𝑐 𝑖𝑐 > 𝑟𝑖 𝑖𝑑 + 𝑟𝑐 𝑖𝑑 (overall preference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (24) The inequalities in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (23) and (24) can be solved by ranking-based loss in RSs, such as Bayesian personalized ranking (BPR) [85], where the disentangled embeddings u𝑐,𝑖 𝑖 , v𝑐,𝑖 𝑗 and the match functions 𝑓 𝑐,𝑖(·, ·) can be learned from R (1) 𝑝𝑛 and R (2) 𝑝𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Finally, we form a rating predictor ˆ𝑟𝑖 𝑗 = 𝑓 𝑖(u𝑖 𝑖, v𝑖 𝑗) + 𝑓 𝑐(u𝑐 𝑖 , v𝑐 𝑗) for future recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='2 Generalizations of DICE DICE disentangles the user intent and promotes the explainability of RSs from the data’s perspective: It partitions the triplets (𝑖, 𝑗, 𝑘) in R 𝑝𝑛 into two disjoint subsets R (1) 𝑝𝑛 and R (2) 𝑝𝑛 based on the relative popularity of the positive and negative items, and shows that the triplets in R (2) 𝑝𝑛 are informative to distinguish the user interests from their conformity to the popularity trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The basic idea of DICE is generalizable to promote explainability for other types of recommendation tasks, if we can find alternative causal explanations to challenge the assumption that the observed positive feedback in these tasks can be attributed solely to user interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' For example, in point-of-interests (PoI) recommendations, the target items are specific point locations that users may find useful or interesting to visit, such as restaurants, grocery stores, and malls [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In this task, the location of a PoI is an important alternative explanation for users’ visits to the PoI other than user interests, because nearby POIs are more convenient to visit than the remote ones [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, to disentangle user interests from potential geographical factors that could causally influence users’ choices, we can take a similar strategy as DICE and partition the user historical visit records according to the distance of positive and negative PoIs to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Then, the disentangled user interest embeddings can be estimated based on the partitioned dataset with the same ranking-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Causal Inference for Recommender Systems: A survey 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='3 Other Works on Explainable RSs Explanable recommendation is a broad topic [87], where disentangling user’s intent based on data partitioning is a small part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' There are also plenty of works that focus on improving the explainability of RSs from the model’s side, where specific disentanglement modules, such as prototype learning [88], context modeling [89], and aspect modeling [90], are designed and integrated with traditional RS models to further enhance their transparency and explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' We refer interested readers to the corresponding papers as well as [91, 92] for further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='3 Causal Generalization of Recommendations After estimating the causal relations from potentially biased and entangled obser- vational datasets, the generalization ability of RSs can be substantially enhanced, because even if the context (or environment) in which we make recommendations changes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', item popularity, user conformity, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' ), we can still basing the rec- ommendations on causal relations that are stable and invariant, while discarding or correcting other undesirable correlations that are transient and susceptible to change [56, 93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In this section, we use the PD algorithm for popularity bias and the DICE algorithm for causal explainability as two examples to show how the generalization of RSs can be improved with causal intervention and disentanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1 Generalization Based on Intervention First, we take the PD algorithm as an example to show how causal intervention can improve the generalization of RSs within a dynamic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In RS, it is generally assumed that user interests can remain unchanged for a certain period of time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', the causal structure 𝑈 → 𝑅 ← 𝑉 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (6) represents the stable user interests on intrinsic item properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' However, the popularity of different items, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', the context in which we make recommendations, can shift rapidly during the same period [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Recall that PD disentangles the causal influences of user interests and item popularity on ratings via the product of two terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', Elu(u𝑇 𝑖 · v𝑗) and 𝑚𝜆 𝑗, as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Suppose 𝑚 𝑗 represents the current popularity level of item 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' If we predict that the popularity of item 𝑗 will change to 𝑚′ 𝑗 in the future [6],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' we can conduct an intervention that sets 𝑀 to the predicted value 𝑚′ 𝑗 in the structural equation 𝑝𝐺(𝑅|𝑈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='𝑉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 𝑀) and predict future ratings 𝑟′ 𝑖 𝑗 via the following formula: 𝑝𝐺(𝑟′ 𝑖 𝑗|u𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' v 𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 𝑑𝑜(𝑚′ 𝑗)) ∝ Elu(u𝑇 𝑖 · v𝑗) ���������������������� stable user interests × (𝑚′ 𝑗)𝜆 ���� future popularity ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (25) 26 Yaochen Zhu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Jing Ma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' and Jundong Li where the user,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' item latent variables u𝑖 and v𝑗 learned from the current time step remain unaltered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' With the influence of future changes in item popularity on ratings considered in the predictions, service providers can make strategic decisions to allo- cate resources for items with different popularity potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In contrast, traditional RSs could mistakenly capture the influence of the current popularity level of items on ratings as user interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, they will not generalize well when the item popularity 𝑚 𝑗 changes to a different level 𝑚′ 𝑗 due to time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='2 Generalization Based on Disentanglement In addition, causal disentanglement can promote the generalization of RSs by iden- tifying and basing recommendations on causes that are more robust to potential changes in the environments [94, 95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' For example, if users’ conformity and interest are disentangled based on their historical behaviors, if a user’s conformity reduces to a low level due to certain reasons, since user interests are assumed to be stable within a certain period of time, we can still use the learned user/item interest variables u𝑖 𝑖, v𝑖 𝑗 to make recommendations based on their interests, where the previously esti- mated unreliable user conformity information can be discarded or down-weighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In contrast, for traditional RSs, different factors that causally influence their histor- ical behaviors are entangled as a single user latent variable u𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, even if some less stable causes of user behaviors are known to change (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', in the PoI RS introduced above, a user could move to a new place where the convenience levels of different PoIs change for the user), these models will still utilize the outdated causes to make recommendations, which could fail to generalize to the new environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 5 Evaluation Strategies for Causal RSs In the previous sections, we have discussed various causal inference techniques that are promising to address multiple types of biases, entanglement, and generalization problems in traditional RSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' However, without a well-designed model evaluation strategy, it is difficult to tell whether the proposed causal RS model is indeed effective, nor can we guarantee that the model will perform reliably after deploying in a real- world environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The evaluation of causal models is particularly difficult, because the groundtruths, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', the causal effects of interest, are usually infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Despite the challenges, there are several strategies that can reliably evaluate causal RSs with biased real-world data, and we will thoroughly discuss them in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In addition, we also compile the available real-world datasets that conduct randomized experiments to eliminate exposure bias to facilitate future causal RS research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Causal Inference for Recommender Systems: A survey 27 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1 Evaluation Strategies for Traditional RSs The assessment of traditional RSs generally follows three steps: First, the observed ratings 𝑟𝑖 𝑗 in the rating matrix R are split into the non-overlapping training set R𝑡𝑟 and test set R𝑡𝑒, usually by randomly holding out a certain percentage of the observed ratings from each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Then, the proposed RS is trained on ratings in R𝑡𝑟 to learn the latent variables and the associated functional models (see Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Finally, the trained RS predicts the missing ratings in R𝑡𝑟 for each user, where the results are compared with the held-out ratings in R𝑡𝑒 to evaluate the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The quality of rating predictions can be measured by accuracy-based metrics such as mean squared error (MSE) and mean absolute error (MAE), and ranking-based metrics such as recall, precision, normalized discounted cumulative gain (NDCG), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' More information on these evaluation metrics can be found in [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='2 Challenges for the Evaluation of Causal RSs The above evaluation strategy, however, is not directly applicable to causal RSs, because ratings in R𝑡𝑒 may have the same spurious correlation and bias as ratings in R𝑡𝑟, which makes the evaluation on R𝑡𝑒 a biased measure of the true model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, to unbiasedly evaluate the effectiveness of causal RSs, it is ideal that we have a biased/entangled training set R𝑏 𝑡𝑟 to learn the model, and an unbiased/disentangled test set R𝑢𝑏 𝑡𝑒 for model evaluation, such that the effectiveness of the causal debiasing/disentangling algorithm can be directly verified from exper- iments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' However, such unbiased/disentangled test set R𝑢𝑏 𝑡𝑒 can be difficult to acquire and expansive to establish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, we first introduce common data simulation strategies for causal RS evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' We then discuss how real-world datasets can be directly utilized to further promote the credibility of causal RS research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='3 Evaluation Based on Simulated Datasets A good dataset simulation strategy to evaluate causal RSs should have the following properties: (1) The generation mechanisms of the bias and entanglement to be studied are clearly identified, credibly designed, and can be adjusted in a flexible manner;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' (2) The available real-world information is utilized as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1 Simulation Based on Generative Models One promising dataset simulation strategy that satisfies the above criteria is to use deep generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Here we take exposure bias as an example to demonstrate how it can be simulated from real-world datasets [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The simulation is composed of 28 Yaochen Zhu, Jing Ma, and Jundong Li two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In the training phase, two variational auto-encoders (VAEs) [22, 73] are trained on the exposure and rating data in a real-world dataset (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', the MovieLens dataset [5]), which results in two decoder networks 𝑓 𝑎 𝑛𝑛 and 𝑓 𝑟 𝑛𝑛 that generate item exposures a𝑖 ∈ {0, 1}𝐽 and user ratings r𝑖 ∈ R𝐽 from 𝐾-dimensional Gaussian user latent variables u𝑎 𝑖 ∼ N (0, I𝐾) and u𝑟 𝑖 ∼ N (0, I𝐾), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The decoders capture the generative distributions of item exposures and user ratings based on the data of real users, where the available real-world information is effectively utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In the generation phase, for each hypothetical user 𝑖′, we draw a confounder c𝑖′ ∼ N (0, I𝐾) that simultaneously affects u𝑎 𝑖′ and u𝑟 𝑖′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Then, to simulate the exposure bias, we set u𝑎 𝑖′ = c𝑖′ and u𝑟 𝑖′ = 𝜆 · c𝑖′ + (1 − 𝜆)𝝐𝑖′ and use 𝑓 𝑎 𝑛𝑛, 𝑓 𝑟 𝑛𝑛 to generate the simulated item exposures a𝑖′ and ratings r𝑖′, where 𝝐𝑖′ ∼ N (0, I𝐾) and hyper- parameter 𝜆 controls the strength of the confounding bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Finally, we mask r𝑖′ with a𝑖′ to form the biased training set R𝑏 𝑡𝑟, and keep the generated ratings r𝑖′ unmasked in the test set R𝑢𝑏 𝑡𝑒 for an unbiased evaluation of model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The advantage of dataset simulation strategies based on generative models is that the true causal mechanisms of interest, such as the rating potential outcomes, are available in the evaluation stage, which is generally impossible for real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, the effectiveness of causal RSs can be easily verified based on the simulated groundtruths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In addition, the simulations are flexible as the strength of biases and entanglements can be set into different levels (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', 𝜆 in the example), where the sensitivity and robustness of causal RSs can be thoroughly investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='2 Test Set Intervention Another reliable dataset simulation strategy is test set intervention, where an in- tervened test set is created from the original test set, such that it has a different bias/entanglement distribution from the training set [56, 60, 97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' For example, to study the popularity bias, we can first select observed ratings from R such that 90% of the interacted items are popular and 10% are unpopular to form the training set R𝑡𝑟 [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' We then select from the remaining ratings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', the original test set R𝑡𝑒, a subset with a different ratio of popular and unpopular items (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', 10% popular and 90% unpopular) to form the intervened test set R𝑖𝑛𝑡 𝑡𝑒 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' If the causal RSs trained on R𝑡𝑟 can still perform well on the intervened test set R𝑖𝑛𝑡 𝑡𝑒 , the model’s invariance to the popularity bias can be supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' A similar test set intervention strategies can be used to evaluate the disentanglement of user interests and conformity for DICE [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The advantage of the test set intervention-based causal RS evaluation strategy is that extra assumptions that cannot be justified by real-world information are mini- mally introduced, because the intervention is usually achieved by selecting samples from the original test set to change the data distribution, which does not introduce extra assumptions of the generative mechanisms or hypothetical users, items, and ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' From this perspective, the evaluation results based on test set intervention may be more credible compared with the generative model-based strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Causal Inference for Recommender Systems: A survey 29 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='4 Evaluation Based on Real-world Datasets 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1 Randomized Experiments For the study of exposure bias, it is feasible to establish-bias free real-world datasets, where ratings for either every item or randomly selected items are collected from a subset of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' This can be extremely expansive and user-unfriendly, but recent years have witnessed a growing interest in causal RS research from the industry, where more such randomized datasets are established and released to facilitate causal RS research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The available real-world datasets are compiled as follows: Coat datasets13 [39] (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The Coat dataset is a small-scale dataset crowd- sourced from the Amazon Mechanical Turkers platform with 300 users and 290 items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Specifically, each Turker is first asked to self-select 24 coats to rate, where the ratings form the biased training set R𝑏 𝑡𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Then each Turker is asked to rate 16 random coats, and these ratings form the unbiased test set R𝑢𝑏 𝑡𝑒 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Yahoo!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' R3 dataset14 [99, 100] (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The Yahoo!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' R3 dataset is collected from the Yahoo!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Music platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The biased training set R𝑏 𝑡𝑟 is composed of 300,000 self-supplied ratings from 15,400 users to 1,000 items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In addition, a subset of 5,400 users is presented with ten randomly selected items to rate, and the ratings are used to create the unbiased test set R𝑢𝑏 𝑡𝑒 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' KuaiRec dataset15 [101] (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The KuaiRec dataset is established based on a popular micro-video sharing platform, KuaiShou, in China (known as Kwai internationally).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The dataset records self-supplied ratings from 7,176 users to 10,728 items as the biased training set R𝑏 𝑡𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The unbiased test set R𝑢𝑏 𝑡𝑒 is composed of a subset of 1,411 users and 3,327 items, where the ratings between these users and items are almost fully observed (with 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='6% density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The statistics of the datasets are summarized in Table 1 for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' There are also randomized datasets for some related topics such as click-through rate prediction [102], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', Criteo Ads datasets16 [103], and bandit-based RS [104], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', Open Bandit dataset17 [105], where the sources are also provided in case the readers are interested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' From Table 1 we can find that, the Coat dataset is small in scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' While for the Yahoo!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' R3 dataset, the training set is comparatively large (15,400 users and 1,000 items), the randomized experiment conducted to establish the unbiased test set is small-scale in comparison (16 and 10 randomly exposed items per user, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, although these ratings are unbiased due to randomization, they may not capture well-rounded user interests and therefore induce a high evaluation variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 13 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
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+page_content='edu/~schnabts/mnar/ 14 https://webscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
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+page_content='datatype=r&did=3 15 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='com/chongminggao/KuaiRec 16 http://cail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='criteo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='com/criteo-uplift-prediction-dataset/ 17 https://research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='zozo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='com/data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='html 30 Yaochen Zhu, Jing Ma, and Jundong Li Dataset # Users # Items Item Type Training Sets Test Sets Coat 300 290 Coat 24 i/u (self-supplied) 16 i/u (random) Yahoo!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' R3 15,400 1,000 Music 300,000 r (self-supplied) 10 i/u (random) for 5,400 u KuaiRec 7,176 10,728 Video 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='3% r (self-supplied) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='6% r for 1,411 u and 3,327 i Table 1: Characteristics of the currently available real-world causal recommendation datasets, where the test sets are devoid of exposure bias either due to randomized item exposures or fully observed ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In the table, terms like 24 i/u mean that every user rates 24 items, the term 300,000 r denotes the number of observed ratings, and terms like 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='3% r represent the density of interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' For the recently released KuaiRec datasets, large-scale experiments are conducted on users to establish the bias-free test set, where the 1,411 users’ ratings for 3,327 items are almost fully collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, it may be a promising new benchmark that allows the evaluation of more complex causal RS models with a lower variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='2 Qualitative Evaluation and Case Study For other types of biases in RSs that cannot be attributed to non-randomized item exposures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', clickbait bias and unfairness), the establishment of bias-free test sets is more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' For example, when studying the clickbait bias, it is difficult to determine whether a user clicked an item due to interests or clickbait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Similarly, when examining the user-oriented fairness of RSs, we cannot know if the generated items are offensive to the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Under such circumstances, we can still conduct case studies for qualitative model evaluations, where we manually select some representative samples from the original test set and observe whether the trained causal RS model would respond as expected to these samples [106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Consider the evaluation of the robustness of a causal RS to clickbait bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' We can select some representative items with low-quality content but highly-attractive exposure features and other items with high-quality content but normal exposure features from the original test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Then, we obtain rating predictions for items from these two groups and draw comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' If the studied causal RS indeed ranks items in the second group higher than those in the first group, we can likely conclude that the model is robust to clickbait bias because the quality of the item content, not its exposure features, is prioritized in recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In addition, to evaluate the user-oriented fairness of a causal RS, we can analyze the generated recommendation for users from certain demographic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' If the recommended items tend to capture the social stereotypes that are negatively associated with user sensitive features, we can conclude that the model is still discriminatory against users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Causal Inference for Recommender Systems: A survey 31 6 Future Directions Despite the recent achievements in marrying causal inference with traditional RSs to address their various limitations of correlational reasoning on observational user data, causal RS research is still in its emerging stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Several promising directions could be pursued to further advance this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In this section, we identify four interesting and important open problems worthy of exploration in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' First, the assumptions required by existing causal RSs could be too strong, which may not hold in reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' For example, most RCM-based causal RSs rely on SUTVA to exclude the interference of item exposures for different users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' However, if users are connected by a social network, they may interact closely with each other or be heavily affected by the influencers in the network [107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Consequently, SUTVA can be violated because the recommendations made to one user may causally affect the ratings of others (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', the spill-over effects [108, 109]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In addition, the positivity assumptions may also be violated if some users never click certain types of items (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', non-compliance and defiers [33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, it is crucial to further weaken the assumptions of causal RSs to make them more practical for real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In addition, there currently lacks a universal causal model for RSs that can be applied for different causal reasoning purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Most SCM-based causal RSs are designed to address one specific type of bias or entanglement problem, where other issues are tacitly assumed to be absent and omitted from the causal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Moreover, even for causal RSs that address the same problem, several varieties of causal graphs that include different sets of variables and relationships can be assumed, which leads to inconsistency between different works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, it would be promising and beneficial to have a generic and widely-accepted causal model that is able to comprehensively address multiple types of causal problems in recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Furthermore, certain types of biases in RSs are double-blade swords, where the positive side is seldom investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Consider the item exposure bias discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' We should note that some items are more likely to be exposed because they have higher quality than other items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, the higher exposure rate of these items can be well justified and may be utilized to further enhance the recommendation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In addition, recent research also found that confounders that spuriously correlate item exposures and user ratings may also help explain the co-occurrence patterns of different items [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, how to properly identify and utilize the positive side of biases while maximally suppressing their negative effects is of great importance and deserves more in-depth investigations in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Finally, although recent years have witnessed the establishment and release of more real-world datasets for causal RS research from the industry, many causal RS models still rely heavily on simulated datasets for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' The simulation can lead to the over-simplification of the problem and is often designed to correspond exactly with the debiasing/disentanglement mechanism of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Therefore, the effectiveness of these methods in more complicated real-world scenarios is still uncertain due to the lack of model deployment and online tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' As such, to more convincingly demonstrate the practical utility of causal RSs, more collaborations with the industry are highly expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' 32 Yaochen Zhu, Jing Ma, and Jundong Li 7 Summary In this survey, we provide a comprehensive overview of recent advances in causal inference for RSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' We start by pointing out issues of traditional RSs that rely on correlations in observed user behaviors and user/item features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' We then introduce two mainstream causal inference frameworks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=', Rubin’s RCM and Pearl’s SCM, which provide deeper insights into these issues and the foundation for moving traditional RSs to the upper rungs of the Ladder of Causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Specifically, we thoroughly discuss several state-of-the-art causal RS models that lead to enhanced robustness to various biases and improved explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' In addition, since causal RSs can base recommendations on causal relationships that are stable and invariant, we also demonstrate that their generalization abilities can be significantly improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Finally, we introduce evaluation strategies for causal RSs, with an emphasis on how to reliably estimate the model performance based on biased real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' We further compile real-world datasets where expensive randomized experiments are conducted on users, which reflects growing attention to causal RSs from the industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Overall, causal RS is still a relatively new and under-explored research topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' More efforts are urgently demanded to systematize the existing works and conduct deeper investigations for further improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Accordingly, we point out four interesting and practically important open problems in causal RSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' We hope that this survey can help readers gain a comprehensive understanding of the main idea of applying causality in RSs and encourage further progress in this promising area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
+page_content=' This work is supported by the National Science Foundation under grants IIS-2006844, IIS-2144209, IIS-2223769, CNS-2154962, and BCS- 2228534, the JP Morgan Chase Faculty Research Award, and the Cisco Faculty Research Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQf_fqZ/content/2301.00910v1.pdf'}
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+Prepared for submission to JHEP
+The Holographic Map of an Evaporating
+Black Hole
+Zsolt Gyongyosia, Timothy J. Hollowooda, S. Prem Kumara, Andrea
+Legramandia,b,c and Neil Talwara
+aDepartment of Physics, Swansea University, Swansea, SA2 8PP, U.K.
+bPitaevskii BEC Center, CNR-INO and Dipartimento di Fisica, Universit`a di Trento, I-
+38123 Trento, Italy
+c INFN-TIFPA, Trento Institute for Fundamental Physics and Applications, Trento, Italy
+E-mail:
+z.gyongyosi.2133547@swansea.ac.uk,t.hollowood@swansea.ac.uk,
+s.p.kumar@swansea.ac.uk, andrea.legramandi@unitn.it,
+n.talwar.2017429@swansea.ac.uk
+Abstract:
+We construct a holographic map that takes the semi-classical state of
+an evaporating black hole and its Hawking radiation to a microscopic model that re-
+flects the scrambling dynamics of the black hole. The microscopic model is given by a
+nested sequence of random unitaries, each one implementing a scrambling time step of
+the black hole evolution. Differently from other models, energy conservation and the
+thermal nature of the Hawking radiation are taken into account. We show that the
+QES formula follows for the entropy of multiple subsets of the radiation and black hole.
+We further show that a version of entanglement wedge reconstruction can be proved by
+computing suitable trace norms and quantum fidelities involving the action of a unitary
+on a subset of Hawking partners. If the Hawking partner is in an island, its unitary
+can be reconstructed by a unitary on the radiation and so the Hawking partners are
+not in any sense behind the horizon of the black hole. We also consider the problem of
+reconstruction for unitaries acting on an infalling system.
+arXiv:2301.08362v1 [hep-th] 19 Jan 2023
+
+Contents
+1
+Introduction
+3
+1.1
+The holographic map
+3
+1.2
+The QES formula
+7
+2
+The model
+11
+2.1
+The refined model
+12
+3
+The microscopic state
+15
+3.1
+The average state
+16
+3.2
+The inner products xΨJ|ΨKy
+17
+4
+Entropies
+19
+4.1
+Refined model
+21
+4.2
+Relation to the island formalism
+24
+5
+Information recovery and reconstruction
+26
+5.1
+Bounding the trace norm
+31
+6
+Reconstruction of the Hawking partners
+31
+7
+Discussion
+35
+A Thermodynamics of free fields
+36
+B Dominant saddles
+38
+B.1 Proof
+39
+– 2 –
+
+1
+Introduction
+Black holes lie at the front line of the struggle to unify quantum mechanics with gravity.
+Recent progress is focused on how this struggle plays out at the level of effective theory
+in a gravitating system like a black hole. In particular, the effective description involves
+techniques that have evolved over many years involving quantum field theory over a
+fixed background spacetime using semi-classical techniques. In a black hole geometry
+this leads to the emission of Hawking radiation and the apparent loss of unitarity [1, 2].
+On the other hand, there is a microscopic level of description, for example provided
+by string theory, in which a black hole is described as a quantum system with a large
+density of states given by the Bekenstein-Hawking (BH) entropy (see [3] for a review).
+1.1 The holographic map
+Recent progress has shed light on how these two levels of description are related and
+how the information-loss paradox is resolved and unitarity is restored [4–7] (also see the
+reviews [8, 9]). A key ingredient is a map, the ‘holographic map’, between the effective
+semi-classical description and the microscopic description
+V :
+Hsc Ñ Hmicro .
+(1.1)
+The idea of such a map between the semi-classical and microscopic descriptions nat-
+urally arises in holography where the semi-classical state describes the state of bulk
+gravitational theory while the microscopic state describes the non-gravitational CFT
+dual [10–15]. It is becoming clear that such a map should apply more generally and
+specifically in spacetimes which are not asymptotically AdS, such as an evaporating
+black hole, where the radiation can escape the AdS bulk. The holographic map has
+been interpreted as the encoding map of a quantum error code and this synergy be-
+tween the two subjects has been very fruitful and has led to a better understanding of
+entanglement wedge reconstruction [16–26]. However, recent work [27, 28]1 has clarified
+certain details and in particular argued that, in the context of a black hole, it is an im-
+portant feature that the map is not isometric, V :V ‰ 1. This means that the relation
+with the standard theory of quantum error correcting codes is not so compelling. The
+non-isometric nature of the map is actually very natural because as the black hole ages
+its Hilbert space becomes too small to accommodate all the Hawking partners of the
+previously emitted radiation and so something has to give. Another key insight of [28]
+1See also [29, 30] for recent developments.
+– 3 –
+
+is that the map does not act on the radiation once it has dispersed away from the black
+hole. This clarifies certain statements that have been made about the radiation, in
+particular it is not possible to change the state of the black hole by making operations
+on the radiation however complicated: there is no long-range non-locality of this kind.
+The purpose of this work is to construct the holographic map V in a very simple
+microscopic model of black hole evaporation defined e.g. in [31, 32] but refined to take
+account of energy conservation leading to thermal states.
+The basic version of the
+model is the block random unitary model (BRU) of [28]. A number of key features
+follow also for this more refined model:
+1 The semi-classical state of the radiation ρsc
+R is precisely the average of the mi-
+croscopic state of the radiation ρR over the quasi-random microscopic scrambling
+dynamics of the black hole.
+2 Past the Page time the quasi-random fluctuations of the microscopic state ρR
+overwhelm the state and it becomes very different from the semi-classical (Hawk-
+ing) state ρsc
+R.
+3 The Quantum Extremal Surface (QES) formula [33, 34] for the entropy of a
+generic number of radiation and black hole subsets is derived in the regime where
+the black hole is evaporating slowly [31, 35].
+4 Unitary actions on an infalling system can be reconstructed on the radiation
+after the Page time showing that the information of the infalling system has been
+teleported out of the black hole realizing the Hayden-Preskill ‘black hole as a
+mirror’ scenario [36].
+5 There is a version of state-specific entanglement wedge reconstruction (of the
+type discussed in [28]): local unitaries acting on the Hawking partners can be
+reconstructed as a unitary acting on the black hole before the Page time and on
+the radiation after the Page time. The discussion is extended for generic subsets
+of the Hawking radiation.
+The last point should not be used to conclude that, past the Page time, one is
+measuring something behind the horizon of the black hole by measuring the radiation:
+there is no such dramatic non-locality. Rather it means that the Hawking partners
+have been teleported out of the black hole and so one is measuring a property that is,
+in any case, of the radiation.
+– 4 –
+
+Let us now put some flesh on the bones. At the semi-classical level, the state of a
+QFT in the black hole background consists of an entangled state between the outgoing
+Hawking radiation R and their partner modes behind the horizon R. The overall state
+is pure
+|ψy “
+! ÿ
+J
+λJ|JyR b |JyR
+)
+b |SyF P Hsc .
+(1.2)
+We have also included the possibility for infalling modes in the state |SyF, including
+the matter that collapsed to form the black hole. We will develop two models: (i) a
+simple one in which the Hilbert space of the radiation is taken to be finite dimensional
+and (1.2) is the maximally entangled state λJ “ 1{?dR and (ii) a more refined one
+for which the radiation and partners are in a thermofield double with a slowly varying
+temperature.
+At the microscopic level, the black hole is described by a finite dimensional Hilbert
+space HB whose dimension is exponential in the BH entropy dB “ eSBH. The black
+hole emits Hawking radiation and at the microscopic level we can write the state of a
+partly evaporated black hole and radiation as
+|Ψy “
+ÿ
+J
+λJ|JyR b |ΨJyB P Hmicro .
+(1.3)
+The two states, the semi-classical |ψy and the microscopic |Ψy are related by the
+holographic map (1.1)
+V :
+HR b HR b HF Ñ HR b HB .
+(1.4)
+It was argued in [28] that the map should act trivially on R because the outgoing
+radiation system is identical in both the semi-classical and microscopic descriptions.
+So V actually only acts non-trivially as HR b HF Ñ HB. This is natural because the
+Hawking partner modes R and the infalling modes F are behind the horizon and so part
+of the black hole whose semi-classical geometry should emerge from the microscopic
+description.2 By comparing (1.2) with (1.3), we have
+V |JyR b |SyF “ |ΨJyB .
+(1.5)
+We leave the dependence on the infalling state implicit.
+2Although this breaks down after the Page time.
+– 5 –
+
+The way that Hawking’s information loss paradox can be resolved now reveals itself.
+In Hawking’s analysis, the state of the radiation is the reduced state, the maximally-
+mixed state in the basic model and a quasi-thermal state in the refined model
+ρsc
+R “
+ÿ
+J
+|λJ|2|JyRxJ| ,
+(1.6)
+since the partner mode states are orthonormal, RxJ|KyR “ δJK. On the other hand,
+at the microscopic level,
+ρR “
+ÿ
+JK
+λK¯λJξKJ|KyRxJ| ,
+ξKJ “ xΨJ|ΨKy .
+(1.7)
+The semi-classical state is devoid of internal correlations, information is lost and uni-
+tarity is violated. The microscopic state, on the other hand, can carry the correlations
+and repair unitarity if the inner products ξJK are non-trivial. The fact that
+xΨJ|ΨKy ‰ δJK ,
+(1.8)
+implies that the holographic map V is non-isometric, a key insight in [28]. It is the
+non-isometric nature of V that allows information to escape out of the black hole in
+the correlations induced by the inner product [37]. Such a release of information would
+presumably be interpreted as being a non-local process to a semi-classical observer.
+This is a major insight but perhaps to be expected when spacetime geometry is an
+emergent concept.
+For a black hole past its Page time, when Srad " SBH, one would expect the states
+|ΨJy to be far from orthogonal because there are order eSradpRq states in a much smaller
+eSBH dimensional Hilbert space. Roughly speaking, as previously argued e.g. in [4, 38],
+we find
+xΨJ|ΨKy “
+#
+1 ` Ope´SBHq
+J “ K ,
+Ope´SBH{2q
+J ‰ K ,
+(1.9)
+so the violation appears to be exponentially small „ e´1{G in the semi-classical limit.
+This seems to suggest that the corrections coming from the microscopic theory will be
+small. However, if we write ξ “ I ` Z, then Z is roughly-speaking a quasi-random
+Hermitian matrix whose elements are order e´SBH{2. It seems, therefore, that the effect
+of Z would be very suppressed. However, if the dimension of the matrix „ eSrad is large
+then its eigenvalues can be expected to lie in a distribution between ˘epSrad´SBHq{2.
+What this indicates is that the fluctuations in Z could be expected to give rise to a
+– 6 –
+
+radical change in the state of the radiation beyond the Page time when Srad " SBH
+and a mechanism to ensure the unitarity of the evaporation. On the other hand, if
+we average the microscopic state over the quasi-random fluctuations Z we recover the
+semi-classical state
+ρR “ ρsc
+R .
+(1.10)
+The fact that ρR ‰ ρsc
+R means that if we were to attempt to interpret the microscopic
+state as a state on the semi-classical geometry, then in the near-horizon region it would
+not be the inertial vacuum and so we could expect there will be non-trivial energy and
+momentum of order e´1{G, as the horizon is approached .
+Another issue that is clarified by the fact that V acts trivially on the radiation
+R is, as already mentioned, that the state of black hole is completely invariant under
+any local action on the radiation.
+In more detail, the most general local action is
+obtained by coupling R to an auxiliary system M and having them interact. On the
+semi-classical state
+|ψy b |∅yM ÝÑ
+ÿ
+α
+Kα|ψy b |αyM ,
+(1.11)
+for some orthonormal states |αy of M and where the operators Kα act on R. This
+defines a quantum channel acting on R and unitarity implies that Kα are Krauss
+operators ř
+α K:
+αKα “ 1. Mapping this to the microscopic state, and using the fact
+that rV, Kαs “ 0, the reduced state on B, after R and M have interacted, is
+ρ1
+B “
+ÿ
+α
+TrR
+!
+Kα|ΨyxΨ|K:
+α
+)
+“ TrR
+!
+|ΨyxΨ|
+ÿ
+α
+K:
+αKα
+)
+“ ρB ,
+(1.12)
+so the state of the black hole is invariant.
+1.2 The QES formula
+One can quantitatively appreciate how ρR differs from ρsc
+R by calculating their von
+Neumann entropies. The entropy of the semi-classical state ρsc
+R, suitably regularized,
+is just the thermal entropy of Hawking radiation familiar from Hawking’s calculation.
+The question is, how to calculate the entropy of the microscopic state ρR? This is
+where the QES, or generalized entropy, formula comes in [14, 33, 39–41]. It relates the
+von Neumann entropy of the microscopic state ρA reduced on some subsystem factor
+e.g. A “ R or B, or some more specific subset of R to the generalised entropy:
+SpρAq “ min ext
+tXju
+! ÿ
+j
+A pXjq
+4G
+` Spρsc
+WpAqq
+)
+.
+(1.13)
+– 7 –
+
+In the above, WpAq is the entanglement wedge of A, some subset of a Cauchy slice3 that
+contains the radiation R near I ` and passes through the Quantum Extremal Surfaces
+(QES) Xj which are the boundaries of WpAq in the gravitating region determined by
+extremization. The first term involves the area of the QES. If A is the radiation, or some
+subset thereof, then the entanglement wedge WpAq consists of A but also, potentially,
+a region disconnected from A known as the ‘entanglement island’, or ‘island’ for short,
+WpAq “ A Y I.
+The formula (1.13) is remarkable in several ways but principally because it allows
+one to calculate the entropy of the microscopic state ρA using only semi-classical tech-
+niques even when the details of the microscopic theory are not known. It does this by
+implicitly averaging over the complex chaotic microscopic dynamics of the black hole
+in the way familiar from statistical mechanics. More precisely, when computed in the
+semi-classical theory, we can think of the left hand side as being equal to the usual
+n Ñ 1 limit of the R´enyi entropies but averaged in the following way
+SpρAq “ lim
+nÑ1
+1
+1 ´ n log ep1´nqSpnqpρAq ,
+(1.14)
+with the average over a suitable ensemble that is a proxy for the underlying complex,
+chaotic microscopic dynamics. Just as in statistical mechanics, the conceptual idea is
+that the average captures the behaviour of a single typical microscopic state because,
+unlike the state itself, the R´enyi entropies are self-averaging quantities.
+For an evaporating black hole, the QES are behind the horizon and when the
+evaporation is slow, which it is for most of the evaporation time apart from the final
+stage, the QES are very close behind the horizon. In fact the QES are completely
+determined within the scope of the slow evaporation approximation [31, 35, 42]. Firstly,
+they have Kruskal-Szekeres (KS) coordinates related via
+UV ∼
+c
+SBH
+! 1 ,
+(1.15)
+where c are the number of (massless) fields. In terms of Eddington-Finkelstein (EF)
+coordinates pu, vq,4 this means
+v “ u ´ ∆tscr ,
+∆tscr “
+1
+2πT log SBH
+c
+.
+(1.16)
+3More precisely the causal diamonds thereof.
+4The KS and EF coordinates are related by an approximately exponential map,
+U
+“
+´ exp
+`
+´2π
+şu Tptqdt
+˘
+and V “ exp
+`
+2π
+şv Tptqdt
+˘
+, where Tptq is the instantaneous temperature of
+the black hole.
+– 8 –
+
+The slow evaporation regime applies precisely when SBH " c so that the QES are
+pressed up against the horizon from within. The time shift above between the infalling
+and outgoing coordinates ∆tscr is identified with the scrambling time of the black hole.
+This is time dependent but only changes slowly as the black hole evaporates.
+The second condition on the QES is that the outgoing EF coordinate of a QES
+uQES (inside the horizon) must be equal to the outgoing EF coordinate of one of the
+endpoints of the radiation uBA (outside the horizon)
+tuQESu Ă tuBAu .
+(1.17)
+This reduces the variation problem to a discrete minimization problem.
+When A is a subset of the radiation and the entanglement wedge WpAq “ A Y I,
+the second term in (1.13) is just the thermal entropy5
+Spρsc
+WpAqq « SradpA a ˜Iq “ πc
+6
+ż
+Aa˜I
+Tpuq du ,
+(1.18)
+where Tpuq is the instantaneous temperature of the black hole as a function of the
+outgoing EF coordinate u on I `.
+Here, ˜I, the ‘island-in-the stream’, is just the
+reflection of the island in the horizon and projected onto I ` [31, 35, 43] . So in terms
+of the outgoing EF coordinate u, I and ˜I are equal, with the former outside the horizon
+and the latter inside. The symmetric difference in (1.18) accounts for the fact that I
+contains purifiers of the radiation. The first term in (1.13) is then approximately equal
+to the Bekenstein-Hawking entropy SBH evaluated at EF outgoing coordinates of the
+QES uBI. Hence, within the slow evaporation approximation, we can write the entropy
+as a discrete minimization problem
+SpAq « min
+I
+! ÿ
+uBI
+SBHpuBIq ` SradpA a ˜Iq
+)
+.
+(1.19)
+This formula can easily be adapted to the case when A includes the black hole itself,
+B Ă A. One simply replaces A by A X R in the second term.6 In section 4 we verify
+5There is a common divergence associated with the end-points of A at I ` which can be regularized.
+The divergences associated to end-points of I, on the other hand, are precisely cancelled by the
+divergences in the QES term in (1.13).
+6Then, if RN R A, the most recent emitted interval of radiation, there must be a QES with a u
+coordinate equal to the u coordinate of the upper end-point of RN, giving a contribution SBHpMNq ”
+SBHpMq to (1.19). On the other hand, if RN P A then it must be that RN Ă I. In the latter case, the
+connected subset of I that includes RN is not strictly-speaking part of the island although it is in the
+entanglement wedge of A.
+– 9 –
+
+this formula in both the basic and refined models using the replica trick. We also find a
+simple formula (4.29) for the island which relates the replica trick and the entanglement
+wedge in quite a direct way.
+The paper is organized as follows. In section 2 we define two simple discrete models
+of a holographic map for an evaporating black hole. There is a basic and refined model.
+Compared with the basic model, the refined model has the nice features that the state
+of a small subsystem is thermal instead of maximally mixed and that the irreversiblity
+of evaporation is naturally incorporated (there is no need to add in ancilla qubits to
+mimic this effect). In section 3 we calculate some of the properties of the microscopic
+state starting with its average over the quasi-random unitary time evolution in order to
+show that the averaged state of the radiation is just the semi-classical state (1.10). We
+then compute the inner products (1.8) and show that they average (over the scrambling
+dynamics) to the delta function, also implying (1.10), but have a non-trivial variance
+consistent with (1.9). In section 4, we compute the R´enyi and von Neumann entropies
+of subsets of the radiation and black hole and derive the minimization problem for the
+generalized entropy of a slowly evaporating black hole (1.19). In section 5, we turn
+to the Hayden-Preskill scenario [36] and consider when the action of a unitary on an
+infalling system be reconstructed (in a state-specific sense) on a subset of the radiation
+or black hole from a ‘decoupling argument’. We find the model reproduces the ‘black
+hole as a mirror’ phenomenon and reconstruction is possible on the radiation when the
+black hole is past the Page time. This problem of reconstruction of operators acting
+on an infalling system was studied in the basic (or BRU) model and a random pairwise
+interaction model (which incorporates the fast scrambling nature of black holes) in [28].
+Our main contributions here are to study this problem in a model which generalises
+the basic model and also to consider when reconstruction is possible not just on the
+radiation or the black hole, but a subset thereof. In section 6 we consider when local
+operations, in the form of a quantum channel, acting on the Hawking partners can be
+reconstructed on a subset of the radiation or black hole. As expected, we find that
+reconstruction on the radiation is possible when the black hole is past the Page time.
+In section 7 we draw some conclusions. In appendix A we review the computation of
+certain thermodynamic quantities for free bosonic and fermionic fields, which is used
+in the refined model. In appendix B we provide a proof of the dominant saddles which
+contribute in the replica trick calculation in the basic model.
+– 10 –
+
+B0 “ F0
+R1
+B1
+U1
+R2
+B2
+U2
+F1
+BN´1
+RN
+B ” BN
+UN
+FN´1
+Figure 1: The model of black hole evaporation consisting of a sequence of random unitaries that
+mimic the scrambling microscopic dynamics. At each time step a small subsystem escapes as the
+Hawking radiation and there can be an infalling system.
+The time steps are of the order of the
+scrambling time of the black hole (1.16) and so the model will appear to be continuous at time scales
+much larger than the scrambling time, including the Page time and the evaporation time.
+2
+The model
+In the model, described in [31], the evaporation at the microscopic level is described
+by a series of discrete time steps identified with the scrambling time of the black hole
+(1.16) shown in the figure 1. During the pth time step the state of the black hole evolves
+by a unitary Up which maps
+Up :
+HBp´1 b HFp´1 ÝÑ HBp b HRp .
+(2.1)
+In the basic model, we have dBp´1dFp´1 “ dBpdRp, whereas in the refined model energy
+conservation is taken into account and Rp is infinite dimensional. In this case, Up is
+an isometric embedding of a microcanonical energy window into HRp b HBp, as we will
+describe later. After N time steps, the state of the black hole and radiation is
+|ΨptNqy “ UN ¨ ¨ ¨ U2U1|SyF P HB b HR ,
+(2.2)
+where
+|SyF “ |s0yF0 b |s1yF1 b ¨ ¨ ¨ b |sN´1yFN´1 ,
+(2.3)
+describes the infalling matter that created the black hole B0 ” F0 as well as matter that
+falls in during each time step Fp as the black hole evaporates. The radiation is split
+into a temporal sequence of subsets R “ Ť
+p Rp. In the above, the remaining black hole
+is B ” BN. A basis of states of the radiation consists of |JyR where J “ tj1, . . . jNu
+– 11 –
+
+and each jp P t1, 2, . . . , dRpu labels the states in the pth time step Rp. In particular, the
+microscopic states defined in (1.3) are
+|ΨJy “
+1
+?λJ
+RxJ|UN ¨ ¨ ¨ U2U1|SyF P HB .
+(2.4)
+Consequently the time evolution of the black hole in the model leads to a concrete
+expression for the holographic map,
+V “
+ÿ
+J
+1
+?λJ
+RxJ| b RxJ|UN ¨ ¨ ¨ U2U1 ,
+(2.5)
+acting on HR b HF. In the refined model the sum here is not well defined and V is
+only defined acting on suitable states such as |ψy.
+The basic model [31] is identified with the block random unitary model of [28].
+What is noteworthy is that the projection in (2.5) onto the maximally-entangled state
+of HR b HR in the basic model, is essentially post selection and manifests the non-
+isometric property of the map and it is this that provides the mechanism for information
+to be teleported out of the black hole.
+At the semi-classical level, the subsets of Hawking radiation Rp and their partners
+behind the horizon Rp are illustrated in the Penrose diagram figure 2.
+2.1 The refined model
+In this section we refine our model of black hole evaporation to take account of energy
+conservation and the thermal nature of the Hawking radiation. We will work in the
+adiabatic, or quasi-static, regime where the black hole is evaporating slowly enough that
+it makes sense to ascribe a slowly varying temperature Tptq to the Hawking radiation
+determined by the thermodynamic equation of the black hole
+1
+T “ dSBH
+dM
+,
+(2.6)
+where M is the black hole mass7. The adiabatic regime is where Hawking’s calculation
+derivation is valid. It is defined by the requirement that
+SBH " c ,
+(2.7)
+7For a Schwarzschild black hole, M is the mass, while for the charged black hole and the black hole
+in JT gravity, M is the mass minus the mass of the extremal black hole.
+– 12 –
+
+I `
+Rp
+Fp
+Rp
+horizon
+Figure 2: Subsets of Hawking modes Rp are their entangled partners Rp behind the horizon and
+infalling modes Fp. The Hawking modes propagate out to null infinity I `. Each Rp and Fp lasts for
+a scrambling time.
+the number of massless fields. In addition, for a semi-classical limit c " 1.
+The time dependence of the energy is determined by the energy flux of the Hawking
+radiation. Since most of the energy loss occurs in the s-wave modes we have effectively
+a 1 ` 1-dimensional relativistic gas. We also ignore the possibility for back-scattering
+of modes and so take a trivial greybody factor. The energy balance equation is then
+dM
+dt “ ´πcT 2
+12
+(2.8)
+and given (2.6) all that it needed to determine the time evolution of M, T and SBH is
+the energy dependence of the BH entropy which depends on the nature of the black
+hole. For example, for Schwarzschild SBH “ 4πGM 2.
+We will model the evaporation in terms as a series of time steps whose size are of
+the order of the scrambling time of the black hole,
+∆t ∼ 1
+T log SBH
+c
+.
+(2.9)
+Note that this is time dependent, so the size of the time steps adapt as the evaporation
+proceeds.
+At each time step, the radiation carries away a small amount of energy in a distri-
+bution that is strongly peaked around an average. Therefore, we can model the state
+– 13 –
+
+of the black hole at each time step as lying in a Hilbert space HBp describing a system
+with energy in a small window Θp “ rMp, Mp`δMs. Implicitly, Mp includes the energy
+of the infalling system Fp. In other words, the black hole is in a microcanonical state.
+The size of the window δM is assumed to be small but for simplicity we will assume
+that it is much larger than the spread of the energy carried away by the radiation at
+each time step. The fact that the BH entropy is so large means that Θp contains a
+vast number of states that forms a quasi-continuum. The dimension of this space is
+exponential in the Bekenstein-Hawking entropy
+dBp “ CδM
+Mp
+eSBHpMpq ∼ eSBHpMpq .
+(2.10)
+In the above, C is some constant which we do not have to specify since SBHpMq is very
+large.
+The picture of the black hole evolving through a sequence of microcanonical states
+is of course an approximation which is justified because the radiation emitted during a
+time step has a sharply defined average energy and a spread that is assumed to be much
+smaller than the width of the windows δM. Let us justify this claim. Since the time
+step, the scrambling time ∆t, is much greater than the thermal scale T ´1, the energy
+and entropy of the Hawking radiation follow from the standard statistical mechanics of
+a relativistic bosonic or fermionic gas (summarized in appendix A). For a bosonic gas
+E “ cV
+ż dω
+2π
+ω
+eω{T ´ 1 “ πcV T 2
+12
+(2.11)
+and the entropy Srad “ πcV T{6, where we identify the volume with the space filled by
+the gas in the scrambling time, i.e. V “ ∆t. In particular, the entropy
+Srad “ πc∆tT
+6
+∼ c log SBH
+c
+" 1 .
+(2.12)
+Hence, the Hawking modes emitted in a time step have a large entropy and so can be
+described thermodynamically. Indeed, the normalized spread of the energy
+∆E
+E
+∼
+1
+?Srad
+! 1 .
+(2.13)
+On the other hand, the radiation is a much smaller system than the black hole because
+SBH " c log SBH
+c
+.
+(2.14)
+We will then assume that this spread is much smaller than the microcanonical energy
+window δM " ∆E justifying the evaporation as a sequence of microcanonical states.
+– 14 –
+
+The semi-classical state is now a thermofield double with a slowly varying tempera-
+ture. Taking the basis states |jpy to be approximate energy eigenstates with eigenvalues
+Ejp, we have
+λJ “ e´ ř
+p Ejp{2Tp
+?
+Z
+,
+(2.15)
+where Z “ ř
+J e´ ř
+p Ejp{Tp is the partition function which provides normalization. The
+temperature Tp is the instantaneous temperature of the Hawking radiation given in
+(2.6) evaluated at E “ Mp. The states |jpy are to be thought of as localized in an
+outgoing shell of thickness ∆tp. This is justified because the modes have characteristic
+momentum Tp and so can be localized on scales T ´1
+p
+which is much smaller than ∆tp.
+3
+The microscopic state
+Black holes are famously fast scramblers so that over the scrambling time Up is essen-
+tially a random unitary. The question of how random time evolution of a black hole is
+an interesting question but one can make the hypothesis that for certain quantities it
+is effectively indistinguishable from a Haar random unitary. In this section, we make
+that assumption and compute some properties of the microscopic state.
+We will need to average quantities over an N ˆ N unitary for which the basic
+results is the integral
+ż
+dU U ˚
+ABUA1B1 “ 1
+N δAA1δBB1 .
+(3.1)
+We will also need the generalization of this involving n replicas:
+ż
+dU
+n
+ź
+j“1
+U ˚
+AjBjUA1
+jB1
+j “
+ÿ
+σ,τPSn
+n
+ź
+j“1
+δAjAσpjqδBjBτpjqWgpστ ´1, Nq ,
+(3.2)
+where Wg is the Weingarten function [44, 45]. Note how the integrals over the replicas
+involves a sum over the elements of the symmetric group σ, τ P Sn that permute the
+replicas. We will only need the behaviour in the limit that N is large, which picks out
+the terms with σ “ τ for which Wgp1, Nq “ 1{N,
+ż
+dU
+n
+ź
+j“1
+U ˚
+AjBjUA1
+jB1
+j “
+1
+N n
+ÿ
+τPSn
+n
+ź
+j“1
+δAjA1
+τpjqδBjB1
+τpjq ` ¨ ¨ ¨ ,
+(3.3)
+– 15 –
+
+3.1 The average state
+Let us consider the microscopic state of the radiation ρR and compute its average over
+the unitaries Up, p “ 1, 2, . . . , N. The ket |Ψy contributes a Up and bra xΨ| a U :
+p. The
+average over Up then knits together the bra and ket.
+Let us focus on the average over Up of its adjoint action on a operator f. Using
+(3.1), we can write this average as
+ż
+dUp UpfU :
+p “ Trpfqρ(mm)
+RpBp .
+(3.4)
+Here, ρ(mm)
+RpBp is the maximally-mixed state on HRp b HBp which in the basic model is,
+ρ(mm)
+RpBp “
+1
+dRpdBp
+.
+(3.5)
+In the refined model it is the maximally-mixed state in the energy window Θp´1 em-
+bedded in HRp b HBp in such a way as to conserve energy,
+ρ(mm)
+RpBp 9 ΠΘp´1 .
+(3.6)
+where ΠΘp´1 is the projector onto the energy window. The following Up`1 average then
+imposes a trace over Bp. In the basic model, that gives
+TrBp
+`
+ρ(mm)
+RpBp
+˘
+“
+1
+dRp
+ÿ
+jp
+|jpyxjp| ,
+(3.7)
+In the refined model, let us denote a basis of energy eigenstates of Rp as |jpy with
+energies Ejp, then
+TrBp
+`
+ρ(mm)
+RpBp
+˘
+“ e´SBHpMp´1q ÿ
+jp
+eSBHpMp´1´Ejpq|jpyxjp| .
+(3.8)
+Implicitly, the sum here is constrained to have Mp´1 ´ Ejp P Θp. We can now follow
+the standard route for deriving the canonical ensemble of a small subsystem of a larger
+system in a microcanonical state [46], in our case the maximally mixed state. Since the
+radiation subsystem is much smaller then the black hole, we can expand SBHpMp´1 ´
+Ejpq « SBHpMp´1q ´ Ejp{Tp where the temperature is defined in the standard way via
+the thermodynamic equation (2.6) for a black hole of mass Mp´1. Then we can extend
+the restricted sum over Ejp to be unrestricted because terms for which Mp´1 ´ Ejp R Θp
+– 16 –
+
+are heavily suppressed. This gives the familiar approximation, namely the canonical
+state
+TrBp
+`
+ρ(mm)
+RpBp
+˘
+«
+ÿ
+jp
+e´Ejp{Tp
+Zp
+|jpyxjp| ,
+(3.9)
+where Zp “ ř
+jp e´Ejp{Tp.
+If we now assemble the expressions (3.7) and (3.9) for all the time steps, to find
+the average state of the radiation
+ρR “ 1
+dR
+ÿ
+J
+|JyxJ|
+(basic) ,
+ρR “
+ÿ
+J
+e´ ř
+p Ejp{Tp
+Z
+|JyxJ|
+(refined) .
+(3.10)
+Hence the averaged microscopic state ρR is precisely the semi-classical state ρsc
+R as
+stated in (1.10).
+3.2 The inner products xΨJ|ΨKy
+In this section, we analyse the inner product of the microscopic states |ΨJy defined in
+(1.5) but only for the basic model. To begin, let us calculate its average. For xΨJ|ΨKy,
+each Up in the ket is matched by a U :
+p in the bra and the integral is given in (3.1). In our
+problem, each index is a compound index A “ pap, jpq where ap “ 1, 2, . . . , dBp and jp “
+1, 2, . . . , dRp, while B “ pap´1, sp´1q with sp´1 “ 1, 2, . . . , dFp´1. It is straightforward to
+see that the average
+xΨJ|ΨKy “ δJK ,
+(3.11)
+and so as in the last section, it follows that on the average the microscopic state ρR is
+equal to the semi-classical state ρsc
+R (1.10). The average removes the subtle correlations
+in the microscopic state.
+On the other hand, we now show that there are fluctuations around the average by
+calculating the variance
+∆2
+JK “ |xΨJ|ΨKy|2 ´
+ˇˇxΨJ|ΨKy
+ˇˇ2 .
+(3.12)
+– 17 –
+
+Now each Up and its conjugate appear twice and the formula we need, to leading order,
+from (3.3)
+ż
+dU U ˚
+A1B1U ˚
+A2B2UA3B3UA4B4
+“
+1
+N 2
+`
+δA1A3δB1B3δA2A4δB2B4 ` δA1A4δB1B4δA2A3δB2B3
+˘
+` ¨ ¨ ¨ ,
+(3.13)
+which is valid at large N. Since we are assuming that all the dimensions are large
+we will ignore the subleading term represented by the ellipsis. The two terms here,
+correspond to the identity e and cyclic permutation η in S2.
+The average over Up takes the form (3.13) with indices A1 “ pap, jpq, B1 “
+pap´1, sp´1q, A2 “ pbp, kpq, B2 “ pbp´1, sp´1q, A3 “ pa1
+p, kpq, B3 “ pa1
+p´1, sp´1q, A4 “
+pb1
+p, jpq and B4 “ pb1
+p´1, sp´1q. There are two terms in (3.13) that we label ϵp “ 1 and
+ϵp “ 0, respectively, where ϵp “ 1 can only occur only if jp “ kp.
+Consider the indices labelled by p. These quantum numbers are affected by the
+delta functions that result from both the Up and Up`1 averages. If ϵp “ ϵp`1 then the
+delta functions enforce either
+ap “ a1
+p , bp “ b1
+p
+or
+ap “ b1
+p , bp “ a1
+p .
+(3.14)
+Given that there are 2 conditions means that the sum over the 4 indices is reduced to
+2 and so the sums over these indices contributes d2
+Bp. On the other hand if ϵp ‰ ϵp`1
+then the delta functions enforce
+ap “ a1
+p “ bp “ b1
+p ,
+(3.15)
+which is 3 conditions. So the sums over these 4 labels contributes only dBp. Using these
+rules, one finds that the final result can be written
+∆2
+JK “
+δj1k1
+ÿ
+ϵ1“0
+¨ ¨ ¨
+δjN kN
+ÿ
+ϵN“0
+N
+ź
+p“1
+1
+d
+|ϵp`1´ϵp|
+Bp
+´ δJK ,
+(3.16)
+with ϵN`1 “ 1. Note that the second term cancels the term with ϵp “ 1, for all p, that
+occurs when J “ K.
+In general the variance is suppressed by various powers of dBp. The least suppressed
+term in the sum is the one with ϵp “ 0, for all p, which is equal to d´1
+B (recall B ” BN`1)
+since dBp ą dB and so we conclude that for a typical element of the ensemble that (1.9)
+holds with dB “ eSBH.
+– 18 –
+
+4
+Entropies
+We can calculate the entropy of the microscopic state reduced on any subset
+A Ă tR1, . . . , RN, Bu .
+(4.1)
+The strategy is to first calculate the R´enyi entropies which can be defined by introducing
+n replicas of the Hilbert space
+ep1´nqSpnqpAq “ Trpρn
+Aq “ TrpnqσrR1s
+1
+¨ ¨ ¨ σrRNs
+N
+τ rBs
+N`1 |ΨyxΨ|bn ,
+(4.2)
+where the σp and τN`1 are elements of the symmetric group Sn. The superscripts,
+e.g. σrRps
+p
+, on these elements indicate which subspace of the replicated Hilbert space
+the element acts on where it is ambiguous. These elements are taken to be either the
+identity element e or the cyclic permutation η according to the definition of the subset
+A
+A “
+␣
+Rp
+ˇˇ σp “ η , p “ 1, 2, . . . , N
+(
+Y
+␣
+B
+ˇˇ τN`1 “ η
+(
+.
+(4.3)
+The R´enyi entropies are known to be self-averaging in the ensemble of the unitaries
+Up (e.g. [47]) and so we will calculate the ensemble average of (4.3) and take this
+to describe a typical element of the ensemble.
+The integrals we need are given in
+(3.3) which capture the leading order behaviour when the Hilbert spaces have a large
+dimension.
+Using (3.3), the average over the unitary Up acting in a replicated Hilbert space at
+large dRpdBp of adjoint action is given by a sum over elements of the symmetric group
+Sn,
+ż
+dUp U : bn
+p
+f U bn
+p
+“
+ÿ
+τpPSn
+!
+Trpnqτ rBp´1Fp´1s
+p
+f
+)
+pτ rRpBps
+p
+q´1 ρ(mm) bn
+RpBp
+` ¨ ¨ ¨ ,
+(4.4)
+for some f in the replicated Hilbert space.
+So each average over Up comes with a
+sum over an element of the symmetric group τp P Sn.
+In the above, ρ(mm)
+RpBp is the
+maximally mixed state of HRp b HBp as in (3.5), while for the refined model, it is the
+subspace with energy in the window Θp´1 as in (3.6). The ellipsis stand for subleading
+corrections, suppressed by inverse powers of dBp´1dFp´1, that we will not keep track of
+in our analysis. Trpnq is the trace defined on the replicated Hilbert space.
+– 19 –
+
+Applying (4.4) for all p, it becomes apparent that the average of (4.2) breaks up
+into a set of building blocks:
+ep1´nqSpnqpAq “
+ÿ
+τ1,...,τNPSn
+Z1 ¨ ¨ ¨ ZN ,
+(4.5)
+where
+Zp “ TrpnqσrRps
+p
+τ rBpFps
+p`1
+pτ rRpBps
+p
+q´1`
+ρ(mm)
+RpBp b ρsc
+Fp
+˘bn ,
+(4.6)
+where ρsc
+Fp “ |spyxsp| is the semi-classical state of the infalling system Fp. In the last
+step p “ N this piece is missing, there is no FN. The traces over HFp are trivial because
+the states ρsc
+Fp are pure and so Trpnqpσρsc bn
+Fp
+q “ 1, for any σ P Sn. This includes the
+initial state in HF0 that collapsed to form the black hole. Hence, the building block
+(4.6) can be written more simply as
+Zp “ TrpnqσrRps
+p
+τ rBps
+p`1 pτ rRpBps
+p
+q´1ρ(mm) bn
+RpBp
+.
+(4.7)
+The expression for the building block Zp can also be interpreted in terms of the equili-
+HRp
+HBp
+Up
+Up`1
+HFp
+τp`1
+ρFp “ |spyxsp|
+τ ´1
+p
+τ ´1
+p
+σp
+τp`1
+ρ(mm)
+RpBp
+Figure 3: Assembling the ingredients for the building block in (4.6).
+bration ansatz of [47] as an alternative to the unitary averages. In this interpretation,
+the pure state of the black hole at time tp´1 equilibrates over the next time step mean-
+ing that for certain observables it is indistinguishable from an equilibrium state, in this
+case precisely the maximally mixed state ρ(mm)
+RpBp (3.5), or (3.6) in the refined model.
+In the basic model, it is then straightforward to evaluate the building block (4.7),
+Zp “ exp
+”
+pkpτp`1τ ´1
+p q ´ nqSBHpMpq ` pkpσpτ ´1
+p q ´ nqSradpRpq
+ı
+,
+(4.8)
+– 20 –
+
+where kpσq is the number cycles of the element σ and with SBHpMpq “ log dBp and
+SradpRpq “ log dRp. Then plugging into (4.5) gives the final result
+ep1´nqSpnqpAq “
+ÿ
+τ1,...,τNPSn
+e
+p1´nqSpnq
+tτpupAq ,
+(4.9)
+where we have defined
+Spnq
+tτpupAq “
+1
+n ´ 1
+N
+ÿ
+p“1
+!
+dpτp`1, τpqSBHpMpq ` dpσp, τpqSradpRpq
+)
+,
+(4.10)
+where dpσ, πq “ n ´ kpσπ´1q is the Cayley distance between elements of Sn.8
+4.1 Refined model
+The refined model is rather more complicated because of the need to enforce energy
+conservation. The R´enyi entropies now involve a sum over both the energies Ejp and
+the elements of the symmetric group τp,
+ep1´nqSpnqpAq “
+ÿ
+tjpu
+ÿ
+tτpuPSn
+n
+ź
+p“1
+Zp “
+ÿ
+tjpu
+ÿ
+tτpuPSn
+e
+p1´nqSpnq
+tτpupAq
+(4.11)
+where the building block is
+Zp “ dRppEjpqkpσpτ ´1
+p
+qdBpMp´1 ` Ep ´ Ejpqkpτp`1τ ´1
+p
+q
+` ř
+jp dRppEjpqdBpMp´1 ` Ep ´ Ejpq
+˘n
+.
+(4.12)
+where Ep is the energy of the infalling system Fp. Note that the mass of the black hole
+depends implicitly on the energy of the radiation emitted up to that point
+Mp “ M0 `
+pÿ
+q“1
+pEq ´ Ejqq ,
+(4.13)
+a point that must be born in mind when we perform the saddle point approximation.
+The denominator in (4.12) can be evaluated by a saddle point approximation where
+the sum is replaced by an integral over a continuous variable Ep. In particular, the
+8Alternatively, the Cayley distance dpσ, πq may be defined as the minimal number of transpositions
+required to go between σ and π.
+– 21 –
+
+radiation can be described thermodynamically in the way summarized in appendix A
+and the entropy
+log dRppEq “ 2
+a
+µpEp
+where
+µp “ πc∆tp
+12
+.
+(4.14)
+Since the saddle point value of the energy is much smaller than the black hole mass,
+the saddle point equation is
+cµp
+Ep
+“ ´dSBHpMp´1 ` Ep ´ Epq
+dEp
+« 1
+Tp
+ùñ
+Ep “ µpT 2
+p ,
+(4.15)
+where Tp defined in (2.6) is precisely the temperature of the Hawking radiation Rp. The
+average energy of the radiation emitted Ep and the infalling energy Ep are assumed to
+be much smaller than the black hole mass . Hence, we have
+ÿ
+jp
+dRppEjpqdBpMp´1 ` Ep ´ Ejpq « dBpMp´1qeSradpRpq{2`Ep{Tp ,
+(4.16)
+where the saddle point value of the entropy is
+SradpRpq “ 2µpTp “ πc∆tpTp
+6
+.
+(4.17)
+This and Ep above are the familiar expressions for the entropy and energy of a volume
+∆tp of a relativistic gas in 1 ` 1 dimensions in a volume V “ ∆tp (as reviewed in
+appendix A). The saddle point approximation is, of course, just the conventional way
+of deriving the Legendre transformation between the internal energy and free energy
+in thermodynamics and is justified precisely because the spread in the energy is small
+(2.13).
+For later use, note that
+dBpMpq “ dBpMp´1 ` Ep ´ Epq « dBpMp´1 ` Epqe´SradpRpq{2
+(4.18)
+and so
+SBHpMp´1 ` Epq ´ SBHpMpq “ SradpRpq
+2
+,
+(4.19)
+which is the familiar relation for a model of black hole evaporation in the s-wave
+approximation and with no back scattering (i.e. grey body factor). Note that it implies
+that the evaporation is irreversible.
+– 22 –
+
+We now proceed to evaluate the sums of the energies in (4.11) by similar saddle
+point approximations. After we replace the sums by integrals over Ep, the exponent of
+the integrand is
+p1 ´ nqSpnq
+tτpupAq “
+N
+ÿ
+p“1
+!
+2pn ´ dpσp, τpqq
+a
+µpEp ´ dpτp`1, τpqSBHpM0q
+´
+´
+n ´
+N
+ÿ
+q“p
+dpτq`1, τqq
+¯Ep
+Tp
+´
+N
+ÿ
+q“p
+dpτq`1, τqqEp
+Tp
+´ n
+2SradpRpq
+)
+.
+(4.20)
+It is now simple to compute the saddle point equations for the energies Ep. In the
+regime of slow evaporation we can ignore the Ep dependence of the temperatures Tp.
+The saddle point values are found to be
+Ep “ µpT 2
+p
+´
+n ´ dpσp, τpq
+n ´ řN
+q“p dpτq`1, τqq
+¯2
+,
+(4.21)
+where for consistency the saddles must have
+n ą
+N
+ÿ
+q“1
+dpτq`1, τqq .
+(4.22)
+The contribution of this saddle to the R´enyi entropy is
+Spnq
+tτpupAq “
+1
+n ´ 1
+N
+ÿ
+p“1
+!
+dpτp`1, τpqSBHpM0q `
+N
+ÿ
+q“p
+dpτq`1, τqqEp
+Tp
+` 1
+2
+´
+n ´
+pn ´ dpσp, τpqq2
+n ´ řN
+q“p dpτq`1, τqq
+¯
+SradpRpq
+)
+.
+(4.23)
+We can re-write this by noting that (4.19) implies
+SBHpMpq “ SBHpM0q `
+pÿ
+q“1
+´Eq
+Tq
+´ SradpRqq
+2
+¯
+,
+(4.24)
+as
+Spnq
+tτpupAq “
+1
+n ´ 1
+N
+ÿ
+p“1
+!
+dpτp`1, τpqSBHpMpq
+`
+2ndpσp, τpq ´ dpσp, τpq2 ´
+` řN
+q“p dpτq`1, τqq
+˘2
+2
+`
+n ´ řN
+q“p dpτq`1, τqq
+˘
+SradpRpq
+)
+,
+(4.25)
+which is the refined model generalization of (4.10).
+– 23 –
+
+A
+˜I
+A a ˜I
+τp
+StτpupAq
+R1 R2 R3 R4 R5 R6 R7 R8
+B
+e
+e
+η
+SBHpM2q
+η
+η
+η
+SradpR5q
+η
+SradpR6q
+η
+e
+SBHpM8q
+Figure 4: An example of a saddle for the model with N “ 8 time steps, with some choice of the
+set A “ R3 Y R4 Y R7 Y R8, as shown, with an island-in-the-stream ˜I. Note that B ˜I Ă BA. The
+contributions to the entropy from each time step are shown and summing these up gives SIpAq “
+SBHpM2q ` SBHpM8q ` SradpR5 Y R6q. Note that the last term is SradpA a ˜Iq.
+4.2 Relation to the island formalism
+We interpret (4.9) as being a sum over saddles of the (Lorentzian) gravitational path
+integral in the semi-classical limit, labelled by the elements tτpu. In this limit, the
+entropies SBHpMpq and SradpRpq are very large. If we avoid the crossover regimes when
+saddles are degenerate, it turns out that only a much smaller number of terms can
+actually dominate in the sum, namely, those for which each τp, p “ 1, . . . , N, is equal
+to e or η only, the identity and cyclic permutations, respectively. This is proved in
+appendix B for the basic model. The te, ηu dominance means that the saddles that
+dominate respect the Zn cyclic symmetry of the replicas mirroring the symmetry of the
+replica wormholes of [4, 5], or, equivalently, we can interpret the average over unitaries
+to be equivalent to the average over baby universe states (see [29, 48]).
+For the refined model, the discussion is very similar. Indeed each element in the
+energy sum in (4.11) behaves like a basic model, and therefore we can again invoke the
+fact that τp is dominated by τp P te, ηu, which will be valid as long we are not in the
+vicinity of a crossover of saddles.9
+The expression for the von Neumann entropy of our chosen subset A Ă R Y B is
+obtained from (4.10) and (4.25) in the limit SpAq “ limnÑ1 SpnqpAq and has the form
+9Notice that we don’t risk of having a crossover at every time step since we assumed that each
+energy window is small (2.13).
+– 24 –
+
+of a minimization problem over the 2N choices τp P te, ηu. Indeed notice that when
+σ, π P te, ηu, we can write
+dpσ, πq “ pn ´ 1qp1 ´ δσπq ,
+(4.26)
+which facilitates the evaluation of the Cayley distances in the n Ñ 1 limit of (4.10)
+and (4.25). In both models, the von Neumann entropy is given by
+SpAq “ min
+tτpu StτpupAq “ min
+tτpu
+! N
+ÿ
+p“1
+p1 ´ δτp`1τpqSBHpMpq ` p1 ´ δσpτpqSradpRpq
+)
+.
+(4.27)
+The resemblance of this equation to the QES formula described in the introduction
+for a slowly evaporating black hole (1.19) becomes more apparent if we set
+SIpAq ” StτpupAq.
+(4.28)
+where I is defined in both models as
+I “
+ď
+pPΦ
+`
+Rp Y Fp´1
+˘
+.
+(4.29)
+with Φ “
+␣
+p | τp “ η
+(
+. The I that minimizes (4.28) is called the ‘entanglement island’
+or ‘island’, for short, will be denoted IpAq. Even if in principle we have 2N possible
+saddles, most of them will not contribute since terms with τp ‰ τp`1 are not favourable
+because of the black hole entropy being big. One can check that the only saddles that
+are not trivially suppressed are the one where τp changes in correspondence with a
+change in σp, which is an analog of the condition (1.17). See figure 4 for an example
+where Φ “ t3, 4, 5, 6, 7, 8u.
+In order to make more transparent the identification of (4.27) with the QES formula
+(1.19) for the A that we have chosen, we can also notice that the second term is a discrete
+version of the continuum expression SradpA a ˜Iq where we identify the island-in-the-
+stream as the reflection of the island I in the horizon and then projected onto I `, so
+each Rp gets mapped to Rp:
+˜I “
+ď
+pPΦ
+Rp .
+(4.30)
+On the other hand, the first term can be written in terms of the BH entropy at the
+outgoing EF coordinates of QES uBI. We can then parametrize the entropy of the
+black hole with its mass at outgoing time u. Notice also that the infalling states in
+– 25 –
+
+(4.29) are shifted by p Ñ p ´ 1. This is how the model accounts for the fact that
+infalling coordinate v of the QES are shifted relative to the outgoing coordinate u by
+the scrambling time (1.16), precisely the size of the time steps in the model.
+In the next sections, we will enforce our definition of the entanglement island (4.29)
+studying when it is possible to reconstruct an unitary acting on the radiation, which
+is equivalent to the well known statement that the island is in the entanglement wedge
+of the radiation. Specifically, since the emitted radiation is in both the semi-classical
+and microscopic descriptions, we include it in its own entanglement wedge
+WpAq “ IpAq Y pA X Rq .
+(4.31)
+Although we call IpAq the island, strictly speaking, this only applies when subsets of
+IpAq are separated from the rest of the entanglement wedge by QES.10
+5
+Information recovery and reconstruction
+In this section, we consider the fate of an infalling system, Hayden and Preskill’s diary
+for instance [36]. We will focus on the single system that falls in during the qth time
+step. For simplicity, we will avoid the case that this is the last time step, in other words
+we will take q ă N. The idea is to consider a family of infalling states W|sqy for a
+unitary W and fixed state |sqy P Fq. This gives a family of microscopic states |ΨpWqy.
+The physical question is, can the effect of the unitary W be achieved by a local action
+on the radiation or the black hole? This will inform us as to when the information in
+Fq has been teleported out of the black hole. More specifically, when can the action of
+W be reconstructed on A “ R or B, or a subset thereof, in the sense that there exists
+a unitary WA acting on A such that
+WA|Ψy
+?“ |ΨpWqy .
+(5.1)
+This is the state-specific notion of reconstruction described in [28]. The above implies
+that WA acts on the reduced state on A via the adjoint action
+ρApWq “ WAρAW :
+A ,
+(5.2)
+while the reduced state on the complement A is invariant
+ρApWq “ ρA .
+(5.3)
+10For example, when A “ B, the black hole before the Page time has IpBq “ WpBq “ R Y F which
+is not an island in the strict sense.
+– 26 –
+
+In fact this decoupling condition on A implies the existence of WA in (5.1). This can
+be seen using the Schmidt decomposition. The decoupling condition implies that if
+|Ψy “ ř
+j
+?pj|jyA|jyA then |ΨpWqy “ ř
+j
+?pj|jy1
+A|jyA. It follows that WA “ ř
+j |jy1
+Axj|
+acting on the subspace of HA spanned by the Schmidt states |jyA although it can
+be extended to a unitary acting on HA. Acting within the subspace, we can write
+explicitly,
+WA “ TrA |ΨpWqyxΨ|ρ´1
+A .
+(5.4)
+The Schmidt basis states depend implicitly on the infalling state |sqy and so the con-
+struction of W is ‘state dependent’ in this sense. It is an interesting question if the
+construction can be extended to any operator acting on any state of the infalling system
+in HFq and thereby be state independent, at least in this limited sense. In fact, the
+construction above can be seen as a special case of the Petz map and, indeed, there is
+a more general state-independent construction [49].
+We cannot expect the conditions (5.1) and (5.3) to hold exactly and approximate
+forms of these conditions are formulated in [28]. However, we will work to leading order
+in the semi-classical limit and we will not need these approximate forms in our analysis.
+The decoupling condition is therefore key to reconstructing that action of W on
+either the radiation or the black hole. Hence, we need to calculate the difference between
+the states ρApWq and ρA. This can be measured by the trace norm11 difference
+ˇˇˇˇσ´ρ
+ˇˇˇˇ
+1
+or the quantum fidelity fpσ, ρq. Both are tractable in our models when averaged over
+the unitary evolution to leading order in the semi-classical limit where they can be
+computed using the replica method and an analytic continuation. For the trace norm
+difference, we take an even number of replicas and then take an analytic continuation,
+ˇˇˇˇσ ´ ρ
+ˇˇˇˇ
+1 “ Tr
+a
+pσ ´ ρq2 “ lim
+nÑ 1
+2
+Trp2nqη
+`
+σ ´ ρ
+˘b2n
+(5.5)
+and similarly for the quantum fidelity,
+fpσ, ρq ” Tr
+b?ρσ?ρ “ lim
+nÑ 1
+2
+Trp2nqη
+`
+σ b ρ
+˘bn .
+(5.6)
+In our context, there is a subtlety in that the analytic continuations must be taken
+after the semi-classical limit has picked out a dominant saddle otherwise saddles would
+become degenerate. We should also emphasize that what we are actually calculating
+11For Hermitian operators the trace norm is equal to
+ˇˇˇˇO
+ˇˇˇˇ
+1 “ ř
+j |λj|, where λj are the eigenvalues
+of O.
+– 27 –
+
+are the unitary averages of the replica expressions before taking the limits n Ñ 1
+2. This
+is in the same spirit as calculating the averages the exponents of the R´enyi entropies as
+in (4.5) before taking the limit n Ñ 1 to recover the von Neumann entropy. In section
+5.1 we compute an upper bound on the trace norm which does not require the n Ñ 1
+2
+limit.
+Let us compute the average of the trace difference in (5.5). The computation is
+similiar to that of the R´enyi entropy via Trρn
+A. In fact, since W acts locally on HFq,
+only the qth time step is modified:
+Zq ÝÑ Trp2nqσrRqs
+q
+τ rBqFqs
+q`1
+pτ rRqBqs
+q
+q´1`
+ρ(mm)
+RqBq b pρsc
+FqpWq ´ ρsc
+Fqq
+˘b2n
+“ Zq Trp2nqτq`1
+`
+ρsc
+FqpWq ´ ρsc
+Fq
+˘b2n ,
+(5.7)
+where we separated out the trace over the replicas of Fq where W acts and the quantity
+Zq is the original quantity in the entropy calculation defined in (4.7). The contribution
+from the other time steps p ‰ q are precisely as for the entropy (4.7). Hence, assembling
+all the pieces gives
+ˇˇˇˇρApWq ´ ρA
+ˇˇˇˇ
+1 “ lim
+nÑ 1
+2
+ÿ
+τ1,...,τNĂte,ηu
+e
+p1´2nqSp2nq
+tτpupAq Trp2nqτq`1
+`
+ρsc
+FqpWq ´ ρsc
+Fq
+˘b2n .
+(5.8)
+Now we have to be careful to take the semi-classical limit before taking the analytic
+continuation n Ñ 1
+2. The semi-classical limit picks out a dominant term in the sum
+over the elements τp and, in particular, fixes τq`1. Hence,
+ˇˇˇˇρApWq ´ ρA
+ˇˇˇˇ
+1 “ lim
+nÑ 1
+2
+Trp2nqτq`1
+`
+ρsc
+FqpWq ´ ρsc
+Fq
+˘b2n
+(5.9)
+One can follow the same steps for the average of the quantum fidelity. Once again
+the contribution comes entirely from the qth time step which is modified as
+Zq ÝÑ Trp2nqσrRqs
+q
+τ rBqFqs
+q`1
+pτ rRqBqs
+q
+q´1`
+ρ(mm)
+RqBq
+˘b2n b
+`
+ρsc
+FqpWq b ρsc
+Fq
+˘bn
+“ Zq Trp2nqτq`1
+`
+ρsc
+FqpWq b ρsc
+Fq
+˘bn ,
+(5.10)
+leading to
+fpρApWq, ρAq “ lim
+nÑ 1
+2
+Trp2nqτq`1
+`
+ρsc
+FqpWq b ρsc
+Fq
+˘bn
+(5.11)
+– 28 –
+
+Let us now evaluate our results above.
+When Fq R WpAq, it follows that the
+dominant saddle has τq`1 “ e.
+For the trace norm difference (5.9), this gives an
+expression that is clearly seen to vanish
+ˇˇˇˇρApWq ´ ρA
+ˇˇˇˇ
+1 “ lim
+nÑ 1
+2
+Trp2nq`
+ρsc
+FqpWq ´ ρsc
+Fq
+˘b2n
+“
+ˇˇTrpρsc
+FqpWq ´ ρsc
+Fqq
+ˇˇ “ 0 .
+(5.12)
+This proves the decoupling condition in terms of the trace norm. On the other hand,
+for the fidelity (5.13),12
+fpρApWq, ρAq “ lim
+nÑ 1
+2
+Trp2nq`
+ρsc
+FqpWq b ρsc
+Fq
+˘bn
+“
+b
+Trρsc
+FqpWq Trρsc
+Fq “ 1 ,
+(5.13)
+which is another expression of decoupling. Note that, if the trace norm difference of
+two states vanishes, then they must have unit quantum fidelity and ρApWq “ ρA.
+On the other hand, when Fq P WpAq, the element τq`1 “ η and the trace norm
+difference (5.9) is
+ˇˇˇˇρApWq ´ ρA
+ˇˇˇˇ
+1 “ lim
+nÑ 1
+2
+Trp2nqη
+`
+ρsc
+FqpWq ´ ρsc
+Fq
+˘b2n
+“
+ˇˇˇˇρsc
+FqpWq ´ ρsc
+Fq
+ˇˇˇˇ
+1 .
+(5.14)
+For the fidelity, we have a similar relation to the semi-classical state
+fpρApWq, ρAq “ lim
+nÑ 1
+2
+Trp2nqη
+`
+ρsc
+FqpWq b ρsc
+Fq
+˘bn
+“ fpρsc
+FqpWq, ρsc
+Fqq .
+(5.15)
+Let us take stock of the results and, in particular, relate them to the state recon-
+struction formula of [50]. This states that if there are two microscopic states ρA and
+12The fidelity plays an important role in quantum hypothesis testing, which is the task of making
+a measurement to distinguish between two quantum states given that the actual state is one of them.
+The fidelity bounds the error on the optimal measurement. We expect that the corrections to (5.13)
+are non-perturbatively suppressed in the semi-classical limit, as in [51, 52]. If so, this would imply
+that whilst it is not possible to distinguish the two states given a single copy of the state, it will be
+possible given sufficiently many copies of the state.
+– 29 –
+
+σA such that the semi-classical saddles that dominate Trpρ2n
+A q and Trpσ2n
+A q are the same
+(and preserve the Zn symmetry of the replicas) then13
+ˇˇˇˇρA ´ σA
+ˇˇˇˇ
+1 “
+ˇˇˇˇρsc
+WpAq ´ σsc
+WpAq
+ˇˇˇˇ
+1 ,
+(5.16)
+up to OpGq corrections, where WpAq is the entanglement wedge of A. The fact that the
+map V preserves the trace norm difference is on the same footing as the preservation
+of the relative entropy [5, 25, 53].
+To relate this to our analysis, we identify σR “ ρRpWq. The saddles associated to
+Trpρ2n
+A q and Trpσ2n
+A q are the ones that determine the R´enyi entropies and are therefore
+associated to the set of elements τp, p “ 1, . . . , N. The fact that they both have the
+same saddle is ensured by the fact that W only acts on a small subset of the infalling
+modes and so cannot alter the dominant saddle.
+Let us consider our results for the case A “ R. Before the Page time, WpRq “ R
+and so Fq R WpRq and the formula (5.16) implies
+ˇˇˇˇρRpWq ´ ρR
+ˇˇˇˇ
+1 “ 0 ,
+(5.17)
+which is the decoupling condition (5.12) with A “ R. This means that W can be
+reconstructed on B. On the other hand, after the Page time, the entanglement wedge
+WpRq “ R Y IpRq, so Fq P WpRq, since the island IpRq contains the outgoing and
+infalling modes IpRq “ R Y F since it lies very close behind the horizon. Hence, (5.16)
+implies
+ˇˇˇˇρRpWq ´ ρR
+ˇˇˇˇ
+1 “
+ˇˇˇˇρsc
+RRFpWq ´ ρsc
+RRF
+ˇˇˇˇ
+1 “
+ˇˇˇˇρsc
+FqpWq ´ ρsc
+Fq
+ˇˇˇˇ
+1 ,
+(5.18)
+which is (5.14) with A “ R. We will see shortly that this is the case when W can be
+reconstructed on R because B decouples.
+Now consider the case A “ B. After the Page time, WpBq “ ∅ and so (5.16)
+predicts decoupling as we found in (5.12).
+This occurs at the same time as (5.18)
+which makes perfect sense as W can be reconstructed on R. On the other hand, before
+the Page time, WpBq “ R Y F, and so (5.16) gives
+ˇˇˇˇρBpWq ´ ρB
+ˇˇˇˇ
+1 “
+ˇˇˇˇρsc
+RFpWq ´ ρsc
+RF
+ˇˇˇˇ
+1 “
+ˇˇˇˇρsc
+FqpWq ´ ρsc
+Fq
+ˇˇˇˇ
+1 .
+(5.19)
+13We have stated the formula in a slightly more general way to include the case when A is any subset
+of the radiation plus the black hole rather than all the radiation as considered in [50]. The condition
+for Zn symmetry is satisfied by our saddles which involve only the elements e or η of Sn.
+– 30 –
+
+But this is precisely (5.14) for A “ B. This is also when R decouples and so W can
+be reconstructed on B. So once again we find precise agreement between our averaged
+results and the formula (5.16).
+5.1 Bounding the trace norm
+The condition for decoupling is that the averaged trace norm difference between ρApWq
+and ρA vanishes in the leading order saddle (5.12). But this is derived with the limits
+in a particular order, first the semi-classical limit picking out a particular saddle and
+then in the replica limit n Ñ 1
+2. Can we trust this? In fact there is standard way to
+bound the averaged trace norm difference,
+ˇˇˇˇρApWq ´ ρA
+ˇˇˇˇ
+1 ď
+b
+dA TrpρApWq ´ ρAq2 .
+(5.20)
+We can evaluate the right-hand side, at least in the case that the subsystem A is finite
+dimensional (so this seems to exclude A “ R, the radiation, in the refined model). The
+average on the right-hand side is just the right-hand side of (5.8) with n Ñ 1, so
+TrpρApWq ´ ρAq2 “
+ÿ
+τ1,...,τNĂte,ηu
+e
+´Sp2q
+tτpupAq Trp2qτq`1
+`
+ρsc
+FqpWq ´ ρsc
+Fq
+˘b2 .
+(5.21)
+If we consider A “ B, so dA „ eSBHpMq, and after the Page time, the sum in (5.21)
+is dominated by the term with τp “ η for which Sp2q
+tηupBq “ αSradpRq, where α “ 1 for
+the basic model and α “ 3
+4, for the refined model.14 Therefore we can bound the trace
+norm difference
+ˇˇˇˇρApWq ´ ρA
+ˇˇˇˇ
+1 Æ Ope
+1
+2 SBHpMq´ α
+2 SradpRqq ! 1 ,
+(5.22)
+after the Page time when SradpRq " SBHpBq.
+6
+Reconstruction of the Hawking partners
+In this section, we consider reconstruction for the Hawking partners which semi-classically
+are behind the horizon and part of the black hole. Conceptually the discussion is very
+14The latter follows from (4.25) with τp “ η, p “ 1, . . . , N ` 1 and σp “ e giving dpτp`1, τpq “ 0 and
+dpσp, τpq “ n ´ 1 giving Spnq
+tηupBq “ pn ` 1qSradpRq{p2nq. This is the R´enyi entropy of the radiation
+(see appendix A) and then taking n “ 2 gives 3
+4SradpRq.
+– 31 –
+
+similar to the reconstruction of the infalling system in the last section but the technical
+details are rather different. The idea is to consider a unitary operator on the Hawking
+partners R and ask if it is possible to reconstruct this on some A Ă R Y B, i.e.
+|ΨpWqy
+?“ WA|Ψy .
+(6.1)
+As in section 5 the condition for such a reconstruction is the decoupling condition for
+the complement
+ρApWq “ ρA ,
+(6.2)
+which can be analysed by calculating the trace norm difference or quantum fidelity.
+In order to proceed, it is useful to deploy the following trick. Exploiting the entan-
+glement between R and R, we can write the action of W on the semi-classical state as
+the action of an operator Ă
+W on R:
+W|ψy “ Ă
+W|ψy ,
+(6.3)
+where
+Ă
+W “ pρsc
+Rq1{2W Tpρsc
+Rq´1{2 .
+(6.4)
+We remark that Ă
+W is not unitary so it is not a physically realizable local action on the
+radiation. It then follows that the reduced state on R is invariant under adjoint action
+by Ă
+W,
+ρsc
+R ÝÑ Ă
+Wρsc
+RĂ
+W : “ pρsc
+Rq1{2`
+W :W
+˘˚pρsc
+Rq1{2 “ ρsc
+R ,
+(6.5)
+as it must be by locality: the action of W on R cannot change the state of R.
+We now compute the trace difference and quantum fidelity of the two states ρApWq
+and ρA using the replica method following the same steps as in section (5). For sim-
+plicity, we will take W to act on just one of the subsets of partner modes Rq. We can
+then use (6.3) to write the action on the Hawking modes Rq by switching W Ñ Ă
+W.
+As for the infalling system, the only effect of W is on the qth time step. For the trace
+norm difference, this time step is modified as
+Zq ÝÑ Trp2nqσrRqs
+q
+τ rBqs
+q`1
+`
+AdĂ
+W ´ 1
+˘b2npτ rRqBqs
+q
+q´1ρ(mm) b2n
+RqBq
+,
+(6.6)
+where AdĂ
+W is the adjoint action of Ă
+W on ρ(mm)
+RqBq. We now assume that the saddle that
+dominates the entropy, and therefore the trace norm difference, has τq`1 “ τq. This
+– 32 –
+
+means that Rq is not just before a QES. One can view this as avoiding an edge effect
+created by having a discrete model. In that case, we can perform the trace over Bq to
+give the semi-classical state ρsc
+Rq “ TrBqρ(mm)
+RqBq:
+Zq ÝÑ Trp2nqσq
+`
+AdĂ
+W ´ 1
+˘b2nτ ´1
+q
+ρsc b2n
+Rq
+.
+(6.7)
+where in the second line we used the fact that all relevant saddles have τq`1 “ τq and
+ρsc
+Rq “ TrBqρ(mm)
+RqBq. Following the same steps as in section 5, and in particular taking the
+semi-classical limit before the analytic continuation in n, gives
+ˇˇˇˇρApWq ´ ρA
+ˇˇˇˇ
+1 “ lim
+nÑ 1
+2
+Trp2nqσq
+`
+AdĂ
+W ´ 1
+˘b2nτ ´1
+q
+ρsc b2n
+Rq
+(6.8)
+where τq is determined by the saddle that dominates the entropy. Similarly, for the
+quantum fidelity
+fpρApWq, ρAq “ lim
+nÑ 1
+2
+Trp2nqσq
+`
+AdĂ
+W b 1
+˘bnτ ´1
+q
+ρsc b2n
+Rq
+(6.9)
+When Rq R WpAq the dominant saddle has τq “ e and then the trace norm differ-
+ence is
+ˇˇˇˇρApWq ´ ρA
+ˇˇˇˇ
+1 “ lim
+nÑ 1
+2
+Trp2nqσq
+`Ă
+Wρsc
+RqĂ
+W : ´ ρsc
+Rq
+˘2n “ 0 ,
+(6.10)
+using the invariance (6.5). We can repeat the analysis for the fidelity,
+fpρApWq, ρAq “ lim
+nÑ 1
+2
+Trp2nqσq
+´
+Ă
+Wρsc
+RqĂ
+W : b ρsc
+Rq
+¯bn
+“ lim
+nÑ 1
+2
+Trp2nqσq ρsc b2n
+Rq
+“ 1 .
+(6.11)
+So decoupling occurs when the partners Rq do not lie in the entanglement wedge of A.
+Under these circumstances, W can be reconstructed on the complement A.
+On the other hand, when Rq P WpAq, the appropriate saddle has τq “ η and (6.10)
+becomes
+ˇˇˇˇρApWq ´ ρA
+ˇˇˇˇ
+1 “ lim
+nÑ 1
+2
+Trp2nqσq
+`
+AdĂ
+W ´ 1
+˘b2nη´1ρsc b2n
+Rq
+.
+(6.12)
+We can now consider this for particular choices for A. For the case A “ R, so after the
+Page time, then σq “ η, and the above becomes
+ˇˇˇˇρRpWq ´ ρR
+ˇˇˇˇ
+1 “ 2
+b
+1 ´
+ˇˇTr
+`
+ρsc
+RqW T˘ˇˇ2
+“
+ˇˇˇˇρsc
+RRpWq ´ ρsc
+RR
+ˇˇˇˇ
+1 .
+(6.13)
+– 33 –
+
+For the case A “ B, so before the Page time, σq “ e, we have
+ˇˇˇˇρBpWq ´ ρB
+ˇˇˇˇ
+1 “ lim
+nÑ 1
+2
+Tr
+`
+W ˚ρsc
+RqW T ´ ρsc
+Rq
+˘2n
+“ lim
+nÑ 1
+2
+Tr
+`
+Wρsc
+RqW : ´ ρsc
+Rq
+˘2n
+“
+ˇˇˇˇρsc
+RpWq ´ ρsc
+R
+ˇˇˇˇ
+1 .
+(6.14)
+Note that (6.14) is not the same as (6.13) because R is entangled with R.
+We can also consider the quantum fidelity. For A “ R (after the Page time),
+fpρRpWq, ρRq “ lim
+nÑ 1
+2
+Trp2nqη
+`
+AdĂ
+W b 1
+˘b2nη´1ρsc b2n
+Rq
+“
+ˇˇTrpρsc
+RqW Tq
+ˇˇ “ fpρsc
+RRpWq, ρsc
+RRq
+(6.15)
+and for A “ B (before the Page time),
+fpρBpWq, ρBq “ lim
+nÑ 1
+2
+Trp2nq`
+AdĂ
+W b 1
+˘b2nη´1ρsc b2n
+Rq
+“ lim
+nÑ 1
+2
+Trp2nqη
+`
+W ˚ρsc
+RqW T b ρsc
+Rq
+˘n
+“ fpρsc
+RpWq, ρsc
+Rq .
+(6.16)
+These expressions are close cousins of the expressions for the trace norm difference in
+(6.13) and (6.14).
+Once again, let us compare our results to the formula (5.16) of [50]. Firstly, let us
+compare the microscopic states ρRpWq and ρR. Before the Page time, Rq R WpRq and
+so (5.16) implies
+ˇˇˇˇρRpWq ´ ρR
+ˇˇˇˇ
+1 “ 0. After the Page time, Rq P WpRq and so (5.16)
+implies
+ˇˇˇˇρRpWq ´ ρR
+ˇˇˇˇ
+1 “
+ˇˇˇˇρsc
+RRFpWq ´ ρsc
+RRF
+ˇˇˇˇ
+1 ,
+(6.17)
+which is precisely (6.13) because F is not entangled with R Y R.
+Now we turn to the states ρBpWq and ρB.
+In this case, after the Page time,
+WpBq “ ∅ and so (5.16) implies
+ˇˇˇˇρBpWq ´ ρB
+ˇˇˇˇ
+1 “ 0. On the other hand, before the
+Page time, WpBq “ R Y F, and so (5.16) implies
+ˇˇˇˇρRpWq ´ ρR
+ˇˇˇˇ
+1 “
+ˇˇˇˇρsc
+RFpWq ´ ρsc
+RF
+ˇˇˇˇ
+1 .
+(6.18)
+This is precisely (6.14) because F is not entangled with R.
+– 34 –
+
+7
+Discussion
+We have defined a simple model that captures the information flow of an evaporating
+black hole. Unitarity is built in and this manifests at the level of the entropy of the
+radiation in the form of a discrete version of the QES variational problem. The model
+then allowed us to investigate in detail entanglement wedge reconstruction for a system
+that falls into the black hole and also for local actions on the Hawking partners. The
+model reproduces the properties of the holographic map that have been proposed in
+[28]; namely, the map acts trivially on the outgoing radiation and non-isometrically on
+the black hole. This latter fact manifests the fact that the Hilbert space of an old black
+hole is not large enough to host all the Hawking partners of the semi-classical state.
+Something must give, the map is non-isometric and as a result the Hawking partners
+have been teleported out into the radiation as subtle features of the microscopic state
+of R. In a sense, when a black hole is past the Page time according to an external
+observer, its inside has been squeezed out into the radiation leaving only a small region
+between the horizon and the QES that could be thought of as being part of the black
+hole.
+Although the proposal of [28] has clarified certain issues, much remains to be
+understood. Of principal interest is the fate of an infalling system. According to our
+model, an infalling system begins to be scrambled immediately. In fact, the infalling
+system will soon enter the entanglement wedge of a late-time observer who collects
+all the radiation, since the QES is very close up behind the horizon meaning that
+the information of the infalling observer is in the radiation available to the late-time
+observer.
+Is this compatible with the idea that the infalling system experiences a
+smooth internal geometry after horizon crossing? We have argued at the microscopic
+level, the state of the radiation is not the inertial vacuum in the neighbourhood of
+the horizon but perhaps the infalling system sees effectively a smooth geometry and
+being thermalized takes some time. The situation seems quite analogous to the same
+questions for the fuzzball paradigm in string theory [54, 55]. In that context, it is argued
+that a macroscopic (i.e. high energy) infalling system would take time to be thermalized
+as it falls into the fuzzball. In a proposal known as fuzzball complementarity, the high
+energy infallling system would not resolve the subtle structure of the microscopic state
+and effectively average it to see a smooth geometry. It seems plausible that the same
+mechanism is at work here, if an observer cannot resolve the fine details of ρR maybe
+it effectively experiences the average ρR “ ρsc
+R, precisely the semi-classical state and a
+smooth horizon, at least for a while.
+– 35 –
+
+Acknowledgments
+TJH, AL and SPK acknowledge support from STFC grant ST/T000813/1. NT and
+ZG acknowledge the support of an STFC Studentship. AL has also received funding
+from the European Research Council (ERC) under the European Union’s Horizon 2020
+research and innovation programme (grant agreement No 804305).
+********************
+For the purpose of open access, the authors have applied a Creative Commons Attribution
+(CC BY) licence to any Author Accepted Manuscript version arising.
+Appendices
+A
+Thermodynamics of free fields
+Consider a set of free fields in 1 ` 1 dimensions. We will consider just the right-moving
+modes. The canonical partition function of a single mode of energy ω is equal to
+Z “
+8
+ÿ
+p“0
+e´pω{T “
+1
+1 ´ e´ω{T ,
+1ÿ
+p“0
+e´pω{T “ 1 ` e´ωT ,
+(A.1)
+for a scalar and spinor field, respectively. Summing over modes in a volume V and
+assuming there are N “ c, 2c fields for bosons/fermions, gives the free energy
+f “ ˘NV T
+ż 8
+0
+dω
+2π logp1 ¯ e´ω{Tq “ ´πcV T 2
+12
+.
+(A.2)
+The average energy
+E “ NV
+ż 8
+0
+dω
+2π
+ω
+eω{T ¯ 1 “ πcV T 2
+12
+(A.3)
+– 36 –
+
+and the entropy
+Srad “ NV
+ż 8
+0
+dω
+2π
+!
+ω
+Tpeω{T ¯ 1q ¯ logp1 ¯ e´ω{Tq
+)
+“ πcV T
+6
+.
+(A.4)
+We can also evaluate the R´enyi entropes,
+p1 ´ nqSpnq
+rad “ NV
+ż 8
+0
+dω
+2π log
+8,1
+ÿ
+p“0
+´e´pω{T
+Z
+¯n
+“ NV
+ż 8
+0
+dω
+2π
+`
+log ZpT{nq ´ n log ZpTq
+˘
+.
+(A.5)
+Hence,
+Spnq
+rad “ nfpTq ´ nfpT{nq
+p1 ´ nqT
+“ 1 ` n
+n
+µT “ 1 ` n
+2n
+Srad .
+(A.6)
+We will need to understand whether the relativistic gas can be described thermo-
+dynamically. We can solve for the entropy in terms of the entropy, Srad “ 2?µE, where
+µ “ πcV {12. In the thermodynamic it should be possible to approximate the canonical
+partition function as a integral over a continuum set of states with energy E and density
+of states eSradpEq, that is
+Z “ e´f{T “
+ż
+dE eSradpEq´E{T .
+(A.7)
+The thermodynamic limit can be understood as when the saddle point approxima-
+tion of this integral is valid. The saddle point equation corresponds to the Legendre
+transformation between the internal energy and free energy:
+f “ ext
+E
+`
+E ´ TSradpEq
+˘
+,
+(A.8)
+and has solution
+E “ µT 2 ,
+(A.9)
+for which the free energy
+f “ ´µT 2 .
+(A.10)
+One can verify that these expressions are are entirely consistent with (A.2) and (A.3).
+The saddle point approximation is valid in the limit that the spread in the energy
+around the saddle point ∆E ! E which is the condition
+∆E
+E
+∼
+1
+?Srad
+! 1 .
+(A.11)
+So when Srad " 1, the gas can be described thermoydnamically.
+– 37 –
+
+B
+Dominant saddles
+In the model, we encounter sums over elements of the symmetric group of the form
+(4.5). This motivates analysing a sum of the form
+Zpnq “
+ÿ
+σPSn
+d´dpσ,τ1q
+1
+d´dpσ,τ2q
+2
+d´dpσ,τ3q
+3
+,
+(B.1)
+where di ě 1, τi P Sn and dpσ, πq is the Cayley distance between elements of Sn. This
+is equal to
+dpσ, πq “ n ´ kpσπ´1q ,
+(B.2)
+where kpσq is the number of cycles the make up σ, e.g. kpeq “ n and kpηq “ 1.
+We are interested in minimising the following ‘free energy’
+fpσq “ x1dpσ, τ1q ` x2dpσ, τ2q ` x3dpσ, τ3q ,
+(B.3)
+where xi “ log di. We first consider the permutations which minimise the free energy
+at the following special regions in the phase diagram (see figure 5), which we may
+parameterise by x1{x3 and x2{x3:
+• for x1{x3 Ñ 0 and x2{x3 Ñ 0: fpσq Ñ x3dpσ, τ3q is minimised for σ “ τ3.
+• for x1{x3 ` x2{x3 “ 1: fpσq “ x1 pdpτ1, σq ` dpσ, τ3qq ` x2 pdpτ2, σq ` dpσ, τ3qq is
+minimised for σ P Γpτ1, τ3q X Γpτ2, τ3q. Here, Γpτi, τjq denotes the set of permuta-
+tions σ which saturate the triangle inequality dpτi, σq ` dpσ, τjq ě dpτi, τjq.
+There are two ` two more regions in the phase diagram where the permutations which
+minimise the free energy can be determined by cyclically permuting the labels in the
+above. Most of the rest of the phase diagram can then be filled in using convexity of
+the free energy. That is, since f is a linear function of the xi, if σ minimises f at two
+points in the phase diagram, then σ also minimises f along the segment joining these
+two points. This argument can only be used to fill in the whole phase diagram if the
+set of permutations Γpτ1, τ2, τ3q – Γpτ1, τ2q X Γpτ2, τ3q X Γpτ3, τ1q which simultaneously
+saturate the three triangle inequalities
+dpτi, σq ` dpσ, τjq ě dpτi, τjq
+for i ‰ j ,
+(B.4)
+– 38 –
+
+is not empty. The argument we have used to find the minima of f by considering
+special regions in the phase diagram and then using convexity to fill in the rest is due
+to [56].
+From the above, we find that:
+• for x1{x3 ` x2{x3 ă 1:
+Zpnq « d´dpτ1,τ3q
+1
+d´dpτ2,τ3q
+2
+.
+(B.5)
+The behaviour of the sum in two other regions may be obtained by cyclically
+permuting the labels in the above.
+• assuming Γpτ1, τ2, τ3q is not empty, for x1{x3 ` x2{x3 ą 1, x2{x1 ` x3{x1 ą 1 and
+x3{x2 ` x1{x2 ą 1:
+Zpnq « |Γpτ1, τ2, τ3q|
+´d1d2
+d3
+¯´dpτ1,τ2q{2´d2d3
+d1
+¯´dpτ2,τ3q{2´d3d1
+d2
+¯´dpτ3,τ1q{2
+.
+(B.6)
+x1{x3
+x2{x3
+1
+1
+Γpτ1, τ2, τ3q
+τ1
+τ2
+τ3
+Figure 5: Phase diagram for the sum (B.1) when Γpτ1, τ2, τ3q is not empty. Along the blue
+lines there are more permutations which can contribute e.g. along x1{x3 `x2{x3 “ 1 the sum
+is dominated by the set of permuations which lie in Γpτ1, τ3q X Γpτ2, τ3q.
+B.1 Proof
+We now prove that when τN P te, ηu the nested sum (4.5) in the simple model:
+ZNpτNq “
+ÿ
+τ0,...,τN´1PSn
+N
+ź
+p“1
+d
+´dpτp´1,τpq
+Bp
+d
+´dpτp´1,σpq
+Rp
+,
+(B.7)
+– 39 –
+
+with dBp, dRp ě 1 and σp P te, ηu, is dominated by the terms with τp´1 P te, ηu for
+each 1 ď p ď N, provided we ignore the crossover regimes. It is useful to notice that
+ZNpτNq satisfies the recursion relation
+ZNpτNq “
+ÿ
+τN´1PSn
+d´dpτN´1,τNq
+BN
+d´dpτN´1,σNq
+RN
+ZN´1pτN´1q ,
+Z0pτ0q “ 1 .
+(B.8)
+First consider
+Z1pτ1q “
+ÿ
+τ0PSn
+d´dpτ0,τ1q
+B1
+d´dpτ0,σ1q
+R1
+.
+(B.9)
+This sum is of the form (B.1) so is dominated by the terms with τ0 P tσ1, τ1u Ă te, η, τ1u.
+Using this fact we see that
+Z2pτ2q “
+ÿ
+τ1PSn
+d´dpτ1,τ2q
+B2
+d´dpτ1,σ2q
+R2
+Z1pτ1q
+«
+ÿ
+τ1PSn
+d´dpτ1,τ2q
+B2
+d´dpτ1,σ2q
+R2
+minpdB1, dR1q´dpτ1,σ1q ,
+(B.10)
+is also of the form (B.1) so is dominated by the terms with τ1 P tσ1, σ2, τ2uYΓpσ1, σ2, τ2q Ă
+te, η, τ2u Y Γpe, η, τ2q. We have assumed that Γpe, η, τ2q is not empty; a fact we will
+verify ex-post facto. Using this, (B.5) and (B.6) it is simple to show that Z3pτ3q is also
+of the form (B.1) so is dominated by the terms with τ2 P te, η, τ3uYΓpe, η, τ3q.15 Again,
+we have assumed that Γpe, η, τ3q is not empty; a fact we will verify ex-post facto. It is
+not too difficult to see that this pattern continues and proceeding with the argument
+we find that, provided Γpe, η, τpq is not empty,
+τp´1 P te, η, τpu Y Γpe, η, τpq
+(B.11)
+for each 1 ď p ď N. However, since τN P te, ηu, this implies that
+τp´1 P te, ηu
+(B.12)
+for each 1 ď p ď N. In particular, each Γpe, η, τpq is not empty, which is consistent
+with our assumption.
+15There is a slight subtlety here as the sum Z3pτ3q can differ from (B.1) by a factor of |Γpσ1, σ2, τ2q|.
+Whilst this factor depends on n, it is independent of dBp and dRp so it is reasonable to expect that
+we can ignore its effect if we are interested in the limit where dBp and dRp are large and eventually
+also the limit n Ñ 1.
+– 40 –
+
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diff --git a/lNE_T4oBgHgl3EQf6By0/content/tmp_files/load_file.txt b/lNE_T4oBgHgl3EQf6By0/content/tmp_files/load_file.txt
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index 0000000000000000000000000000000000000000..275424e95e15810edcbd72c05a4352fb8aff9f89
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+page_content='Prepared for submission to JHEP The Holographic Map of an Evaporating Black Hole Zsolt Gyongyosia, Timothy J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Hollowooda, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Prem Kumara, Andrea Legramandia,b,c and Neil Talwara aDepartment of Physics, Swansea University, Swansea, SA2 8PP, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' bPitaevskii BEC Center, CNR-INO and Dipartimento di Fisica, Universit`a di Trento, I- 38123 Trento, Italy c INFN-TIFPA, Trento Institute for Fundamental Physics and Applications, Trento, Italy E-mail: z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='gyongyosi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='2133547@swansea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='uk,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='hollowood@swansea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='uk, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='kumar@swansea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='uk, andrea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='legramandi@unitn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='it, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='talwar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='2017429@swansea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='uk Abstract: We construct a holographic map that takes the semi-classical state of an evaporating black hole and its Hawking radiation to a microscopic model that re- flects the scrambling dynamics of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The microscopic model is given by a nested sequence of random unitaries, each one implementing a scrambling time step of the black hole evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Differently from other models, energy conservation and the thermal nature of the Hawking radiation are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We show that the QES formula follows for the entropy of multiple subsets of the radiation and black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We further show that a version of entanglement wedge reconstruction can be proved by computing suitable trace norms and quantum fidelities involving the action of a unitary on a subset of Hawking partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' If the Hawking partner is in an island, its unitary can be reconstructed by a unitary on the radiation and so the Hawking partners are not in any sense behind the horizon of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We also consider the problem of reconstruction for unitaries acting on an infalling system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='08362v1 [hep-th] 19 Jan 2023 Contents 1 Introduction 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1 The holographic map 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='2 The QES formula 7 2 The model 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1 The refined model 12 3 The microscopic state 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1 The average state 16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='2 The inner products xΨJ|ΨKy 17 4 Entropies 19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1 Refined model 21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='2 Relation to the island formalism 24 5 Information recovery and reconstruction 26 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1 Bounding the trace norm 31 6 Reconstruction of the Hawking partners 31 7 Discussion 35 A Thermodynamics of free fields 36 B Dominant saddles 38 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1 Proof 39 – 2 – 1 Introduction Black holes lie at the front line of the struggle to unify quantum mechanics with gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Recent progress is focused on how this struggle plays out at the level of effective theory in a gravitating system like a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In particular, the effective description involves techniques that have evolved over many years involving quantum field theory over a fixed background spacetime using semi-classical techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In a black hole geometry this leads to the emission of Hawking radiation and the apparent loss of unitarity [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' On the other hand, there is a microscopic level of description, for example provided by string theory, in which a black hole is described as a quantum system with a large density of states given by the Bekenstein-Hawking (BH) entropy (see [3] for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1 The holographic map Recent progress has shed light on how these two levels of description are related and how the information-loss paradox is resolved and unitarity is restored [4–7] (also see the reviews [8, 9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' A key ingredient is a map, the ‘holographic map’, between the effective semi-classical description and the microscopic description V : Hsc Ñ Hmicro .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1) The idea of such a map between the semi-classical and microscopic descriptions nat- urally arises in holography where the semi-classical state describes the state of bulk gravitational theory while the microscopic state describes the non-gravitational CFT dual [10–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' It is becoming clear that such a map should apply more generally and specifically in spacetimes which are not asymptotically AdS, such as an evaporating black hole, where the radiation can escape the AdS bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The holographic map has been interpreted as the encoding map of a quantum error code and this synergy be- tween the two subjects has been very fruitful and has led to a better understanding of entanglement wedge reconstruction [16–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' However, recent work [27, 28]1 has clarified certain details and in particular argued that, in the context of a black hole, it is an im- portant feature that the map is not isometric, V :V ‰ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This means that the relation with the standard theory of quantum error correcting codes is not so compelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The non-isometric nature of the map is actually very natural because as the black hole ages its Hilbert space becomes too small to accommodate all the Hawking partners of the previously emitted radiation and so something has to give.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Another key insight of [28] 1See also [29, 30] for recent developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' – 3 – is that the map does not act on the radiation once it has dispersed away from the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This clarifies certain statements that have been made about the radiation, in particular it is not possible to change the state of the black hole by making operations on the radiation however complicated: there is no long-range non-locality of this kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The purpose of this work is to construct the holographic map V in a very simple microscopic model of black hole evaporation defined e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' in [31, 32] but refined to take account of energy conservation leading to thermal states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The basic version of the model is the block random unitary model (BRU) of [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' A number of key features follow also for this more refined model: 1 The semi-classical state of the radiation ρsc R is precisely the average of the mi- croscopic state of the radiation ρR over the quasi-random microscopic scrambling dynamics of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' 2 Past the Page time the quasi-random fluctuations of the microscopic state ρR overwhelm the state and it becomes very different from the semi-classical (Hawk- ing) state ρsc R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' 3 The Quantum Extremal Surface (QES) formula [33, 34] for the entropy of a generic number of radiation and black hole subsets is derived in the regime where the black hole is evaporating slowly [31, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' 4 Unitary actions on an infalling system can be reconstructed on the radiation after the Page time showing that the information of the infalling system has been teleported out of the black hole realizing the Hayden-Preskill ‘black hole as a mirror’ scenario [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' 5 There is a version of state-specific entanglement wedge reconstruction (of the type discussed in [28]): local unitaries acting on the Hawking partners can be reconstructed as a unitary acting on the black hole before the Page time and on the radiation after the Page time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The discussion is extended for generic subsets of the Hawking radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The last point should not be used to conclude that, past the Page time, one is measuring something behind the horizon of the black hole by measuring the radiation: there is no such dramatic non-locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Rather it means that the Hawking partners have been teleported out of the black hole and so one is measuring a property that is, in any case, of the radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' – 4 – Let us now put some flesh on the bones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' At the semi-classical level, the state of a QFT in the black hole background consists of an entangled state between the outgoing Hawking radiation R and their partner modes behind the horizon R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The overall state is pure |ψy “ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' ÿ J λJ|JyR b |JyR ) b |SyF P Hsc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='2) We have also included the possibility for infalling modes in the state |SyF, including the matter that collapsed to form the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We will develop two models: (i) a simple one in which the Hilbert space of the radiation is taken to be finite dimensional and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='2) is the maximally entangled state λJ “ 1{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='dR and (ii) a more refined one for which the radiation and partners are in a thermofield double with a slowly varying temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' At the microscopic level, the black hole is described by a finite dimensional Hilbert space HB whose dimension is exponential in the BH entropy dB “ eSBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The black hole emits Hawking radiation and at the microscopic level we can write the state of a partly evaporated black hole and radiation as |Ψy “ ÿ J λJ|JyR b |ΨJyB P Hmicro .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='3) The two states, the semi-classical |ψy and the microscopic |Ψy are related by the holographic map (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1) V : HR b HR b HF Ñ HR b HB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='4) It was argued in [28] that the map should act trivially on R because the outgoing radiation system is identical in both the semi-classical and microscopic descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' So V actually only acts non-trivially as HR b HF Ñ HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This is natural because the Hawking partner modes R and the infalling modes F are behind the horizon and so part of the black hole whose semi-classical geometry should emerge from the microscopic description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='2 By comparing (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='2) with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='3), we have V |JyR b |SyF “ |ΨJyB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='5) We leave the dependence on the infalling state implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' 2Although this breaks down after the Page time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' – 5 – The way that Hawking’s information loss paradox can be resolved now reveals itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In Hawking’s analysis, the state of the radiation is the reduced state, the maximally- mixed state in the basic model and a quasi-thermal state in the refined model ρsc R “ ÿ J |λJ|2|JyRxJ| , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='6) since the partner mode states are orthonormal, RxJ|KyR “ δJK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' On the other hand, at the microscopic level, ρR “ ÿ JK λK¯λJξKJ|KyRxJ| , ξKJ “ xΨJ|ΨKy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='7) The semi-classical state is devoid of internal correlations, information is lost and uni- tarity is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The microscopic state, on the other hand, can carry the correlations and repair unitarity if the inner products ξJK are non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The fact that xΨJ|ΨKy ‰ δJK , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='8) implies that the holographic map V is non-isometric, a key insight in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' It is the non-isometric nature of V that allows information to escape out of the black hole in the correlations induced by the inner product [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Such a release of information would presumably be interpreted as being a non-local process to a semi-classical observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This is a major insight but perhaps to be expected when spacetime geometry is an emergent concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' For a black hole past its Page time, when Srad " SBH, one would expect the states |ΨJy to be far from orthogonal because there are order eSradpRq states in a much smaller eSBH dimensional Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Roughly speaking, as previously argued e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' in [4, 38], we find xΨJ|ΨKy “ # 1 ` Ope´SBHq J “ K , Ope´SBH{2q J ‰ K , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='9) so the violation appears to be exponentially small „ e´1{G in the semi-classical limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This seems to suggest that the corrections coming from the microscopic theory will be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' However, if we write ξ “ I ` Z, then Z is roughly-speaking a quasi-random Hermitian matrix whose elements are order e´SBH{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' It seems, therefore, that the effect of Z would be very suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' However, if the dimension of the matrix „ eSrad is large then its eigenvalues can be expected to lie in a distribution between ˘epSrad´SBHq{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' What this indicates is that the fluctuations in Z could be expected to give rise to a – 6 – radical change in the state of the radiation beyond the Page time when Srad " SBH and a mechanism to ensure the unitarity of the evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' On the other hand, if we average the microscopic state over the quasi-random fluctuations Z we recover the semi-classical state ρR “ ρsc R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='10) The fact that ρR ‰ ρsc R means that if we were to attempt to interpret the microscopic state as a state on the semi-classical geometry, then in the near-horizon region it would not be the inertial vacuum and so we could expect there will be non-trivial energy and momentum of order e´1{G, as the horizon is approached .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Another issue that is clarified by the fact that V acts trivially on the radiation R is, as already mentioned, that the state of black hole is completely invariant under any local action on the radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In more detail, the most general local action is obtained by coupling R to an auxiliary system M and having them interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' On the semi-classical state |ψy b |∅yM ÝÑ ÿ α Kα|ψy b |αyM , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='11) for some orthonormal states |αy of M and where the operators Kα act on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This defines a quantum channel acting on R and unitarity implies that Kα are Krauss operators ř α K: αKα “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Mapping this to the microscopic state, and using the fact that rV, Kαs “ 0, the reduced state on B, after R and M have interacted, is ρ1 B “ ÿ α TrR !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Kα|ΨyxΨ|K: α ) “ TrR !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' |ΨyxΨ| ÿ α K: αKα ) “ ρB , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='12) so the state of the black hole is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='2 The QES formula One can quantitatively appreciate how ρR differs from ρsc R by calculating their von Neumann entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The entropy of the semi-classical state ρsc R, suitably regularized, is just the thermal entropy of Hawking radiation familiar from Hawking’s calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The question is, how to calculate the entropy of the microscopic state ρR?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This is where the QES, or generalized entropy, formula comes in [14, 33, 39–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' It relates the von Neumann entropy of the microscopic state ρA reduced on some subsystem factor e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' A “ R or B, or some more specific subset of R to the generalised entropy: SpρAq “ min ext tXju !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' ÿ j A pXjq 4G ` Spρsc WpAqq ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='13) – 7 – In the above, WpAq is the entanglement wedge of A, some subset of a Cauchy slice3 that contains the radiation R near I ` and passes through the Quantum Extremal Surfaces (QES) Xj which are the boundaries of WpAq in the gravitating region determined by extremization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The first term involves the area of the QES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' If A is the radiation, or some subset thereof, then the entanglement wedge WpAq consists of A but also, potentially, a region disconnected from A known as the ‘entanglement island’, or ‘island’ for short, WpAq “ A Y I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='13) is remarkable in several ways but principally because it allows one to calculate the entropy of the microscopic state ρA using only semi-classical tech- niques even when the details of the microscopic theory are not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' It does this by implicitly averaging over the complex chaotic microscopic dynamics of the black hole in the way familiar from statistical mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' More precisely, when computed in the semi-classical theory, we can think of the left hand side as being equal to the usual n Ñ 1 limit of the R´enyi entropies but averaged in the following way SpρAq “ lim nÑ1 1 1 ´ n log ep1´nqSpnqpρAq , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='14) with the average over a suitable ensemble that is a proxy for the underlying complex, chaotic microscopic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Just as in statistical mechanics, the conceptual idea is that the average captures the behaviour of a single typical microscopic state because, unlike the state itself, the R´enyi entropies are self-averaging quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' For an evaporating black hole, the QES are behind the horizon and when the evaporation is slow, which it is for most of the evaporation time apart from the final stage, the QES are very close behind the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In fact the QES are completely determined within the scope of the slow evaporation approximation [31, 35, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Firstly, they have Kruskal-Szekeres (KS) coordinates related via UV ∼ c SBH !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' 1 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='15) where c are the number of (massless) fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In terms of Eddington-Finkelstein (EF) coordinates pu, vq,4 this means v “ u ´ ∆tscr , ∆tscr “ 1 2πT log SBH c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='16) 3More precisely the causal diamonds thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' 4The KS and EF coordinates are related by an approximately exponential map, U “ ´ exp ` ´2π şu Tptqdt ˘ and V “ exp ` 2π şv Tptqdt ˘ , where Tptq is the instantaneous temperature of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' – 8 – The slow evaporation regime applies precisely when SBH " c so that the QES are pressed up against the horizon from within.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The time shift above between the infalling and outgoing coordinates ∆tscr is identified with the scrambling time of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This is time dependent but only changes slowly as the black hole evaporates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The second condition on the QES is that the outgoing EF coordinate of a QES uQES (inside the horizon) must be equal to the outgoing EF coordinate of one of the endpoints of the radiation uBA (outside the horizon) tuQESu Ă tuBAu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='17) This reduces the variation problem to a discrete minimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' When A is a subset of the radiation and the entanglement wedge WpAq “ A Y I, the second term in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='13) is just the thermal entropy5 Spρsc WpAqq « SradpA a ˜Iq “ πc 6 ż Aa˜I Tpuq du , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='18) where Tpuq is the instantaneous temperature of the black hole as a function of the outgoing EF coordinate u on I `.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Here, ˜I, the ‘island-in-the stream’, is just the reflection of the island in the horizon and projected onto I ` [31, 35, 43] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' So in terms of the outgoing EF coordinate u, I and ˜I are equal, with the former outside the horizon and the latter inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The symmetric difference in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='18) accounts for the fact that I contains purifiers of the radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The first term in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='13) is then approximately equal to the Bekenstein-Hawking entropy SBH evaluated at EF outgoing coordinates of the QES uBI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Hence, within the slow evaporation approximation, we can write the entropy as a discrete minimization problem SpAq « min I !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' ÿ uBI SBHpuBIq ` SradpA a ˜Iq ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='19) This formula can easily be adapted to the case when A includes the black hole itself, B Ă A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' One simply replaces A by A X R in the second term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='6 In section 4 we verify 5There is a common divergence associated with the end-points of A at I ` which can be regularized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The divergences associated to end-points of I, on the other hand, are precisely cancelled by the divergences in the QES term in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' 6Then, if RN R A, the most recent emitted interval of radiation, there must be a QES with a u coordinate equal to the u coordinate of the upper end-point of RN, giving a contribution SBHpMNq ” SBHpMq to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' On the other hand, if RN P A then it must be that RN Ă I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In the latter case, the connected subset of I that includes RN is not strictly-speaking part of the island although it is in the entanglement wedge of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' – 9 – this formula in both the basic and refined models using the replica trick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We also find a simple formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='29) for the island which relates the replica trick and the entanglement wedge in quite a direct way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In section 2 we define two simple discrete models of a holographic map for an evaporating black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' There is a basic and refined model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Compared with the basic model, the refined model has the nice features that the state of a small subsystem is thermal instead of maximally mixed and that the irreversiblity of evaporation is naturally incorporated (there is no need to add in ancilla qubits to mimic this effect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In section 3 we calculate some of the properties of the microscopic state starting with its average over the quasi-random unitary time evolution in order to show that the averaged state of the radiation is just the semi-classical state (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We then compute the inner products (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='8) and show that they average (over the scrambling dynamics) to the delta function, also implying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='10), but have a non-trivial variance consistent with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In section 4, we compute the R´enyi and von Neumann entropies of subsets of the radiation and black hole and derive the minimization problem for the generalized entropy of a slowly evaporating black hole (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In section 5, we turn to the Hayden-Preskill scenario [36] and consider when the action of a unitary on an infalling system be reconstructed (in a state-specific sense) on a subset of the radiation or black hole from a ‘decoupling argument’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We find the model reproduces the ‘black hole as a mirror’ phenomenon and reconstruction is possible on the radiation when the black hole is past the Page time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This problem of reconstruction of operators acting on an infalling system was studied in the basic (or BRU) model and a random pairwise interaction model (which incorporates the fast scrambling nature of black holes) in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Our main contributions here are to study this problem in a model which generalises the basic model and also to consider when reconstruction is possible not just on the radiation or the black hole, but a subset thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In section 6 we consider when local operations, in the form of a quantum channel, acting on the Hawking partners can be reconstructed on a subset of the radiation or black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' As expected, we find that reconstruction on the radiation is possible when the black hole is past the Page time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In section 7 we draw some conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In appendix A we review the computation of certain thermodynamic quantities for free bosonic and fermionic fields, which is used in the refined model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In appendix B we provide a proof of the dominant saddles which contribute in the replica trick calculation in the basic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' – 10 – B0 “ F0 R1 B1 U1 R2 B2 U2 F1 BN´1 RN B ” BN UN FN´1 Figure 1: The model of black hole evaporation consisting of a sequence of random unitaries that mimic the scrambling microscopic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' At each time step a small subsystem escapes as the Hawking radiation and there can be an infalling system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The time steps are of the order of the scrambling time of the black hole (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='16) and so the model will appear to be continuous at time scales much larger than the scrambling time, including the Page time and the evaporation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' 2 The model In the model, described in [31], the evaporation at the microscopic level is described by a series of discrete time steps identified with the scrambling time of the black hole (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='16) shown in the figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' During the pth time step the state of the black hole evolves by a unitary Up which maps Up : HBp´1 b HFp´1 ÝÑ HBp b HRp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1) In the basic model, we have dBp´1dFp´1 “ dBpdRp, whereas in the refined model energy conservation is taken into account and Rp is infinite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In this case, Up is an isometric embedding of a microcanonical energy window into HRp b HBp, as we will describe later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' After N time steps, the state of the black hole and radiation is |ΨptNqy “ UN ¨ ¨ ¨ U2U1|SyF P HB b HR , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='2) where |SyF “ |s0yF0 b |s1yF1 b ¨ ¨ ¨ b |sN´1yFN´1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='3) describes the infalling matter that created the black hole B0 ” F0 as well as matter that falls in during each time step Fp as the black hole evaporates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The radiation is split into a temporal sequence of subsets R “ Ť p Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In the above, the remaining black hole is B ” BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' A basis of states of the radiation consists of |JyR where J “ tj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' jNu – 11 – and each jp P t1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' , dRpu labels the states in the pth time step Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In particular, the microscopic states defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='3) are |ΨJy “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='λJ RxJ|UN ¨ ¨ ¨ U2U1|SyF P HB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='4) Consequently the time evolution of the black hole in the model leads to a concrete expression for the holographic map, V “ ÿ J 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='λJ RxJ| b RxJ|UN ¨ ¨ ¨ U2U1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='5) acting on HR b HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In the refined model the sum here is not well defined and V is only defined acting on suitable states such as |ψy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The basic model [31] is identified with the block random unitary model of [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' What is noteworthy is that the projection in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='5) onto the maximally-entangled state of HR b HR in the basic model, is essentially post selection and manifests the non- isometric property of the map and it is this that provides the mechanism for information to be teleported out of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' At the semi-classical level, the subsets of Hawking radiation Rp and their partners behind the horizon Rp are illustrated in the Penrose diagram figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1 The refined model In this section we refine our model of black hole evaporation to take account of energy conservation and the thermal nature of the Hawking radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We will work in the adiabatic, or quasi-static, regime where the black hole is evaporating slowly enough that it makes sense to ascribe a slowly varying temperature Tptq to the Hawking radiation determined by the thermodynamic equation of the black hole 1 T “ dSBH dM , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='6) where M is the black hole mass7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The adiabatic regime is where Hawking’s calculation derivation is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' It is defined by the requirement that SBH " c , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='7) 7For a Schwarzschild black hole, M is the mass, while for the charged black hole and the black hole in JT gravity, M is the mass minus the mass of the extremal black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' – 12 – I ` Rp Fp Rp horizon Figure 2: Subsets of Hawking modes Rp are their entangled partners Rp behind the horizon and infalling modes Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The Hawking modes propagate out to null infinity I `.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Each Rp and Fp lasts for a scrambling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' the number of massless fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In addition, for a semi-classical limit c " 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The time dependence of the energy is determined by the energy flux of the Hawking radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Since most of the energy loss occurs in the s-wave modes we have effectively a 1 ` 1-dimensional relativistic gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We also ignore the possibility for back-scattering of modes and so take a trivial greybody factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The energy balance equation is then dM dt “ ´πcT 2 12 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='8) and given (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='6) all that it needed to determine the time evolution of M, T and SBH is the energy dependence of the BH entropy which depends on the nature of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' For example, for Schwarzschild SBH “ 4πGM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We will model the evaporation in terms as a series of time steps whose size are of the order of the scrambling time of the black hole, ∆t ∼ 1 T log SBH c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='9) Note that this is time dependent, so the size of the time steps adapt as the evaporation proceeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' At each time step, the radiation carries away a small amount of energy in a distri- bution that is strongly peaked around an average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Therefore, we can model the state – 13 – of the black hole at each time step as lying in a Hilbert space HBp describing a system with energy in a small window Θp “ rMp, Mp`δMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Implicitly, Mp includes the energy of the infalling system Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In other words, the black hole is in a microcanonical state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The size of the window δM is assumed to be small but for simplicity we will assume that it is much larger than the spread of the energy carried away by the radiation at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The fact that the BH entropy is so large means that Θp contains a vast number of states that forms a quasi-continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The dimension of this space is exponential in the Bekenstein-Hawking entropy dBp “ CδM Mp eSBHpMpq ∼ eSBHpMpq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='10) In the above, C is some constant which we do not have to specify since SBHpMq is very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The picture of the black hole evolving through a sequence of microcanonical states is of course an approximation which is justified because the radiation emitted during a time step has a sharply defined average energy and a spread that is assumed to be much smaller than the width of the windows δM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Let us justify this claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Since the time step, the scrambling time ∆t, is much greater than the thermal scale T ´1, the energy and entropy of the Hawking radiation follow from the standard statistical mechanics of a relativistic bosonic or fermionic gas (summarized in appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' For a bosonic gas E “ cV ż dω 2π ω eω{T ´ 1 “ πcV T 2 12 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='11) and the entropy Srad “ πcV T{6, where we identify the volume with the space filled by the gas in the scrambling time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' V “ ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In particular, the entropy Srad “ πc∆tT 6 ∼ c log SBH c " 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='12) Hence, the Hawking modes emitted in a time step have a large entropy and so can be described thermodynamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Indeed, the normalized spread of the energy ∆E E ∼ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='Srad !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='13) On the other hand, the radiation is a much smaller system than the black hole because SBH " c log SBH c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='14) We will then assume that this spread is much smaller than the microcanonical energy window δM " ∆E justifying the evaporation as a sequence of microcanonical states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' – 14 – The semi-classical state is now a thermofield double with a slowly varying tempera- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Taking the basis states |jpy to be approximate energy eigenstates with eigenvalues Ejp, we have λJ “ e´ ř p Ejp{2Tp ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Z , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='15) where Z “ ř J e´ ř p Ejp{Tp is the partition function which provides normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The temperature Tp is the instantaneous temperature of the Hawking radiation given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='6) evaluated at E “ Mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The states |jpy are to be thought of as localized in an outgoing shell of thickness ∆tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This is justified because the modes have characteristic momentum Tp and so can be localized on scales T ´1 p which is much smaller than ∆tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' 3 The microscopic state Black holes are famously fast scramblers so that over the scrambling time Up is essen- tially a random unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The question of how random time evolution of a black hole is an interesting question but one can make the hypothesis that for certain quantities it is effectively indistinguishable from a Haar random unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In this section, we make that assumption and compute some properties of the microscopic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We will need to average quantities over an N ˆ N unitary for which the basic results is the integral ż dU U ˚ ABUA1B1 “ 1 N δAA1δBB1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1) We will also need the generalization of this involving n replicas: ż dU n ź j“1 U ˚ AjBjUA1 jB1 j “ ÿ σ,τPSn n ź j“1 δAjAσpjqδBjBτpjqWgpστ ´1, Nq , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='2) where Wg is the Weingarten function [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Note how the integrals over the replicas involves a sum over the elements of the symmetric group σ, τ P Sn that permute the replicas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We will only need the behaviour in the limit that N is large, which picks out the terms with σ “ τ for which Wgp1, Nq “ 1{N, ż dU n ź j“1 U ˚ AjBjUA1 jB1 j “ 1 N n ÿ τPSn n ź j“1 δAjA1 τpjqδBjB1 τpjq ` ¨ ¨ ¨ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='3) – 15 – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1 The average state Let us consider the microscopic state of the radiation ρR and compute its average over the unitaries Up, p “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The ket |Ψy contributes a Up and bra xΨ| a U : p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The average over Up then knits together the bra and ket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Let us focus on the average over Up of its adjoint action on a operator f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1), we can write this average as ż dUp UpfU : p “ Trpfqρ(mm) RpBp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='4) Here, ρ(mm) RpBp is the maximally-mixed state on HRp b HBp which in the basic model is, ρ(mm) RpBp “ 1 dRpdBp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='5) In the refined model it is the maximally-mixed state in the energy window Θp´1 em- bedded in HRp b HBp in such a way as to conserve energy, ρ(mm) RpBp 9 ΠΘp´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='6) where ΠΘp´1 is the projector onto the energy window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The following Up`1 average then imposes a trace over Bp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In the basic model, that gives TrBp ` ρ(mm) RpBp ˘ “ 1 dRp ÿ jp |jpyxjp| , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='7) In the refined model, let us denote a basis of energy eigenstates of Rp as |jpy with energies Ejp, then TrBp ` ρ(mm) RpBp ˘ “ e´SBHpMp´1q ÿ jp eSBHpMp´1´Ejpq|jpyxjp| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='8) Implicitly, the sum here is constrained to have Mp´1 ´ Ejp P Θp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We can now follow the standard route for deriving the canonical ensemble of a small subsystem of a larger system in a microcanonical state [46], in our case the maximally mixed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Since the radiation subsystem is much smaller then the black hole, we can expand SBHpMp´1 ´ Ejpq « SBHpMp´1q ´ Ejp{Tp where the temperature is defined in the standard way via the thermodynamic equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='6) for a black hole of mass Mp´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Then we can extend the restricted sum over Ejp to be unrestricted because terms for which Mp´1 ´ Ejp R Θp – 16 – are heavily suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This gives the familiar approximation, namely the canonical state TrBp ` ρ(mm) RpBp ˘ « ÿ jp e´Ejp{Tp Zp |jpyxjp| , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='9) where Zp “ ř jp e´Ejp{Tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' If we now assemble the expressions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='9) for all the time steps, to find the average state of the radiation ρR “ 1 dR ÿ J |JyxJ| (basic) , ρR “ ÿ J e´ ř p Ejp{Tp Z |JyxJ| (refined) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='10) Hence the averaged microscopic state ρR is precisely the semi-classical state ρsc R as stated in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='2 The inner products xΨJ|ΨKy In this section, we analyse the inner product of the microscopic states |ΨJy defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='5) but only for the basic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' To begin, let us calculate its average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' For xΨJ|ΨKy, each Up in the ket is matched by a U : p in the bra and the integral is given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In our problem, each index is a compound index A “ pap, jpq where ap “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' , dBp and jp “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' , dRp, while B “ pap´1, sp´1q with sp´1 “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' , dFp´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' It is straightforward to see that the average xΨJ|ΨKy “ δJK , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='11) and so as in the last section, it follows that on the average the microscopic state ρR is equal to the semi-classical state ρsc R (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The average removes the subtle correlations in the microscopic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' On the other hand, we now show that there are fluctuations around the average by calculating the variance ∆2 JK “ |xΨJ|ΨKy|2 ´ ˇˇxΨJ|ΨKy ˇˇ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='12) – 17 – Now each Up and its conjugate appear twice and the formula we need, to leading order, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='3) ż dU U ˚ A1B1U ˚ A2B2UA3B3UA4B4 “ 1 N 2 ` δA1A3δB1B3δA2A4δB2B4 ` δA1A4δB1B4δA2A3δB2B3 ˘ ` ¨ ¨ ¨ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='13) which is valid at large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Since we are assuming that all the dimensions are large we will ignore the subleading term represented by the ellipsis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The two terms here, correspond to the identity e and cyclic permutation η in S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The average over Up takes the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='13) with indices A1 “ pap, jpq, B1 “ pap´1, sp´1q, A2 “ pbp, kpq, B2 “ pbp´1, sp´1q, A3 “ pa1 p, kpq, B3 “ pa1 p´1, sp´1q, A4 “ pb1 p, jpq and B4 “ pb1 p´1, sp´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' There are two terms in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='13) that we label ϵp “ 1 and ϵp “ 0, respectively, where ϵp “ 1 can only occur only if jp “ kp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Consider the indices labelled by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' These quantum numbers are affected by the delta functions that result from both the Up and Up`1 averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' If ϵp “ ϵp`1 then the delta functions enforce either ap “ a1 p , bp “ b1 p or ap “ b1 p , bp “ a1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='14) Given that there are 2 conditions means that the sum over the 4 indices is reduced to 2 and so the sums over these indices contributes d2 Bp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' On the other hand if ϵp ‰ ϵp`1 then the delta functions enforce ap “ a1 p “ bp “ b1 p , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='15) which is 3 conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' So the sums over these 4 labels contributes only dBp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Using these rules, one finds that the final result can be written ∆2 JK “ δj1k1 ÿ ϵ1“0 ¨ ¨ ¨ δjN kN ÿ ϵN“0 N ź p“1 1 d |ϵp`1´ϵp| Bp ´ δJK , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='16) with ϵN`1 “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Note that the second term cancels the term with ϵp “ 1, for all p, that occurs when J “ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In general the variance is suppressed by various powers of dBp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The least suppressed term in the sum is the one with ϵp “ 0, for all p, which is equal to d´1 B (recall B ” BN`1) since dBp ą dB and so we conclude that for a typical element of the ensemble that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='9) holds with dB “ eSBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' – 18 – 4 Entropies We can calculate the entropy of the microscopic state reduced on any subset A Ă tR1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' , RN, Bu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1) The strategy is to first calculate the R´enyi entropies which can be defined by introducing n replicas of the Hilbert space ep1´nqSpnqpAq “ Trpρn Aq “ TrpnqσrR1s 1 ¨ ¨ ¨ σrRNs N τ rBs N`1 |ΨyxΨ|bn , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='2) where the σp and τN`1 are elements of the symmetric group Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The superscripts, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' σrRps p , on these elements indicate which subspace of the replicated Hilbert space the element acts on where it is ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' These elements are taken to be either the identity element e or the cyclic permutation η according to the definition of the subset A A “ ␣ Rp ˇˇ σp “ η , p “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' , N ( Y ␣ B ˇˇ τN`1 “ η ( .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='3) The R´enyi entropies are known to be self-averaging in the ensemble of the unitaries Up (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' [47]) and so we will calculate the ensemble average of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='3) and take this to describe a typical element of the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The integrals we need are given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='3) which capture the leading order behaviour when the Hilbert spaces have a large dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='3), the average over the unitary Up acting in a replicated Hilbert space at large dRpdBp of adjoint action is given by a sum over elements of the symmetric group Sn, ż dUp U : bn p f U bn p “ ÿ τpPSn !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Trpnqτ rBp´1Fp´1s p f ) pτ rRpBps p q´1 ρ(mm) bn RpBp ` ¨ ¨ ¨ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='4) for some f in the replicated Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' So each average over Up comes with a sum over an element of the symmetric group τp P Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In the above, ρ(mm) RpBp is the maximally mixed state of HRp b HBp as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='5), while for the refined model, it is the subspace with energy in the window Θp´1 as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The ellipsis stand for subleading corrections, suppressed by inverse powers of dBp´1dFp´1, that we will not keep track of in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Trpnq is the trace defined on the replicated Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' – 19 – Applying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='4) for all p, it becomes apparent that the average of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='2) breaks up into a set of building blocks: ep1´nqSpnqpAq “ ÿ τ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=',τNPSn Z1 ¨ ¨ ¨ ZN , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='5) where Zp “ TrpnqσrRps p τ rBpFps p`1 pτ rRpBps p q´1` ρ(mm) RpBp b ρsc Fp ˘bn , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='6) where ρsc Fp “ |spyxsp| is the semi-classical state of the infalling system Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In the last step p “ N this piece is missing, there is no FN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The traces over HFp are trivial because the states ρsc Fp are pure and so Trpnqpσρsc bn Fp q “ 1, for any σ P Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This includes the initial state in HF0 that collapsed to form the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Hence, the building block (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='6) can be written more simply as Zp “ TrpnqσrRps p τ rBps p`1 pτ rRpBps p q´1ρ(mm) bn RpBp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='7) The expression for the building block Zp can also be interpreted in terms of the equili- HRp HBp Up Up`1 HFp τp`1 ρFp “ |spyxsp| τ ´1 p τ ´1 p σp τp`1 ρ(mm) RpBp Figure 3: Assembling the ingredients for the building block in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' bration ansatz of [47] as an alternative to the unitary averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In this interpretation, the pure state of the black hole at time tp´1 equilibrates over the next time step mean- ing that for certain observables it is indistinguishable from an equilibrium state, in this case precisely the maximally mixed state ρ(mm) RpBp (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='5), or (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='6) in the refined model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In the basic model, it is then straightforward to evaluate the building block (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='7), Zp “ exp ” pkpτp`1τ ´1 p q ´ nqSBHpMpq ` pkpσpτ ´1 p q ´ nqSradpRpq ı , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='8) – 20 – where kpσq is the number cycles of the element σ and with SBHpMpq “ log dBp and SradpRpq “ log dRp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Then plugging into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='5) gives the final result ep1´nqSpnqpAq “ ÿ τ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=',τNPSn e p1´nqSpnq tτpupAq , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='9) where we have defined Spnq tτpupAq “ 1 n ´ 1 N ÿ p“1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' dpτp`1, τpqSBHpMpq ` dpσp, τpqSradpRpq ) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='10) where dpσ, πq “ n ´ kpσπ´1q is the Cayley distance between elements of Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1 Refined model The refined model is rather more complicated because of the need to enforce energy conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The R´enyi entropies now involve a sum over both the energies Ejp and the elements of the symmetric group τp, ep1´nqSpnqpAq “ ÿ tjpu ÿ tτpuPSn n ź p“1 Zp “ ÿ tjpu ÿ tτpuPSn e p1´nqSpnq tτpupAq (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='11) where the building block is Zp “ dRppEjpqkpσpτ ´1 p qdBpMp´1 ` Ep ´ Ejpqkpτp`1τ ´1 p q ` ř jp dRppEjpqdBpMp´1 ` Ep ´ Ejpq ˘n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='12) where Ep is the energy of the infalling system Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Note that the mass of the black hole depends implicitly on the energy of the radiation emitted up to that point Mp “ M0 ` pÿ q“1 pEq ´ Ejqq , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='13) a point that must be born in mind when we perform the saddle point approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The denominator in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='12) can be evaluated by a saddle point approximation where the sum is replaced by an integral over a continuous variable Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In particular, the 8Alternatively, the Cayley distance dpσ, πq may be defined as the minimal number of transpositions required to go between σ and π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' – 21 – radiation can be described thermodynamically in the way summarized in appendix A and the entropy log dRppEq “ 2 a µpEp where µp “ πc∆tp 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='14) Since the saddle point value of the energy is much smaller than the black hole mass, the saddle point equation is cµp Ep “ ´dSBHpMp´1 ` Ep ´ Epq dEp « 1 Tp ùñ Ep “ µpT 2 p , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='15) where Tp defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='6) is precisely the temperature of the Hawking radiation Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The average energy of the radiation emitted Ep and the infalling energy Ep are assumed to be much smaller than the black hole mass .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Hence, we have ÿ jp dRppEjpqdBpMp´1 ` Ep ´ Ejpq « dBpMp´1qeSradpRpq{2`Ep{Tp , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='16) where the saddle point value of the entropy is SradpRpq “ 2µpTp “ πc∆tpTp 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='17) This and Ep above are the familiar expressions for the entropy and energy of a volume ∆tp of a relativistic gas in 1 ` 1 dimensions in a volume V “ ∆tp (as reviewed in appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The saddle point approximation is, of course, just the conventional way of deriving the Legendre transformation between the internal energy and free energy in thermodynamics and is justified precisely because the spread in the energy is small (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' For later use, note that dBpMpq “ dBpMp´1 ` Ep ´ Epq « dBpMp´1 ` Epqe´SradpRpq{2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='18) and so SBHpMp´1 ` Epq ´ SBHpMpq “ SradpRpq 2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='19) which is the familiar relation for a model of black hole evaporation in the s-wave approximation and with no back scattering (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' grey body factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Note that it implies that the evaporation is irreversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' – 22 – We now proceed to evaluate the sums of the energies in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='11) by similar saddle point approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' After we replace the sums by integrals over Ep, the exponent of the integrand is p1 ´ nqSpnq tτpupAq “ N ÿ p“1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' 2pn ´ dpσp, τpqq a µpEp ´ dpτp`1, τpqSBHpM0q ´ ´ n ´ N ÿ q“p dpτq`1, τqq ¯Ep Tp ´ N ÿ q“p dpτq`1, τqqEp Tp ´ n 2SradpRpq ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='20) It is now simple to compute the saddle point equations for the energies Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In the regime of slow evaporation we can ignore the Ep dependence of the temperatures Tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The saddle point values are found to be Ep “ µpT 2 p ´ n ´ dpσp, τpq n ´ řN q“p dpτq`1, τqq ¯2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='21) where for consistency the saddles must have n ą N ÿ q“1 dpτq`1, τqq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='22) The contribution of this saddle to the R´enyi entropy is Spnq tτpupAq “ 1 n ´ 1 N ÿ p“1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' dpτp`1, τpqSBHpM0q ` N ÿ q“p dpτq`1, τqqEp Tp ` 1 2 ´ n ´ pn ´ dpσp, τpqq2 n ´ řN q“p dpτq`1, τqq ¯ SradpRpq ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='23) We can re-write this by noting that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='19) implies SBHpMpq “ SBHpM0q ` pÿ q“1 ´Eq Tq ´ SradpRqq 2 ¯ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='24) as Spnq tτpupAq “ 1 n ´ 1 N ÿ p“1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' dpτp`1, τpqSBHpMpq ` 2ndpσp, τpq ´ dpσp, τpq2 ´ ` řN q“p dpτq`1, τqq ˘2 2 ` n ´ řN q“p dpτq`1, τqq ˘ SradpRpq ) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='25) which is the refined model generalization of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' – 23 – A ˜I A a ˜I τp StτpupAq R1 R2 R3 R4 R5 R6 R7 R8 B e e η SBHpM2q η η η SradpR5q η SradpR6q η e SBHpM8q Figure 4: An example of a saddle for the model with N “ 8 time steps, with some choice of the set A “ R3 Y R4 Y R7 Y R8, as shown, with an island-in-the-stream ˜I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Note that B ˜I Ă BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The contributions to the entropy from each time step are shown and summing these up gives SIpAq “ SBHpM2q ` SBHpM8q ` SradpR5 Y R6q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Note that the last term is SradpA a ˜Iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='2 Relation to the island formalism We interpret (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='9) as being a sum over saddles of the (Lorentzian) gravitational path integral in the semi-classical limit, labelled by the elements tτpu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In this limit, the entropies SBHpMpq and SradpRpq are very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' If we avoid the crossover regimes when saddles are degenerate, it turns out that only a much smaller number of terms can actually dominate in the sum, namely, those for which each τp, p “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' , N, is equal to e or η only, the identity and cyclic permutations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This is proved in appendix B for the basic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The te, ηu dominance means that the saddles that dominate respect the Zn cyclic symmetry of the replicas mirroring the symmetry of the replica wormholes of [4, 5], or, equivalently, we can interpret the average over unitaries to be equivalent to the average over baby universe states (see [29, 48]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' For the refined model, the discussion is very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Indeed each element in the energy sum in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='11) behaves like a basic model, and therefore we can again invoke the fact that τp is dominated by τp P te, ηu, which will be valid as long we are not in the vicinity of a crossover of saddles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='9 The expression for the von Neumann entropy of our chosen subset A Ă R Y B is obtained from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='10) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='25) in the limit SpAq “ limnÑ1 SpnqpAq and has the form 9Notice that we don’t risk of having a crossover at every time step since we assumed that each energy window is small (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' – 24 – of a minimization problem over the 2N choices τp P te, ηu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Indeed notice that when σ, π P te, ηu, we can write dpσ, πq “ pn ´ 1qp1 ´ δσπq , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='26) which facilitates the evaluation of the Cayley distances in the n Ñ 1 limit of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='10) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In both models, the von Neumann entropy is given by SpAq “ min tτpu StτpupAq “ min tτpu !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' N ÿ p“1 p1 ´ δτp`1τpqSBHpMpq ` p1 ´ δσpτpqSradpRpq ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='27) The resemblance of this equation to the QES formula described in the introduction for a slowly evaporating black hole (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='19) becomes more apparent if we set SIpAq ” StτpupAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='28) where I is defined in both models as I “ ď pPΦ ` Rp Y Fp´1 ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='29) with Φ “ ␣ p | τp “ η ( .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The I that minimizes (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='28) is called the ‘entanglement island’ or ‘island’, for short, will be denoted IpAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Even if in principle we have 2N possible saddles, most of them will not contribute since terms with τp ‰ τp`1 are not favourable because of the black hole entropy being big.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' One can check that the only saddles that are not trivially suppressed are the one where τp changes in correspondence with a change in σp, which is an analog of the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' See figure 4 for an example where Φ “ t3, 4, 5, 6, 7, 8u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In order to make more transparent the identification of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='27) with the QES formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='19) for the A that we have chosen, we can also notice that the second term is a discrete version of the continuum expression SradpA a ˜Iq where we identify the island-in-the- stream as the reflection of the island I in the horizon and then projected onto I `, so each Rp gets mapped to Rp: ˜I “ ď pPΦ Rp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='30) On the other hand, the first term can be written in terms of the BH entropy at the outgoing EF coordinates of QES uBI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We can then parametrize the entropy of the black hole with its mass at outgoing time u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Notice also that the infalling states in – 25 – (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='29) are shifted by p Ñ p ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This is how the model accounts for the fact that infalling coordinate v of the QES are shifted relative to the outgoing coordinate u by the scrambling time (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='16), precisely the size of the time steps in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In the next sections, we will enforce our definition of the entanglement island (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='29) studying when it is possible to reconstruct an unitary acting on the radiation, which is equivalent to the well known statement that the island is in the entanglement wedge of the radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Specifically, since the emitted radiation is in both the semi-classical and microscopic descriptions, we include it in its own entanglement wedge WpAq “ IpAq Y pA X Rq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='31) Although we call IpAq the island, strictly speaking, this only applies when subsets of IpAq are separated from the rest of the entanglement wedge by QES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='10 5 Information recovery and reconstruction In this section, we consider the fate of an infalling system, Hayden and Preskill’s diary for instance [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We will focus on the single system that falls in during the qth time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' For simplicity, we will avoid the case that this is the last time step, in other words we will take q ă N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The idea is to consider a family of infalling states W|sqy for a unitary W and fixed state |sqy P Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This gives a family of microscopic states |ΨpWqy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The physical question is, can the effect of the unitary W be achieved by a local action on the radiation or the black hole?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This will inform us as to when the information in Fq has been teleported out of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' More specifically, when can the action of W be reconstructed on A “ R or B, or a subset thereof, in the sense that there exists a unitary WA acting on A such that WA|Ψy ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='“ |ΨpWqy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1) This is the state-specific notion of reconstruction described in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The above implies that WA acts on the reduced state on A via the adjoint action ρApWq “ WAρAW : A , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='2) while the reduced state on the complement A is invariant ρApWq “ ρA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='3) 10For example, when A “ B, the black hole before the Page time has IpBq “ WpBq “ R Y F which is not an island in the strict sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' – 26 – In fact this decoupling condition on A implies the existence of WA in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This can be seen using the Schmidt decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The decoupling condition implies that if |Ψy “ ř j ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='pj|jyA|jyA then |ΨpWqy “ ř j ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='pj|jy1 A|jyA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' It follows that WA “ ř j |jy1 Axj| acting on the subspace of HA spanned by the Schmidt states |jyA although it can be extended to a unitary acting on HA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Acting within the subspace, we can write explicitly, WA “ TrA |ΨpWqyxΨ|ρ´1 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='4) The Schmidt basis states depend implicitly on the infalling state |sqy and so the con- struction of W is ‘state dependent’ in this sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' It is an interesting question if the construction can be extended to any operator acting on any state of the infalling system in HFq and thereby be state independent, at least in this limited sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In fact, the construction above can be seen as a special case of the Petz map and, indeed, there is a more general state-independent construction [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We cannot expect the conditions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='3) to hold exactly and approximate forms of these conditions are formulated in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' However, we will work to leading order in the semi-classical limit and we will not need these approximate forms in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The decoupling condition is therefore key to reconstructing that action of W on either the radiation or the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Hence, we need to calculate the difference between the states ρApWq and ρA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This can be measured by the trace norm11 difference ˇˇˇˇσ´ρ ˇˇˇˇ 1 or the quantum fidelity fpσ, ρq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Both are tractable in our models when averaged over the unitary evolution to leading order in the semi-classical limit where they can be computed using the replica method and an analytic continuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' For the trace norm difference, we take an even number of replicas and then take an analytic continuation, ˇˇˇˇσ ´ ρ ˇˇˇˇ 1 “ Tr a pσ ´ ρq2 “ lim nÑ 1 2 Trp2nqη ` σ ´ ρ ˘b2n (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='5) and similarly for the quantum fidelity, fpσ, ρq ” Tr b?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='ρσ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='ρ “ lim nÑ 1 2 Trp2nqη ` σ b ρ ˘bn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='6) In our context, there is a subtlety in that the analytic continuations must be taken after the semi-classical limit has picked out a dominant saddle otherwise saddles would become degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We should also emphasize that what we are actually calculating 11For Hermitian operators the trace norm is equal to ˇˇˇˇO ˇˇˇˇ 1 “ ř j |λj|, where λj are the eigenvalues of O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' – 27 – are the unitary averages of the replica expressions before taking the limits n Ñ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This is in the same spirit as calculating the averages the exponents of the R´enyi entropies as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='5) before taking the limit n Ñ 1 to recover the von Neumann entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1 we compute an upper bound on the trace norm which does not require the n Ñ 1 2 limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Let us compute the average of the trace difference in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The computation is similiar to that of the R´enyi entropy via Trρn A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In fact, since W acts locally on HFq, only the qth time step is modified: Zq ÝÑ Trp2nqσrRqs q τ rBqFqs q`1 pτ rRqBqs q q´1` ρ(mm) RqBq b pρsc FqpWq ´ ρsc Fqq ˘b2n “ Zq Trp2nqτq`1 ` ρsc FqpWq ´ ρsc Fq ˘b2n , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='7) where we separated out the trace over the replicas of Fq where W acts and the quantity Zq is the original quantity in the entropy calculation defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The contribution from the other time steps p ‰ q are precisely as for the entropy (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Hence, assembling all the pieces gives ˇˇˇˇρApWq ´ ρA ˇˇˇˇ 1 “ lim nÑ 1 2 ÿ τ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=',τNĂte,ηu e p1´2nqSp2nq tτpupAq Trp2nqτq`1 ` ρsc FqpWq ´ ρsc Fq ˘b2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='8) Now we have to be careful to take the semi-classical limit before taking the analytic continuation n Ñ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The semi-classical limit picks out a dominant term in the sum over the elements τp and, in particular, fixes τq`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Hence, ˇˇˇˇρApWq ´ ρA ˇˇˇˇ 1 “ lim nÑ 1 2 Trp2nqτq`1 ` ρsc FqpWq ´ ρsc Fq ˘b2n (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='9) One can follow the same steps for the average of the quantum fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Once again the contribution comes entirely from the qth time step which is modified as Zq ÝÑ Trp2nqσrRqs q τ rBqFqs q`1 pτ rRqBqs q q´1` ρ(mm) RqBq ˘b2n b ` ρsc FqpWq b ρsc Fq ˘bn “ Zq Trp2nqτq`1 ` ρsc FqpWq b ρsc Fq ˘bn , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='10) leading to fpρApWq, ρAq “ lim nÑ 1 2 Trp2nqτq`1 ` ρsc FqpWq b ρsc Fq ˘bn (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='11) – 28 – Let us now evaluate our results above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' When Fq R WpAq, it follows that the dominant saddle has τq`1 “ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' For the trace norm difference (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='9), this gives an expression that is clearly seen to vanish ˇˇˇˇρApWq ´ ρA ˇˇˇˇ 1 “ lim nÑ 1 2 Trp2nq` ρsc FqpWq ´ ρsc Fq ˘b2n “ ˇˇTrpρsc FqpWq ´ ρsc Fqq ˇˇ “ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='12) This proves the decoupling condition in terms of the trace norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' On the other hand, for the fidelity (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='13),12 fpρApWq, ρAq “ lim nÑ 1 2 Trp2nq` ρsc FqpWq b ρsc Fq ˘bn “ b Trρsc FqpWq Trρsc Fq “ 1 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='13) which is another expression of decoupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Note that, if the trace norm difference of two states vanishes, then they must have unit quantum fidelity and ρApWq “ ρA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' On the other hand, when Fq P WpAq, the element τq`1 “ η and the trace norm difference (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='9) is ˇˇˇˇρApWq ´ ρA ˇˇˇˇ 1 “ lim nÑ 1 2 Trp2nqη ` ρsc FqpWq ´ ρsc Fq ˘b2n “ ˇˇˇˇρsc FqpWq ´ ρsc Fq ˇˇˇˇ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='14) For the fidelity, we have a similar relation to the semi-classical state fpρApWq, ρAq “ lim nÑ 1 2 Trp2nqη ` ρsc FqpWq b ρsc Fq ˘bn “ fpρsc FqpWq, ρsc Fqq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='15) Let us take stock of the results and, in particular, relate them to the state recon- struction formula of [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This states that if there are two microscopic states ρA and 12The fidelity plays an important role in quantum hypothesis testing, which is the task of making a measurement to distinguish between two quantum states given that the actual state is one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The fidelity bounds the error on the optimal measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We expect that the corrections to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='13) are non-perturbatively suppressed in the semi-classical limit, as in [51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' If so, this would imply that whilst it is not possible to distinguish the two states given a single copy of the state, it will be possible given sufficiently many copies of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' – 29 – σA such that the semi-classical saddles that dominate Trpρ2n A q and Trpσ2n A q are the same (and preserve the Zn symmetry of the replicas) then13 ˇˇˇˇρA ´ σA ˇˇˇˇ 1 “ ˇˇˇˇρsc WpAq ´ σsc WpAq ˇˇˇˇ 1 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='16) up to OpGq corrections, where WpAq is the entanglement wedge of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The fact that the map V preserves the trace norm difference is on the same footing as the preservation of the relative entropy [5, 25, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' To relate this to our analysis, we identify σR “ ρRpWq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The saddles associated to Trpρ2n A q and Trpσ2n A q are the ones that determine the R´enyi entropies and are therefore associated to the set of elements τp, p “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The fact that they both have the same saddle is ensured by the fact that W only acts on a small subset of the infalling modes and so cannot alter the dominant saddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Let us consider our results for the case A “ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Before the Page time, WpRq “ R and so Fq R WpRq and the formula (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='16) implies ˇˇˇˇρRpWq ´ ρR ˇˇˇˇ 1 “ 0 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='17) which is the decoupling condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='12) with A “ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This means that W can be reconstructed on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' On the other hand, after the Page time, the entanglement wedge WpRq “ R Y IpRq, so Fq P WpRq, since the island IpRq contains the outgoing and infalling modes IpRq “ R Y F since it lies very close behind the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Hence, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='16) implies ˇˇˇˇρRpWq ´ ρR ˇˇˇˇ 1 “ ˇˇˇˇρsc RRFpWq ´ ρsc RRF ˇˇˇˇ 1 “ ˇˇˇˇρsc FqpWq ´ ρsc Fq ˇˇˇˇ 1 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='18) which is (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='14) with A “ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We will see shortly that this is the case when W can be reconstructed on R because B decouples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Now consider the case A “ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' After the Page time, WpBq “ ∅ and so (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='16) predicts decoupling as we found in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This occurs at the same time as (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='18) which makes perfect sense as W can be reconstructed on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' On the other hand, before the Page time, WpBq “ R Y F, and so (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='16) gives ˇˇˇˇρBpWq ´ ρB ˇˇˇˇ 1 “ ˇˇˇˇρsc RFpWq ´ ρsc RF ˇˇˇˇ 1 “ ˇˇˇˇρsc FqpWq ´ ρsc Fq ˇˇˇˇ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='19) 13We have stated the formula in a slightly more general way to include the case when A is any subset of the radiation plus the black hole rather than all the radiation as considered in [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The condition for Zn symmetry is satisfied by our saddles which involve only the elements e or η of Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' – 30 – But this is precisely (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='14) for A “ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This is also when R decouples and so W can be reconstructed on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' So once again we find precise agreement between our averaged results and the formula (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1 Bounding the trace norm The condition for decoupling is that the averaged trace norm difference between ρApWq and ρA vanishes in the leading order saddle (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' But this is derived with the limits in a particular order, first the semi-classical limit picking out a particular saddle and then in the replica limit n Ñ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Can we trust this?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In fact there is standard way to bound the averaged trace norm difference, ˇˇˇˇρApWq ´ ρA ˇˇˇˇ 1 ď b dA TrpρApWq ´ ρAq2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='20) We can evaluate the right-hand side, at least in the case that the subsystem A is finite dimensional (so this seems to exclude A “ R, the radiation, in the refined model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The average on the right-hand side is just the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='8) with n Ñ 1, so TrpρApWq ´ ρAq2 “ ÿ τ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=',τNĂte,ηu e ´Sp2q tτpupAq Trp2qτq`1 ` ρsc FqpWq ´ ρsc Fq ˘b2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='21) If we consider A “ B, so dA „ eSBHpMq, and after the Page time, the sum in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='21) is dominated by the term with τp “ η for which Sp2q tηupBq “ αSradpRq, where α “ 1 for the basic model and α “ 3 4, for the refined model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='14 Therefore we can bound the trace norm difference ˇˇˇˇρApWq ´ ρA ˇˇˇˇ 1 Æ Ope 1 2 SBHpMq´ α 2 SradpRqq !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' 1 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='22) after the Page time when SradpRq " SBHpBq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' 6 Reconstruction of the Hawking partners In this section, we consider reconstruction for the Hawking partners which semi-classically are behind the horizon and part of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Conceptually the discussion is very 14The latter follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='25) with τp “ η, p “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' , N ` 1 and σp “ e giving dpτp`1, τpq “ 0 and dpσp, τpq “ n ´ 1 giving Spnq tηupBq “ pn ` 1qSradpRq{p2nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This is the R´enyi entropy of the radiation (see appendix A) and then taking n “ 2 gives 3 4SradpRq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' – 31 – similar to the reconstruction of the infalling system in the last section but the technical details are rather different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The idea is to consider a unitary operator on the Hawking partners R and ask if it is possible to reconstruct this on some A Ă R Y B, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' |ΨpWqy ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='“ WA|Ψy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1) As in section 5 the condition for such a reconstruction is the decoupling condition for the complement ρApWq “ ρA , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='2) which can be analysed by calculating the trace norm difference or quantum fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In order to proceed, it is useful to deploy the following trick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Exploiting the entan- glement between R and R, we can write the action of W on the semi-classical state as the action of an operator Ă W on R: W|ψy “ Ă W|ψy , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='3) where Ă W “ pρsc Rq1{2W Tpρsc Rq´1{2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='4) We remark that Ă W is not unitary so it is not a physically realizable local action on the radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' It then follows that the reduced state on R is invariant under adjoint action by Ă W, ρsc R ÝÑ Ă Wρsc RĂ W : “ pρsc Rq1{2` W :W ˘˚pρsc Rq1{2 “ ρsc R , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='5) as it must be by locality: the action of W on R cannot change the state of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We now compute the trace difference and quantum fidelity of the two states ρApWq and ρA using the replica method following the same steps as in section (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' For sim- plicity, we will take W to act on just one of the subsets of partner modes Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We can then use (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='3) to write the action on the Hawking modes Rq by switching W Ñ Ă W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' As for the infalling system, the only effect of W is on the qth time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' For the trace norm difference, this time step is modified as Zq ÝÑ Trp2nqσrRqs q τ rBqs q`1 ` AdĂ W ´ 1 ˘b2npτ rRqBqs q q´1ρ(mm) b2n RqBq , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='6) where AdĂ W is the adjoint action of Ă W on ρ(mm) RqBq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We now assume that the saddle that dominates the entropy, and therefore the trace norm difference, has τq`1 “ τq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This – 32 – means that Rq is not just before a QES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' One can view this as avoiding an edge effect created by having a discrete model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In that case, we can perform the trace over Bq to give the semi-classical state ρsc Rq “ TrBqρ(mm) RqBq: Zq ÝÑ Trp2nqσq ` AdĂ W ´ 1 ˘b2nτ ´1 q ρsc b2n Rq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='7) where in the second line we used the fact that all relevant saddles have τq`1 “ τq and ρsc Rq “ TrBqρ(mm) RqBq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Following the same steps as in section 5, and in particular taking the semi-classical limit before the analytic continuation in n, gives ˇˇˇˇρApWq ´ ρA ˇˇˇˇ 1 “ lim nÑ 1 2 Trp2nqσq ` AdĂ W ´ 1 ˘b2nτ ´1 q ρsc b2n Rq (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='8) where τq is determined by the saddle that dominates the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Similarly, for the quantum fidelity fpρApWq, ρAq “ lim nÑ 1 2 Trp2nqσq ` AdĂ W b 1 ˘bnτ ´1 q ρsc b2n Rq (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='9) When Rq R WpAq the dominant saddle has τq “ e and then the trace norm differ- ence is ˇˇˇˇρApWq ´ ρA ˇˇˇˇ 1 “ lim nÑ 1 2 Trp2nqσq `Ă Wρsc RqĂ W : ´ ρsc Rq ˘2n “ 0 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='10) using the invariance (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We can repeat the analysis for the fidelity, fpρApWq, ρAq “ lim nÑ 1 2 Trp2nqσq ´ Ă Wρsc RqĂ W : b ρsc Rq ¯bn “ lim nÑ 1 2 Trp2nqσq ρsc b2n Rq “ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='11) So decoupling occurs when the partners Rq do not lie in the entanglement wedge of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Under these circumstances, W can be reconstructed on the complement A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' On the other hand, when Rq P WpAq, the appropriate saddle has τq “ η and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='10) becomes ˇˇˇˇρApWq ´ ρA ˇˇˇˇ 1 “ lim nÑ 1 2 Trp2nqσq ` AdĂ W ´ 1 ˘b2nη´1ρsc b2n Rq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='12) We can now consider this for particular choices for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' For the case A “ R, so after the Page time, then σq “ η, and the above becomes ˇˇˇˇρRpWq ´ ρR ˇˇˇˇ 1 “ 2 b 1 ´ ˇˇTr ` ρsc RqW T˘ˇˇ2 “ ˇˇˇˇρsc RRpWq ´ ρsc RR ˇˇˇˇ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='13) – 33 – For the case A “ B, so before the Page time, σq “ e, we have ˇˇˇˇρBpWq ´ ρB ˇˇˇˇ 1 “ lim nÑ 1 2 Tr ` W ˚ρsc RqW T ´ ρsc Rq ˘2n “ lim nÑ 1 2 Tr ` Wρsc RqW : ´ ρsc Rq ˘2n “ ˇˇˇˇρsc RpWq ´ ρsc R ˇˇˇˇ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='14) Note that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='14) is not the same as (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='13) because R is entangled with R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We can also consider the quantum fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' For A “ R (after the Page time), fpρRpWq, ρRq “ lim nÑ 1 2 Trp2nqη ` AdĂ W b 1 ˘b2nη´1ρsc b2n Rq “ ˇˇTrpρsc RqW Tq ˇˇ “ fpρsc RRpWq, ρsc RRq (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='15) and for A “ B (before the Page time), fpρBpWq, ρBq “ lim nÑ 1 2 Trp2nq` AdĂ W b 1 ˘b2nη´1ρsc b2n Rq “ lim nÑ 1 2 Trp2nqη ` W ˚ρsc RqW T b ρsc Rq ˘n “ fpρsc RpWq, ρsc Rq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='16) These expressions are close cousins of the expressions for the trace norm difference in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='13) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Once again, let us compare our results to the formula (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='16) of [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Firstly, let us compare the microscopic states ρRpWq and ρR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Before the Page time, Rq R WpRq and so (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='16) implies ˇˇˇˇρRpWq ´ ρR ˇˇˇˇ 1 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' After the Page time, Rq P WpRq and so (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='16) implies ˇˇˇˇρRpWq ´ ρR ˇˇˇˇ 1 “ ˇˇˇˇρsc RRFpWq ´ ρsc RRF ˇˇˇˇ 1 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='17) which is precisely (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='13) because F is not entangled with R Y R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Now we turn to the states ρBpWq and ρB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In this case, after the Page time, WpBq “ ∅ and so (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='16) implies ˇˇˇˇρBpWq ´ ρB ˇˇˇˇ 1 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' On the other hand, before the Page time, WpBq “ R Y F, and so (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='16) implies ˇˇˇˇρRpWq ´ ρR ˇˇˇˇ 1 “ ˇˇˇˇρsc RFpWq ´ ρsc RF ˇˇˇˇ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='18) This is precisely (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='14) because F is not entangled with R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' – 34 – 7 Discussion We have defined a simple model that captures the information flow of an evaporating black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Unitarity is built in and this manifests at the level of the entropy of the radiation in the form of a discrete version of the QES variational problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The model then allowed us to investigate in detail entanglement wedge reconstruction for a system that falls into the black hole and also for local actions on the Hawking partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The model reproduces the properties of the holographic map that have been proposed in [28];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' namely, the map acts trivially on the outgoing radiation and non-isometrically on the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This latter fact manifests the fact that the Hilbert space of an old black hole is not large enough to host all the Hawking partners of the semi-classical state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Something must give, the map is non-isometric and as a result the Hawking partners have been teleported out into the radiation as subtle features of the microscopic state of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In a sense, when a black hole is past the Page time according to an external observer, its inside has been squeezed out into the radiation leaving only a small region between the horizon and the QES that could be thought of as being part of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Although the proposal of [28] has clarified certain issues, much remains to be understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Of principal interest is the fate of an infalling system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' According to our model, an infalling system begins to be scrambled immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In fact, the infalling system will soon enter the entanglement wedge of a late-time observer who collects all the radiation, since the QES is very close up behind the horizon meaning that the information of the infalling observer is in the radiation available to the late-time observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Is this compatible with the idea that the infalling system experiences a smooth internal geometry after horizon crossing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We have argued at the microscopic level, the state of the radiation is not the inertial vacuum in the neighbourhood of the horizon but perhaps the infalling system sees effectively a smooth geometry and being thermalized takes some time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The situation seems quite analogous to the same questions for the fuzzball paradigm in string theory [54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In that context, it is argued that a macroscopic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' high energy) infalling system would take time to be thermalized as it falls into the fuzzball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In a proposal known as fuzzball complementarity, the high energy infallling system would not resolve the subtle structure of the microscopic state and effectively average it to see a smooth geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' It seems plausible that the same mechanism is at work here, if an observer cannot resolve the fine details of ρR maybe it effectively experiences the average ρR “ ρsc R, precisely the semi-classical state and a smooth horizon, at least for a while.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' – 35 – Acknowledgments TJH, AL and SPK acknowledge support from STFC grant ST/T000813/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' NT and ZG acknowledge the support of an STFC Studentship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' AL has also received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 804305).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' ******************** For the purpose of open access, the authors have applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Appendices A Thermodynamics of free fields Consider a set of free fields in 1 ` 1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We will consider just the right-moving modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The canonical partition function of a single mode of energy ω is equal to Z “ 8 ÿ p“0 e´pω{T “ 1 1 ´ e´ω{T , 1ÿ p“0 e´pω{T “ 1 ` e´ωT , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1) for a scalar and spinor field, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Summing over modes in a volume V and assuming there are N “ c, 2c fields for bosons/fermions, gives the free energy f “ ˘NV T ż 8 0 dω 2π logp1 ¯ e´ω{Tq “ ´πcV T 2 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='2) The average energy E “ NV ż 8 0 dω 2π ω eω{T ¯ 1 “ πcV T 2 12 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='3) – 36 – and the entropy Srad “ NV ż 8 0 dω 2π !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' ω Tpeω{T ¯ 1q ¯ logp1 ¯ e´ω{Tq ) “ πcV T 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='4) We can also evaluate the R´enyi entropes, p1 ´ nqSpnq rad “ NV ż 8 0 dω 2π log 8,1 ÿ p“0 ´e´pω{T Z ¯n “ NV ż 8 0 dω 2π ` log ZpT{nq ´ n log ZpTq ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='5) Hence, Spnq rad “ nfpTq ´ nfpT{nq p1 ´ nqT “ 1 ` n n µT “ 1 ` n 2n Srad .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='6) We will need to understand whether the relativistic gas can be described thermo- dynamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We can solve for the entropy in terms of the entropy, Srad “ 2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='µE, where µ “ πcV {12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In the thermodynamic it should be possible to approximate the canonical partition function as a integral over a continuum set of states with energy E and density of states eSradpEq, that is Z “ e´f{T “ ż dE eSradpEq´E{T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='7) The thermodynamic limit can be understood as when the saddle point approxima- tion of this integral is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The saddle point equation corresponds to the Legendre transformation between the internal energy and free energy: f “ ext E ` E ´ TSradpEq ˘ , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='8) and has solution E “ µT 2 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='9) for which the free energy f “ ´µT 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='10) One can verify that these expressions are are entirely consistent with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='2) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The saddle point approximation is valid in the limit that the spread in the energy around the saddle point ∆E !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' E which is the condition ∆E E ∼ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='Srad !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='11) So when Srad " 1, the gas can be described thermoydnamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' – 37 – B Dominant saddles In the model, we encounter sums over elements of the symmetric group of the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This motivates analysing a sum of the form Zpnq “ ÿ σPSn d´dpσ,τ1q 1 d´dpσ,τ2q 2 d´dpσ,τ3q 3 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1) where di ě 1, τi P Sn and dpσ, πq is the Cayley distance between elements of Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This is equal to dpσ, πq “ n ´ kpσπ´1q , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='2) where kpσq is the number of cycles the make up σ, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' kpeq “ n and kpηq “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We are interested in minimising the following ‘free energy’ fpσq “ x1dpσ, τ1q ` x2dpσ, τ2q ` x3dpσ, τ3q , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='3) where xi “ log di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We first consider the permutations which minimise the free energy at the following special regions in the phase diagram (see figure 5), which we may parameterise by x1{x3 and x2{x3: for x1{x3 Ñ 0 and x2{x3 Ñ 0: fpσq Ñ x3dpσ, τ3q is minimised for σ “ τ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' for x1{x3 ` x2{x3 “ 1: fpσq “ x1 pdpτ1, σq ` dpσ, τ3qq ` x2 pdpτ2, σq ` dpσ, τ3qq is minimised for σ P Γpτ1, τ3q X Γpτ2, τ3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Here, Γpτi, τjq denotes the set of permuta- tions σ which saturate the triangle inequality dpτi, σq ` dpσ, τjq ě dpτi, τjq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' There are two ` two more regions in the phase diagram where the permutations which minimise the free energy can be determined by cyclically permuting the labels in the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Most of the rest of the phase diagram can then be filled in using convexity of the free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' That is, since f is a linear function of the xi, if σ minimises f at two points in the phase diagram, then σ also minimises f along the segment joining these two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' This argument can only be used to fill in the whole phase diagram if the set of permutations Γpτ1, τ2, τ3q – Γpτ1, τ2q X Γpτ2, τ3q X Γpτ3, τ1q which simultaneously saturate the three triangle inequalities dpτi, σq ` dpσ, τjq ě dpτi, τjq for i ‰ j , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='4) – 38 – is not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' The argument we have used to find the minima of f by considering special regions in the phase diagram and then using convexity to fill in the rest is due to [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' From the above, we find that: for x1{x3 ` x2{x3 ă 1: Zpnq « d´dpτ1,τ3q 1 d´dpτ2,τ3q 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='5) The behaviour of the sum in two other regions may be obtained by cyclically permuting the labels in the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' assuming Γpτ1, τ2, τ3q is not empty, for x1{x3 ` x2{x3 ą 1, x2{x1 ` x3{x1 ą 1 and x3{x2 ` x1{x2 ą 1: Zpnq « |Γpτ1, τ2, τ3q| ´d1d2 d3 ¯´dpτ1,τ2q{2´d2d3 d1 ¯´dpτ2,τ3q{2´d3d1 d2 ¯´dpτ3,τ1q{2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='6) x1{x3 x2{x3 1 1 Γpτ1, τ2, τ3q τ1 τ2 τ3 Figure 5: Phase diagram for the sum (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1) when Γpτ1, τ2, τ3q is not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Along the blue lines there are more permutations which can contribute e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' along x1{x3 `x2{x3 “ 1 the sum is dominated by the set of permuations which lie in Γpτ1, τ3q X Γpτ2, τ3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1 Proof We now prove that when τN P te, ηu the nested sum (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='5) in the simple model: ZNpτNq “ ÿ τ0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=',τN´1PSn N ź p“1 d ´dpτp´1,τpq Bp d ´dpτp´1,σpq Rp , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='7) – 39 – with dBp, dRp ě 1 and σp P te, ηu, is dominated by the terms with τp´1 P te, ηu for each 1 ď p ď N, provided we ignore the crossover regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' It is useful to notice that ZNpτNq satisfies the recursion relation ZNpτNq “ ÿ τN´1PSn d´dpτN´1,τNq BN d´dpτN´1,σNq RN ZN´1pτN´1q , Z0pτ0q “ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='8) First consider Z1pτ1q “ ÿ τ0PSn d´dpτ0,τ1q B1 d´dpτ0,σ1q R1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='9) This sum is of the form (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1) so is dominated by the terms with τ0 P tσ1, τ1u Ă te, η, τ1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Using this fact we see that Z2pτ2q “ ÿ τ1PSn d´dpτ1,τ2q B2 d´dpτ1,σ2q R2 Z1pτ1q « ÿ τ1PSn d´dpτ1,τ2q B2 d´dpτ1,σ2q R2 minpdB1, dR1q´dpτ1,σ1q , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='10) is also of the form (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1) so is dominated by the terms with τ1 P tσ1, σ2, τ2uYΓpσ1, σ2, τ2q Ă te, η, τ2u Y Γpe, η, τ2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' We have assumed that Γpe, η, τ2q is not empty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' a fact we will verify ex-post facto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Using this, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='5) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='6) it is simple to show that Z3pτ3q is also of the form (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1) so is dominated by the terms with τ2 P te, η, τ3uYΓpe, η, τ3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='15 Again, we have assumed that Γpe, η, τ3q is not empty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' a fact we will verify ex-post facto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' It is not too difficult to see that this pattern continues and proceeding with the argument we find that, provided Γpe, η, τpq is not empty, τp´1 P te, η, τpu Y Γpe, η, τpq (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='11) for each 1 ď p ď N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' However, since τN P te, ηu, this implies that τp´1 P te, ηu (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='12) for each 1 ď p ď N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' In particular, each Γpe, η, τpq is not empty, which is consistent with our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' 15There is a slight subtlety here as the sum Z3pτ3q can differ from (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content='1) by a factor of |Γpσ1, σ2, τ2q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Whilst this factor depends on n, it is independent of dBp and dRp so it is reasonable to expect that we can ignore its effect if we are interested in the limit where dBp and dRp are large and eventually also the limit n Ñ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' – 40 – References [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
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+page_content=' Hawking, “Particle Creation by Black Holes,” Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
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+page_content=' Hartman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE_T4oBgHgl3EQf6By0/content/2301.08362v1.pdf'}
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+Criticality of quantum energy teleportation at phase transition points in quantum
+field theory
+Kazuki Ikeda1, 2, ∗
+1Co-design Center for Quantum Advantage, Department of Physics and Astronomy,
+Stony Brook University, Stony Brook, New York 11794-3800, USA
+2Center for Nuclear Theory, Department of Physics and Astronomy,
+Stony Brook University, Stony Brook, New York 11794-3800, USA
+Quantum field theory can be a new medium for communication through quantum energy telepor-
+tation. We performed a demonstration of quantum energy teleportation with a relativistic fermionic
+field theory of self-coupled fermions, called the massive Thirring model. Our results reveal that there
+is a close relation between the amount of energy teleported and the phase diagram of the theory. In
+particular, it is shown that the teleported energy peaks near the phase transition points. The results
+provide new implications for phase diagrams of field theory in terms of quantum communication
+and quantum computing.
+I.
+INTRODUCTION
+Quantum field theory (QFT) has been quite success-
+ful in explaining quantum many-body systems.
+From
+condensed matter physics, such as superconductors and
+topological insulators, to the Standard Model of elemen-
+tary particles as a low-energy effective theory of high-
+energy physics, QFT can explain a wide variety of ex-
+perimental results with extremely high precision.
+The
+approach to non-perturbative phenomena is a remaining
+challenge for QFT, which has been explored by various
+methods such as first-principles calculations and lattice
+QCD. In addition, with the advent of quantum comput-
+ers, we are able to perform real-time non-perturbative
+quantum simulations of many-body systems. One of the
+key challenges in studying QFT is the complexity of the
+calculations involved.
+Simulating these systems using
+classical computers can be computationally expensive, as
+the complexity of the calculations increases rapidly with
+the size of the system. Quantum computers, on the other
+hand, have the potential to perform these simulations
+much more efficiently. In addition to this, the develop-
+ment of quantum algorithms and quantum computers has
+greatly contributed to the fundamental understanding of
+quantum mechanics, including the control of quantum
+states and the measurement of quantum states.
+As such, understanding the behavior of quantum
+many-body systems through quantum simulations has
+been the primary focus of recent cross-disciplinary in-
+terest in physics and computer science, but for physics,
+the connection to quantum science and technology is not
+limited to quantum computation.
+Regarding the con-
+nection between QFT and quantum information theory,
+there are active studies on entanglement entropy and
+black holes [1–3].
+These studies are mainly concerned
+with high-energy physics at the Planck scale.
+While
+such attempts have been extremely successful, new efforts
+∗kazuki7131@gmail.com
+to reveal the nature of quantum systems and spacetime
+through measurement have been active in recent years
+in a wide range of fields, including high-energy physics,
+condensed matter physics and quantum computation [4–
+14].
+Quantum energy teleportation (QET) is a protocol for
+the study of local energies that takes advantage of the en-
+tanglement nature of the ground state of quantum many-
+body systems
+[15–21]. Just as quantum teleportation
+can transfer quantum states to remote locations [22–26],
+it is expected that QET can transfer energy to remote
+locations using local operation and classical communica-
+tion (LOCC) only. The role of QET in physics and infor-
+mation engineering is largely unexplored, as the theory
+has not received much attention for long time since it
+was proposed about 15 years ago. An interesting prop-
+erty of QET is that multiple people in different locations,
+who share the same ground state initially, can simultane-
+ously lower the energy of their local systems by applying
+conditional operations. This is only possible when the
+sender and receivers of the energy conduct the appro-
+priate LOCC, and cannot be obtained by any unitary
+operation or random conditional operations. Therefore
+QET will not only help to enhance our understanding
+of fundamental issues in quantum statistical mechanics,
+condensed matter physics, and high-energy physics but
+will also provide interesting perspectives for engineering
+applications of quantum computation and quantum com-
+munication.
+The purpose of this paper is to investigate the role and
+properties of QET in field theory. From the viewpoint of
+quantum computer applications, we simulate QET using
+the massive Thirring model (low dimensional quantum
+electrodynamics (QED)), which is one of the most widely
+used (1+1) dimensional models of QFT. First, we esti-
+mate the phase diagram of the massive Thrring model
+using entanglement entropy and chiral condensate. The
+main result of this paper is the identification of a sharp
+peak in teleported energy near the phase transition point.
+We also analyze the time-evolution of the entanglement
+entropy difference ∆SAB using Alice’s post-measurement
+arXiv:2301.11712v1 [hep-th] 27 Jan 2023
+
+2
+FIG. 1: Protocol of quantum energy teleportation [Left] and the corresponding quantum circuits [Right]. First, Alice measures
+her local operator XnA and tells her result (µ ∈ {+1, −1}) to Bob. At this point, Alice’s local energy is excited EnA > 0.
+Then, to obtain energy, Bob applies conditional operation UnB(µ) to his local qubit and measures the corresponding terms
+of his local Hamiltonian HnB. Statistically he will observe ⟨HnB⟩ = Tr[ρQETHnB] < 0 and gain EnB = −⟨HnB⟩ through his
+measurement device.
+state and show numerically how the entanglement en-
+tropy lost in Alice’s measurement is recovered over time
+due to particle-particle interactions in the system if Bob
+does nothing after Alice’s measurement.
+Some of the
+results in this paper are based on simulations of quan-
+tum gate operations using qasm simulator provided by
+IBM, and we confirm that all of these results are fully
+consistent with those obtained by exact diagonalization.
+These results provide new insights into local operations
+of quantum fields based on remote communication and
+non-trivial energy flow mediated by many-body quantum
+systems.
+II.
+LOW DIMENSIONAL QFT
+The (1+1) dimensional QFTs are of significant in-
+terest since they are simpler and more tractable than
+higher-dimensional QFT, and they have rich mathemat-
+ical structures that have been studied extensively from
+various motivations, including condensed matter physics,
+high energy physics, statistical mechanics and mathemat-
+ical physics [27]. Some of the models have a number of
+interesting properties, including confinement and the chi-
+ral anomaly therefore they are useful toy models of QCD.
+Typical models preferred in studies of (1+1) dimensional
+QFTs are the Thirring model and the Schwinger model.
+In particular, they are attractive models in terms of quan-
+tum simulation and quantum computation [28–33].
+The Thirring model is a simplified version of quantum
+electrodynamics (QED) in (1+1) dimensions, which was
+introduced by Walter Thirring in 1958 [34]. It is a the-
+ory of a self–coupled Dirac field, and it can be used to
+describe a variety of physical systems, such as supercon-
+ductors [35], statistical mechanics, high energy physics
+and mathematical physics [36].
+While the Thirring model and the Schwinger model are
+models for fermions, there is a significant (1+1) dimen-
+sional model for bosons, called sine-Gordon model, which
+is of significant interest in theoretical physics due to
+its integrability, soliton solutions, and relations to other
+models such as the Thirring model, massive Schwinger
+model, and to the XY -model. The sine-Gordon model
+is a (1+1) dimensional field theory that is described by
+the sine-Gordon equation, which is a nonlinear partial
+differential equation. The soliton solution of this model
+describes a kink or anti-kink solution which is a topolog-
+ical mode in the field that can be interpreted as particle
+like excitation [37]. The topological nature of the solitons
+ensures the stability and the solitons retain their shape
+even during collision.
+It has been widely known that both models are related
+by the bosonization. By representing the fermionic fields
+in terms of bosonic fields, the bosonized version of the
+Thirring model becomes the sine-Gordon model. This is
+known as the S-duality between the two models. More
+detailed theoretical descriptions are given in Appendix B.
+Throughout
+this
+work,
+we
+consider
+the
+massive
+Thirring model, whose Lagrangian is
+LTh = ψ(iγµ∂µ − m)ψ − g
+2ψγµψψγµψ,
+(1)
+where m is the fermion mass, g is the dimensionless four-
+fermion coupling constant and ψ = ψ(x) is a spinor
+field with two components ψ1(x) and ψ2(x). It is widely
+known that the massive Thirring model is dual to the
+sine-Gordon model and the classical two-dimensional XY
+model [38].
+For example, a Kosterlitz-Thouless phase
+transition at T ∼ Kπ/2 in the XY model corresponds
+to a critical point g ∼ −π/2, called Coleman’s instability
+point, in the Thirring model. They are also related with
+a critical point at t ∼ 8π in the sine-Gordon model.
+It turns out that the spin representation of the massive
+
+(+1)
+D(-1)
+0(+1)
+0-1)
+(+1)
+-1)3
+Thirring model is
+HTh = − 1
+4a
+N−2
+�
+n=0
+(XnXn+1 + YnYn+1)
++ m
+2
+N
+�
+n=0
+(−1)n+1Zn
++ ∆(g)
+a
+N−2
+�
+n=0
+�Zn + 1
+2
+� �Zn+1 + 1
+2
+�
+,
+(2)
+where ∆(g) = cos
+� π−g
+2
+�
+, a is the lattice spacing [38–43].
+The theoretical background of the lattice Hamiltonian is
+described in Appendix C.
+III.
+SIMULATION OF QUANTUM ENERGY
+TELEPORTATION
+To facilitate clarity of results, we add a constant ϵi to
+every local Hamiltonian of the Thirring model
+HTh =
+�
+n
+Hn
+(3)
+where Hn is the local Hamiltonian including the nearest
+neighbor interactions and each ϵn should be chosen in
+such a way that
+⟨g| HTh |g⟩ = ⟨g| Hn |g⟩ = 0, ∀i ∈ E
+(4)
+where |g⟩ is the ground state of the total Hamiltonian
+HTh. Note that, in general, |g⟩ is not the ground state
+of local Hn. The explicit form of Bob’s local Hamilto-
+nian and the details of computation are given in Sup-
+plemental Information (eq. (A9)). It is important that
+non-trivial local manipulations, including measurement
+of the ground state, yield excited states and thus increase
+the energy expectation value. The increase in energy is
+supplied by the experimental apparatus. Moreover, our
+ground state |g⟩ is an entangled state in general.
+The QET protocol is as follows.
+First, Alice mea-
+sures her Pauli operator σnA by PnA(µ) = 1
+2(1 + µσnA)
+and obtains either µ = −1 or +1. Local measurement
+of the quantum state at a subsystem A destroys this
+ground state entanglement.
+At the same time, energy
+EA from the device making the measurement is injected
+into the entire system. The injected energy EA is local-
+ized around the subsystem A in the very early stages of
+time-evolution, however, it is not possible for Alice to ex-
+tract EA from the system by her operations alone at nA.
+This is because information about EA is also stored in
+remote locations other than nA due to the entanglement
+that exists prior to the measurement. In other words, Al-
+ice’s energy EA can be partially extracted at any location
+other than nA. Now let us consider taking advantage of
+the quantum many-body nature of the quantum many-
+body system to extract energy from a different location
+other than nA. This can be accomplished by LOCC, as
+shown below.
+Via a classical channel, Alice sends her measurement
+result µ to Bob, who applies an operation UnB(µ) to his
+qubit and measures his local operators XnB, YnB, ZnB
+independently. The density matrix ρQET after Bob oper-
+ates UnB(µ) to PnA(µ) |g⟩ is where ρQET is
+ρQET =
+�
+µ∈{−1,1}
+UnB(µ)PnA(µ) |g⟩ ⟨g| PnA(µ)U †
+nB(µ).
+(5)
+Using ρQET, the expected local energy at Bob’s local
+system is evaluated as ⟨EnB⟩ = Tr[ρQETHnB], which
+is negative in general.
+Due to the conservation of en-
+ergy, EB = −⟨EnB⟩(> 0) is extracted from the system
+by the device that operates UnB(µ) [44].
+In this way,
+Alice and Bob can transfer the energy of the quantum
+system only by operations on their own local system and
+classical communication (LOCC). Those are summarized
+in Fig. 1.
+It should be noted that the Thirring model is a rela-
+tivistic field theory in performing QET, which could be a
+problem if the particle is massless since the speed of clas-
+sical communication does not exceed the speed of light.
+We will consider a massive particle and assume that Bob
+can receive energy faster than the time evolution rate of
+the system.
+In what follows we give the details about the operations
+of Alice and Bob. We define UnB(µ) by
+UnB(µ) = cos θI − iµ sin θσnB,
+(6)
+where θ obeys
+cos(2θ) =
+ξ
+�
+ξ2 + η2
+(7)
+sin(2θ) = −
+η
+�
+ξ2 + η2
+(8)
+where
+ξ = ⟨g| σnBHσnB |g⟩
+(9)
+η = ⟨g| σnA ˙σnB |g⟩
+(10)
+with
+˙
+σnB = i[H, σnB]. The local Hamiltonian should be
+chosen so that [H, σnB] = [HnB, σnB]. The average quan-
+tum state ρQET is obtained after Bob operates UnB(µ)
+to PnA(µ) |g⟩. Then the average energy Bob measures is
+⟨EnB⟩ = Tr[ρQETHnB] = 1
+2
+�
+ξ −
+�
+ξ2 + η2
+�
+,
+(11)
+which is negative if η ̸= 0.
+If there is no energy dis-
+sipation, the positive energy of −⟨EnB⟩ is transferred to
+Bob’s device after the measurement due to energy conser-
+vation. Based on the quantum circuit in Fig. 1, we per-
+formed a quantum simulation of QET for N = 6, 10, 14
+at ∆(g) = −0.2, a = 0.2 and results are shown in
+Fig. 2 (F). Dashed lines correspond to exact results.
+
+4
+FIG. 2: (A): Heat map of entanglement entropy at N = 6. The Thirring model has three distinct phases, which can be clearly
+read off the diagram at N = 6. (B): Heat map of entanglement entropy difference ∆SAB. (C): Heat map of teleported energy
+⟨HnB⟩ at N = 6. It is crucial that the value of the teleported energy peaks at the phase transition points, showing a clear
+correspondence to the phase diagram. (D) and (E): Time-evolution of entanglement entropy difference. This is due to the
+natural time evolution of the system, as seen when Bob does not perform any operations on his system after Alice’s local
+operations. Decreasing 1− ∆SAB
+SAB
+in the early stages of time evolution means that entanglements broken by Alice’s observations
+are recreated by the interactions in the system. (F): Simulation results of expected energy of Bob’s local system obtained by
+QET. Error bars indicate statistical errors.
+In this work, we put Alice and Bob near the boundary
+nA = 1, nB = N − 2. Bob’s local energy can be calcu-
+lated by the explicit form of his local Hamiltonian given
+in eq.(A9). The simulation results are given in Table I in
+Sec. A of Suplimental Information.
+We next study the entanglement entropy between
+two subsystems A, B such that A ∩ B = ∅, A ∪ B =
+{1, 2, · · · , N}. Let ρ be a density operator on the entire
+system A ∪ B. Then the entanglement entropy between
+A and B are defined by
+S(ρ) = −TrA(ρA log ρA),
+(12)
+where ρA is defined by tracing out the Hilbert space of B:
+ρA = TrBρ. In this study we choose ρ as the ground state
+|g⟩ of the Hamiltonian (ρ = |g⟩ ⟨g|). Fig. 2 (A) shows
+the entanglement entropy between the left and right half
+subsystems, i.e., A = {0, · · · , N
+2 }.
+The figure exhibits
+sharp peaks at the critical points of phase transitions
+that can be understood by the phase diagram of chiral
+condensate in Fig. 4 in Appendix D. Fig. 2 (C) shows the
+teleported energy Tr[ρQETHnB] to Bob’s local system.
+It is significant that the teleported energy is enhanced
+along the critical points of the phase transition.
+This
+will be understood by a relation between Bob’s energy
+Tr[ρQETHnB] and the entanglement entropy difference
+∆SSA, which is shown in Fig. 2 (B).
+The change in entropy before and after the measure-
+ment by Alice can be evaluated as follows
+∆SAB = SAB −
+�
+µ
+pµSAB(µ)
+(13)
+where pµ is the probability distribution of µ, SAB(µ) is
+the entanglement entropy after the measurement, ξ =
+arctan
+� k
+h
+�
+. After Alice’s post-measurement, the state is
+mapped to
+|A(µ)⟩ =
+1
+√pµ
+PnA(µ) |g⟩ .
+(14)
+Then SAB(µ) is calculated with the density matrix
+|A(µ)⟩ ⟨A(µ)|.
+As discussed in [16, 45], ∆SAB is bounded below by
+a function f(ξ, η) in such a way that
+∆SAB ≥ f(ξ, η)EB.
+(15)
+This indicates that the transferring energy involves a
+commensurate consumption of entropy. Similar to the
+Maxwell Demon argument [46, 47], Bob’s conditional op-
+erations reduce the entropy of the local system. If Bob
+
+Entanglement entropy
+Entanglement entropy difference Asaz
+Teleported energy
+00000.
+0.6
+0.0005
+0.0010
+04
+0.0015
+-0.0020
+60
+0.0025
+-02
+90 t0
+0.2
+0.0030
+01
+-@1
+-0.0035
+@0
+02 04 06 08
+Teleported energy
+0L6
+0.60
+025
+m=0.5
+0.002
+0.55
+50
+0.001
+0.50
+附=2
+0.000
+@4
+9899338388090000005
+0.45
+0.001
+0.40
+0.002
+se
+0.2
+0.003
+0.30
+五
+.1
+0.25
+N=10
+N=14
+0.005
+Q5
+15
+25
+@5
+15
+ON
+2
+0.6
+12
+16
+18
+t5
+does nothing after Alice’s measurement, Figs 2 (D) and
+(E) illustrate how the entanglement entropy is recreated
+by the natural time evolution of the system. Moreover,
+the maximal energy that Bob would receive is bounded
+below by the difference in entropy:
+max
+U1(µ) EB ≥ h(ξ, η)∆SAB,
+(16)
+where h(ξ, η) is a certain function.
+Although it is difficult to analytically obtain the con-
+crete forms of functions f and g, the results of this study
+show that there is a clear correspondence between the en-
+ergy obtained by QET and the phase diagram of QFT.
+Acknowledgement
+I thank Adrien Florio, David Frenklakh, Sebastian
+Grieninger,
+Fangcheng He,
+Masahiro Hotta,
+Dmitri
+Kharzeev, Yuta Kikuchi, Vladimir Korepin, Qiang Li,
+Adam Lowe, Ren´e Meyer, Shuzhe Shi, Hiroki Sukeno,
+Tzu-Chieh Wei, Kwangmin Yu and Ismail Zahed for
+fruitful communication and collaboration.
+I thank
+Megumi Ikeda for providing the cartoons.
+I acknowl-
+edge the use of IBM quantum computers and simulators.
+I was supported by the U.S. Department of Energy, Of-
+fice of Science, National Quantum Information Science
+Research Centers, Co-design Center for Quantum Ad-
+vantage (C2QA) under Contract No.DESC0012704.
+Author contribution
+All work was performed by the author.
+Competing interests
+The author declares that there is no competing finan-
+cial interests.
+Appendix A: Quantum Gates and Measurement
+The goal of this section is to describe how to compute
+Bob’s local energy ⟨HnB⟩ gained by quantum energy tele-
+portation, using the quantum circuit shown in Fig. 1. For
+this we provide a self-contained description of the back-
+ground knowledge used in the main text.
+We use the
+following one-qubit operators whose matrix representa-
+tions are given as
+X =
+�
+0 1
+1 0
+�
+, Y =
+�
+0 −i
+i
+0
+�
+, Z =
+�
+1
+0
+0 −1
+�
+,
+S =
+�
+1 0
+0 i
+�
+, H =
+1
+√
+2
+�
+1
+1
+1 −1
+�
+.
+(A1)
+We use |0⟩ =
+�1
+0
+�
+, |1⟩ =
+�0
+1
+�
+for the computational
+basis states, which are eigenstates of Z:
+Z |0⟩
+=
+|0⟩ , Z |1⟩ = − |1⟩.
+We also work with another ba-
+sis vectors |±⟩ =
+|0⟩±|1⟩
+√
+2
+.
+They are eignestates of X:
+X |−⟩ = − |−⟩ , X |+⟩ = − |+⟩. Note that |±⟩ are created
+by applying H to |0⟩ and |1⟩; H |0⟩ = |+⟩ , H |1⟩ = |−⟩.
+For example, Alice finds µ = ±1 by observing the eigen-
+values ±1 of her local Pauli X operator.
+The rotation of X, Y, Z is defined by
+RX(α) = e−i α
+2 X, RY (α) = e−i α
+2 Y , RZ(α) = e−i α
+2 Z.
+(A2)
+We use two-qubit gate operations. In general, a control
+U operation Λ(U) is defined by
+Λ(U) = |0⟩ ⟨0| ⊗ I + |1⟩ ⟨1| ⊗ U
+(A3)
+and the corresponding diagram is drwan as
+control U=
+U
+One of the most frequently used controlled gates is a
+CNOT gate CNOT = Λ(X), whose diagram is especially
+drawn as
+CNOT=
+It is convenient to define an anti-control gate, which is
+activated when the control bit is in state |0⟩: |1⟩ ⟨1|⊗I +
+|0⟩ ⟨0| ⊗ U, whose diagram is drawn as
+Anti-control U=
+U
+=
+X
+X
+U
+With those operators, we can draw time evolution of
+XX, Y Y, ZZ type interactions of spins as
+
+6
+m
+0.5
+1
+1.5
+2
+⟨ZN−2⟩
+N = 14
+0.2303 ± 0.0010
+0.3459 ± 0.0009
+0.4111 ± 0.0009
+0.5050 ± 0.0009
+N = 10
+0.2453 ± 0.0010
+0.3021 ± 0.0010
+0.4125 ± 0.0009
+0.5069 ± 0.0009
+N = 6
+0.2132 ± 0.0010
+0.3245 ± 0.0010
+0.4256 ± 0.0009
+0.5141 ± 0.0009
+⟨XN−3XN−2⟩
+N = 14
+0.5281 ± 0.0008
+0.5211 ± 0.0009
+0.5710 ± 0.0008
+0.5550 ± 0.0008
+N = 10
+0.5376 ± 0.0008
+0.5717 ± 0.0008
+0.5697 ± 0.0008
+0.5535 ± 0.0008
+N = 6
+0.5270 ± 0.0008
+0.5454 ± 0.0008
+0.5522 ± 0.0008
+0.5436 ± 0.0008
+⟨XN−2XN−1⟩
+N = 14
+0.7977 ± 0.0006
+0.7572 ± 0.0007
+0.6757 ± 0.0007
+0.6304 ± 0.0008
+N = 10
+0.7813 ± 0.0006
+0.7271 ± 0.0007
+0.6776 ± 0.0007
+0.6308 ± 0.0008
+N = 6
+0.8005 ± 0.0006
+0.7412 ± 0.0007
+0.6840 ± 0.0007
+0.6330 ± 0.0008
+⟨YN−3YN−2⟩
+N = 14
+0.5287 ± 0.0008
+0.5218 ± 0.0009
+0.5712 ± 0.0008
+0.5548 ± 0.0008
+N = 10
+0.5375 ± 0.0008
+0.5728 ± 0.0008
+0.5687 ± 0.0008
+0.5541 ± 0.0008
+N = 6
+0.5334 ± 0.0008
+0.5531 ± 0.0008
+0.5579 ± 0.0008
+0.5502 ± 0.0008
+⟨YN−2YN−1⟩
+N = 14
+0.7958 ± 0.0006
+0.7578 ± 0.0006
+0.6763 ± 0.0007
+0.6302 ± 0.0008
+N = 10
+0.7811 ± 0.0006
+0.7270 ± 0.0007
+0.6773 ± 0.0008
+0.6301 ± 0.0008
+N = 6
+0.8018 ± 0.0006
+0.7427 ± 0.0007
+0.6849 ± 0.0007
+0.6319 ± 0.0008
+⟨ZN−3ZN−2⟩
+N = 14 −0.2173 ± 0.0010 −0.2426 ± 0.0010 −0.4943 ± 0.0009 −0.5658 ± 0.0008
+N = 10 −0.2026 ± 0.0010 −0.4092 ± 0.0009 −0.4957 ± 0.0009 −0.5663 ± 0.0008
+N = 6 −0.2764 ± 0.0010 −0.3749 ± 0.0009 −0.4724 ± 0.0009 −0.5569 ± 0.0008
+⟨ZN−2ZN−1⟩
+N = 14 −0.6429 ± 0.0008 −0.6646 ± 0.0007 −0.7268 ± 0.0007 −0.7637 ± 0.0006
+N = 10 −0.5915 ± 0.0008 −0.6934 ± 0.0007 −0.7283 ± 0.0006 −0.7661 ± 0.0006
+N = 6 −0.7083 ± 0.0007 −0.7196 ± 0.0007 −0.7413 ± 0.0007 −0.7723 ± 0.0006
+TABLE I: Expectation values of operators evaluated by 106 sampling data with a simulator. ∆(g) and the lattice spacing a are
+fixed to −0.2 and 0.2. By substituting those values into Bob’s expected energy ⟨HnB⟩, one can recover the result of Fig. 2 (F).
+FIG. 3: Teleported energy for a different system size N = 6, 10, 14.
+e−i α
+2 (XnXn+1+YnYn+1)=
+H
+R(n)
+Z (α)
+H
+R(n+1)
+Z
+(−α)
+e−i α
+2 ZnZn+1=
+R(n+1)
+Z
+(α)
+A Hamiltonian HZ = �N
+n=1 anZn containing only local
+Zns is implemented by
+e−iαHZ =
+N
+�
+n=1
+R(n)
+Z (2anα),
+(A4)
+
+Teleported energy N = 6
+Teleported energy N = 10
+Teleported energy N = 14
+0.0000
+0.0000
+0.00000
+0.0005
+0.00005
+0.0002
+0.0010
+0.00010
+0.0015
+0.0004
+338632
+0.00015
+0.0020
+E 10 -
+E
+- 6'0
+- 8.0
+0.0025
+0.0006
+388842
+0.00020
+0.0030
+20
+0.00025
+0.0008
+0.3
+0.0035
+1
+0.00030
+0.07
+L00
+0.0 -
+0987654m21
+0987654321
+(g)
+(g)
+A(g)7
+Now we describe the measurement of quantum opera-
+tors. Measurement of Zn is done by the following circuit
+The output of the measurement is a bit string bn ∈
+{0, 1}.
+Since the eigenvalues of Z are −1, 1, we con-
+vert the bit string into 1 − 2bn. Let nshot be the num-
+ber of repetitions of the circuit, and countsb0···bN−1 be
+the number of times b0, · · · , bN−1 are detected. There-
+fore
+countsb0···bN−1
+nshots
+is the probability that a bit string of
+b0 · · · bN−1 is obtained. Then the expectation value of Zn
+is computed by the formula
+⟨Zn⟩ =
+�
+b0,··· ,bN−1
+(1 − 2bn)countsb0···bN−1
+nshots
+.
+(A5)
+Measurement of XnXn+1 is done by the following cir-
+cuit
+H
+H
+Note that H maps |0⟩ , |1⟩ to |+⟩ , |−⟩, which are eigen-
+vectors of X. The output is again a bit string bnbn+1 ∈
+{00, 01, 10, 11}. They are converted to the eigenvalues of
+XnXn+1 by (1 − 2bn)(1 − 2bn+1). Then the expectation
+value of XnXn+1 is computed by the formula
+⟨XnXn+1⟩ =
+�
+bn,bn+1
+(1 − 2bn)(1 − 2bn+1)countsbnbn+1
+nshots
+.
+(A6)
+Similarly, measurement of YnYn+1 is possible by the
+following circuit
+S†
+H
+S†
+H
+in such a way that
+⟨YnYn+1⟩ =
+�
+bn,bn+1
+(1 − 2bn)(1 − 2bn+1)countsbnbn+1
+nshots
+.
+(A7)
+Now let us discuss how to compute the expectation
+energy of the massive Thirring model (eqs. (2) and (3)).
+Once we obatin ⟨Zn⟩, ⟨ZnZn+1⟩, ⟨XnXn+1⟩, ⟨YnYn+1⟩ as
+shown in Table I, then we can compute the expectation
+value of teleported energy by
+HTh =
+N−2
+�
+n=0
+Hn
+⟨Hn⟩ = ⟨H±,n⟩ + ⟨Hm,n⟩ + ⟨HZZ,n⟩ + ϵ.
+(A8)
+For example Bob’s local Hamiltonian HnB is given as
+follows
+HnB =H±,B + Hm,B + HZZ,B + ϵn
+H±,B = − 1
+4a[XN−2(XN−3 + XN−1)
++ YN−2(YN−3 + YN−1)],
+Hm,B =m
+2 (−1)N−1ZN−2
+HZZ,B =∆(g)
+8
+ZN−2(ZN−3 + ZN−1) + ∆(g)
+2a ZN−2.
+(A9)
+Now it will be clear that [HTh, σnB] = [HnB, σnB] since
+σnB is a local operator. The teleported energy ⟨HnB⟩ to
+Bob’s local system is shown in Fig. 2 (F) in the main
+text. Bob will receive −⟨HnB⟩ through his measurement
+device.
+Appendix B: Sine-Gordon Model and Thirring
+Model
+The sine-Gordon model is a theory of a single scalar
+field φ(x) whose Lagrangian is written as
+LSG = 1
+2∂µφ(x)∂µφ(x) + α
+β2 (cos(βφ(x)) − 1),
+(B1)
+where α and β are real positive parameters. This model
+is invariant under
+φ(x) → φ′(x) = φ(x) + 2πn
+β , n ∈ Z.
+(B2)
+The Thirring model is a theory of a self–coupled Dirac
+field
+LTh = ψ(iγµ∂µ − m)ψ − g
+2ψγµψψγµψ,
+(B3)
+where m is the fermion mass, g is the dimensionless four-
+fermion coupling constant and ψ = ψ(x) is a spinor
+field with two components ψ1(x) and ψ2(x). It is widely
+known that the massive Thirring model is dual to the
+sine-Gordon model and the classical two-dimensional XY
+model [35, 38]. For example a Kosterlitz-Thouless phase
+transition at T ∼ Kπ/2 in the XY model corresponds
+to a critical point g ∼ −π/2, called Coleman’s instability
+point, in the Thirring model. They are also related with
+a critical point at t ∼ 8π in the sine-Gordon model.
+The model is obviously invariant under U(1)V group
+ψ(x) → ψ′(x) = eiαV ψ(x).
+(B4)
+
+8
+The corresponding conserved current is
+jµ = ψγµψ.
+(B5)
+If m = 0, the Thirring model is also invariant under
+the chiral group U(1)V × U(1)A
+ψ(x) → ψ′(x) = eiαV ψ(x)
+ψ(x) → ψ′(x) = eiαAγ5ψ(x),
+(B6)
+under which the pseudo-vector current
+j5
+µ = ϵµνjν (ϵ01 = 1 − ϵ10 = 1)
+(B7)
+also conserves ∂µj5
+µ = 0.
+Bosonization of the interaction term is described by
+LSG
+int (x) = α
+β2 cos(βφ(x)) =
+α
+2β2 (A+(x) + A−(x))
+LTh
+int(x) = −mψ(x)ψ(x) = −m(σ+(x) + σ−(x)),
+(B8)
+where A±(x) = e±iβφ(x) and σ±(x) = 1
+2ψ(x)(1±γ5)ψ(x)
+According to the Coleman’s prescription, those two
+models are related with
+ψ(x)ψ(x) = −M cos(βφ(x)) + m
+g
+iψ(x)γ5ψ(x) = −M sin(βφ(x))
+g
+mψ(x)
+�1 ∓ γ5
+2
+�
+ψ(x) = − α
+2β2 e±iβφ(x) + m2
+2g .
+(B9)
+The massive Thirring model and the sine-Gordon model
+are equivalent only if β2 < 8π.
+Appendix C: Hamiltonian Formalism of Thirring
+Model
+The Hamiltonian including quantum effects in the
+energy-momentum tensor at the operator level can be
+written as
+HTh =
+�
+dx
+�
+− iZψ(g) ¯ψγ1∂1ψ + m ¯ψψ
++ g
+4( ¯ψγ0ψ)2 − ˜g
+2( ¯ψγ1ψ)2
+�
+,
+(C1)
+where Zψ(g) is the wavefunction renormalization con-
+stant and ˜g = g
+2
+�
+1 + 2g
+π
+�−1 [48–50].
+ψ1(x) →
+1
+√aχ2n, ψ2(x) →
+1
+√aχ2n+1
+(C2)
+We address this Hamiltonian on a lattice one-dimensional
+lattice with open boundary condition.
+Then the dis-
+cretized Hamiltonian is
+HTh = − i
+2aZψ(g)
+N−1
+�
+j=1
+�
+χ†
+jχj+1 + h.c.
+�
++ m
+N
+�
+j=1
+(−1)nnj + g
+2a
+N−1
+�
+j=1
+njnj+1,
+(C3)
+where nj = χ†
+jχj is the fermion number operator.
+For the purpose of quantum computation, we convert
+the Hamiltonian into the corresponding spin Hamilto-
+nian. We will work with
+γ0 = X, γ1 = −iY, γ5 = −iXY
+(C4)
+We first write down the discrete Hamiltonian by
+means of staggered fermion.
+Based on the established
+method [39, 51], we can replace Zψ(g) and g with
+Zψ(g) →
+π − g
+π sin
+� π−g
+2
+�
+g → 2(π − g)
+π
+cot
+�π − g
+2
+�
+(C5)
+As a result we obtain the Hamiltonian of the massive
+Thirring model we used in this work
+HTh =
+2γ
+aπ sin(γ)H,
+(C6)
+where the explicit representation of H is
+H = − i
+2a
+N−1
+�
+n=1
+(χ†
+nχn+1 − χ†
+n+1χn)
++ m
+N
+�
+n=1
+(−1)nχ†
+nχn + ∆(g)
+a
+N−1
+�
+n=1
+χ†
+nχnχ†
+n+1χn+1,
+(C7)
+where ∆(g) = cos
+� π−g
+2
+�
+[38–43]
+The Jordan-Wigner transformation maps fermionic op-
+erators to spin operators [52]. It is commonly used in the
+study of quantum many-body systems and plays a key
+role in the study of strongly correlated electron systems.
+After the Jordan-Winger transformation
+χn = Xn − iYn
+2
+n−1
+�
+m=1
+(−iZm),
+χ†
+n = Xn + iYn
+2
+n−1
+�
+i=m
+(iZm),
+(C8)
+we arrive at the spin representation of the massive
+Thirring model
+Hspin = − 1
+4a
+N−1
+�
+n=1
+(XnXn+1 + YnYn+1)
++ m
+2
+N
+�
+n=1
+(−1)n(Zn + 1)
++ ∆(g)
+a
+N−1
+�
+n=1
+�Zn + 1
+2
+� �Zn+1 + 1
+2
+�
+.
+(C9)
+Those correspondences are summarized in the follow-
+ing dictionary.
+
+9
+Dirac
+Staggerd
+Pauli
+ψψ
+(−1)n
+a
+χ†
+nχn
+(−1)n
+2a
+(Zn + 1)
+ψγ0ψ
+1
+aχ†
+nχn
+1
+2a(Zn + 1)
+ψγ1ψ
+1
+2a(χ†
+nχn+1 + χ†
+n+1χn)
+1
+4a(XnYn+1 − YnXn+1)
+ψγ5ψ
+(−1)n
+2a
+(χ†
+nχn+1 − χ†
+n+1χn) − i(−1)n
+4a
+(XnXn+1 + YnYn+1)
+ψγ1∂1ψ −
+1
+2a2 (χ†
+nχn+1 − χ†
+n+1χn)
+−
+i
+4a2 (XnXn+1 + YnYn+1)
+TABLE II: Correspondence among the three different repre-
+sentations of fermionic fields.
+Appendix D: Adiabatic Ground State Preparation
+1.
+General Remark
+One of the most difficult problems in performing quan-
+tum computation is how to design an initial state since
+it can affect the overall outcome of the computation. A
+good initial state is one that is easy to prepare and that
+allows for efficient quantum gates and measurements to
+be applied to it. One common initial state used in quan-
+tum computation is the ground state of a Hamiltonian.
+However a difficulty in preparing a good initial state
+is the quality of the qubits and the limitation of gate
+depth. If the qubits are prone to errors, initializing them
+with high fidelity is very challenging. A commonly used
+method to solve such problems is the adiabatic quan-
+tum computation (AQC), which is a model of quantum
+computation based on the adiabatic theorem of quantum
+mechanics. It can be used to solve certain optimization
+problems by encoding the problem into the energy lev-
+els of a time-dependent Hamiltonian, and then slowly
+varying the Hamiltonian over time to drive the system
+through a series of states that correspond to the solu-
+tion of the problem.
+One such quantum computation
+method is quantum annealing, which is a heuristic opti-
+mization method that is used to find the global minimum
+of a given spin Hamiltonian [53] and practically used for
+various purposes [54–56].
+A well-known example is the Quantum Adiabatic Al-
+gorithm (QAA), it is based on the Hamiltonian of the
+problem to be solved, the initial Hamiltonian is chosen
+such that its ground state is easy to prepare and known
+and the final Hamiltonian is such that its ground state is
+the solution of the problem. The system is then slowly
+evolved to the final Hamiltonian and the final state is the
+ground state of the final Hamiltonian. One of the main
+requirements for the QAA to work correctly is that the
+adiabatic condition holds, this condition states that the
+adiabatic evolution should be slow enough so that the
+system remains in the ground state at all times, if this
+condition is not met the system may end up in excited
+state and the final solution will not be correct. One chal-
+lenge that arises during AQC is the trade-off between
+the speed of the evolution and the accuracy of the fi-
+nal solution. The adiabatic theorem states that the sys-
+tem remains in the ground state if the evolution is slow
+enough. However, if the evolution is too slow, the com-
+putation may become infeasible due to the long running
+time. On the other hand, if the evolution is too fast, the
+system may not stay in the ground state and the final
+solution may be inaccurate.
+Another challenge is the presence of a phase transition,
+which is a point where the ground state of the Hamilto-
+nian changes. At these points, the energy gap between
+the ground state and the first excited state can become
+small, which makes it more difficult to maintain the sys-
+tem in the ground state. This can also lead to critical
+slowing down, in which the system slows down drastically
+as it evolves through the phase transition point. In the
+context of adiabatic quantum computation, a phase dia-
+gram could be used to understand how the performance
+of the computation changes as a function of parameters
+such as the duration of the computation or the strength
+of the interactions between the quantum particles.
+Additionally, AQC is also sensitive to noise and errors
+in the system, as small fluctuations can cause the system
+to leave the ground state.
+This can lead to incorrect
+solutions or a loss of quantum coherence.
+2.
+Method
+Here we describe a way to prepare the ground state
+by a quantum computer. For this, we use the method
+so-called adiabatic state preparation.
+The essence of
+the method for studying phase transitions with adiabatic
+quantum calculations has already been established in
+many cases with quantum annealing [53, 57, 58]. We first
+prepare a state |vac0⟩ which is a known ground state of an
+initial Hamiltonian Hinitial. We use the time-dependent
+Hamiltonian H(t) which is equal to the target Hamilto-
+nian Htarget at the end of computation:
+H(t) = tHtarget + (1 − t)Hinitial.
+(D1)
+By adiabatically changing parameter t, we expect to ob-
+tain the ground state of the target Hamiltonian
+|vac⟩ = lim
+T →∞ T exp
+�
+−i
+� T
+0
+dtH(t)
+�
+|vac⟩0 .
+(D2)
+We have two reasonable choices of initial Hamiltonian.
+The first one is
+Hm = m
+2
+N
+�
+n=1
+(−1)nZn,
+m > 0
+(D3)
+whose
+ground
+state
+is
+|vac⟩0
+=
+|0101 · · · 01⟩
+=
+�N/2
+i=1 X2i |0 · · · 0⟩, with Z |0⟩ = |0⟩ and Z |1⟩ = − |1⟩.
+The other choice is
+Hg = ∆(g)
+a
+N−1
+�
+n=1
+�Zn + 1
+2
+� �Zn+1 + 1
+2
+�
+,
+∆(g) < 0
+(D4)
+
+10
+whose ground state is |vac⟩0 = |0 · · · 0⟩. Each of those
+states can be easily prepared but we will choose one of
+them so that any possible errors including become small.
+The matrix representation of the chiral condensate is
+given as
+N
+�
+n=1
+¯ψψn = 1
+2a
+N
+�
+n=1
+(−1)nZn.
+(D5)
+Its vacuum expectation value is − N
+2a with the ground
+state |0101 · · · 01⟩ of an initial Hamiltonian (D3)
+⟨01 · · · 01| ¯ψψ |01 · · · 01⟩ = − N
+2a
+(D6)
+Those two states corresponds to the ground states of the
+Hamiltonian (2) in the limit: limm→∞ 1
+m(Hspin − Hm) =
+0 and lim|∆(g)|→∞
+1
+|∆(g)|(Hspin − Hg) = 0, respectively.
+We chose an initial state and parameters are chosen so
+that the effect of the phase transition on the probability
+distribution and statistical errors are relatively small. In
+the rest of this section, we describe how we manage an
+adiabatic state preparation.
+FIG. 4: Chiral condensate
+In the adiabatic process, we decompose the time-
+dependent Hamiltonian H(t) = Hstatic +Hdynamic(t) into
+the static part and the dynamic part. For example, if the
+Hamiltonian (D3) is used for the initial Hamiltonian, the
+static part is
+Hstatic = m
+2
+N
+�
+n=1
+(−1)nZn
+(D7)
+and the dynamic part is
+Hdynamic(t) = − w(t)
+4a
+N−1
+�
+n=1
+(XnXn+1 + YnYn+1)
++ g(t)
+a
+N−1
+�
+n=1
+�Zn + 1
+2
+� �Zn+1 + 1
+2
+�
+(D8)
+with
+∆(g)(t) = t
+T
+�∆(gtarget)t
+T
++ ∆(g0)
+�
+1 − t
+T
+��
+,
+w(t) = t
+T ,
+(D9)
+where ∆(g0) is an initial coupling that should be cho-
+sen in a way that statistical errors become small, and
+∆(gtarget) is the target coupling parameter.
+3.
+Study on Phase Diagram by Adiabatic State
+Preparation
+It will be interesting to explore the phase diagram of
+the massive Thirring model by the adiabatic algorithm,
+in particular for the purpose of quantum computation
+since we have to prepare the ground state anyway. By
+using two different initial Hamiltonians above, one can
+draw the phase diagram of the massive Thirring model.
+In Fig. 5, we illustrate a path we take for the adiabatic
+ground state preparation. Suppose we fix an initial mass
+m and ∆(g) to m = m0, ∆(g) = 0, which corresponds
+to the start of
+1○. One can use the initial ground state
+|vac⟩0 = |0101 · · · 01⟩ of the Hamiltonian eq.(D3).
+As
+long as the target mass mtarget and ∆(gtarget) remain in
+the same phase of the initial state, one can efficiently
+obtain the value without changing paths.
+However, if
+there is a critical point of a 1st order phase transition on
+the dashed line, one needs to change paths to avoid a 1st
+order phase transition. For this, we draw an additional
+path consisting of
+2○,
+3○, and
+4○. In the end of track
+2○, one reaches to a point close to m = 0, ∆(g) < 0,
+to avoid a strong effect of first order phase transition.
+Throughout the track
+3○, ∆(g) gradually approaches to
+the target ∆(gtarget) and the adiabatic process should
+remain in (or at least close to) the ground state of
+2○.
+
+Chiral Condensate
+0
+2
+-4
+19882
+-6
+8-
+-10
+0.0
+6
+m
+N
+1 0
+Q
+0
+0
+i0
+Q
+Q
+Q
+Q
+Q
+Q
+Q
+Q
+(g)11
+FIG. 5: Schematic design of path to compute the vacuum ex-
+pectation value of chiral condensate by adiabatic state prepa-
+ration. Paths should be selected so that results are not af-
+fected by phase transitions.
+For quantum simulation, we decompose the time-
+dependent Hamiltonian (D1) as
+H(t) = H±(t) + HZZ(t) + HZ(t)
+H±(t) = −w(t)
+4a
+N−1
+�
+n=1
+(XnXn+1 + YnYn+1)
+HZZ(t) = ∆(g)(t)
+4a
+N
+�
+n=1
+ZnZn+1
+HZ(t) = m(t)
+2
+N
+�
+n=1
+(−1)nZn + ∆(g)(t)
+4
+N−1
+�
+n=1
+(Zn + Zn+1),
+(D10)
+where
+time-dependent
+coefficients
+w(t), m(t), ∆(g)(t)
+should be defined so that they agree with a path of com-
+putation drawn in Fig. 5. The scalar term (N−1)∆(g)(t)
+4
+dose not contribute to time-evolution, thereby we neglect
+it. For example, in this study we use the following time-
+dependent parameters:
+w(t) =
+�
+t/T
+1○
+1
+2○, 3○, 4○
+(D11)
+m(t) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+m0
+1○
+m1 t
+T + m0
+�
+1 − t
+T
+�
+2○
+m1
+3○
+m2 t
+T + m1
+�
+1 − t
+T
+�
+4○
+(D12)
+∆(g)(t) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+∆(g0) t2
+T 2
+1○
+∆(g0)
+2○
+∆(g2) t
+T + ∆(g1)
+�
+1 − t
+T
+�
+3○
+∆(g2)
+4○
+(D13)
+Let T be the computational time, M be a large positive
+integer (Trotter number), δt = T/M be a time-step. We
+implement the integral (D2) in the following way. First,
+the unitary time-evolution for k time steps can be given
+U(kδt) = e−iδtH±(kδt)e−iδtHZZ(kδt)e−iδtHZ(kδt)
+(D14)
+and the state at t = kδt (k = 1, 2, · · · , T) is
+|vac(kδt)⟩ = U(kδt) · · · U(2δt)U(δt) |vac⟩0 .
+(D15)
+The quantum circuits we implemented are shown in
+Sec. A. Therefore at the end of computation (t = T =
+Mδt), the state evolves into
+|vac⟩ =
+M
+�
+k=1
+e−iδtH±(kδt)e−iδtHZZ(kδt)e−iδtHZ(kδt) |vac⟩0 .
+(D16)
+Throughout this work we chose N = 10, T = 100, M =
+1000, δt = T/M = 0.1 and fix the lattice spacing a to 1.
+As a demonstration, we design a path in Fig. 5, which
+consists of four tracks whose initial and target values are
+given in Table III.
+1○
+2○
+3○
+4○
+initial (∆(g), m)
+(0,0.7)
+(-0.1,0.7) (-0.1,0)
+(-0.6,0)
+target (∆(g), m) (-0.1,0.7)
+(-0.1,0)
+(-0.6,0) (-0.6,0.3)
+TABLE III: Initial and target values for adiabatic state prepa-
+ration. Each number corresponds to the number in Fig. 5,
+respectively.
+FIG. 6: Time-evolution of chiral condensate with the ground
+state eq. (D15) prepared by adiabatic state preparation. Path
+1 consists of paths
+1○ · · ·
+4○, and Path 2 is the dashed line
+in Fig. 5. For each path, the average of 10000 samplings is
+shown.
+For comparison, let us consider a linear path connect-
+ing (m1, ∆(g1)) and (m2, ∆(g2)):
+∆(g)(m) = ∆(g2) − ∆(g1)
+m2 − m1
+(m − m1) + ∆(g1).
+(D17)
+
+Time-evolution of chiral condensate
+0
+Path 1
+Path 2
+-1
+-2
+2
+-3
+-4
+(1
+-5
+0
+50
+100
+150
+200
+250
+300
+350
+400
+112
+As a function of t, m and ∆(g) change as follows
+m(t) = m1
+�
+1 − t
+T
+�
++ m2
+t
+T
+∆(g)(t) = ∆(g1)
+�
+1 − t
+T
+�
++ ∆(g2) t
+T
+(D18)
+The results are shown in Fig. 6. It is clear that results
+of Path 1 is consistent with Fig. 4. Path 2 continues to
+take the same values as Path 1 when there is no phase
+transition, while after crossing the phase transition point
+it deviates significantly from the exact theoretical values.
+Note that the time steps for Path 1 and Path 2 are the
+same.
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf,len=910
+page_content='Criticality of quantum energy teleportation at phase transition points in quantum field theory Kazuki Ikeda1, 2, ∗ 1Co-design Center for Quantum Advantage, Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA 2Center for Nuclear Theory, Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA Quantum field theory can be a new medium for communication through quantum energy telepor- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' We performed a demonstration of quantum energy teleportation with a relativistic fermionic field theory of self-coupled fermions, called the massive Thirring model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Our results reveal that there is a close relation between the amount of energy teleported and the phase diagram of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' In particular, it is shown that the teleported energy peaks near the phase transition points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The results provide new implications for phase diagrams of field theory in terms of quantum communication and quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' INTRODUCTION Quantum field theory (QFT) has been quite success- ful in explaining quantum many-body systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' From condensed matter physics, such as superconductors and topological insulators, to the Standard Model of elemen- tary particles as a low-energy effective theory of high- energy physics, QFT can explain a wide variety of ex- perimental results with extremely high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The approach to non-perturbative phenomena is a remaining challenge for QFT, which has been explored by various methods such as first-principles calculations and lattice QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' In addition, with the advent of quantum comput- ers, we are able to perform real-time non-perturbative quantum simulations of many-body systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' One of the key challenges in studying QFT is the complexity of the calculations involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Simulating these systems using classical computers can be computationally expensive, as the complexity of the calculations increases rapidly with the size of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Quantum computers, on the other hand, have the potential to perform these simulations much more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' In addition to this, the develop- ment of quantum algorithms and quantum computers has greatly contributed to the fundamental understanding of quantum mechanics, including the control of quantum states and the measurement of quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' As such, understanding the behavior of quantum many-body systems through quantum simulations has been the primary focus of recent cross-disciplinary in- terest in physics and computer science, but for physics, the connection to quantum science and technology is not limited to quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Regarding the con- nection between QFT and quantum information theory, there are active studies on entanglement entropy and black holes [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' These studies are mainly concerned with high-energy physics at the Planck scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' While such attempts have been extremely successful, new efforts ∗kazuki7131@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='com to reveal the nature of quantum systems and spacetime through measurement have been active in recent years in a wide range of fields, including high-energy physics, condensed matter physics and quantum computation [4– 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Quantum energy teleportation (QET) is a protocol for the study of local energies that takes advantage of the en- tanglement nature of the ground state of quantum many- body systems [15–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Just as quantum teleportation can transfer quantum states to remote locations [22–26], it is expected that QET can transfer energy to remote locations using local operation and classical communica- tion (LOCC) only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The role of QET in physics and infor- mation engineering is largely unexplored, as the theory has not received much attention for long time since it was proposed about 15 years ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' An interesting prop- erty of QET is that multiple people in different locations, who share the same ground state initially, can simultane- ously lower the energy of their local systems by applying conditional operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' This is only possible when the sender and receivers of the energy conduct the appro- priate LOCC, and cannot be obtained by any unitary operation or random conditional operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Therefore QET will not only help to enhance our understanding of fundamental issues in quantum statistical mechanics, condensed matter physics, and high-energy physics but will also provide interesting perspectives for engineering applications of quantum computation and quantum com- munication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The purpose of this paper is to investigate the role and properties of QET in field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' From the viewpoint of quantum computer applications, we simulate QET using the massive Thirring model (low dimensional quantum electrodynamics (QED)), which is one of the most widely used (1+1) dimensional models of QFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' First, we esti- mate the phase diagram of the massive Thrring model using entanglement entropy and chiral condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The main result of this paper is the identification of a sharp peak in teleported energy near the phase transition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' We also analyze the time-evolution of the entanglement entropy difference ∆SAB using Alice’s post-measurement arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='11712v1 [hep-th] 27 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 1: Protocol of quantum energy teleportation [Left] and the corresponding quantum circuits [Right].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' First, Alice measures her local operator XnA and tells her result (µ ∈ {+1, −1}) to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' At this point, Alice’s local energy is excited EnA > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Then, to obtain energy, Bob applies conditional operation UnB(µ) to his local qubit and measures the corresponding terms of his local Hamiltonian HnB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Statistically he will observe ⟨HnB⟩ = Tr[ρQETHnB] < 0 and gain EnB = −⟨HnB⟩ through his measurement device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' state and show numerically how the entanglement en- tropy lost in Alice’s measurement is recovered over time due to particle-particle interactions in the system if Bob does nothing after Alice’s measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Some of the results in this paper are based on simulations of quan- tum gate operations using qasm simulator provided by IBM, and we confirm that all of these results are fully consistent with those obtained by exact diagonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' These results provide new insights into local operations of quantum fields based on remote communication and non-trivial energy flow mediated by many-body quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' LOW DIMENSIONAL QFT The (1+1) dimensional QFTs are of significant in- terest since they are simpler and more tractable than higher-dimensional QFT, and they have rich mathemat- ical structures that have been studied extensively from various motivations, including condensed matter physics, high energy physics, statistical mechanics and mathemat- ical physics [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Some of the models have a number of interesting properties, including confinement and the chi- ral anomaly therefore they are useful toy models of QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Typical models preferred in studies of (1+1) dimensional QFTs are the Thirring model and the Schwinger model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' In particular, they are attractive models in terms of quan- tum simulation and quantum computation [28–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The Thirring model is a simplified version of quantum electrodynamics (QED) in (1+1) dimensions, which was introduced by Walter Thirring in 1958 [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' It is a the- ory of a self–coupled Dirac field, and it can be used to describe a variety of physical systems, such as supercon- ductors [35], statistical mechanics, high energy physics and mathematical physics [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' While the Thirring model and the Schwinger model are models for fermions, there is a significant (1+1) dimen- sional model for bosons, called sine-Gordon model, which is of significant interest in theoretical physics due to its integrability, soliton solutions, and relations to other models such as the Thirring model, massive Schwinger model, and to the XY -model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The sine-Gordon model is a (1+1) dimensional field theory that is described by the sine-Gordon equation, which is a nonlinear partial differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The soliton solution of this model describes a kink or anti-kink solution which is a topolog- ical mode in the field that can be interpreted as particle like excitation [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The topological nature of the solitons ensures the stability and the solitons retain their shape even during collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' It has been widely known that both models are related by the bosonization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' By representing the fermionic fields in terms of bosonic fields, the bosonized version of the Thirring model becomes the sine-Gordon model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' This is known as the S-duality between the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' More detailed theoretical descriptions are given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Throughout this work, we consider the massive Thirring model, whose Lagrangian is LTh = ψ(iγµ∂µ − m)ψ − g 2ψγµψψγµψ, (1) where m is the fermion mass, g is the dimensionless four- fermion coupling constant and ψ = ψ(x) is a spinor field with two components ψ1(x) and ψ2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' It is widely known that the massive Thirring model is dual to the sine-Gordon model and the classical two-dimensional XY model [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' For example, a Kosterlitz-Thouless phase transition at T ∼ Kπ/2 in the XY model corresponds to a critical point g ∼ −π/2, called Coleman’s instability point, in the Thirring model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' They are also related with a critical point at t ∼ 8π in the sine-Gordon model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' It turns out that the spin representation of the massive (+1) D(-1) 0(+1) 0-1) (+1) 1)3 Thirring model is HTh = − 1 4a N−2 � n=0 (XnXn+1 + YnYn+1) + m 2 N � n=0 (−1)n+1Zn + ∆(g) a N−2 � n=0 �Zn + 1 2 � �Zn+1 + 1 2 � , (2) where ∆(g) = cos � π−g 2 � , a is the lattice spacing [38–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The theoretical background of the lattice Hamiltonian is described in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' SIMULATION OF QUANTUM ENERGY TELEPORTATION To facilitate clarity of results, we add a constant ϵi to every local Hamiltonian of the Thirring model HTh = � n Hn (3) where Hn is the local Hamiltonian including the nearest neighbor interactions and each ϵn should be chosen in such a way that ⟨g| HTh |g⟩ = ⟨g| Hn |g⟩ = 0, ∀i ∈ E (4) where |g⟩ is the ground state of the total Hamiltonian HTh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Note that, in general, |g⟩ is not the ground state of local Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The explicit form of Bob’s local Hamilto- nian and the details of computation are given in Sup- plemental Information (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (A9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' It is important that non-trivial local manipulations, including measurement of the ground state, yield excited states and thus increase the energy expectation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The increase in energy is supplied by the experimental apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Moreover, our ground state |g⟩ is an entangled state in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The QET protocol is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' First, Alice mea- sures her Pauli operator σnA by PnA(µ) = 1 2(1 + µσnA) and obtains either µ = −1 or +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Local measurement of the quantum state at a subsystem A destroys this ground state entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' At the same time, energy EA from the device making the measurement is injected into the entire system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The injected energy EA is local- ized around the subsystem A in the very early stages of time-evolution, however, it is not possible for Alice to ex- tract EA from the system by her operations alone at nA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' This is because information about EA is also stored in remote locations other than nA due to the entanglement that exists prior to the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' In other words, Al- ice’s energy EA can be partially extracted at any location other than nA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Now let us consider taking advantage of the quantum many-body nature of the quantum many- body system to extract energy from a different location other than nA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' This can be accomplished by LOCC, as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Via a classical channel, Alice sends her measurement result µ to Bob, who applies an operation UnB(µ) to his qubit and measures his local operators XnB, YnB, ZnB independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The density matrix ρQET after Bob oper- ates UnB(µ) to PnA(µ) |g⟩ is where ρQET is ρQET = � µ∈{−1,1} UnB(µ)PnA(µ) |g⟩ ⟨g| PnA(µ)U † nB(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (5) Using ρQET, the expected local energy at Bob’s local system is evaluated as ⟨EnB⟩ = Tr[ρQETHnB], which is negative in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Due to the conservation of en- ergy, EB = −⟨EnB⟩(> 0) is extracted from the system by the device that operates UnB(µ) [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' In this way, Alice and Bob can transfer the energy of the quantum system only by operations on their own local system and classical communication (LOCC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Those are summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' It should be noted that the Thirring model is a rela- tivistic field theory in performing QET, which could be a problem if the particle is massless since the speed of clas- sical communication does not exceed the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' We will consider a massive particle and assume that Bob can receive energy faster than the time evolution rate of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' In what follows we give the details about the operations of Alice and Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' We define UnB(µ) by UnB(µ) = cos θI − iµ sin θσnB, (6) where θ obeys cos(2θ) = ξ � ξ2 + η2 (7) sin(2θ) = − η � ξ2 + η2 (8) where ξ = ⟨g| σnBHσnB |g⟩ (9) η = ⟨g| σnA ˙σnB |g⟩ (10) with ˙ σnB = i[H, σnB].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The local Hamiltonian should be chosen so that [H, σnB] = [HnB, σnB].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The average quan- tum state ρQET is obtained after Bob operates UnB(µ) to PnA(µ) |g⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Then the average energy Bob measures is ⟨EnB⟩ = Tr[ρQETHnB] = 1 2 � ξ − � ξ2 + η2 � , (11) which is negative if η ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' If there is no energy dis- sipation, the positive energy of −⟨EnB⟩ is transferred to Bob’s device after the measurement due to energy conser- vation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Based on the quantum circuit in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 1, we per- formed a quantum simulation of QET for N = 6, 10, 14 at ∆(g) = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='2, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='2 and results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 2 (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Dashed lines correspond to exact results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 2: (A): Heat map of entanglement entropy at N = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The Thirring model has three distinct phases, which can be clearly read off the diagram at N = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (B): Heat map of entanglement entropy difference ∆SAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (C): Heat map of teleported energy ⟨HnB⟩ at N = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' It is crucial that the value of the teleported energy peaks at the phase transition points, showing a clear correspondence to the phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (D) and (E): Time-evolution of entanglement entropy difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' This is due to the natural time evolution of the system, as seen when Bob does not perform any operations on his system after Alice’s local operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Decreasing 1− ∆SAB SAB in the early stages of time evolution means that entanglements broken by Alice’s observations are recreated by the interactions in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (F): Simulation results of expected energy of Bob’s local system obtained by QET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Error bars indicate statistical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' In this work, we put Alice and Bob near the boundary nA = 1, nB = N − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Bob’s local energy can be calcu- lated by the explicit form of his local Hamiltonian given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='(A9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The simulation results are given in Table I in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' A of Suplimental Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' We next study the entanglement entropy between two subsystems A, B such that A ∩ B = ∅, A ∪ B = {1, 2, · · · , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Let ρ be a density operator on the entire system A ∪ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Then the entanglement entropy between A and B are defined by S(ρ) = −TrA(ρA log ρA), (12) where ρA is defined by tracing out the Hilbert space of B: ρA = TrBρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' In this study we choose ρ as the ground state |g⟩ of the Hamiltonian (ρ = |g⟩ ⟨g|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 2 (A) shows the entanglement entropy between the left and right half subsystems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=', A = {0, · · · , N 2 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The figure exhibits sharp peaks at the critical points of phase transitions that can be understood by the phase diagram of chiral condensate in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 4 in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 2 (C) shows the teleported energy Tr[ρQETHnB] to Bob’s local system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' It is significant that the teleported energy is enhanced along the critical points of the phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' This will be understood by a relation between Bob’s energy Tr[ρQETHnB] and the entanglement entropy difference ∆SSA, which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 2 (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The change in entropy before and after the measure- ment by Alice can be evaluated as follows ∆SAB = SAB − � µ pµSAB(µ) (13) where pµ is the probability distribution of µ, SAB(µ) is the entanglement entropy after the measurement, ξ = arctan � k h � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' After Alice’s post-measurement, the state is mapped to |A(µ)⟩ = 1 √pµ PnA(µ) |g⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (14) Then SAB(µ) is calculated with the density matrix |A(µ)⟩ ⟨A(µ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' As discussed in [16, 45], ∆SAB is bounded below by a function f(ξ, η) in such a way that ∆SAB ≥ f(ξ, η)EB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (15) This indicates that the transferring energy involves a commensurate consumption of entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Similar to the Maxwell Demon argument [46, 47], Bob’s conditional op- erations reduce the entropy of the local system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' If Bob Entanglement entropy Entanglement entropy difference Asaz Teleported energy 00000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0010 04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0020 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0025 02 90 t0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0030 01 @1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0035 @0 02 04 06 08 Teleported energy 0L6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='60 025 m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='55 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='50 附=2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='000 @4 9899338388090000005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='002 se 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='30 五 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='25 N=10 N=14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='005 Q5 15 25 @5 15 ON 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='6 12 16 18 t5 does nothing after Alice’s measurement, Figs 2 (D) and (E) illustrate how the entanglement entropy is recreated by the natural time evolution of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Moreover, the maximal energy that Bob would receive is bounded below by the difference in entropy: max U1(µ) EB ≥ h(ξ, η)∆SAB, (16) where h(ξ, η) is a certain function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Although it is difficult to analytically obtain the con- crete forms of functions f and g, the results of this study show that there is a clear correspondence between the en- ergy obtained by QET and the phase diagram of QFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Acknowledgement I thank Adrien Florio, David Frenklakh, Sebastian Grieninger, Fangcheng He, Masahiro Hotta, Dmitri Kharzeev, Yuta Kikuchi, Vladimir Korepin, Qiang Li, Adam Lowe, Ren´e Meyer, Shuzhe Shi, Hiroki Sukeno, Tzu-Chieh Wei, Kwangmin Yu and Ismail Zahed for fruitful communication and collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' I thank Megumi Ikeda for providing the cartoons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' I acknowl- edge the use of IBM quantum computers and simulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' I was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Department of Energy, Of- fice of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Ad- vantage (C2QA) under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='DESC0012704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Author contribution All work was performed by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Competing interests The author declares that there is no competing finan- cial interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Appendix A: Quantum Gates and Measurement The goal of this section is to describe how to compute Bob’s local energy ⟨HnB⟩ gained by quantum energy tele- portation, using the quantum circuit shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' For this we provide a self-contained description of the back- ground knowledge used in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' We use the following one-qubit operators whose matrix representa- tions are given as X = � 0 1 1 0 � , Y = � 0 −i i 0 � , Z = � 1 0 0 −1 � , S = � 1 0 0 i � , H = 1 √ 2 � 1 1 1 −1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (A1) We use |0⟩ = �1 0 � , |1⟩ = �0 1 � for the computational basis states, which are eigenstates of Z: Z |0⟩ = |0⟩ , Z |1⟩ = − |1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' We also work with another ba- sis vectors |±⟩ = |0⟩±|1⟩ √ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' They are eignestates of X: X |−⟩ = − |−⟩ , X |+⟩ = − |+⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Note that |±⟩ are created by applying H to |0⟩ and |1⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' H |0⟩ = |+⟩ , H |1⟩ = |−⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' For example, Alice finds µ = ±1 by observing the eigen- values ±1 of her local Pauli X operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The rotation of X, Y, Z is defined by RX(α) = e−i α 2 X, RY (α) = e−i α 2 Y , RZ(α) = e−i α 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (A2) We use two-qubit gate operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' In general,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' a control U operation Λ(U) is defined by Λ(U) = |0⟩ ⟨0| ⊗ I + |1⟩ ⟨1| ⊗ U (A3) and the corresponding diagram is drwan as control U= U One of the most frequently used controlled gates is a CNOT gate CNOT = Λ(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' whose diagram is especially drawn as CNOT= It is convenient to define an anti-control gate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' which is activated when the control bit is in state |0⟩: |1⟩ ⟨1|⊗I + |0⟩ ⟨0| ⊗ U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' whose diagram is drawn as Anti-control U= U = X X U With those operators,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' we can draw time evolution of XX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Y Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' ZZ type interactions of spins as 6 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5 2 ⟨ZN−2⟩ N = 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='2303 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='3459 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='4111 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5050 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0009 N = 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='2453 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='3021 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='4125 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5069 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0009 N = 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='2132 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='3245 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='4256 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5141 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0009 ⟨XN−3XN−2⟩ N = 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5281 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5211 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5710 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5550 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 N = 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5376 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5717 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5697 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5535 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 N = 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5270 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5454 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5522 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5436 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 ⟨XN−2XN−1⟩ N = 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='7977 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='7572 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='6757 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='6304 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 N = 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='7813 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='7271 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='6776 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='6308 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 N = 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='8005 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='7412 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='6840 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='6330 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 ⟨YN−3YN−2⟩ N = 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5287 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5218 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5712 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5548 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 N = 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5375 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5728 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5687 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5541 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 N = 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5334 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5531 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5579 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5502 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 ⟨YN−2YN−1⟩ N = 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='7958 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='7578 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='6763 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='6302 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 N = 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='7811 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='7270 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='6773 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='6301 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 N = 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='8018 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='7427 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='6849 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='6319 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 ⟨ZN−3ZN−2⟩ N = 14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='2173 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0010 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='2426 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0010 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='4943 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0009 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5658 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 N = 10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='2026 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0010 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='4092 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0009 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='4957 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0009 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5663 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 N = 6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='2764 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0010 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='3749 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0009 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='4724 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0009 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5569 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 ⟨ZN−2ZN−1⟩ N = 14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='6429 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='6646 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0007 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='7268 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0007 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='7637 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0006 N = 10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='5915 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='6934 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0007 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='7283 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0006 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='7661 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0006 N = 6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='7083 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0007 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='7196 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0007 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='7413 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0007 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='7723 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0006 TABLE I: Expectation values of operators evaluated by 106 sampling data with a simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' ∆(g) and the lattice spacing a are fixed to −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' By substituting those values into Bob’s expected energy ⟨HnB⟩, one can recover the result of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 2 (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 3: Teleported energy for a different system size N = 6, 10, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' e−i α 2 (XnXn+1+YnYn+1)= H R(n) Z (α) H R(n+1) Z (−α) e−i α 2 ZnZn+1= R(n+1) Z (α) A Hamiltonian HZ = �N n=1 anZn containing only local Zns is implemented by e−iαHZ = N � n=1 R(n) Z (2anα), (A4) Teleported energy N = 6 Teleported energy N = 10 Teleported energy N = 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='00000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='00005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='00010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0004 338632 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='00015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content="0020 E 10 - E 6'0 8." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0006 388842 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='00020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0030 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='00025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0035 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='00030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='07 L00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0 - 0987654m21 0987654321 (g) (g) A(g)7 Now we describe the measurement of quantum opera- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Measurement of Zn is done by the following circuit The output of the measurement is a bit string bn ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Since the eigenvalues of Z are −1, 1, we con- vert the bit string into 1 − 2bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Let nshot be the num- ber of repetitions of the circuit, and countsb0···bN−1 be the number of times b0, · · · , bN−1 are detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' There- fore countsb0···bN−1 nshots is the probability that a bit string of b0 · · · bN−1 is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Then the expectation value of Zn is computed by the formula ⟨Zn⟩ = � b0,··· ,bN−1 (1 − 2bn)countsb0···bN−1 nshots .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (A5) Measurement of XnXn+1 is done by the following cir- cuit H H Note that H maps |0⟩ , |1⟩ to |+⟩ , |−⟩, which are eigen- vectors of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The output is again a bit string bnbn+1 ∈ {00, 01, 10, 11}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' They are converted to the eigenvalues of XnXn+1 by (1 − 2bn)(1 − 2bn+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Then the expectation value of XnXn+1 is computed by the formula ⟨XnXn+1⟩ = � bn,bn+1 (1 − 2bn)(1 − 2bn+1)countsbnbn+1 nshots .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (A6) Similarly, measurement of YnYn+1 is possible by the following circuit S† H S† H in such a way that ⟨YnYn+1⟩ = � bn,bn+1 (1 − 2bn)(1 − 2bn+1)countsbnbn+1 nshots .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (A7) Now let us discuss how to compute the expectation energy of the massive Thirring model (eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (2) and (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Once we obatin ⟨Zn⟩, ⟨ZnZn+1⟩, ⟨XnXn+1⟩, ⟨YnYn+1⟩ as shown in Table I, then we can compute the expectation value of teleported energy by HTh = N−2 � n=0 Hn ⟨Hn⟩ = ⟨H±,n⟩ + ⟨Hm,n⟩ + ⟨HZZ,n⟩ + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (A8) For example Bob’s local Hamiltonian HnB is given as follows HnB =H±,B + Hm,B + HZZ,B + ϵn H±,B = − 1 4a[XN−2(XN−3 + XN−1) + YN−2(YN−3 + YN−1)], Hm,B =m 2 (−1)N−1ZN−2 HZZ,B =∆(g) 8 ZN−2(ZN−3 + ZN−1) + ∆(g) 2a ZN−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (A9) Now it will be clear that [HTh, σnB] = [HnB, σnB] since σnB is a local operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The teleported energy ⟨HnB⟩ to Bob’s local system is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 2 (F) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Bob will receive −⟨HnB⟩ through his measurement device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Appendix B: Sine-Gordon Model and Thirring Model The sine-Gordon model is a theory of a single scalar field φ(x) whose Lagrangian is written as LSG = 1 2∂µφ(x)∂µφ(x) + α β2 (cos(βφ(x)) − 1), (B1) where α and β are real positive parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' This model is invariant under φ(x) → φ′(x) = φ(x) + 2πn β , n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (B2) The Thirring model is a theory of a self–coupled Dirac field LTh = ψ(iγµ∂µ − m)ψ − g 2ψγµψψγµψ, (B3) where m is the fermion mass, g is the dimensionless four- fermion coupling constant and ψ = ψ(x) is a spinor field with two components ψ1(x) and ψ2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' It is widely known that the massive Thirring model is dual to the sine-Gordon model and the classical two-dimensional XY model [35, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' For example a Kosterlitz-Thouless phase transition at T ∼ Kπ/2 in the XY model corresponds to a critical point g ∼ −π/2, called Coleman’s instability point, in the Thirring model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' They are also related with a critical point at t ∼ 8π in the sine-Gordon model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The model is obviously invariant under U(1)V group ψ(x) → ψ′(x) = eiαV ψ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (B4) 8 The corresponding conserved current is jµ = ψγµψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (B5) If m = 0, the Thirring model is also invariant under the chiral group U(1)V × U(1)A ψ(x) → ψ′(x) = eiαV ψ(x) ψ(x) → ψ′(x) = eiαAγ5ψ(x), (B6) under which the pseudo-vector current j5 µ = ϵµνjν (ϵ01 = 1 − ϵ10 = 1) (B7) also conserves ∂µj5 µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Bosonization of the interaction term is described by LSG int (x) = α β2 cos(βφ(x)) = α 2β2 (A+(x) + A−(x)) LTh int(x) = −mψ(x)ψ(x) = −m(σ+(x) + σ−(x)), (B8) where A±(x) = e±iβφ(x) and σ±(x) = 1 2ψ(x)(1±γ5)ψ(x) According to the Coleman’s prescription, those two models are related with ψ(x)ψ(x) = −M cos(βφ(x)) + m g iψ(x)γ5ψ(x) = −M sin(βφ(x)) g mψ(x) �1 ∓ γ5 2 � ψ(x) = − α 2β2 e±iβφ(x) + m2 2g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (B9) The massive Thirring model and the sine-Gordon model are equivalent only if β2 < 8π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Appendix C: Hamiltonian Formalism of Thirring Model The Hamiltonian including quantum effects in the energy-momentum tensor at the operator level can be written as HTh = � dx � − iZψ(g) ¯ψγ1∂1ψ + m ¯ψψ + g 4( ¯ψγ0ψ)2 − ˜g 2( ¯ψγ1ψ)2 � , (C1) where Zψ(g) is the wavefunction renormalization con- stant and ˜g = g 2 � 1 + 2g π �−1 [48–50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' ψ1(x) → 1 √aχ2n, ψ2(x) → 1 √aχ2n+1 (C2) We address this Hamiltonian on a lattice one-dimensional lattice with open boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Then the dis- cretized Hamiltonian is HTh = − i 2aZψ(g) N−1 � j=1 � χ† jχj+1 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' � + m N � j=1 (−1)nnj + g 2a N−1 � j=1 njnj+1, (C3) where nj = χ† jχj is the fermion number operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' For the purpose of quantum computation, we convert the Hamiltonian into the corresponding spin Hamilto- nian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' We will work with γ0 = X, γ1 = −iY, γ5 = −iXY (C4) We first write down the discrete Hamiltonian by means of staggered fermion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Based on the established method [39,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 51],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' we can replace Zψ(g) and g with Zψ(g) → π − g π sin � π−g 2 � g → 2(π − g) π cot �π − g 2 � (C5) As a result we obtain the Hamiltonian of the massive Thirring model we used in this work HTh = 2γ aπ sin(γ)H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (C6) where the explicit representation of H is H = − i 2a N−1 � n=1 (χ† nχn+1 − χ† n+1χn) + m N � n=1 (−1)nχ† nχn + ∆(g) a N−1 � n=1 χ† nχnχ† n+1χn+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (C7) where ∆(g) = cos � π−g 2 � [38–43] The Jordan-Wigner transformation maps fermionic op- erators to spin operators [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' It is commonly used in the study of quantum many-body systems and plays a key role in the study of strongly correlated electron systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' After the Jordan-Winger transformation χn = Xn − iYn 2 n−1 � m=1 (−iZm), χ† n = Xn + iYn 2 n−1 � i=m (iZm), (C8) we arrive at the spin representation of the massive Thirring model Hspin = − 1 4a N−1 � n=1 (XnXn+1 + YnYn+1) + m 2 N � n=1 (−1)n(Zn + 1) + ∆(g) a N−1 � n=1 �Zn + 1 2 � �Zn+1 + 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (C9) Those correspondences are summarized in the follow- ing dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 9 Dirac Staggerd Pauli ψψ (−1)n a χ† nχn (−1)n 2a (Zn + 1) ψγ0ψ 1 aχ† nχn 1 2a(Zn + 1) ψγ1ψ 1 2a(χ† nχn+1 + χ† n+1χn) 1 4a(XnYn+1 − YnXn+1) ψγ5ψ (−1)n 2a (χ† nχn+1 − χ† n+1χn) − i(−1)n 4a (XnXn+1 + YnYn+1) ψγ1∂1ψ − 1 2a2 (χ† nχn+1 − χ† n+1χn) − i 4a2 (XnXn+1 + YnYn+1) TABLE II: Correspondence among the three different repre- sentations of fermionic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Appendix D: Adiabatic Ground State Preparation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' General Remark One of the most difficult problems in performing quan- tum computation is how to design an initial state since it can affect the overall outcome of the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' A good initial state is one that is easy to prepare and that allows for efficient quantum gates and measurements to be applied to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' One common initial state used in quan- tum computation is the ground state of a Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' However a difficulty in preparing a good initial state is the quality of the qubits and the limitation of gate depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' If the qubits are prone to errors, initializing them with high fidelity is very challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' A commonly used method to solve such problems is the adiabatic quan- tum computation (AQC), which is a model of quantum computation based on the adiabatic theorem of quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' It can be used to solve certain optimization problems by encoding the problem into the energy lev- els of a time-dependent Hamiltonian, and then slowly varying the Hamiltonian over time to drive the system through a series of states that correspond to the solu- tion of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' One such quantum computation method is quantum annealing, which is a heuristic opti- mization method that is used to find the global minimum of a given spin Hamiltonian [53] and practically used for various purposes [54–56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' A well-known example is the Quantum Adiabatic Al- gorithm (QAA), it is based on the Hamiltonian of the problem to be solved, the initial Hamiltonian is chosen such that its ground state is easy to prepare and known and the final Hamiltonian is such that its ground state is the solution of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The system is then slowly evolved to the final Hamiltonian and the final state is the ground state of the final Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' One of the main requirements for the QAA to work correctly is that the adiabatic condition holds, this condition states that the adiabatic evolution should be slow enough so that the system remains in the ground state at all times, if this condition is not met the system may end up in excited state and the final solution will not be correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' One chal- lenge that arises during AQC is the trade-off between the speed of the evolution and the accuracy of the fi- nal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The adiabatic theorem states that the sys- tem remains in the ground state if the evolution is slow enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' However, if the evolution is too slow, the com- putation may become infeasible due to the long running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' On the other hand, if the evolution is too fast, the system may not stay in the ground state and the final solution may be inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Another challenge is the presence of a phase transition, which is a point where the ground state of the Hamilto- nian changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' At these points, the energy gap between the ground state and the first excited state can become small, which makes it more difficult to maintain the sys- tem in the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' This can also lead to critical slowing down, in which the system slows down drastically as it evolves through the phase transition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' In the context of adiabatic quantum computation, a phase dia- gram could be used to understand how the performance of the computation changes as a function of parameters such as the duration of the computation or the strength of the interactions between the quantum particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Additionally, AQC is also sensitive to noise and errors in the system, as small fluctuations can cause the system to leave the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' This can lead to incorrect solutions or a loss of quantum coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Method Here we describe a way to prepare the ground state by a quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' For this, we use the method so-called adiabatic state preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The essence of the method for studying phase transitions with adiabatic quantum calculations has already been established in many cases with quantum annealing [53, 57, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' We first prepare a state |vac0⟩ which is a known ground state of an initial Hamiltonian Hinitial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' We use the time-dependent Hamiltonian H(t) which is equal to the target Hamilto- nian Htarget at the end of computation: H(t) = tHtarget + (1 − t)Hinitial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (D1) By adiabatically changing parameter t, we expect to ob- tain the ground state of the target Hamiltonian |vac⟩ = lim T →∞ T exp � −i � T 0 dtH(t) � |vac⟩0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (D2) We have two reasonable choices of initial Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The first one is Hm = m 2 N � n=1 (−1)nZn, m > 0 (D3) whose ground state is |vac⟩0 = |0101 · · · 01⟩ = �N/2 i=1 X2i |0 · · · 0⟩, with Z |0⟩ = |0⟩ and Z |1⟩ = − |1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The other choice is Hg = ∆(g) a N−1 � n=1 �Zn + 1 2 � �Zn+1 + 1 2 � , ∆(g) < 0 (D4) 10 whose ground state is |vac⟩0 = |0 · · · 0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Each of those states can be easily prepared but we will choose one of them so that any possible errors including become small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The matrix representation of the chiral condensate is given as N � n=1 ¯ψψn = 1 2a N � n=1 (−1)nZn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (D5) Its vacuum expectation value is − N 2a with the ground state |0101 · · · 01⟩ of an initial Hamiltonian (D3) ⟨01 · · · 01| ¯ψψ |01 · · · 01⟩ = − N 2a (D6) Those two states corresponds to the ground states of the Hamiltonian (2) in the limit: limm→∞ 1 m(Hspin − Hm) = 0 and lim|∆(g)|→∞ 1 |∆(g)|(Hspin − Hg) = 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' We chose an initial state and parameters are chosen so that the effect of the phase transition on the probability distribution and statistical errors are relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' In the rest of this section, we describe how we manage an adiabatic state preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 4: Chiral condensate In the adiabatic process, we decompose the time- dependent Hamiltonian H(t) = Hstatic +Hdynamic(t) into the static part and the dynamic part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' For example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' if the Hamiltonian (D3) is used for the initial Hamiltonian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' the static part is Hstatic = m 2 N � n=1 (−1)nZn (D7) and the dynamic part is Hdynamic(t) = − w(t) 4a N−1 � n=1 (XnXn+1 + YnYn+1) + g(t) a N−1 � n=1 �Zn + 1 2 � �Zn+1 + 1 2 � (D8) with ∆(g)(t) = t T �∆(gtarget)t T + ∆(g0) � 1 − t T �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' w(t) = t T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (D9) where ∆(g0) is an initial coupling that should be cho- sen in a way that statistical errors become small,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' and ∆(gtarget) is the target coupling parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Study on Phase Diagram by Adiabatic State Preparation It will be interesting to explore the phase diagram of the massive Thirring model by the adiabatic algorithm, in particular for the purpose of quantum computation since we have to prepare the ground state anyway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' By using two different initial Hamiltonians above, one can draw the phase diagram of the massive Thirring model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 5, we illustrate a path we take for the adiabatic ground state preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Suppose we fix an initial mass m and ∆(g) to m = m0, ∆(g) = 0, which corresponds to the start of 1○.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' One can use the initial ground state |vac⟩0 = |0101 · · · 01⟩ of the Hamiltonian eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (D3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' As long as the target mass mtarget and ∆(gtarget) remain in the same phase of the initial state, one can efficiently obtain the value without changing paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' However, if there is a critical point of a 1st order phase transition on the dashed line, one needs to change paths to avoid a 1st order phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' For this, we draw an additional path consisting of 2○, 3○, and 4○.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' In the end of track 2○, one reaches to a point close to m = 0, ∆(g) < 0, to avoid a strong effect of first order phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Throughout the track 3○, ∆(g) gradually approaches to the target ∆(gtarget) and the adiabatic process should remain in (or at least close to) the ground state of 2○.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Chiral Condensate 0 2 4 19882 6 8- 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='0 6 m N 1 0 Q 0 0 i0 Q Q Q Q Q Q Q Q (g)11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 5: Schematic design of path to compute the vacuum ex- pectation value of chiral condensate by adiabatic state prepa- ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Paths should be selected so that results are not af- fected by phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' For quantum simulation, we decompose the time- dependent Hamiltonian (D1) as H(t) = H±(t) + HZZ(t) + HZ(t) H±(t) = −w(t) 4a N−1 � n=1 (XnXn+1 + YnYn+1) HZZ(t) = ∆(g)(t) 4a N � n=1 ZnZn+1 HZ(t) = m(t) 2 N � n=1 (−1)nZn + ∆(g)(t) 4 N−1 � n=1 (Zn + Zn+1), (D10) where time-dependent coefficients w(t), m(t), ∆(g)(t) should be defined so that they agree with a path of com- putation drawn in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' The scalar term (N−1)∆(g)(t) 4 dose not contribute to time-evolution, thereby we neglect it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' For example, in this study we use the following time- dependent parameters: w(t) = � t/T 1○ 1 2○, 3○, 4○ (D11) m(t) = � � � � � � � � � m0 1○ m1 t T + m0 � 1 − t T � 2○ m1 3○ m2 t T + m1 � 1 − t T � 4○ (D12) ∆(g)(t) = � � � � � � � � � ∆(g0) t2 T 2 1○ ∆(g0) 2○ ∆(g2) t T + ∆(g1) � 1 − t T � 3○ ∆(g2) 4○ (D13) Let T be the computational time, M be a large positive integer (Trotter number), δt = T/M be a time-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' We implement the integral (D2) in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' First, the unitary time-evolution for k time steps can be given U(kδt) = e−iδtH±(kδt)e−iδtHZZ(kδt)e−iδtHZ(kδt) (D14) and the state at t = kδt (k = 1, 2, · · · , T) is |vac(kδt)⟩ = U(kδt) · · · U(2δt)U(δt) |vac⟩0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (D15) The quantum circuits we implemented are shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Therefore at the end of computation (t = T = Mδt), the state evolves into |vac⟩ = M � k=1 e−iδtH±(kδt)e−iδtHZZ(kδt)e−iδtHZ(kδt) |vac⟩0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (D16) Throughout this work we chose N = 10, T = 100, M = 1000, δt = T/M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='1 and fix the lattice spacing a to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' As a demonstration, we design a path in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 5, which consists of four tracks whose initial and target values are given in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 1○ 2○ 3○ 4○ initial (∆(g), m) (0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='7) (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='7) (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='1,0) (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='6,0) target (∆(g), m) (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='7) (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='1,0) (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='6,0) (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='6,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='3) TABLE III: Initial and target values for adiabatic state prepa- ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Each number corresponds to the number in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 6: Time-evolution of chiral condensate with the ground state eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (D15) prepared by adiabatic state preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Path 1 consists of paths 1○ · · · 4○, and Path 2 is the dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' For each path, the average of 10000 samplings is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' For comparison, let us consider a linear path connect- ing (m1, ∆(g1)) and (m2, ∆(g2)): ∆(g)(m) = ∆(g2) − ∆(g1) m2 − m1 (m − m1) + ∆(g1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' (D17) Time-evolution of chiral condensate 0 Path 1 Path 2 1 2 2 3 4 (1 5 0 50 100 150 200 250 300 350 400 112 As a function of t, m and ∆(g) change as follows m(t) = m1 � 1 − t T � + m2 t T ∆(g)(t) = ∆(g1) � 1 − t T � + ∆(g2) t T (D18) The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' It is clear that results of Path 1 is consistent with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Path 2 continues to take the same values as Path 1 when there is no phase transition, while after crossing the phase transition point it deviates significantly from the exact theoretical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Note that the time steps for Path 1 and Path 2 are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
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+page_content=' Snizhko, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
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+page_content=' Van Loock, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
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+page_content=' Kikuchi, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content=' Tanizaki, Progress of Theoretical and Experimental Physics 2022 (2022), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='1093/ptep/ptac007, 033B01, https://academic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='oup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='com/ptep/article- pdf/2022/3/033B01/42782471/ptac007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
+page_content='pdf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
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+page_content=' Thompson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFKT4oBgHgl3EQfDi1s/content/2301.11712v1.pdf'}
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+arXiv:2301.01632v1 [math.PR] 4 Jan 2023
+Subcritical sharpness for multiscale Boolean
+percolation
+Barbara Dembin1
+1D-MATH, ETH Zürich, Switzerland.
+Abstract
+We consider a multiscale Boolean percolation on Rd with radius dis-
+tribution µ on [1, +∞), d ≥ 2.
+The model is defined by superposing
+the original Boolean percolation model with radius distribution µ with
+a countable number of scaled independent copies.
+The n-th copy is a
+Boolean percolation with radius distribution µ|[1,κ] rescaled by κn. We
+prove that under some regularity assumption on µ, the subcritical phase
+of the multiscale model is sharp for κ large enough. Moreover, we prove
+that the existence of an unbounded connected component depends only
+on the fractal part (and not of the balls with radius larger than 1).
+1
+Introduction
+Overview
+Boolean percolation was introduced by Gilbert in [6] as a continu-
+ous version of Bernoulli percolation, introduced by Broadbent and Hammersley
+[2]. We consider a Poisson point process of intensity λ > 0 on Rd and on each
+point, we center a ball of potentially random radius. In Boolean percolation we
+are interested in the connectivity properties of the occupied set: it is defined
+as the subset of Rd consisting of all the points covered by at least one ball.
+This model undergoes a phase transition in λ for the existence of an unbounded
+connected component of balls. For λ < λc, all the connected components are
+bounded, and for λ > λc, there exists at least one unbounded connected com-
+ponent.
+Boolean model
+Let d ≥ 2. Denote by ∥ · ∥ the ℓ2-norm on Rd. For r > 0
+and x ∈ Rd, set
+Bx
+r := {y ∈ Rd : ∥y − x∥ ≤ r}
+and
+∂Bx
+r := {y ∈ Rd : ∥y − x∥ = r}
+for the closed ball of radius r centered at x and its boundary. For short, we will
+write Br for B0
+r. For a subset η of Rd × R+, we define
+O(η) :=
+�
+(z,r)∈η
+Bz
+r.
+1
+
+Let µ be a distribution on R+ representing the distribution on the radius. Let η
+be a Poisson point process of intensity λdz⊗µ where dz is the Lebesgue measure
+on Rd. Write Pλ,µ for the law of η and Eλ,µ for the expectation under the law
+Pλ,µ.
+We say that two points x and y in Rd are connected by η, if there exists a
+continuous path in O(η) that joins x to y. We say that two sets A and B are
+connected if there exists x ∈ A and y ∈ B such that a and b are connected by
+η. We denote by {A ←→ B} this event.
+Define for every λ ≥ 0 and µ, the probability of percolation
+θµ(λ) := lim
+r→∞ Pλ,µ (0 ←→ ∂Br) .
+We define the critical parameter associated to the existence of an infinite con-
+nected component:
+λc(µ) := sup {λ ≥ 0 : θµ(λ) = 0} .
+We will work with measures µ such that
+� ∞
+0
+tddµ(t) < ∞.
+(1.1)
+Hall proved in [11] that this condition is necessary to avoid that all the space
+is covered.
+Under the minimal assumption (1.1), Gouéré proved in [8] that
+0 < λc(µ) < ∞. We also define the following critical parameter:
+�λc(µ) := inf
+�
+λ ≥ 0 : inf
+r>0 Pλ,µ(Br ←→ ∂B2r) > 0
+�
+.
+Knowing that λ ≤ �λc(µ) enables to do renormalization arguments and deduce
+a lot of properties (see [4, 10]). Hence, the equality �λc(µ) = λc(µ) implies that
+we have a good control on the subcritical regime. If the equality occurs, we
+say that we have subcritical sharpness. This equality has been proved under
+moment condition on µ (see [1, 4, 16]) and for almost all power-law distributions
+(see [3]).
+Multiscale Boolean percolation
+The model of multiscale Boolean perco-
+lation consists of an infinite superposition of independent copies of Boolean
+percolation at different scales. Let µ be a finite distribution on [1, +∞) that
+satisfies (1.1). Let κ > 1. Let λ > 0. For a set E ⊂ Rd × R+, write E/κ for the
+set {x/κ, x ∈ E}. We denote by
+ηκ(λ) := η(0)(λ) ∪
+∞
+�
+i=1
+1
+κi (η(i)(λ) ∩ (Rd × [1, κ]))
+where (η(i)(λ))i≥1 are i.i.d. Poisson point process of intensity λ dz ⊗ µ. Note
+that every point in O(ηκ(λ)) is almost surely covered. Yet, it does not necessar-
+ily imply that there exists an unbounded connected component as it does not
+prevent the existence of a blocking surface of null Lebesgue measure.
+We are interested in the percolation properties of O(ηκ(λ)). Let µκ be the
+distribution such that ηκ(λ) is a Poisson point process of intensity λdz ⊗ µκ.
+2
+
+The distribution µκ has an infinite mass but is σ-finite. We will explicit its
+expression later.
+We will here work under the following assumption
+∃κ0 > 1
+∀κ ≥ κ0
+sup
+a≥κ
+sup
+r≥1
+adµ([ar, aκ])
+µ([r, κ])
+≤ 1
+(1.2)
+with the convention 0/0 = 0. This assumption is in particular satisfied for distri-
+butions with compact support or distributions of the form f(r)r−(d+1+δ)1r≥1dr
+where f is a non-increasing function such that 0 < inf f < sup f < ∞ and δ > 0.
+The following theorem is the main result of the paper. It states that there is
+subcritical sharpness for the fractal distribution µκ and that the existence of an
+unbounded connected component does not depend on the large balls.
+Theorem 1.1. Let µ that satisfies assumption (1.2). Let κ0 be as in (1.2). For
+any κ ≥ κ0, we have
+λc(µκ) = �λc(µκ) = λc(µκ|[0,1]).
+Idea of the proof
+The proof relies on the following key observation. Thanks
+to condition (1.2), for κ ≥ κ0, we can prove that the Poisson model with in-
+tensity λdz ⊗ µκ|[0,1] stochastically dominates the Poisson model with intensity
+λdz ⊗ µκ|[0,κj] rescaled by κj. Since the support of the distribution µκ|[0,1] is
+bounded, it is possible to prove subcritical sharpness for this distribution using
+the standard ϕp(S) argument introduced by Duminil-Copin–Tassion in [5] in
+the context of standard percolation and generalized in the context of Boolean
+percolation by Ziesche [16].
+Using this argument, we can prove that when
+λ < λc(µκ|[0,1]), there is exponential decay of the probability of connection. To-
+gether with the stochastic domination, we can prove that when λ < λc(µκ|[0,1])
+we have
+inf
+r>0 Pλ,µκ(Br ←→ ∂B2r) = 0
+and λ < �λc(µκ). This yields λc(µκ|[0,1]) ≤ �λc(µκ) ≤ λc(µκ). The coincidence
+of these three critical points follows from the previous inequality together with
+λc(µκ|[0,1]) ≥ λc(µκ).
+Background
+In previous works on multiscale Boolean percolation, a slightly
+different definition was used. Define for κ ≥ 1
+�ηκ(λ) := η(0)(λ) ∪
+∞
+�
+i=1
+η(i)(λ)
+κi
+where (η(i)(λ))i≥1 are i.i.d. Poisson point process of intensity λ dz ⊗ µ. Let
+�µκ be the distribution such that �ηκ(λ) is a Poisson point process of intensity
+λdz ⊗ �µκ. With this definition, the range of the radius of the different scaled
+copies are no longer disjoint, the condition (1.1) is not enough to ensure that
+the multiscale Boolean model exhibits a non-trivial phase transition. Gouéré
+proved in [9] that λc(�µκ) > 0 if and only if
+�
+t≥1
+td log(t)dµ(t) < ∞.
+(1.3)
+3
+
+If this condition is not satisfied, the balls with radius greater than 1 have an
+infinite mass and λc(�µκ) = 0.
+Remark 1.2. Note in our definition of multiscale percolation, the range of
+radius among the different scaled copies are disjoint. This enables to remove
+assumption (1.3).
+The Boolean multiscale model was first studied for the distribution µ = δ1
+by Menshikov–Popov–Vachkovskaia in [14]. They proved that for λ < λc(δ1)
+and κ large enough the multiscale model does not percolate.
+They later extended in [15] their result to more general distribution µ that
+satisfy the following self-similarity condition
+lim
+a→∞ sup
+r≥1
+adµ([ar, +∞))
+µ([r, +∞))
+= 0
+and for λ > 0 such that
+lim
+r→∞ rdPλ,µ(Br ←→ ∂B2r) = 0.
+(1.4)
+Note that the condition (1.4) is quite restrictive since for distributions µ with
+an infinite 2d-moment, there exists no such positive λ.
+The condition (1.4) was relaxed later by Gouéré in [7], who proved that
+under the assumption (1.3), for λ < �λc(µ) and κ large enough, the multiscale
+model does not percolate.
+2
+Proofs
+2.1
+Proof of Theorem 1.1
+In this section, we prove the main theorem. We will need the two following
+propositions. This proposition is an adaptation of [16], the only difference is
+that the intensity is not finite but locally finite.
+Proposition 2.1. Let κ > 1 and λ < λc(µκ|[0,1]). There exists cκ > 0 depend-
+ing on κ and λ such that
+Pλ,µκ|(0,1](B1 ←→ ∂Bl) ≤ exp(−cκl)
+(2.1)
+The following proposition is the key observation to prove subcritical sharp-
+ness.
+Proposition 2.2. Let µ that satisfies hypothesis (1.2). Let κ ≥ κ0. We have
+for any j ≥ 1, l > 1, λ ≥ 0
+Pλ,µκ|[0,κj ](Bκj ←→ ∂Blκj) ≤ Pλ,µκ|(0,1](B1 ←→ ∂Bl).
+Before proving these two propositions, let us prove the main theorem.
+Proof of Theorem 1.1. Let λ < λc(µκ|(0,1]). Let j, l ≥ 1. We have
+Pλ,µκ(Blκj ←→ ∂B2lκj) ≤ Pλ,µκ|(0,κj ](Blκj ←→ ∂B2lκj)
++ Pλ,µκ
+�
+∃(x, r) ∈ ηκ(λ) : r ≥ κj, Bx
+r ∩ B2lκj ̸= ∅
+�
+.
+(2.2)
+4
+
+Let us start by estimating the second term in the inequality:
+Pλ,µκ
+�
+∃(x, r) ∈ ηκ(λ) : r ≥ κj, Bx
+r ∩ B2lκj ̸= ∅
+�
+= 1 − exp(−λdz ⊗ µ(E))
+where E := {(x, r) : ∥x∥2 ≤ 2lκj + r, r ≥ κj}. We have
+dz ⊗ µ(E) =
+�
+r≥κj αd(2lκj + r)ddµ(r) ≤ αd(4l)d
+�
+r≥κj rddµ(r)
+where αd is the volume of the unit ball in Rd. It yields that
+Pλ,µκ
+�
+∃(x, r) ∈ ηκ(λ) : r ≥ κj, Bx
+r ∩ B2lκj ̸= ∅
+�
+≤ λαd(4l)d
+�
+r≥κj rddµ(r).
+(2.3)
+Let us now control the first term. There exists a constant cd depending only on
+d such that we can cover ∂Blκj by at most cdld−1 balls of radius κj centered at
+∂Blκj. By union bound, we get
+Pλ,µκ|(0,κj ](Blκj ←→ ∂B2lκj) ≤ cdld−1Pλ,µκ|(0,κj ](Bκj ←→ ∂Blκj)
+≤ cdld−1 exp(−cκl)
+(2.4)
+where we use in the last inequality Propositions 2.2 and 2.1. Combining in-
+equalities (2.2), (2.3) and (2.4), we obtain
+Pλ,µκ(Blκj ←→ ∂B2lκj) ≤ cdld−1 exp(−cκl) + λαd(4l)d
+�
+r≥κj rddµ(r).
+Let ε > 0. We first choose l large enough depending on cκ and ε and then j
+large enough depending on κ, ε and l so that
+Pλ,µκ(Blκj ←→ ∂B2lκj) ≤ ε
+where we recall that since µ has a finite d-moment
+lim
+j→∞
+�
+r≥κj rddµ(r) = 0.
+It follows that
+inf
+r>0 Pλ,µκ(Br ←→ ∂B2r) = 0
+and λ ≤ �λc(µκ). Hence,
+�λc(µκ) ≥ λc(µκ|(0,1]) ≥ λc(µκ).
+The result follows from the fact that �λc(µκ) ≤ λc(µκ).
+2.2
+Proof of Propositions 2.1 and 2.2
+Let m > 0. Set hm be the contraction by m that is hm(x) := x/m for x ∈ R.
+Set
+Tmµ := mdhm ∗ µ
+where hm ∗ µ is the pushforward of µ by hm. We will need the following Lemma
+that characterized the distribution of a contracted in space Poisson point pro-
+cess.
+5
+
+Lemma 2.3. Let m > 0 and λ > 0. Let ν be a distribution on R+. Let η be a
+Poisson point process of intensity λdz ⊗ν. Then η/m is a Poisson point process
+of intensity λdz ⊗ Tmν.
+From this lemma, we can deduce the following straightforward corollary.
+Corollary 2.4. Let κ ≥ 1. We have
+µκ = µ +
+∞
+�
+j=1
+Tκjµ|[1,κ].
+Proof of Lemma 2.3. It is clear that η/m is still a Poisson point process, we
+only need to prove that its intensity is λdz ⊗ Tmν. Let E ⊂ Rd × R+. We claim
+that
+(dz ⊗ ν)(mE) = (dz ⊗ Tmν)(E).
+(2.5)
+Indeed, we have
+(dz ⊗ ν)(mE) =
+�
+(z,r)∈mE
+dzdν(r) =
+�
+(mz,mr)∈mE
+mddzdν(r/m)
+=
+�
+(z,r)∈E
+dzdTmν(r) = (dz ⊗ Tmν)(E).
+Thanks to Corollary 2.4, we can now prove Proposition 2.2.
+Proof of Proposition 2.2. Thanks to Lemma 2.3, we have for l > 1 and j ≥ 0
+Pλ,µκ|(0,κj ](Bκj ←→ ∂Blκj) = Pλ,Tκj µκ|(0,κj ](B1 ←→ ∂Bl).
+To complete the proof, let us prove the following inequality
+Pλ,Tκj µκ|(0,κj ](B1 ←→ ∂Bl) ≤ Pλ,µκ|(0,1](B1 ←→ ∂Bl).
+Using Corollary 2.4, we have
+Tκjµκ|(0,κj] = Tκjµ|[1,κj] +
+∞
+�
+k=1
+TκjTκkµ|[1,κ]
+=
+j
+�
+k=1
+TκkTκj−kµ|[κj−k,κj−k+1] +
+∞
+�
+k=j+1
+Tκkµ|[1,κ]
+Let us prove that for any k ≥ 1 Tκkµ|[κk,κkk+1] ⪯ µ|[1,κ] where we write µ ⪰ ν
+when µ stochastically dominates ν (for every r > 0, we have µ([r, +∞)) ≥
+ν([r, +∞))). Let κ0 be as in hypothesis 1.2. Let κ ≥ κ0. By hypothesis (1.2),
+we have for r ∈ [1, κ]
+Tκkµ|[κk,κk+1]([r, κ]) = κdkµ([κkr, κk+1] ≤ µ([r, κ]).
+It yields that
+Tκjµκ|(0,κj] ⪯
+j
+�
+k=1
+Tκkµ|[1,κ] +
+∞
+�
+k=j+1
+Tκkµ|[1,κ] = µκ|(0,1].
+6
+
+Hence, we have
+Pλ,Tκj µκ|(0,κj ](B1 ←→ ∂Bl) ≤ Pλ,µκ|(0,1](B1 ←→ ∂Bl).
+This yields the proof.
+Finally, let us explain how the proof of Ziesche [16] can be extended in the
+general case of σ-finite measure (Proposition 2.1).
+Sketch of the proof of Proposition 2.1. First note that λds⊗µκ is a s-finite mea-
+sure on Rd × R+ \ {0} (hence σ- finite), that is, it can be written as a countable
+sum of finite measures. The Mecke equation (see Theorem 4.1 in [13]) and the
+Margulis-Russo formula (see [12]) both hold for intensity measures that are s-
+finite. Denote by B(Rd) the Borelian subsets of Rd. For each S ∈ B(Rd) such
+that B1 ⊂ S, we define
+ϕλ(S) := λ
+�
+r∈(0,1]
+�
+z∈Rd 1Bzr∩∂S̸=∅ Pλ,µ|(0,1]
+�
+B1
+O({(w,s)∈η:Bw
+s ⊂S})
+←→
+Bz
+r
+�
+dz dµκ(r).
+(2.6)
+This corresponds to the expected number of open balls intersecting the boundary
+of S that are connected to B1 inside S. The arguments of Ziesche hold in that
+context, in particular, when λ < λc(µκ|[0,1]), there exists S ∈ B(Rd) such that
+B1 ⊂ S and ϕλ(S) < 1. We conclude the existence of cκ > 0 depending on κ
+and λ such that inequality (2.1) holds.
+Acknowledgements
+The author would like to thank Vincent Tassion for
+fruitful discussions that initiated this project. This project has received fund-
+ing from the European Research Council (ERC) under the European Union’s
+Horizon 2020 research and innovation program (grant agreement No 851565).
+References
+[1] Daniel Ahlberg, Vincent Tassion, and Augusto Teixeira. Sharpness of the
+phase transition for continuum percolation in R2. Probab. Theory Related
+Fields, 172(1-2):525–581, 2018.
+[2] S. R. Broadbent and J. M. Hammersley. Percolation processes. I. Crystals
+and mazes. Proc. Cambridge Philos. Soc., 53:629–641, 1957.
+[3] Barbara Dembin and Vincent Tassion. Almost sharp sharpness for poisson
+boolean percolation, 2022.
+[4] Hugo Duminil-Copin, Aran Raoufi, and Vincent Tassion. Subcritical phase
+of d-dimensional Poisson–Boolean percolation and its vacant set. Annales
+Henri Lebesgue, 3:677–700, 2020.
+[5] Hugo Duminil-Copin and Vincent Tassion. A new proof of the sharpness of
+the phase transition for Bernoulli percolation and the Ising model. Comm.
+Math. Phys., 343(2):725–745, 2016.
+[6] E. N. Gilbert. Random plane networks. Journal of the Society for Industrial
+and Applied Mathematics, 9(4):533–543, 1961.
+7
+
+[7] Jean-Baptiste Gouéré. Percolation in a multiscale Boolean model. ALEA
+Lat. Am. J. Probab. Math. Stat., 11(1):281–297, 2014.
+[8] Jean-Baptiste Gouéré. Subcritical regimes in the Poisson Boolean model
+of continuum percolation. The Annals of Probability, 36(4):1209 – 1220,
+2008.
+[9] Jean-Baptiste Gouéré. Subcritical regimes in some models of continuum
+percolation. The Annals of Applied Probability, 19(4):1292 – 1318, 2009.
+[10] Jean-Baptiste Gouéré and Marie Théret. Equivalence of some subcritical
+properties in continuum percolation. Bernoulli, 25(4B):3714 – 3733, 2019.
+[11] Peter Hall.
+On Continuum Percolation.
+The Annals of Probability,
+13(4):1250 – 1266, 1985.
+[12] Günter Last.
+Perturbation analysis of Poisson processes.
+Bernoulli,
+20(2):486 – 513, 2014.
+[13] Günter Last and Mathew Penrose. Lectures on the Poisson Process. In-
+stitute of Mathematical Statistics Textbooks. Cambridge University Press,
+2017.
+[14] M. V. Menshikov, S. Yu. Popov, and M. Vachkovskaia. On the connectivity
+properties of the complementary set in fractal percolation models. Probab.
+Theory Related Fields, 119(2):176–186, 2001.
+[15] M. V. Menshikov, S. Yu. Popov, and M. Vachkovskaia. On a multiscale
+continuous percolation model with unbounded defects. volume 34, pages
+417–435. 2003. Sixth Brazilian School in Probability (Ubatuba, 2002).
+[16] Sebastian Ziesche. Sharpness of the phase transition and lower bounds for
+the critical intensity in continuum percolation on Rd. Annales de l’Institut
+Henri Poincaré, Probabilités et Statistiques, 54(2):866 – 878, 2018.
+8
+
diff --git a/odAzT4oBgHgl3EQfqv0U/content/tmp_files/load_file.txt b/odAzT4oBgHgl3EQfqv0U/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..9dbc9b5e5661e55bec7d6fadf05bc8ee56c5916a
--- /dev/null
+++ b/odAzT4oBgHgl3EQfqv0U/content/tmp_files/load_file.txt
@@ -0,0 +1,282 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf,len=281
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='01632v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='PR] 4 Jan 2023 Subcritical sharpness for multiscale Boolean percolation Barbara Dembin1 1D-MATH, ETH Zürich, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Abstract We consider a multiscale Boolean percolation on Rd with radius dis- tribution µ on [1, +∞), d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' The model is defined by superposing the original Boolean percolation model with radius distribution µ with a countable number of scaled independent copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' The n-th copy is a Boolean percolation with radius distribution µ|[1,κ] rescaled by κn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' We prove that under some regularity assumption on µ, the subcritical phase of the multiscale model is sharp for κ large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Moreover, we prove that the existence of an unbounded connected component depends only on the fractal part (and not of the balls with radius larger than 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' 1 Introduction Overview Boolean percolation was introduced by Gilbert in [6] as a continu- ous version of Bernoulli percolation, introduced by Broadbent and Hammersley [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' We consider a Poisson point process of intensity λ > 0 on Rd and on each point, we center a ball of potentially random radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' In Boolean percolation we are interested in the connectivity properties of the occupied set: it is defined as the subset of Rd consisting of all the points covered by at least one ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' This model undergoes a phase transition in λ for the existence of an unbounded connected component of balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' For λ < λc, all the connected components are bounded, and for λ > λc, there exists at least one unbounded connected com- ponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Boolean model Let d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Denote by ∥ · ∥ the ℓ2-norm on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' For r > 0 and x ∈ Rd, set Bx r := {y ∈ Rd : ∥y − x∥ ≤ r} and ∂Bx r := {y ∈ Rd : ∥y − x∥ = r} for the closed ball of radius r centered at x and its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' For short, we will write Br for B0 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' For a subset η of Rd × R+, we define O(η) := � (z,r)∈η Bz r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' 1 Let µ be a distribution on R+ representing the distribution on the radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Let η be a Poisson point process of intensity λdz⊗µ where dz is the Lebesgue measure on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Write Pλ,µ for the law of η and Eλ,µ for the expectation under the law Pλ,µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' We say that two points x and y in Rd are connected by η, if there exists a continuous path in O(η) that joins x to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' We say that two sets A and B are connected if there exists x ∈ A and y ∈ B such that a and b are connected by η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' We denote by {A ←→ B} this event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Define for every λ ≥ 0 and µ, the probability of percolation θµ(λ) := lim r→∞ Pλ,µ (0 ←→ ∂Br) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' We define the critical parameter associated to the existence of an infinite con- nected component: λc(µ) := sup {λ ≥ 0 : θµ(λ) = 0} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' We will work with measures µ such that � ∞ 0 tddµ(t) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='1) Hall proved in [11] that this condition is necessary to avoid that all the space is covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Under the minimal assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='1), Gouéré proved in [8] that 0 < λc(µ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' We also define the following critical parameter: �λc(µ) := inf � λ ≥ 0 : inf r>0 Pλ,µ(Br ←→ ∂B2r) > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Knowing that λ ≤ �λc(µ) enables to do renormalization arguments and deduce a lot of properties (see [4, 10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Hence, the equality �λc(µ) = λc(µ) implies that we have a good control on the subcritical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' If the equality occurs, we say that we have subcritical sharpness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' This equality has been proved under moment condition on µ (see [1, 4, 16]) and for almost all power-law distributions (see [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Multiscale Boolean percolation The model of multiscale Boolean perco- lation consists of an infinite superposition of independent copies of Boolean percolation at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Let µ be a finite distribution on [1, +∞) that satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Let κ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Let λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' For a set E ⊂ Rd × R+, write E/κ for the set {x/κ, x ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' We denote by ηκ(λ) := η(0)(λ) ∪ ∞ � i=1 1 κi (η(i)(λ) ∩ (Rd × [1, κ])) where (η(i)(λ))i≥1 are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Poisson point process of intensity λ dz ⊗ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Note that every point in O(ηκ(λ)) is almost surely covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Yet, it does not necessar- ily imply that there exists an unbounded connected component as it does not prevent the existence of a blocking surface of null Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' We are interested in the percolation properties of O(ηκ(λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Let µκ be the distribution such that ηκ(λ) is a Poisson point process of intensity λdz ⊗ µκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' 2 The distribution µκ has an infinite mass but is σ-finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' We will explicit its expression later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' We will here work under the following assumption ∃κ0 > 1 ∀κ ≥ κ0 sup a≥κ sup r≥1 adµ([ar, aκ]) µ([r, κ]) ≤ 1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='2) with the convention 0/0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' This assumption is in particular satisfied for distri- butions with compact support or distributions of the form f(r)r−(d+1+δ)1r≥1dr where f is a non-increasing function such that 0 < inf f < sup f < ∞ and δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' The following theorem is the main result of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' It states that there is subcritical sharpness for the fractal distribution µκ and that the existence of an unbounded connected component does not depend on the large balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Let µ that satisfies assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Let κ0 be as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' For any κ ≥ κ0, we have λc(µκ) = �λc(µκ) = λc(µκ|[0,1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Idea of the proof The proof relies on the following key observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Thanks to condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='2), for κ ≥ κ0, we can prove that the Poisson model with in- tensity λdz ⊗ µκ|[0,1] stochastically dominates the Poisson model with intensity λdz ⊗ µκ|[0,κj] rescaled by κj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Since the support of the distribution µκ|[0,1] is bounded, it is possible to prove subcritical sharpness for this distribution using the standard ϕp(S) argument introduced by Duminil-Copin–Tassion in [5] in the context of standard percolation and generalized in the context of Boolean percolation by Ziesche [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Using this argument, we can prove that when λ < λc(µκ|[0,1]), there is exponential decay of the probability of connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' To- gether with the stochastic domination, we can prove that when λ < λc(µκ|[0,1]) we have inf r>0 Pλ,µκ(Br ←→ ∂B2r) = 0 and λ < �λc(µκ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' This yields λc(µκ|[0,1]) ≤ �λc(µκ) ≤ λc(µκ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' The coincidence of these three critical points follows from the previous inequality together with λc(µκ|[0,1]) ≥ λc(µκ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Background In previous works on multiscale Boolean percolation, a slightly different definition was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Define for κ ≥ 1 �ηκ(λ) := η(0)(λ) ∪ ∞ � i=1 η(i)(λ) κi where (η(i)(λ))i≥1 are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Poisson point process of intensity λ dz ⊗ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Let �µκ be the distribution such that �ηκ(λ) is a Poisson point process of intensity λdz ⊗ �µκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' With this definition, the range of the radius of the different scaled copies are no longer disjoint, the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='1) is not enough to ensure that the multiscale Boolean model exhibits a non-trivial phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Gouéré proved in [9] that λc(�µκ) > 0 if and only if � t≥1 td log(t)dµ(t) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='3) 3 If this condition is not satisfied, the balls with radius greater than 1 have an infinite mass and λc(�µκ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Note in our definition of multiscale percolation, the range of radius among the different scaled copies are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' This enables to remove assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' The Boolean multiscale model was first studied for the distribution µ = δ1 by Menshikov–Popov–Vachkovskaia in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' They proved that for λ < λc(δ1) and κ large enough the multiscale model does not percolate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' They later extended in [15] their result to more general distribution µ that satisfy the following self-similarity condition lim a→∞ sup r≥1 adµ([ar, +∞)) µ([r, +∞)) = 0 and for λ > 0 such that lim r→∞ rdPλ,µ(Br ←→ ∂B2r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='4) Note that the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='4) is quite restrictive since for distributions µ with an infinite 2d-moment, there exists no such positive λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' The condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='4) was relaxed later by Gouéré in [7], who proved that under the assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='3), for λ < �λc(µ) and κ large enough, the multiscale model does not percolate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' 2 Proofs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='1 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='1 In this section, we prove the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' We will need the two following propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' This proposition is an adaptation of [16], the only difference is that the intensity is not finite but locally finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Let κ > 1 and λ < λc(µκ|[0,1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' There exists cκ > 0 depend- ing on κ and λ such that Pλ,µκ|(0,1](B1 ←→ ∂Bl) ≤ exp(−cκl) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='1) The following proposition is the key observation to prove subcritical sharp- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Let µ that satisfies hypothesis (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Let κ ≥ κ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' We have for any j ≥ 1, l > 1, λ ≥ 0 Pλ,µκ|[0,κj ](Bκj ←→ ∂Blκj) ≤ Pλ,µκ|(0,1](B1 ←→ ∂Bl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Before proving these two propositions, let us prove the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Let λ < λc(µκ|(0,1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Let j, l ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' We have Pλ,µκ(Blκj ←→ ∂B2lκj) ≤ Pλ,µκ|(0,κj ](Blκj ←→ ∂B2lκj) + Pλ,µκ � ∃(x, r) ∈ ηκ(λ) : r ≥ κj, Bx r ∩ B2lκj ̸= ∅ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='2) 4 Let us start by estimating the second term in the inequality: Pλ,µκ � ∃(x, r) ∈ ηκ(λ) : r ≥ κj, Bx r ∩ B2lκj ̸= ∅ � = 1 − exp(−λdz ⊗ µ(E)) where E := {(x, r) : ∥x∥2 ≤ 2lκj + r, r ≥ κj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' We have dz ⊗ µ(E) = � r≥κj αd(2lκj + r)ddµ(r) ≤ αd(4l)d � r≥κj rddµ(r) where αd is the volume of the unit ball in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' It yields that Pλ,µκ � ∃(x, r) ∈ ηκ(λ) : r ≥ κj, Bx r ∩ B2lκj ̸= ∅ � ≤ λαd(4l)d � r≥κj rddµ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='3) Let us now control the first term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' There exists a constant cd depending only on d such that we can cover ∂Blκj by at most cdld−1 balls of radius κj centered at ∂Blκj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' By union bound, we get Pλ,µκ|(0,κj ](Blκj ←→ ∂B2lκj) ≤ cdld−1Pλ,µκ|(0,κj ](Bκj ←→ ∂Blκj) ≤ cdld−1 exp(−cκl) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='4) where we use in the last inequality Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Combining in- equalities (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='4), we obtain Pλ,µκ(Blκj ←→ ∂B2lκj) ≤ cdld−1 exp(−cκl) + λαd(4l)d � r≥κj rddµ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Let ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' We first choose l large enough depending on cκ and ε and then j large enough depending on κ, ε and l so that Pλ,µκ(Blκj ←→ ∂B2lκj) ≤ ε where we recall that since µ has a finite d-moment lim j→∞ � r≥κj rddµ(r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' It follows that inf r>0 Pλ,µκ(Br ←→ ∂B2r) = 0 and λ ≤ �λc(µκ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Hence, �λc(µκ) ≥ λc(µκ|(0,1]) ≥ λc(µκ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' The result follows from the fact that �λc(µκ) ≤ λc(µκ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='2 Proof of Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='2 Let m > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Set hm be the contraction by m that is hm(x) := x/m for x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Set Tmµ := mdhm ∗ µ where hm ∗ µ is the pushforward of µ by hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' We will need the following Lemma that characterized the distribution of a contracted in space Poisson point pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' 5 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Let m > 0 and λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Let ν be a distribution on R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Let η be a Poisson point process of intensity λdz ⊗ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Then η/m is a Poisson point process of intensity λdz ⊗ Tmν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' From this lemma, we can deduce the following straightforward corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Let κ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' We have µκ = µ + ∞ � j=1 Tκjµ|[1,κ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' It is clear that η/m is still a Poisson point process, we only need to prove that its intensity is λdz ⊗ Tmν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Let E ⊂ Rd × R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' We claim that (dz ⊗ ν)(mE) = (dz ⊗ Tmν)(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='5) Indeed, we have (dz ⊗ ν)(mE) = � (z,r)∈mE dzdν(r) = � (mz,mr)∈mE mddzdν(r/m) = � (z,r)∈E dzdTmν(r) = (dz ⊗ Tmν)(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Thanks to Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='4, we can now prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Thanks to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='3, we have for l > 1 and j ≥ 0 Pλ,µκ|(0,κj ](Bκj ←→ ∂Blκj) = Pλ,Tκj µκ|(0,κj ](B1 ←→ ∂Bl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' To complete the proof, let us prove the following inequality Pλ,Tκj µκ|(0,κj ](B1 ←→ ∂Bl) ≤ Pλ,µκ|(0,1](B1 ←→ ∂Bl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Using Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='4, we have Tκjµκ|(0,κj] = Tκjµ|[1,κj] + ∞ � k=1 TκjTκkµ|[1,κ] = j � k=1 TκkTκj−kµ|[κj−k,κj−k+1] + ∞ � k=j+1 Tκkµ|[1,κ] Let us prove that for any k ≥ 1 Tκkµ|[κk,κkk+1] ⪯ µ|[1,κ] where we write µ ⪰ ν when µ stochastically dominates ν (for every r > 0, we have µ([r, +∞)) ≥ ν([r, +∞))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Let κ0 be as in hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Let κ ≥ κ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' By hypothesis (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='2), we have for r ∈ [1, κ] Tκkµ|[κk,κk+1]([r, κ]) = κdkµ([κkr, κk+1] ≤ µ([r, κ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' It yields that Tκjµκ|(0,κj] ⪯ j � k=1 Tκkµ|[1,κ] + ∞ � k=j+1 Tκkµ|[1,κ] = µκ|(0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' 6 Hence, we have Pλ,Tκj µκ|(0,κj ](B1 ←→ ∂Bl) ≤ Pλ,µκ|(0,1](B1 ←→ ∂Bl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' This yields the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Finally, let us explain how the proof of Ziesche [16] can be extended in the general case of σ-finite measure (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Sketch of the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' First note that λds⊗µκ is a s-finite mea- sure on Rd × R+ \\ {0} (hence σ- finite), that is, it can be written as a countable sum of finite measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' The Mecke equation (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='1 in [13]) and the Margulis-Russo formula (see [12]) both hold for intensity measures that are s- finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Denote by B(Rd) the Borelian subsets of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' For each S ∈ B(Rd) such that B1 ⊂ S, we define ϕλ(S) := λ � r∈(0,1] � z∈Rd 1Bzr∩∂S̸=∅ Pλ,µ|(0,1] � B1 O({(w,s)∈η:Bw s ⊂S}) ←→ Bz r � dz dµκ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='6) This corresponds to the expected number of open balls intersecting the boundary of S that are connected to B1 inside S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' The arguments of Ziesche hold in that context, in particular, when λ < λc(µκ|[0,1]), there exists S ∈ B(Rd) such that B1 ⊂ S and ϕλ(S) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' We conclude the existence of cκ > 0 depending on κ and λ such that inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content='1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Acknowledgements The author would like to thank Vincent Tassion for fruitful discussions that initiated this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' This project has received fund- ing from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No 851565).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' References [1] Daniel Ahlberg, Vincent Tassion, and Augusto Teixeira.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Sharpness of the phase transition for continuum percolation in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
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+page_content=' [11] Peter Hall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' On Continuum Percolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' The Annals of Probability, 13(4):1250 – 1266, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' [12] Günter Last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Perturbation analysis of Poisson processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Bernoulli, 20(2):486 – 513, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' [13] Günter Last and Mathew Penrose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Lectures on the Poisson Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' In- stitute of Mathematical Statistics Textbooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Cambridge University Press, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Menshikov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Popov, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Vachkovskaia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' On the connectivity properties of the complementary set in fractal percolation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Theory Related Fields, 119(2):176–186, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' [15] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Menshikov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Popov, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Vachkovskaia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' On a multiscale continuous percolation model with unbounded defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' volume 34, pages 417–435.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Sixth Brazilian School in Probability (Ubatuba, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' [16] Sebastian Ziesche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Sharpness of the phase transition and lower bounds for the critical intensity in continuum percolation on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 54(2):866 – 878, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
+page_content=' 8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfqv0U/content/2301.01632v1.pdf'}
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+Differentiable Simulations for Enhanced Sampling of Rare Events
+Martin ˇS´ıpka 1 2 Johannes C. B. Dietschreit 1 Rafael G´omez-Bombarelli 1
+Abstract
+We develop a novel approach to enhanced sam-
+pling of chemically reactive events using differ-
+entiable simulations. We merge the reaction path
+discovery and biasing potential computation into
+one end-to-end problem and solve it by path-
+integral optimization. The techniques developed
+contribute directly to the understanding and us-
+ability of differentiable simulations as we intro-
+duce new approaches and prove the stability prop-
+erties of our method.
+1. Introduction
+The idea to differentiate through simulations comes natu-
+rally from the optimization of path-dependent quantities. If
+the minimization of a loss function cannot be formulated
+separately for every frame in a trajectory, but only for the
+entire path, then optimization has to include the whole path
+leading up to each frame. For gradient descent, we must ob-
+tain the gradient of the loss with respect to the controllable
+parameters along the entire path. Simulations that are fully
+differentiable have been developed for optimization, con-
+trol, and learning of motion.(Degrave et al., 2016; de Avila
+Belbute-Peres et al., 2018; Hu et al., 2019; 2020) but also
+for the learning and optimization of quantities of interest
+in molecular dynamics (Wang et al., 2020; Ingraham et al.,
+2019; Greener & Jones, 2021). While the results are often
+promising, it is well known and summarized in (Metz et al.,
+2021) that na¨ıvely backpropagated gradients may not lead
+to a useful parameter update. Gradient vanishing, or on the
+other hand, explosions, do not allow their efficient use and
+are an open challenge. The problem is well-known to be
+associated with the spectrum of the system’s Jacobian (Metz
+et al., 2021; Galimberti et al., 2021) and closely connected
+to the chaotic nature of the simulated equations. Therefore,
+in order to employ path differentiation, we need to find
+solutions to these limitations.
+1Massachusetts Institute of Technology, 77 Massachusetts
+Ave, Cambridge, MA 02139 2Mathematical Institute, Faculty of
+Mathematics and Physics, Charles University, Sokolovsk´a 83,
+186 75 Prague. Correspondence to: Rafael G´omez-Bombarelli
+.
+In this paper, we modify the framework of differentiable
+simulations for the investigation of reactive chemical events.
+A chemical reaction can be viewed as a transition from one
+depression (reactant) on the potential energy surface (PES)
+to another (product). The two energy wells are separated by
+a potential energy barrier that the system has to surmount.
+It is only possible to estimate said reaction barrier, and
+therefore the likelihood of the reaction, once the transition
+mechanism has been unveiled. The major obstacle in deter-
+mining the mechanism, i.e., the most likely path connecting
+reactant and product, lays in its high dimensional nature.
+Typical reactive systems under study might have degrees
+of freedom on the orders of hundreds, thousands, or even
+hundreds of thousands in the case of large solvated sys-
+tems. On high dimensional PESs, it is impossible to explore
+paths blindly, and it is necessary to introduce some form
+of intuition. Extensive sampling of regions with high free
+energies(Chipot & Pohorille, 2007; Chipot, 2014) is needed
+and simple sampling algorithms such as e.g., Molecular dy-
+namics (MD) or Monte-Carlo (MC) often remain trapped in
+(meta)stable regions, leading to a non-ergodic sampling of
+configuration space.
+The problem has been commonly addressed by splitting it
+into two seemingly easier sub-tasks. First, the so-called
+collective variables (CVs) are identified, which act as a di-
+mensionality reduction technique. Due to the exponential
+growth of computational cost, known as the curse of dimen-
+sionality,(Bellman, 1967; K¨oppen, 2000) one picks one to
+three degrees of freedom (DoF) that represent the reaction,
+while all others are expected to be noise. In other words,
+the CVs should be selected as those DoF, which are much
+slower than all others and which fully describe the rare tran-
+sition event. Once equipped with a low dimensional map,
+we study the event by means of enhanced sampling(Torrie
+et al., 1977; Darve & Pohorille, 2001; Laio & Parrinello,
+2002; Abrams & Bussi, 2013; Spiwok et al., 2015; Valsson
+et al., 2016) usually by introducing a biasing potential. This
+biasing potential is a function of the identified CVs and
+modifies the original Hamiltonian by lowering the reaction
+barrier and enhancing exploration along the chosen degrees
+of freedom. If CVs and enhanced sampling are chosen
+well, the biased dynamics will show reactive events, and
+subsequent analysis of the biasing potential will unravel the
+properties of the reactive dynamics.
+arXiv:2301.03480v1 [physics.chem-ph] 9 Jan 2023
+
+DiffSim for Rare Events
+While there are standard packages and general, widely used
+methods helping us to solve the second problem, the first
+part - collective variables identification is still largely a
+manual task with automatic tools just emerging. (Sultan
+& Pande, 2018; Mendels et al., 2018; Wehmeyer & No´e,
+2018; Wang et al., 2019; Bonati et al., 2020; Wang & Ti-
+wary, 2021; Sun et al., 2022; ˇS´ıpka et al., 2022) The use
+of machine learning for the automatic identification of im-
+portant degrees of freedom in complicated and unstructured
+systems seems like a natural choice, yet it comes with many
+pitfalls and sharp edges. The main problem of the method is
+referred to as a chicken-and-egg situation. To properly learn
+the correct CVs for the reaction mechanism, training data
+close to the transition state is needed. However, it is difficult
+to obtain such data when the transition itself is what one is
+trying to discover in the first place. To solve this problem,
+iterative improvements of the CVs were introduced. (Chen
+et al., 2018b) Another potential issue of the CVs identifica-
+tion a priori is the inability to correct the collective variables
+on the fly. This means that once we have identified some
+CVs, we must be certain that the CV function is properly
+defined and well behaved in all regions. This places a sig-
+nificant burden on the CV identification method as it needs
+to be constructed in a way that any point from the transition
+path does not result in unexpected CV values and stays well
+withing the desired range. A hard task, for example, for a
+neural network.
+Employing differentiable simulations, we aim to create a
+method merging two steps of reaction barrier exploration.
+We define a differentiable loss function that, when mini-
+mized, results in a biasing potential promoting desired re-
+active events. The loss gradient behaviour is thoroughly
+investigated, and a mechanism to control its fluctuations
+and magnitude is proposed. The manuscript is structured
+as follows. In Section 2 we define molecular dynamics
+simulations biased with a learnable potential, introduce a
+formalism to describe chemical reactions using path inte-
+grals, and outline the concept of differentiable simulations.
+We discuss the challenges and limitations of using differen-
+tiable simulations in their current state in Section 3. In the
+same section we propose two novel techniques to resolve
+the outlined limitations. Firstly, partially detaching the state
+variable from the computational graph and thereby pruning
+the computational graph, which enables the linearization of
+the equations, making it possible to keep the arising gradi-
+ents provably finite. Secondly, graph mini-batching further
+stabilizes the training and allows working with lower learn-
+ing rates and a higher number of small updates. Finally, we
+demonstrate the usefulness of differentiable simulations in
+the context of chemical reactions by training the bias func-
+tion promoting barrier crossing for low dimensional numeri-
+cal examples as well as the prototypical alanine-dipeptide
+molecule.
+2. Problem Definition
+Molecular dynamics is commonly used to explore reaction
+processes on a detailed level of individual atoms. Let the
+column vector x ∈ RN denote the mass-weighted coordi-
+nates of the system and p the conjugated momenta. The
+particle motion is simulated using Hamiltonian equations
+with potential energy function U0(x).
+˙x(t) = p(t)
+˙p(t) = −∂U0(x(t))
+∂x
+(1)
+These equations conserve energy and are purely reversible
+with respect to time. However, it is common in molecular
+modeling not to work with the micro-canonical ensemble
+but rather with the canonical ensemble that conserves tem-
+perature (Callen & Scott, 1998). To do so, a thermostat
+is coupled to the system, keeping it close to a preset tem-
+perature. In this work, we choose the Langevin thermostat
+because of its implementational simplicity and its favorable
+properties with respect to differentiating along the computa-
+tional graph, as will be shown later. In Langevin dynamics,
+the thermostat is coupled to the system through the friction
+constant γ.
+The presence of a thermostat alone, however, does not guar-
+antee that reaction events can be observed on typical simu-
+lation scales, as transitions are extremely rare. Therefore, to
+enable barrier crossing we modify the PES with a learnable
+bias term B(x, θ)
+U(x, θ) = U0(x) + B(x, θ),
+(2)
+where the biasing function is parameterized by θ, which
+we aim to train to increase the frequency of reaction events.
+The biased dynamics evolve according to
+˙x(t) = p(t)
+˙p(t) = −∂U(x(t)
+∂x
+− γp(t) +
+�
+2γkBTR(t),
+(3)
+where kb is the Boltzmann constant, T the absolute temper-
+ature of the bath, and R(t) a Gaussian process.
+2.1. Process of chemical reactions
+It is often suitable to use general curvilinear coordinates
+and not simply Cartesian or mass-weighted coordinates to
+describe reactions. Common are internal coordinates such
+as interatomic distances, angles, or dihedrals, as they are
+invariant with respect to system rotation and translation.
+These special coordinates are denoted with ξ(x) ∈ RM and
+M ≤ N.
+The wells W of reactant (-1) and product (1), divided by a
+reaction barrier, are characterized by the set of points Γα
+
+DiffSim for Rare Events
+(α = −1, 1), which correspond to the equilibrium configu-
+rations in reactants and products, i.e., we expect an unbiased
+simulation on the PES U0(x), to stay in these basins with a
+very high probability. To model which points belong to the
+basin, we approximate the wells with a multivariate normal
+distribution. From short, unbiased simulations, we estimate
+mean µα and covariance matrix Σα. We consider a point to
+be part of a well if the probability of the point belonging to
+the distribution is above some chosen probability threshold.
+Wα =
+�
+x | (ξ(x) − µα)T Σ−1
+α (ξ(x) − µα) < ϵ
+�
+, (4)
+where epsilon can be obtained from χ2 distribution. The
+indicator function for a well is
+1α(x) =
+�
+1
+for
+x ∈ Wα
+0
+for
+x /∈ Wα.
+(5)
+In this manuscript, we only consider transitions between two
+wells, W−1 and W1. Additional basins would be handled
+analogously. The (escape) probability pα, within a specified
+time interval (t0, te) of a transition W−α → Wα is defined
+as
+pα = P
+�� te
+t0
+1α(x(t)) dt > 0
+���� x(t0) ∈ W−α
+�
+,
+(6)
+where t0 is the start and te the end time of the trajectory X.
+This can be understood as the probability of finding at least
+one point in Wα of a trajectory that has started in W−α. Our
+objective is to increase both p1 and p−1 simultaneously to
+a level where both events can be observed frequently on a
+typical simulation time scale.
+2.2. Optimizing the probability
+The form of the probability in (6) is not usable for differen-
+tiable optimization and needs to be recast to a differentiable,
+continuous form. Under suitable regularity conditions, we
+can replace the expression of (6) with
+pα = P
+�
+sup
+t 0
+���� x(t0) ∈ W−α
+�
+.
+(7)
+We can then define a soft loss function that is continuous
+everywhere and differentiable for any trajectory X with
+x(t0) ∈ W−α as
+L = Lξα =
+�
+0
+if ∃ x(t) ∈ Wα
+min
+t00 min
+i
+λi(x(τ)).
+(16)
+Then for every γ that fulfills: (dτλmin + γ) = ϵ > 0, it
+holds:
+∀τ > 0 : ∥a(τ)∥2 ≤ ∥a(0)∥2 e−2ϵτ
+(17)
+We proof the theorem in the Appendix A. There is a useful
+corollary of the above
+Corollary 3.4. Under the assumptions of the theorem 3.3,
+∥a(τ)∥ ∈ Lr(0, τe), r ∈ [1, ∞].
+Proof. Case r = ∞ is trivial as the square root of the
+upper bound (17) is still finite ∀τ. Let us now consider only
+r ∈ [1, ∞).
+∥a(τ)∥r
+Lr(0,τe) =
+� τe
+0
+∥a(τ)∥r ≤ ∥a(0)∥r
+� τe
+0
+e−ϵ r τ
+= −∥a(0)∥r
+ϵ r
+�
+e−ϵ r τ�τe
+0
+(18)
+Which is finite for every value of τe including ∞.
+Proof of the finite gradient update 1. Since we know that
+a(τ) ∈ L1(0, τe) from the previous corollary and that
+∂f(z(τ),θ)
+∂θ
+∈ L∞(Ωx × Ωp) from the assumption, it is now
+trivial to show
+∂L(ztL)
+∂θ
+=
+� τe
+0
+a(τ)f(z(τ), θ)
+∂θ
+dτ
+≤
+sup
+z(τ)∈T
+����
+∂f(z(τ), θ)
+∂θ
+����
+� τe
+0
+a(τ)dτ,
+(19)
+which is finite.
+This property allows us to backpropagate the dynamics with-
+out exploding gradients as long as γ is chosen large enough.
+The exponential scaling of the adjoints also indicates that
+once we identify the point where the loss function will be cal-
+culated, we only need to consider a handful of points before
+a(τ) essentially vanishes. Any further adjoint propagation
+does not significantly contribute to the gradient update. This
+is intuitively desirable, as for the noisy equation (3) the loss
+function information becomes diluted as we backpropagate.
+Keeping only recent data points thus introduces a natural
+cutoff to the information we use for optimization.
+The theorem also gives more insight into when such back-
+propagation may lead to exploding gradients. If the expres-
+sion (dτλmin + γ) = ϵ < 0, then the upper bound may
+
+DiffSim for Rare Events
+not hold, and gradients can increase exponentially. Strongly
+negative λi of the hessian indicates a concave part in the
+potential landscape, which is generally problematic for con-
+trol.
+3.3. Mini batching the graph
+One of the problems associated with differentiable simu-
+lation is the low number of updates. Usually, only one
+gradient step is taken per trajectory, making the gradients
+averaged across the entire path and necessitating rather large
+learning rates to train the network in just a few updates. The
+problem can be alleviated by a technique we call graph mini
+batching. The idea is to calculate trajectory depended gra-
+dients first (the adjoints a) in one pass and then split them
+to mini batches. The adjoints are then used as vectors in
+Jacobi-vector products (12) during backpropagation of the
+bias function evaluated in batches. The approach stabilizes
+learning and allows for much lower learning rates, better
+suited for training neural networks. An example of a use
+case is more thoroughly discussed in Appendix B.
+3.4. Summary
+The use of the Langevin thermostat with reasonable γ cre-
+ates finite memory dynamics and therefore decaying ad-
+joints. Employing also the .detach() operator ensures that
+the adjoints do vanish smoothly, without high frequency
+oscillations, thus making them bounded (solving Item 1)
+and ignoring fast motion, helping with Item 2. Such a fi-
+nite memory system is likely to be less chaotic, addressing
+Item 3. Splitting the loss gradient into random mini-batches
+obviously solves the point 4.
+4. Practical implementation
+It is important to promote the transition across the barrier
+equally. If only one direction is sampled, then one may
+end up with a ”landslide” potential strongly tilted towards
+one minimum and not a diffusive behaviour. Therefore, we
+choose the following approach.
+1. Create a batch of 2l starting configurations, with l in
+each well respectively.
+2. Run all trajectories simultaneously for a fixed number
+of time steps.
+3. Collect the loss 8 after all simulations have ended. Af-
+ter N initial steps, which serve as equilibration, we also
+calculate the minimal distance from the start to encour-
+age the eventual return to the starting well and, thereby,
+true diffusive behavior. Thus, we have two losses: A
+forward loss Lf(xLf ) and start loss Ls(xLs).
+4. Calculate adjoints and optimize the bias function
+Figure 2: Log-density of simulated points before (left) and
+after the training (right) of bias function by differentiable
+simulations. The right plot shows how well all important
+regions are sampled after training. The background of the
+Figure is the UMB(x, y), the underlying Muller-Brown po-
+tential.
+B(x, θ) using the graph mini batching technique. Re-
+peat from step 1 until convergence.
+By running a large number of simulations concurrently, one
+can leverage the vectorization of the operations and reduce
+computational time.
+5. Results
+In this Section, we present the results of our differentiable
+simulation setup. First, we apply it to a commonly used,
+notorious two dimensional PES, the Muller-Brown poten-
+tial(M¨uller & Brown, 1979), where any linear combination
+of the Cartesian coordinates does not yield a good CV. Then
+we lift this example to five dimensions by introducing three
+noisy DoFs demonstrating the efficiency and basic function-
+ality of the approach.
+Second, we investigate the golden standard for enhanced
+sampling in molecular systems, alanine dipeptide (amino
+acid alanine capped at both ends). The two collective vari-
+ables describing the metastable states are well known in
+the community, the backbone dihedrals φ and ψ. We will
+assume no such knowledge and generate the enhanced sam-
+pling simulation from all backbone dihedral angles as can-
+didates in an end-to-end process.
+5.1. 2D Muller-Brown potential
+The parameters of the commonly investigated 2D Muller-
+Brown PES (M¨uller & Brown, 1979; Sun et al., 2022) are
+given in the Appendix C. The simulation details are reported
+in Appendix C. For the bias potential, B(x, h), we employ
+a grid of Gaussian functions, controlling their individual
+
+104
+103
+Biased sampling - log-density
+103
+102
+102
+101
+101DiffSim for Rare Events
+0
+20
+40
+60
+80
+100
+Iteration
+100
+200
+300
+400
+500
+600
+Average Loss
+0%
+20%
+40%
+60%
+80%
+Success rate [%]
+−2.0
+−1.5
+−1.0
+−0.5
+0.0
+0.5
+1.0
+1.5
+CV
+0
+2
+4
+6
+8
+10
+Potential [kcal/mol]
+Original potential
+Potential + Bias
+Figure 3: Postprocessing of the converged trajectory. left: Loss functions and the probability of barrier crossing during
+the training progresses. middle: Variational Autoencoder producing a collective variable by training on a fully diffusive
+trajectory. right: Potential energy along the VAE collective variable with and without bias.
+height. The biasing function is
+B(x, h) =
+n2
+g
+�
+i=1
+hi exp
+�
+−
+�
+x − x0
+i
+�2
+2σ2
+�
+(20)
+with trainable h. Means x0
+i are evenly distributed in the
+computational domain. In two dimensions, we can afford
+such a setup; with a reasonable number ng of Gaussians
+along each dimension, we only need to calculate n2
+g contribu-
+tions to the total bias. As this approach scales exponentially
+with the dimensionality of the problem, it cannot be used
+with the 5D version of the potential (Appendix D). After
+training the bias via differentiable simulation with parame-
+ters reported in Appendix F, we obtain biased dynamics that
+generate a lot of successful transitions between reactants
+and products along the transition path. The log-density of
+the points along the transition path is also much more level
+Figure 2. The evolution of loss function and success rate
+during training are shown in Figure 3.
+Following our philosophy of having CVs determined from
+well sampled transitions, we construct the collective variable
+by dimensionality reduction of converged diffusive trajecto-
+ries. To obtain a single value collective variable describing
+the path, we use a Variational Autoencoder (Kingma &
+Welling, 2013). A very simple setup is employed with a two
+hidden layer encoder and a two hidden layer decoder, both
+with 50 hidden neurons and a Softplus activation function.
+The resulting CV is visualized in Figure 3. The CV distin-
+guishes well and interpolates smoothly between products
+and reactants. Using the CV, the unbiased and biased PES
+are plotted as averages along the CV. It is easy to see in
+Figure 3 how effectively the PES has been flattened by the
+bias function.
+5.2. 5D Generalization
+The situation is more complicated when we include addi-
+tional degrees of freedom that are harmonic and identical
+0
+50
+100
+150
+200
+250
+300
+Iteration
+2
+4
+6
+8
+10
+12
+14
+Average Loss
+0%
+10%
+20%
+30%
+40%
+50%
+60%
+70%
+80%
+Success rate [%]
+Figure 4: top row: Metrics for the alanine dipeptide run.
+Total loss function and transition success rate. bottom row:
+Log-density of simulated points before the training of bias
+by differentiable simulations (left) and after the training
+converged (right).
+for both reactants and products. One may consider them
+to be, e.g., quickly oscillating hydrogen atoms that do not
+influence the reaction (see Appendix D for details regarding
+extended potential). In five dimensions, the grid of Gaus-
+sian basis functions is not feasible due to the exponential
+computational complexity. Instead, we employed a fully
+connected neural network as a function of all five variables,
+which made training significantly harder. For the results
+postprocessing we employ the same approach as in the two
+dimensional case. See Figure 7.
+5.3. Alanine dipeptide
+Alanine dipeptide is a simple model system exhibiting typi-
+cal protein dihedral dynamics. Therefore, it has become an
+
+1.5
+30 -
+1.0
+0.5
+25
+0.0
+Y20:
+-0.5
+15
+--1.0
+-1.5
+10 -
+-2.0
+20
+25
+30
+35
+40
+X104
+I sampling - log-density
+log-density
+103
+103
+rad
+U
+102
+102
+Unbiased
+Biased
+101
+101
+-2
+2
+-2
+2
+3
+Φ [rad]
+Φ [rad]DiffSim for Rare Events
+Figure 5: left: Average bias potential projected on the Ra-
+machandran plane. White regions are without sufficient
+sampling to calculate bias potential. right: PMF of the
+φ-ψ-plane.
+important benchmark to test and verify free energy calcula-
+tion methods. The collective variables, the dihedral angles
+φ and ψ, are well known and the PES is rather complex
+with relatively low barriers. (Vymˇetal & Vondr´aˇsek, 2010;
+Mironov et al., 2019).
+We use this system to test the ability of our method to bias
+the dynamics along the important DoFs. As candidate de-
+grees of freedom, we choose four dihedral angles along
+the backbone of alanine dipeptide and one dihedral angle
+involving the methyl group that we expect to be correlated
+with ψ. In each dihedral angle, we define a Gaussian ba-
+sis set, which accounts for periodicity. These expanded
+dihedrals are then input to a fully connected network that
+calculates the bias function. The detailed settings are listed
+in Appendix F.
+As φ and ψ are known, the results are reported as Ramachan-
+dran plots. No additional dimensionality reduction via VAEs
+is performed.
+By comparing the averaged bias potential with a potential
+of mean force (PMF) obtained with Metadynamics (see
+Appendix E) we can see that their shapes are similar. This
+demonstrates that our differentiable simulations can unravel
+the transition paths and reaction barriers in the same way as
+a collective variable based method would, except without
+requiring prior knowledge of ideal CVs.
+The numerical tool used for the differentiable simulation
+of alanine dipeptide contained components modified from
+TorchMD library (Doerr et al., 2021). We, with some modi-
+fications, used mainly the Amber forcefield adapter written
+in PyTorch and some of the utilities for simulation setup.
+The forcefield used was Amber ff19SB (Tian et al., 2020).
+We simulated in vacuum.
+The progress of the training and log-density of points are re-
+ported in Figure 4. The comparison of the obtained potential
+of the mean force is in the Figure 5.
+6. Discussion
+The proposed method can solve the problem of finding reac-
+tion paths and estimating barriers without prior knowledge
+of important degrees of freedom. In this section we discuss
+its main contributions to the topic of differentiable simu-
+lations, summarize theoretical results and outline possible
+limitations and further steps.
+Path integral optimization: We formulate the loss func-
+tion optimization as a minimization of a path dependent
+integral. The usefulness of differentiable simulations for
+such problems can be best understood on simple examples
+that contain path dependence explicitly. One such example,
+the famous Brachistochrone curve, the shortest path in a
+gravitational field, is included in the Appendix G. In our
+case, this means we can define the total loss for the simu-
+lated trajectory and translate how it can be minimized by
+modifying the bias in the preceding points.
+Transition state identification: According to the Transi-
+tion State Theory developed by Eyring and Polanyi (Laidler
+& King, 1983) the reaction path passes a saddle point on
+the potential energy surface (PES) called transition state.
+A successful method used to study reactions is expected
+to find the transition state at least approximately. We do
+not explicitly include any terms in the loss function to find
+the saddle point, although this would be a possibility. For
+the studied systems, it was sufficient to progressively de-
+cay the learning rate with an increasing success rate of the
+transition and therefore accumulate only just enough bias to
+cross freely. The transition state is then found naturally as a
+region with the lowest barrier for reactions.
+Simulations with vanishing information: A vital concept
+we develop is the principle of finite time horizon influenc-
+ing the loss function. We employ the detached Langevin
+dynamics and show that the friction term in the momen-
+tum equation gradually replaces the initial information with
+noise. This translates to adjoints exponentially vanishing
+and being effectively zero after a certain number of steps.
+In this paper, we focused on method development and ana-
+lyzed properties of differentiable simulations applied to the
+study of chemical systems. We have demonstrated that dif-
+ferentiable simulations with our innovations can handle not
+only model systems but also complex molecular motions. In
+the future, we intend to extend the tool to more challenging
+reactions with complicated transition paths, such as protein
+motion and chemical reactions with multiple intermediate
+steps or competing reaction paths.
+Acknowledgements
+M.S. was supported by project No. START/SCI/053 of
+Charles University Research program. J.C.B.D. is thank-
+
+5.0
+2.5
+0.0
+Bias [kcal/mol]
+1
+2.5
+[rad]
+0
+-5.0
+-11
+-7.5
+-2
+-10.0
+12.5
+-3
+-3
+-2
+-1
+0
+1
+2
+3
+Φ [rad]30
+2
+25
+radian
+20
+mol
+kcal
+15
+10
+-2
+5
+2
+0
+2
+Φ/ radianDiffSim for Rare Events
+ful for the support of the Leopoldina Fellowship Program,
+German National Academy of Sciences Leopoldina, grant
+number LPDS 2021-08. R.G.-B. acknowledges support
+from the Jeffrey Cheah Career Development Chair.
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+hancing
+Important
+Fluctuations:
+Rare
+Events
+and Metadynamics from a Conceptual Viewpoint.
+http://dx.doi.org/10.1146/annurev-physchem-040215-
+112229, 67:159–184, 5 2016. ISSN 0066426X. doi:
+10.1146/ANNUREV-PHYSCHEM-040215-112229.
+URL
+https://www.annualreviews.
+org/doi/abs/10.1146/
+annurev-physchem-040215-112229.
+Vymˇetal, J. and Vondr´aˇsek, J. Metadynamics As a Tool
+for Mapping the Conformational and Free-Energy Space
+of Peptides — The Alanine Dipeptide Case Study. The
+Journal of Physical Chemistry B, 114(16):5632–5642, 4
+2010. ISSN 1520-6106. doi: 10.1021/jp100950w.
+Wang, D. and Tiwary, P. State predictive information bottle-
+neck. The Journal of Chemical Physics, 154(13):134111,
+4 2021.
+ISSN 0021-9606.
+doi: 10.1063/5.0038198.
+URL https://aip.scitation.org/doi/abs/
+10.1063/5.0038198.
+Wang, W., Axelrod, S., and G´omez-Bombarelli, R. Differ-
+entiable Molecular Simulations for Control and Learning.
+2 2020.
+Wang, Y., Ribeiro, J. M. L., and Tiwary, P. Past–future
+information bottleneck for sampling molecular reac-
+tion coordinate simultaneously with thermodynamics
+and kinetics. Nature Communications 2019 10:1, 10
+(1):1–8, 8 2019.
+ISSN 2041-1723.
+doi: 10.1038/
+s41467-019-11405-4. URL https://www.nature.
+com/articles/s41467-019-11405-4.
+Wehmeyer, C. and No´e, F.
+Time-lagged autoencoders:
+Deep learning of slow collective variables for molec-
+ular kinetics.
+The Journal of Chemical Physics, 148
+(24):241703, 2018.
+doi: 10.1063/1.5011399.
+URL
+https://doi.org/10.1063/1.5011399.
+Zhang, S.-X., Wan, Z.-Q., and Yao, H. Automatic Dif-
+ferentiable Monte Carlo: Theory and Application. 11
+2019.
+
+DiffSim for Rare Events
+A. Proof of adjoint convergence theorem
+To prove Theorem 3.3 we need to state one more lemma.
+Lemma A.1. Let x ∈ RN, U(x) scalar, real, C2(RN) function. Consider a hessian computed at x0:
+∂2U(x0)
+∂x2
+with
+minimum and maximum eigenvalues λmin and λmax respectively. Then for any vector v ∈ RN
+λmin ∥v∥2 ≤ vT · ∂U(x2
+0)
+∂x2
+v ≤ λmax ∥v∥2 .
+(21)
+Proof. We note that a hessian of a real continuous function is a symmetric matrix. Such a matrix is orthogonally diagonaliz-
+able and has real eigenvalues. The rest of the proof is a part of most standard linear algebra textbooks.
+We can now prove the Theorem 3.3.
+Proof. We start by multiplying (14) by 2a. This yields
+2a(τ) · ˙a(τ) = 2dτaT (τ) · ∂2U(x(τ))
+∂2x
+a(τ) − 2γ ∥a(τ)∥2
+(22)
+and can be recast using 2aT (τ) · ˙a(τ) =
+˙
+∥a(τ)∥2 (the norm is a standard vector 2-norm) to
+˙
+∥a(τ)∥2 = −2dτaT (τ) · ∂2U(x(τ))
+∂2x
+a(τ) − 2γ ∥a(τ)∥2
+(23)
+Using Lemma A.1 and, subsequently, the assumption of the theorem, we can estimate the upper bound of the time derivative
+as
+˙
+∥a(τ)∥2 ≤ −2 (dτλmin + γ) ∥a(τ)∥2 = −2ϵ ∥a(τ)∥2
+(24)
+Using Gromwall lemma we can now estimate a(τ) easily as
+∥a(τ)∥2 ≤ ∥a(0)∥2 exp
+�
+−2
+� τ
+0
+ϵ dt
+�
+= ∥a(0)∥2 e−2ϵτ
+(25)
+and since ϵ > 0, the ∥a(τ)∥2 is bounded for all τ.
+B. Graph minibatching and adjoints
+To better explain the graph minibatching technique, let us consider a simple differential equation with trainable parameters θ
+˙z = f(z, θ)
+(26)
+Let us discretize the equation using a simple Forward Euler method such that it becomes
+zn+1 = zn + dtf(zn, θ).
+(27)
+For simplicity consider a three step differentiable simulation (z0, z1, z2) such that
+z2 = z1 + dtf(z1, θ)
+z1 = z0 + dtf(z0, θ)
+where a loss function is defined for the last point L(z2). Our goal is to find the gradient of ∂L(z2)
+∂θ
+. Let us derive
+∂L(z2)
+∂θ
+= ∂L(z2)
+∂z2
+∂z2
+∂θ
+∂z2
+∂θ = ∂z1
+∂θ + dt∂f(z1, θ)
+∂θ
+= ∂z1
+∂θ + dt
+�∂f(z1, θ)
+∂z1
+∂z1
+∂θ + ∂f(z1, θ)
+∂θ
+�
+∂z1
+∂θ = ∂z0
+∂θ + dt∂f(z0, θ)
+∂θ
+= dt∂f(z0, θ)
+∂θ
+.
+
+DiffSim for Rare Events
+0
+50
+100
+150
+200
+250
+300
+Iteration
+0
+200
+400
+600
+800
+1000
+1200
+1400
+Loss
+T
+ot. loss - non-batched
+T
+ot. loss - batched
+0
+50
+100
+150
+200
+250
+300
+Iteration
+0%
+20%
+40%
+60%
+80%
+100%
+Success rate [%]
+Non-batched
+Batched
+Figure 6: Comparison of the batched and one-time update of the weights in the 2D example from 5.1. The learning rate for
+the unbatched example was set approximately a number of batches times larger than for the batched run. The convergence is
+clearly more stable and even faster in the batched case. This was also observed for any other setting we tried during the
+development.
+Put together,
+∂L(z2)
+∂θ
+= ∂L(z2)
+∂z2
+�
+dt
+�
+1 + dt∂f(z1, θ)
+∂z1
+� ∂f(z0, θ)
+∂θ
++ dt∂f(z1, θ)
+∂θ
+�
+.
+(28)
+Meaning, when we optimize the biased function f(zn, θ) We can split the derivative into two parts
+�∂L(z2)
+∂z2
+dt
+�
+1 + dt∂f(z1, θ)
+∂z1
+�� ∂f(z0, θ)
+∂θ
+�∂L(z2)
+∂z2
+dt
+� ∂f(z1, θ)
+∂θ
+(29)
+More steps can be obtained by continuing the iterations. One can easily see that the vectors we put into square brackets
+are actually the adjoints a(zn) from (11). By saving these vectors, we can then take zn, feed forward through f(zn, θ) and
+backpropagate using the vector jacobian product. This can be done in one gradient update, accumulating a gradient with
+respect to θ and updating it after going through all adjoints, or we can update weights in batches as it is common in neural
+network training. The latter is shown to be the more stable and faster converging of the methods (see Figure 6).
+C. 2D Muller-Brown potential
+The equation of the PES:
+UMB(x, y) = B
+4
+�
+i=1
+Ai exp
+�
+αi(x − x0)2 + βi(x − x0)(y − y0) + γi(y − y0)2�
+(30)
+The parameters used in this work are:
+i
+Ai
+αi
+βi
+γi
+x0
+y0
+1
+-1.73
+0
+-0.39
+-3.91
+48
+8
+2
+-0.87
+0
+-0.39
+-3.91
+32
+16
+3
+-1.47
+4.3
+-2.54
+-2.54
+24
+31
+4
+0.13
+0.23
+0.273
+0.273
+16
+24
+The barrier parameter B = 10 kcal/mol
+D. 5D Generalization
+We consider a generalization of the Muller-Brown potential. By adding three harmonic DoFs we complicate the problem
+and make it necessary to use a general form of a biasing potential, dependent on all degrees of freedom, as we do not know
+
+DiffSim for Rare Events
+0
+50
+100
+150
+200
+250
+300
+Iteration
+50
+100
+150
+200
+250
+300
+Average Loss
+0%
+10%
+20%
+30%
+40%
+50%
+60%
+70%
+Success rate [%]
+−3
+−2
+−1
+0
+1
+2
+3
+CV
+0
+2
+4
+6
+8
+10
+Potential [kcal/mol]
+Original potential
+Potential + Bias
+Figure 7: Results for the 5D extension of the Muller-Brown potential. left: Evolution of loss value and probability of barrier
+crossing during the training progresses. middle: CV determined with a Variational Autoencoder trained on a fully diffusive
+trajectory. The collective variables are not sharp around the transition region due to the high variance of the other noisy
+DoFs. This could be improved by more data, and more refined dimensionality reduction techniques that include temporal
+data such as e.g TiCA (Schwantes & Pande, 2015) or time-lagged autoencoders (Wehmeyer & No´e, 2018). right: Average
+potential energy along the VAE collective variable with and without bias. The barriers were lowered to the level where they
+could be crossed with high probability.
+which of them defines the reaction. The resulting potential has the form:
+U5D(x1, x2, x3, x4, x5) = UMB(x1, x3) + κ(x2
+2 + x2
+4 + x2
+5)
+(31)
+The parameters for UMB(x1, x3) are identical to the 2D-case, the new parameter κ = 0.1 The results for the 5D case are
+visualized in the figure Figure 3.
+E. Alanine dipeptide simulation settings
+The PMF of alanine dipeptide was obtained by means of well-tempered metadynamics(Barducci et al., 2008). The deposited
+Gaussians had an initial height of 1 kcal/mol, a width of 10◦ along both dihedrals, and were deposited every 50 fs. The
+WTMetaD temperature was 4000 K a The simulation time step was 1 fs and the temperature was kept at 300 K with the
+Langevin thermostat with a friction constant of 1 ps−1. The total simulation time was 50 ns.
+F. Differentiable simulation parameters
+The equations we simulate are (3), discretized by the Leapfrog algorithm. The method is symplectic and conserves energy.
+The constants and parameters of the method were chosen as follows:
+case
+m [g/mol]
+γ [ps−1]
+T [K]
+dt [fs]
+timesteps
+epochs
+2D
+0.1
+0.1
+10
+1
+6000
+101
+5D
+0.01
+1.0
+300
+1
+20 000
+65
+Ala2
+-
+0.1
+300
+1
+10 000
+301
+The column ”timesteps” lists for how many steps we propagate a single simulation in each epoch. The update of parameters
+then represents an epoch. For the backward dynamics, we use 190 adjoints directly before the point where the loss function
+is calculated.
+Batches of Parallel MDs These batches refer to the number of replicas that are simulated simultaneously. Using GPUs,
+we can parallelize the computation of forces and time step integration and thus are able to run 600 systems at once with a
+similar speed of running just one. Accumulating the simulated data from so many systems allows us to increase the number
+of adjoints obtained and enables us to use a lower learning rate, making the training more stable.
+The setup of the bias function differed for every test case:
+2D Muller-Brown: In this case, we use the setup described in (20) with 50 times 50 basis functions.
+5d Muller-brown: We use a fully connected network with all five degrees of freedom used as five continuous input neurons.
+
+35
+3
+30
+2
+25
+-1
+20
+0
+2
+15
+-1
+10
+-2
+5
+-3
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+XDiffSim for Rare Events
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+x
+−2.0
+−1.5
+−1.0
+−0.5
+0.0
+y
+Learned path
+True path
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+x
+−2.0
+−1.5
+−1.0
+−0.5
+0.0
+y
+Learned path
+True path
+Figure 8: left Initial state. The path is initialized as an almost straight path. right After 200 iterations of differentiable
+simulations training, the path approximates the true path. The difference between the true curve and the one obtained by
+training is likely in the numerical scheme used to evaluate the integral.
+The network has four hidden layers, each 150 neurons with SiLU as activation functions. The final layer has a single output
+neuron - the bias - and no activation.
+Alanine dipeptide: In the case of real molecules, the bias function gets more complicated. We define the Gaussian basis set
+in every single degree of freedom represented by a basis vector ej and form a vector
+v(x) =
+ndof
+�
+j=1
+ng
+�
+i=1
+exp
+�
+−(x − x0
+ij)2
+2σ2
+�
+ej.
+(32)
+ndof represents the total number of candidate CVs or degrees of freedom considered. In our case, this was 5. ng is the
+total number of basis functions defined separately for every candidate CV. In our case, this was 50 and since we described
+dihedral angles, centers x0
+ij were distributed uniformly from 0 to 2π, respecting the periodicity of dihedrals. The flattened
+vector v(x) with size ndof · ng is then used as an input to a fully connected neural network with three hidden layers, each
+150 neurons with SiLU as an activation function. The final layer has one output neuron without an activation function.
+Graph-Minibatching batch size This batch size refers to the mini-batching of the computational graph illustrated in
+Appendix B. Here we split the accumulated adjoints into smaller batches and train the network sequentially. The mini-batch
+of 120 was used for all systems. We use the learning rate as a learning factor divided by the number of replicas to make it
+independent of the number of systems simulated simultaneously. The learning factor is chosen as 10 for the 2D case, 1
+5 for
+the 5D case and 3 for the Alanine dipeptide. This, with 300 replicas running from reactant to the product and 300 the other
+way, gives us learning rates on the orders 10−2 to 10−3. We use Adam optimizer for all our cases.
+G. Brachistochrone curve
+Here we exemplify how one can employ differentiable simulations and their capabilities to optimize path dependent integrals
+and solve the Brachistochrone problem. The problem is formulated as follows: Given a mass freely sliding on a curve
+y = y(x) in the gravitational field g, find the curve from point A to lower point B for which the sliding time is the shortest.
+We assume no friction or air resistance and assume that B does not lie directly below A. For simplicity, we choose A to
+be the origin of the coordinate system. The solution, the cyclone curve, of this famous problem was obtained by Leibniz,
+L’Hospital, Newton, and Bernoulli brothers (Boyer & Merzbach Uta, 1991). A modified version, where we allow for an
+arbitrary difference in height between the two points and only prescribe their horizontal distance ∆x was solved by Lagrange
+and much later summarized and written in the modern language of variational formalism by (Mertens & Mingramm, 2008).
+In this case, a solution is also a cyclone with some parameters fixed. We prescribe the horizontal ∆x to be π and search for
+a solution using differentiable simulations. A simple fully connected neural network f(x) serves as a derivative of the curve
+f(x) = dy(x)
+dx , so that y(x) is then obtained by the path integration of the neural network. After integration, we numerically
+evaluate the time from the simulated path
+t =
+� π
+0
+dx
+v =
+� π
+0
+�
+�
+�
+�1 +
+�
+dy(x)
+dx
+�2
+−2gy(x)
+dx
+(33)
+
+DiffSim for Rare Events
+and minimize it. The formula can be easily derived from the conservation of kinetic energy and from a Pythagorean
+expression ds2 = dx2 + dy2. For the path construction and backpropagation we employ the torchdiffeq python package
+shipped with the paper (Chen et al., 2018a).
+In this example, we present a problem that could not be solved with just a point-wise neural network optimization but
+requires consideration of a full path.
+
diff --git a/qtE1T4oBgHgl3EQf2gXF/content/tmp_files/load_file.txt b/qtE1T4oBgHgl3EQf2gXF/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..97a7ebf8dfec740761cc230b34fc51f0b5884474
--- /dev/null
+++ b/qtE1T4oBgHgl3EQf2gXF/content/tmp_files/load_file.txt
@@ -0,0 +1,1115 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf,len=1114
+page_content='Differentiable Simulations for Enhanced Sampling of Rare Events Martin ˇS´ıpka 1 2 Johannes C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=' Dietschreit 1 Rafael G´omez-Bombarelli 1 Abstract We develop a novel approach to enhanced sam- pling of chemically reactive events using differ- entiable simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=' We merge the reaction path discovery and biasing potential computation into one end-to-end problem and solve it by path- integral optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=' The techniques developed contribute directly to the understanding and us- ability of differentiable simulations as we intro- duce new approaches and prove the stability prop- erties of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=' Introduction The idea to differentiate through simulations comes natu- rally from the optimization of path-dependent quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=' If the minimization of a loss function cannot be formulated separately for every frame in a trajectory, but only for the entire path, then optimization has to include the whole path leading up to each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=' For gradient descent, we must ob- tain the gradient of the loss with respect to the controllable parameters along the entire path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=' Simulations that are fully differentiable have been developed for optimization, con- trol, and learning of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=' (Degrave et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=' de Avila Belbute-Peres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=' 2020) but also for the learning and optimization of quantities of interest in molecular dynamics (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=' Ingraham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=' Greener & Jones, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=' While the results are often promising, it is well known and summarized in (Metz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=', 2021) that na¨ıvely backpropagated gradients may not lead to a useful parameter update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=' Gradient vanishing, or on the other hand, explosions, do not allow their efficient use and are an open challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=' The problem is well-known to be associated with the spectrum of the system’s Jacobian (Metz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=' Galimberti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=', 2021) and closely connected to the chaotic nature of the simulated equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=' Therefore, in order to employ path differentiation, we need to find solutions to these limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=' 1Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139 2Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Sokolovsk´a 83, 186 75 Prague.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE1T4oBgHgl3EQf2gXF/content/2301.03480v1.pdf'}
+page_content=' Correspondence to: Rafael G´omez-Bombarelli 0.
+Remark 3.4 (Well-posedness and Poincar´e inequality).
+The well-posedness of (11)-(12) means that a strong solu-
+tion ¯X exists with a well-defined density {¯pt}t≥0. To show
+well-posedness, apart from the standard Lipschitz condi-
+tion on the drift terms a(·) and σB(·), the main technical
+condition is that the posterior density pt (of Xt) satis-
+fies the Poincar´e inequality (PI), and
+�
+|h(x)|2pt(x)dx <
+∞ (Laugesen et al., 2015, Theorem 2.2). (A probability
+density ρ = e−V satisfies the PI if xT∇V (x) ≥ α|x| for
+|x| ≥ R where α and R are positive constants (Bakry et al.,
+2008, Cor. 1.6). This condition is true, e.g., whenever ρ
+has a Gaussian tail.) An explanation of the relevance of
+the PI for the well-posedness (existence, uniqueness, and
+regularity) of the solution φ of the Poisson equation (12) is
+deferred to Sec. 4, where algorithms for its approximation
+are also described. Once a solution φ of the Poisson equa-
+tion is obtained together with necessary apriori estimates,
+well posedness of ¯X follows from the standard theory of
+mean-field SDEs (Carmona et al., 2018).
+Although the
+general case remains open, it has been possible to prove
+the PI under certain additional conditions on the filtering
+model (1) (Pathiraja et al., 2021, Lemma 5.1), (Laugesen
+et al., 2015, Prop 2.1).
+We next describe the finite-N algorithm which is how
+the FPF is implemented in practice.
+CIPS: The particles {Xi
+t : t ≥ 0, 1 ≤ i ≤ N} evolve
+according to:
+dXi
+t = a(Xi
+t)dt + σ(Xi
+t)dBi
+t
++ K(N)
+t
+(Xi
+t) ◦ (dZt − h(Xi
+t) + h(N)
+t
+2
+dt),
+Xi
+0
+i.i.d
+∼ p0,
+i = 1, . . . N,
+(13)
+6
+
+where {Bi
+t}t≥, for i = 1, 2, . . . , N, are mutually indepen-
+dent W.P., h(N)
+t
+:= N −1 �N
+i=1 h(Xi
+t), and K(N)
+t
+is the out-
+put of an algorithm that is used to approximates the so-
+lution to the Poisson equation (12):
+K(N)
+t
+:= Algorithm({Xi
+t}N
+i=1; h).
+The notation is suggestive of the fact that algorithm is
+adapted to the ensemble {Xi
+t}N
+i=1 and the function h; the
+density ¯pt is not known in an explicit form. Before de-
+scribing the algorithms for gain function approximation in
+(the following) Sec. 4, we discuss the linear Gaussian case.
+The main computational challenge to simulate the finite-
+N FPF (13) is the computation of the gain function. The
+difficulty arises because, for a general nonlinear observa-
+tion function h and a non-Gaussian density ρ, there are no
+known closed-form solutions of the Poisson equation (12).
+In the linear Gaussian special case, with a linear obser-
+vation function h(x) = Hx and a Gaussian density, the
+Poisson equation admits an explicit solution whereby the
+gain function is given by the Kalman gain:
+Proposition 3.5 (Lem. 3.4, Yang et al. (2013b)). Con-
+sider the Poisson equation (12). Suppose ρ is a Gaussian
+density N(m, Σ) and h(x) = Hx. Then its unique solution
+is given by:
+φ(x) = (HΣ)(x − m),
+x ∈ Rd.
+Consequently, the gain function ∇φ(x) = ΣH T is the
+Kalman gain.
+Using the Kalman gain, the FPF algorithm simplifies to
+a square-root form of the ensemble Kalman filter (EnKF)
+algorithm. This is described next.
+3.3. Ensemble Kalman filter
+In the linear Gaussian case, upon replacing the gain
+function with the Kalman gain, the mean-field FPF (11)
+is the Itˆo-SDE
+d ¯Xt = A ¯Xtdt + σBd ¯Bt + ¯ΣtH
+T(dZt − H ¯Xt + H ¯mt
+2
+dt),
+(14)
+where
+¯mt = E[ ¯Xt|Zt],
+¯Σt = E[( ¯Xt − ¯mt)( ¯Xt − ¯mt)
+T|Zt].
+As a corollary of Thm. 3.3, the mean-field process ¯X is
+exact which, in the linear Gaussian case, means that the
+conditional density of ¯Xt is Gaussian whose mean ¯mt and
+the covariance matrix ¯Σt evolve according to the Kalman
+filter (8).
+A direct proof showing (14) is exact appears
+in Sec. 5.1.
+The finite-N FPF is obtained as follows:
+dXi
+t = AXi
+tdt+σBdBi
+t+Σ(N)
+t
+H
+T(dZt − HXi
+t + Hm(N)
+t
+2
+dt),
+(15a)
+where the mean-field terms in (14) are approximated em-
+pirically as follows:
+m(N)
+t
+:= 1
+N
+N
+�
+j=1
+Xi
+t,
+(15b)
+Σ(N)
+t
+:=
+1
+N − 1
+N
+�
+j=1
+(Xi
+t − m(N)
+t
+)(Xi
+t − m(N)
+t
+)
+T.
+(15c)
+The linear Gaussian FPF (15) is identical to the square-
+root form of the ensemble Kalman filter (Bergemann and
+Reich, 2012, Eq. 3.3).
+Remark 3.6 (Historical context for EnKF). The EnKF
+algorithm was first introduced in Evensen (1994), in the
+discrete-time setting of the filtering problem.
+At the
+time, the algorithm was introduced as an alternative to
+the extended Kalman filter (EKF). As already mentioned
+in Sec. 1, a major reason for using an EnKF is that, un-
+like EKF, it does not require an explicit solution of the
+DRE (Van Leeuwen and Evensen, 1996; Burgers et al.,
+1998; Houtekamer and Mitchell, 1998).
+Since its intro-
+duction, a number of distinct types of EnKF algorithms
+have appeared in the literature. Amongst these, the most
+well-known types are as follows: (i) EnKF based on per-
+turbed observation (Evensen, 2003); and (ii) The square
+root EnKF (Anderson, 2001; Whitaker and Hamill, 2002;
+Bishop et al., 2001). The details for these algorithms can
+be found in (Reich and Cotter, 2015, Ch.
+6-7).
+The
+two aforementioned types of the EnKF algorithm have also
+been extended to the continuous-time setting (Bergemann
+and Reich, 2012). In these settings, the EnKF is usually
+referred to as the ensemble Kalman-Bucy filter (EnKBF).
+A review of the EnKBF algorithm and its connection to
+the FPF algorithm can be found in (Taghvaei et al., 2018).
+The EnKBF algorithm and the linear FPF admits several
+extensions: (i) EnKBF with perturbed observation (Berge-
+mann and Reich, 2012) (Del Moral and Tugaut, 2018);
+(ii) Stochastic linear FPF (Yang et al., 2016, Eq. (26))
+which is same as the square root EnKBF (Bergemann and
+Reich, 2012);(iii) Deterministic linear FPF (Taghvaei and
+Mehta, 2016, Eq. (15)) (de Wiljes et al., 2018). EnKF
+was recently extended to the case with correlated observa-
+tion noise (Ertel and Stannat, 2022). An excellent recent
+survey on this topic appears in Calvello et al. (2022).
+Remark 3.7 (Current research on EnKF). Error analysis
+of the EnKF algorithm remains an active area of research.
+For the discrete-time EnKF algorithm, these results ap-
+pear in (Le Gland et al., 2009; Mandel et al., 2011; Tong
+et al., 2016; Kelly et al., 2014; Kwiatkowski and Mandel,
+2015). The analysis for continuous-time EnKF is more re-
+cent (Del Moral and Tugaut, 2018; Bishop and Del Moral,
+2018; Taghvaei and Mehta, 2018; Del Moral et al., 2017;
+de Wiljes et al., 2018; Bishop and Del Moral, 2020; Chen
+et al., 2021). Typically, one is interested in obtaining a
+7
+
+uniform error bound as follows:
+E[∥m(N)
+t
+− mt∥2] + E[∥Σ(N)
+t
+− Σt∥2] ≤
+C
+√
+N
+,
+(16)
+where (mt, Σt) are the solutions of the Kalman filter (8)
+and (m(N)
+t
+, Σ(N)
+t
+) are obtained from simulating an EnKF;
+and C > 0 is a time-independent constant. In the most
+recent result (Bishop and Del Moral, 2020), (16) is shown
+under the assumption that H TH is a positive-definite ma-
+trix. It is expected that (16) also holds under the weaker
+condition of the pair (A, H) being detectable, which is the
+condition for the stability of the Kalman filter. However, a
+complete resolution is still open. A comprehensive review
+of recent developments in this area can be found in Bishop
+and Del Moral (2020).
+3.4. Comparison
+In this section, we provide an analytical comparison of
+the FPF with the importance sampling-based particle fil-
+ter.
+For this purpose, consider a parameter estimation
+example with a fully observed model as follows:
+dXt = 0,
+X0 ∼ N(0, σ2
+0Id) = p0,
+dZt = Xtdt + σwdWt,
+(17)
+where the time t ∈ [0, 1], σW , σ0 > 0, and Id is the d × d
+identity matrix. The posterior p1 at time t = 1 is a Gaus-
+sian N(m1, Σ1) with m1 =
+σ2
+0
+σ2
+0+σ2
+W Z1 and Σ1 =
+σ2
+0σ2
+w
+σ2
+0+σ2w Id.
+Let {Xi
+0}N
+i=1 be N i.i.d samples from the prior p0. The
+importance sampling-based particle filter yields an empir-
+ical approximation of the posterior p1 as follows:
+π(N)
+PF (f) :=
+N
+�
+i=1
+W i
+1f(Xi
+0),
+W i
+1 =
+e
+−
+|Z1−Xi
+0|2
+2σ2w
+�N
+i=1 e
+−
+|Z1−Xi
+0|2
+2σ2w
+.
+(18)
+In contrast, given the initial samples {Xi
+0}N
+i=1, the FPF
+approximates the posterior by implementing a feedback
+control law as follows:
+π(N)
+FPF(f) := 1
+N
+N
+�
+i=1
+f(Xi
+1), dXi
+t = Σ(N)
+t
+σ2w
+(dZt−Xi
+t + m(N)
+t
+2
+dt),
+(19)
+where the mean m(N)
+t
+and covariance Σ(N)
+t
+are empirically
+approximated using (15b) and (15c), respectively.
+The MSE in estimating the conditional expectation of a
+given function f is defined as follows:
+MSE∗(f) := E[|π(N)
+∗
+(f) − ⟨p1, f⟩|2],
+where the subscript ∗ is either the PF or the FPF.
+For f(x) =
+1
+√
+d1Tx, a numerically computed plot of the
+level-sets of MSE, as a function of N and d, is depicted
+in Figure 1-(a)-(b). The expectation is approximated by
+averaging over M = 1000 independent simulations. It is
+observed that, in order to have the same error, the im-
+portance sampling-based approach requires the number of
+samples N to grow exponentially with the dimension d,
+whereas the growth using the FPF for this numerical ex-
+ample is O(d
+1
+2 ). This conclusion is consistent with other
+numerical studies reported in the literature (Surace et al.,
+2019; Stano et al., 2014; Berntorp, 2015).
+For the purposes of the analysis, a modified form of the
+particle filter is considered whereby the denominator is
+replaced by its exact form:
+π(N)
+PF (f) :=
+N
+�
+i=1
+¯W i
+1f(Xi
+0),
+¯W i
+1 =
+e
+−
+|Z1−Xi
+0|2
+2σ2w
+NE[e
+− |Z1−X0|2
+2σ2w
+|Z1]
+.
+(20)
+Proposition 3.8 (Prop. 4 in (Taghvaei and Mehta,
+2020)). Consider the filtering problem (17) with state di-
+mension d. Suppose σ0 = σw = σ > 0 and f(x) = aTx
+where a ∈ Rd with |a| = 1. Then:
+1. The MSE. for the modified importance sampling esti-
+mator (20) is given by
+MSEPF(f) = σ2
+N
+�
+3(2d) − 1
+2
+�
+≥ σ2
+N 2d+1.
+2. The MSE for the FPF estimator (19) is bounded as
+MSEFPF(f) ≤ σ2
+N (3d2 + 2d).
+(21)
+Remark 3.9 (Curse of Dimensionality (CoD)). In the
+limit as d → ∞, the performance of the importance
+sampling-based particle filters is studied in the litera-
+ture (Bickel et al., 2008; Bengtsson et al., 2008; Snyder
+et al., 2008; Rebeschini et al., 2015). The main focus of
+these studies is on the particle degeneracy (or the weight
+collapse) issue: it is shown that if log N log d
+d
+→ 0 then the
+largest weight max1≤i≤N W i
+t → 1 in probability. Conse-
+quently, in order to prevent the weight collapse, the number
+of particles must grow exponentially with the dimension.
+This phenomenon is referred to as the curse of dimension-
+ality for the particle filters.
+In contrast, the weights in
+an FPF are uniform by design (see (19)). Therefore, the
+FPF does not suffer from the weight collapse issue and, in
+particular, does not require resampling. A complete com-
+parison of the two types of particle filters remains open
+(see (Abedi et al., 2022) for recent progress on this topic).
+Remark 3.10 (Scaling with the dimension). The scal-
+ing with dimension depicted in Fig. 1 (b) suggests that the
+O(d2) bound in (21) is loose.
+This is the case because,
+in deriving the bound, the inequality ∥ · ∥2 ≤ ∥ · ∥F is
+used (Taghvaei and Mehta, 2020, Appendix E). The in-
+equality is loose particularly so as the dimension grows.
+Also, it is observed that the MSE for the particle filter
+8
+
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+5
+6
+7
+8
+9
+log(N)
+d
+N
+2d
+0.01
+0.02
+0.05
+0.10
+m.s.e (PF)
+(a)
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+5.0
+5.5
+6.0
+6.5
+7.0
+7.5
+8.0
+8.5
+log(N)
+d
+N
+d
+1
+2
+0.001
+0.002
+0.005
+0.010
+m.s.e (FPF)
+(b)
+Figure 1: Numerical comparison for the filtering model (17). Level sets of the MSE. using: (a) importance sampling-based algorithm (18)
+and (b) the FPF (19). As the state dimension d grows, in order to have same performance (MSE), the number of particles N must increase
+as 2d for (18) while they increase as d
+1
+2 for (19).
+grows slightly slower than the lower-bound 2d. This is be-
+cause the lower-bound is obtained for the modified particle
+filter (20), while the MSE is numerically evaluated for the
+standard particle filter (18). The correlation between the
+numerator and denominator in (18) reduces the MSE.
+3.5. Extensions of FPF
+In deriving the FPF, the main modeling assumption is
+the nature of observation model (1b). (Such a model is
+referred to as the white noise observation model.) In sev-
+eral follow on works, the basic FPF is extended to handle
+more general types of models for the state process. These
+extensions are briefly described next.
+1) FPF on Riemannian manifolds. The feedback control
+form of the FPF formula (11) holds not only for the
+Euclidean state-space but also for the cases where the
+state {Xt}t≥0 evolves on a Riemannian manifold, such as
+the matrix Lie groups.
+These extensions are described
+in (Zhang et al., 2016b,a, 2017a,b). In these papers, the
+FPF is shown to provide an intrinsic description of the fil-
+ter that automatically satisfies the geometric constraints
+of the manifold. The gain is expressed as grad φ and ob-
+tained as a solution of the Poisson equation. It is shown
+that the gain is also intrinsic that furthermore does not
+depend upon the choice of the Riemannian metric. For
+the special case when the manifold is a matrix Lie group,
+explicit formulae for the filter are derived, using the ma-
+trix coordinates.
+Filters for two example problems are
+presented: the attitude estimation problem on SO(3) and
+the robot localization problem in SE(3). Comparisons are
+also provided between the FPF and popular algorithms
+for attitude estimation, namely the multiplicative EKF,
+the invariant EKF, the unscented quaternion estimator,
+the invariant ensemble Kalman filter, and the bootstrap
+particle filter. Specifically, under a certain assumption of
+a “concentrated distribution”, the evolution equations for
+the mean and the covariance are shown to be identical to
+the left invariant EKF algorithm.
+2) FPF on discrete state-space. In Yang et al. (2015), FPF
+is extended to the filtering problem where the hidden state
+{Xt}t≥0 is a continuous-time Markov process that evolves
+on a finite state-space. (For this model, the optimal non-
+linear filter is called the Wonham filter.) A standard algo-
+rithm to simulate a Markov process is based on the use of
+Poisson counters to simulate transitions between discrete
+states. In order to define the process ¯X, a control process
+U is introduced that serves to modulate the rates of these
+counters based on causal observations of data Z. An ex-
+plicit formula for the FPF feedback control law is derived
+and shown to be exact.
+Similar to (11), the formula is
+in the form of “gain times error” where the gain is now
+obtained by solving a certain linear matrix problem. The
+linear matrix problem is the finite state-space counterpart
+of the Poisson equation (12).
+3) FPF with data association and model uncertainty. In
+applications such as multiple target tracking, the filter-
+ing problem often involves additional uncertainties in the
+state model (1a) and the observation model (1b). In the
+classical linear Gaussian settings, algorithms based on the
+Kalman filter have been developed to provide a solution
+to these problems.
+These algorithms are referred to as
+the interacting multiple model (IMM) filter (Blom, 2013)
+and the probabilistic data association (PDA) filter (Bar-
+Shalom et al., 2009). In the PDA filter, the Kalman gain is
+allowed to vary based on an estimate of the instantaneous
+uncertainty in the observations. In the IMM filter, mul-
+tiple Kalman filters are run in parallel and their outputs
+combined to form an estimate.
+9
+
+Like the Kalman filter, the FPF is easily extended to
+handle additional uncertainties in the observation and sig-
+nal models: These extensions, namely, the probabilistic
+data association (PDA)-FPF and the interacting multiple
+model (IMM)-FPF are derived in our prior works (Yang
+et al., 2012, 2013a; Yang and Mehta, 2018). Structurally,
+the FPF based implementations are similar to the classical
+algorithms based on the Kalman filter. In the linear Gaus-
+sian settings, the equations for the mean and the variance
+of the FPF-based filters evolve according the classical PDA
+and IMM filters.
+4) Collective inference FPF. The term “collective infer-
+ence” is used to describe filtering problems with a large
+number of aggregate and anonymized data (Sheldon and
+Dietterich, 2011; Singh et al., 2020).
+Some of these
+problems have gained in importance recently because of
+COVID-19. Indeed, the spread of COVID-19 involves dy-
+namically evolving hidden processes (e.g., number of in-
+fected, number of asymptomatic etc..) that must be de-
+duced from noisy and partially observed data (e.g., num-
+ber of tested positive, number of deaths, number of hos-
+pitalized etc.). In carrying out data assimilation for such
+problems, one typically only has aggregate observations.
+For example, while the number of daily tested positives
+is available, the information on the disease status of any
+particular agent in the population is not known.
+In Kim et al. (2021), the FPF algorithm is extended
+for a model with M agents and M observations. The M
+observations are non-agent specific. Therefore, in its ba-
+sic form, the problem is characterized by data association
+uncertainty whereby the association between the observa-
+tions and agents must be deduced in addition to the agent
+state. In Kim et al. (2021), the large-M limit is interpreted
+as a problem of collective inference. This viewpoint is used
+to derive the equation for the empirical distribution of the
+hidden agent states. An FPF algorithm for this problem is
+presented and illustrated via numerical simulations. For-
+mulae are described for both the Euclidean and the finite
+state-space case.
+The classical FPF algorithm is shown
+to be the special case (with M = 1) of these more gen-
+eral results. The simulations help show that the algorithm
+well approximates the empirical distribution of the hidden
+states for large M.
+Before
+closing
+this
+section,
+we
+remark
+on
+the
+Stratonovich form of the mean-field FPF SDE (11). The
+FPF is expressed in this form because of two reasons:
+1. The feedback control law is “gain times error” which is
+appealing to control engineers, and structurally sim-
+ilar to the update formula in a Kalman filter. More-
+over, for the linear Gaussian model, the gain is the
+Kalman gain.
+2. Expressed in its Stratonovich form, the gain times er-
+ror formula carries over to the Riemannian manifolds
+settings. This is because of the intrinsic nature of the
+Stratonovich form (Zhang et al., 2017b, Remark 1).
+Notably, for the linear Gaussian model, the gain function is
+a constant (i.e., does not depend upon x) and therefore the
+Stratonovich form and the Itˆo form are the same. For the
+general case, the Itˆo form involves a Wong-Zakai correction
+term as described in the following remark.
+Remark 3.11 (Itˆo form of FPF). In its Itˆo form, the
+mean-field FPF (11) is expressed as
+d ¯Xt =a( ¯Xt)dt + σ( ¯Xt)d ¯Bt + Kt( ¯Xt)(dZt − h( ¯Xt) + ¯ht
+2
+dt)
++ 1
+4
+m
+�
+j=1
+∇|K(j)
+t ( ¯Xt)|2dt,
+where 1
+4
+�m
+j=1 ∇|K(j)
+t ( ¯Xt)|2 is the Wong-Zakai correction
+term. The Itˆo-Stratonovich relationship discussed here is
+based on interpreting Kt(x) as a function of space x and
+time t, and interpreting the ◦ in the Stratonovich form only
+with respect to the space x. In a recent paper (Pathiraja
+et al., 2021, Sec. 3), the gain function is defined and in-
+terpreted as a function of space x and the density. This is
+natural because the dependence upon time t comes because
+of the changes in density (¯pt) as the time evolves. Because
+the density is a stochastic process, it is argued that the
+appropriate interpretation of ◦ in the Stratonovich form
+should involve both space x and the density. Using such an
+interpretation, the Stratonovich form involves extra-terms
+that are solutions to accompanying Poisson equations.
+4. Algorithms for gain function approximation
+The exact gain K is a d × m matrix-valued function,
+where the j-th column of K is the solution of the Poisson
+equation (12) for j = 1, . . . , m. For the ease of presen-
+tation, the exposition in this section is restricted to the
+scalar-valued observation setting, i.e. m = 1, so that K
+becomes a d-dimensional vector-valued function and the
+superscript j is dropped from the Poisson equation (12).
+In practice, the Poisson equation must be solved numeri-
+cally. The numerical gain function approximation problem
+is as follows:
+input:
+samples {Xi : 1 ≤ i ≤ N}
+i.i.d.
+∼ ρ, h(·)
+output:
+gain function {Ki : 1 ≤ i ≤ N}
+where ρ is the (posterior) density and Ki := K(Xi). The
+explicit dependence on time t is suppressed in this section.
+An illustration of the gain function approximation problem
+appears in Fig. 2.
+4.1. Motivation and overview of approaches
+The Poisson equation is a linear PDE. In order to mo-
+tivate the various solution approaches, it is useful to first
+consider a finite-dimensional counterpart
+Ax = b,
+(22)
+10
+
+Figure 2: Gain function approximation problem in the feedback par-
+ticle filter. The exact gain function K(x) = ∇φ(x) where φ solves
+the Poisson equation (12). The numerical problem is to approximate
+Ki = ∇φ(x)|x=Xi using only the particles {Xi : 1 ≤ i ≤ N} sam-
+pled from density ρ (depicted as shaded region). The dashed line
+indicates the constant gain approximation, where the gain function
+is approximated by its expected value according to (26).
+where A is a n × n (strictly) positive-definite symmetric
+matrix and the righthand-side b is a given n × 1 vector.
+The problem is to compute the unknown n × 1 vector x.
+For this purpose, the following equivalent formulations of
+the finite-dimensional problem are first introduced:
+1. x is the solution of the weak form
+y
+TAx = y
+Tb,
+∀ y ∈ Rn.
+2. For some chosen positive ϵ, x is the solution to the
+fixed-point equation
+x = e−ϵAx +
+� ϵ
+0
+e−sAb ds.
+3. x is the solution of an optimization problem
+x = arg min
+z∈Rn
+1
+2z
+TAz − z
+Tb.
+When n is large, these formulations are useful to numeri-
+cally approximate the solution of (22):
+1. For each fixed y ∈ Rn, the weak form is a single equa-
+tion. By restricting y to a suitable low-dimensional
+subspace S ⊂ Rn, the number of linear equations is
+reduced for the purposes of obtaining an approximate
+solution (possibly also in S).
+2. The fixed-point equation is useful because e−ϵA is
+a strict contraction for ϵ > 0 (because A is strictly
+positive-definite). So, a good initial guess for x can
+readily be improved by using the Banach iteration.
+3. The optimization form is useful to develop alternate
+(e.g., search type) algorithms to obtain the solution.
+With this background, we turn our attention to the Pois-
+son equation (12) expressed succinctly as
+−∆ρφ = (h − ¯h),
+where ¯h :=
+�
+h(x)ρ(x)dx and ∆ρ := 1
+ρ∇ · (ρ∇). The lin-
+ear operator ∆ρ is referred to as the probability weighted
+Laplacian. Functional analytic considerations require in-
+troduction of the function spaces: L2(ρ) is the space of
+square integrable functions with respect to ρ with inner
+product ⟨f, g⟩ :=
+�
+f(x)g(x)ρ(x)dx; H1(ρ) is the Hilbert
+space of functions in L2(ρ) whose first derivative, defined
+in the weak sense, is the also in L2(ρ); and H1
+0(ρ) = {ψ ∈
+H1(ρ)|
+�
+ψ(x)ρ(x)dx = 0}.
+These definitions are important because H1
+0(ρ) is the
+natural space for the solution φ of the Poisson equa-
+tion (12). The operator −∆ρ is symmetric (self-adjoint)
+and positive definite because
+−⟨f, ∆ρg⟩ = ⟨∇f, ∇g⟩ = −⟨∆ρf, g⟩,
+∀f, g ∈ H1
+0(ρ).
+In the infinite-dimensional settings, one requires an addi-
+tional technical condition—the Poincar´e inequality (PI)—
+to conclude that the operator is in fact strictly positive-
+definite (Taghvaei et al., 2020, Sec. 2.2). Assuming the
+PI holds, it is also readily shown that ∆−1
+ρ
+is well de-
+fined, i.e., a unique solution φ ∈ H1
+0(ρ) exists for any given
+h ∈ L2(ρ) (Yang et al., 2016, Thm. 2).
+For the purposes of numerical approximation, entirely
+analogous to the finite-dimensional case, the following
+equivalent formulations of the Poisson equation are intro-
+duced:
+1. φ is a solution of the weak form
+⟨∇ψ, ∇φ⟩ = ⟨ψ, h − ¯h⟩
+∀ ψ ∈ H1
+0(ρ).
+(23)
+2. For some chosen positive ϵ, φ is a solution of the fixed-
+point equation
+φ = eϵ∆ρφ +
+� ϵ
+0
+es∆ρ(h − ¯h)ds.
+(24)
+The notation eϵ∆ρ is used to denote the semigroup as-
+sociated with ∆ρ (Bakry et al., 2013). The semigroup
+is readily shown to be a Markov operator.
+3. φ is the solution of an optimization problem
+φ = arg min
+f∈H1
+0(ρ)
+1
+2⟨∇f, ∇f⟩ + ⟨f, h − ¯h⟩.
+(25)
+Each of the three formulations has been used to develop
+numerical algorithms for gain function approximation. A
+review of the resulting constructions appears in the follow-
+ing three subsections:
+4.2. Galerkin and constant gain approximation
+The starting point is the weak form (23). A relaxation is
+considered whereby ψ ∈ S = span{ψ1, . . . , ψM}, a finite-
+dimensional subspace of H1
+0(ρ). The functions ψ1, . . . , ψM
+need to be picked and are referred to as the basis functions.
+The resulting algorithm is referred to as the Galerkin algo-
+rithm (Yang et al., 2016, Sec 3.3). The algorithm is given
+in Table 4.2.
+11
+
+Algorithm 1 Synthesis of the gain function: Galerkin
+approximation
+Input: {Xi}N
+i=1, {h(Xi)}N
+i=1, basis functions {ψl(x)}L
+l=1.
+Output: {Ki}N
+i=1.
+1: Calculate h(N) = 1
+N
+�N
+i=1 h(Xi
+t).
+2: Calculate bk = 1
+N
+�N
+i=1(h(Xi
+t) − h(N))ψk(Xi
+t).
+3: Calculate Akl = 1
+N
+�N
+i=1 ∇ψl(Xi
+t)T∇ψk(Xi
+t).
+4: Solve the linear matrix equation Aκ = b for κ, where
+A = [Akl] and b = [bk].
+5: Ki = �L
+l=1 κl∇ψl(Xi
+t).
+Algorithm 2 Synthesis of the gain function: constant
+gain approximation
+Input: {Xi}N
+i=1, {h(Xi)}N
+i=1.
+Output: {Ki}N
+i=1.
+1: Calculate ˆh(N) = 1
+N
+�N
+i=1 h(Xi
+t).
+2: Ki = 1
+N
+�N
+j=1 Xj
+t
+�
+h(Xj
+t ) − ˆh(N)�
+The most important special case of the Galerkin al-
+gorithm is obtained upon picking S to be the subspace
+spanned by the d coordinate functions {x1, x2, . . . , xd}.
+The special case yields the constant gain approximation
+of the gain K as its expected value. Remarkably, the ex-
+pected value admits a closed-form expression which is then
+readily approximated empirically using the particles:
+K(cnst. apprx.) :=
+�
+∇φ(x)ρ(x)dx =
+�
+(h(x) − ¯h)xρ(x)dx
+≈ 1
+N
+N
+�
+i=1
+(h(Xi) − h(N))Xi,
+(26)
+where h(N) := N −1 �
+i h(Xi).
+(See Fig. 2 for an illus-
+tration of the constant gain approximation.)
+With the
+constant gain approximation, the FPF algorithm is a non-
+linear EnKF algorithm (Taghvaei et al., 2018).
+While
+its derivation starting from an FPF is novel, the for-
+mula (26) has been used as a heuristic in the EnKF liter-
+ature (Evensen, 2006; Bergemann and Reich, 2012).
+The main issue with the Galerkin approximation is that
+it is in general very difficult to pick the basis functions.
+There have been a number of studies to refine and improve
+upon this formula (Yang et al., 2016, 2013b; Berntorp and
+Grover, 2016; Matsuura et al., 2016; Radhakrishnan et al.,
+2016; Radhakrishnan and Meyn, 2018; Berntorp, 2018). In
+the following two subsections, we describe two approxima-
+tions which appear to be more promising approaches in
+general settings.
+4.3. Diffusion map-based algorithm
+The starting point is the fixed-point equation (24) based
+on the Markov semigroup eϵ∆ρ. For small values of ϵ, there
+is a well known approximation of eϵ∆ρ in terms of the so-
+called diffusion map (which too is a Markov operator):
+(Tϵf)(x) :=
+1
+nϵ(x)
+�
+Rd
+gϵ(|x − y|)
+��
+gϵ(|y − z|)ρ(z)dz
+f(y)ρ(y)dy,
+(27)
+where gϵ(z) := e− z2
+4ϵ is the Gaussian kernel in R and nϵ(x)
+is the normalization factor chosen so that
+�
+(Tϵ1)(x)dx =
+1 (Coifman and Lafon, 2006). A representative approxi-
+mation result is as follows:
+Proposition 4.1 (Prop. 3.4 in (Taghvaei et al., 2020)).
+Let n ∈ N, t0 < ∞, and t ∈ (0, t0) with ϵ = t
+n. Then, for
+all functions f such that f, ∇f ∈ L4(ρ):
+∥(T n
+t
+n − et∆ρ)f∥L2(ρ) ≤ t
+√
+t
+n C(∥f∥L4(ρ) + ∥∇f∥L4(ρ)),
+where the constant C depends only on t0 and ρ.
+Because the diffusion map (27) is defined using Gaussian
+kernels, its empirical approximation is straightforward:
+(T (N)
+ϵ
+f)(x) =
+1
+n(N)
+ϵ
+(x)
+N
+�
+i=1
+gϵ(|x − Xi|)
+��N
+j=1 gϵ(|Xi − Xj|)
+f(Xi),
+where n(N)
+ϵ
+(x) is the normalization factor. The nature of
+the approximation is as follows:
+Proposition 4.2 (Prop. 3.5 in Taghvaei et al. (2020)).
+Consider the diffusion map kernel Tϵ and its empirical ap-
+proximation {T (N)
+ϵ
+}N∈N. Then for any bounded continu-
+ous function f ∈ Cb(Rd):
+1. (Almost sure convergence) For all x ∈ Rd
+lim
+N→∞(T (N)
+ϵ
+f)(x) = (Tϵf)(x),
+a.s.
+2. (Convergence rate) For any δ ∈ (0, 1), in the asymp-
+totic limit as N → ∞,
+�
+|(T (N)
+ϵ
+f)(x) − (Tϵf)(x)|2ρ(x)dx ≤ O(log( N
+δ )
+Nϵd ),
+with probability higher than 1 − δ.
+With
+these
+approximations,
+the
+fixed-point
+equa-
+tion (24) is approximated in two steps:
+1. The semigroup eϵ∆ρ is approximated by the diffusion
+map Tϵ:
+(step 1)
+φϵ = Tϵφϵ + ϵ(h − ¯hϵ),
+(28a)
+where
+¯hϵ
+=
+�
+h(x)ρ(ϵ)(x)dx
+with
+ρ(ϵ)(x)
+=
+nϵ(x)ρ(x)
+�
+nϵ(x)ρ(x)dx.
+2. Tϵ is approximated by its empirical approximation
+T (N)
+ϵ
+:
+(step 2)
+φ(N)
+ϵ
+= T (N)
+ϵ
+φ(N)
+ϵ
++ ϵ(h − ¯h(N)
+ϵ
+),
+(28b)
+where ¯h(N)
+ϵ
+=
+�
+h(x)ρ(N)
+ϵ
+(x)dx with ρ(N)
+ϵ
+(x)
+=
+�N
+i=1 nϵ(x)δXi
+�N
+i=1 nϵ(Xi) .
+12
+
+3
+2
+1
+0
+1
+2
+3
+x
+0
+2
+4
+6
+8
+10
+K
+const. gain
+exact
+= 0.02
+= 0.10
+= 0.50
+= 2.00
+(a)
+variance
+dominates
+bias
+dominates
+diffusion map
+constant gain
+(b)
+Figure 3: Bias variance trade-off in the diffusion map-based gain function approximation algorithm: (a) Gain function computed for different
+values of ϵ with N = 200 particles. The dashed line is the constant gain solution (26). As ϵ gets larger, the diffusion map gain converges to
+the constant gain. (b) Plot of the MSE as a function of ϵ. The shaded area in the background of part (a) is the density ρ which is taken as
+sum of two Gaussians N(−1, σ2) and N(+1, σ2) with σ2 = 0.2. The exact gain function K(x) is computed for h(x) = x by using an (exact)
+integral formula forr the solution (Taghvaei et al., 2020, Eq. 4.6). In part (b), the MSE is computed as an empirical approximation of the
+lefthand-side of (29) by averaging over 1000 simulation runs.
+Algorithm 3 Synthesis of the gain function: diffusion
+map-based algorithm
+Input: {Xi}N
+i=1, {h(Xi)}N
+i=1, Φprev, ϵ, L.
+Output: {Ki}N
+i=1.
+1: Calculate gij := e− |Xi−Xj |2
+4ϵ
+for i, j = 1 to N.
+2: Calculate kij :=
+gij
+√�
+l gil√�
+l gjl for i, j = 1 to N.
+3: Calculate di = �
+j kij for i = 1 to N.
+4: Calculate Tij := kij
+di for i, j = 1 to N.
+5: Calculate πi =
+di
+�
+j dj for i = 1 to N.
+6: Calculate ˆh = �N
+i=1 πjh(Xi).
+7: Initialize Φ = Φprev.
+8: for t = 1 to L do
+9:
+Φi = �N
+j=1 TijΦj + ϵ(h − ˆh) for i = 1 to N.
+10: end for
+11: Calculate ri = Φi + ϵhi for i = 1 to N.
+12: Calculate sij =
+1
+2ϵTij(rj − �N
+k=1 Tikrk) for i, j = 1 to
+N.
+13: Calculate Ki = �
+j sijXj for i = 1 to N.
+Based
+on
+the
+finite-dimensional
+fixed-point
+equa-
+tion (28b), an algorithm for gain function approximation
+is given in Table 3.
+Error analysis. The error in diffusion map approximation
+comes from two sources:
+1. The bias error due to the diffusion map approximation
+of the semigroup (step 1); and
+2. The variance error due to empirical approximation in
+terms of particles (step 2).
+The error is analyzed in (Taghvaei et al., 2020) where the
+following result is proved:
+Proposition 4.3 (Thm. 4.3 and 4.4 in (Taghvaei et al.,
+2020) ). Consider the fixed-point formulation of the Pois-
+son equation (24), its diffusion-map approximation (28a),
+and its empirical approximation (28b).
+1. For each fixed ϵ > 0, there exists a unique solu-
+tion to (28a) with a uniform bound ∥φϵ∥L2(ρϵ) ≤
+C∥h∥L2(ρϵ). In the asymptotic limit as ϵ → 0
+∥φϵ − φ∥L2(ρϵ) ≤ O(ϵ).
+2. The operator T (N)
+ϵ
+is a strict contraction on L2
+0(ρ(N)
+ϵ
+)
+and the fixed-point equation (28b) admits a unique
+solution. The approximate solution φ(N)
+ϵ
+converges to
+the kernel solution φϵ
+lim
+N→∞ ∥φ(N)
+ϵ
+− φϵ∥L∞(Ω) = 0,
+a.s.
+The following diagram illustrates the convergence and
+the respective types of errors:
+φ(N)
+ϵ
+N↑∞
+−→
+(variance) φϵ
+ϵ↓0
+−→
+(bias) φ.
+A quantitative bound on the mean-squared error (MSE)
+is obtained in the asymptotic limit as ϵ ↓ 0 and N → ∞
+as follows:
+�
+E[ 1
+N
+N
+�
+i=1
+|Ki − ∇φ(Xi)|2]
+�
+�
+��
+�
+MSE
+≤ O(ϵ2)
+� �� �
+bias
++ O(
+1
+ϵ(2+d)N )
+�
+��
+�
+variance
+,
+(29)
+where {Ki}N
+i=1 is computed from the Algorithm (Table 3)
+and ∇φ is the exact gain function from solving the Poisson
+13
+
+const. gain
+-0.16d - 0.3
+o( )
+m.s.e
+(a)
+const. gain
+m.s.e
+(b)
+Figure 4: Bias-variance trade-off as a function of (a) the state dimension d ∈ {1, 2, 5, 10} (for a fixed N = 1000); and (b) the number of
+particles N ∈ {100, 200, 500, 1000} (for a fixed d = 1). In the vector case, ρ(x) = ρb(x1) �d
+n=2 ρg(xn) where ρb is the bimodal density (same
+as in Fig. 3) and ρg is the Gaussian density.
+equation (12). The error due to bias converges to zero as
+ϵ → 0 and the error due to variance converges to zero as
+N → ∞. There is trade-off between the two errors: To
+reduce bias, one must reduce ϵ. However, for any fixed
+value of N, one can reduce ϵ only up to a point where
+the variance starts increasing. The bais-variance trade-off
+is illustrated with the aid of a scalar (d = 1) example in
+Fig. 3: If ϵ is large, the error due to bias dominates, while
+if ϵ is small, the error due to variance dominates.
+An
+numerical illustration of scalings with N and d appears in
+Fig. 4. Additional details on both these examples can be
+found in (Taghvaei et al., 2020, Sec. 5).
+Remark 4.4 (Relationship to the constant gain for-
+mula (26)). There is a remarkable and somewhat unex-
+pected relationship between the diffusion map and the con-
+stant gain approximation (Taghvaei et al., 2020, Prop.
+4.7). In particular, in the limit as ϵ → ∞, the diffusion
+map gain converges to the constant gain (26). This sug-
+gests a systematic procedure to improve upon the constant
+gain by de-tuning the value of ϵ away from the [ϵ = ∞]
+limit. For any fixed N, a finite value of ϵ is chosen to
+minimize the MSE according to the bias variance trade-
+off.
+Based on this, a rule of thumb for choosing the ϵ
+value appears in (Taghvaei et al., 2020, Remark 5.1).
+Remark 4.5 (Analysis of FPF with diffusion map approx-
+imation). An analysis of the finite-N FPF using the diffu-
+sion map approximation appears in (Pathiraja and Stan-
+nat, 2021). Under mild technical conditions on the drift
+a(·), σ(·), h(·), it is shown that the finite-N FPF is well-
+posed, i.e., a strong solution exists for all time t (Pathiraja
+and Stannat, 2021, Thm. 1.1). Based on a propagation of
+chaos type analysis, convergence estimates are derived to
+relate the finite-N system to its mean-field limit (Pathiraja
+and Stannat, 2021, Thm. 1.2). These estimates are shown
+to hold up to a certain stopping time. For arbitrary time t,
+well-posedness and convergence remains an open problem.
+4.4. Variational approximation
+The starting point is the variational form (25).
+The
+objective function is denoted by J(f) with its empirical
+approximation is obtained as
+J(N)(f) := 1
+N
+N
+�
+i=1
+1
+2|∇f(Xi)|2 − f(Xi)(h(Xi) − h(N))
+The problem of minimizing the empirical approximation
+over all functions is ill-posed: the minimum is unbounded
+and minimizer does not exist. (Abstractly, this is because
+the empirical probability distribution does not satisfy the
+Poincar´e inequality.) Therefore, we consider
+min
+fθ∈FΘ J(N)(fθ)
+where FΘ is a parameterized class of functions. A function
+in the class FΘ is denoted by fθ(x) or f(x; θ) where θ ∈ Θ
+is the parameter, and Θ is the parameter set. The two
+main examples are as follows:
+1. FΘ
+=
+{�M
+j=1 θjψj;
+ψj
+∈
+H1
+0, θj
+∈
+R for j
+=
+1, . . . , M} is a linear combination of selected basis
+functions. With a linear parametrization, the solution
+of the empirical optimization problem is given by the
+Galerkin algorithm (Yang et al., 2016, Remark 5).
+2. FΘ is a neural network where the parameters θ are
+the weights in the network.
+In practice, it is not possible to solve the optimization
+problem exactly, but up to some optimization gap. In par-
+ticular, let φ(N)
+θ
+be the output of an optimization algorithm
+that solves the problem up to ϵ error, i.e.,
+J(φ(N)
+θ
+) ≤ min
+f∈H1
+0
+J(f) + ϵ.
+The good news is that it is possible to upper-bound the
+error in approximating the gain function in terms of this
+optimization gap.
+14
+
+3
+2
+1
+0
+1
+2
+3
+x
+0
+1
+2
+3
+4
+5
+6
+7
+exact
+Iter = 100
+Iter = 200
+Iter = 300
+Iter = 1000
+0
+200
+400
+600
+800
+1000
+iterations
+10
+1
+100
+101
+optimization gap
+Figure 5: Results of the variational gain function approximation using a neural network parameterization: Plot of (a) the gain function; and
+(b) the optimization gap as the number of iterations of the Adam algorithm. The problem setup is the same as Fig. 3.
+Proposition 4.6 (Prop. 1 in Olmez et al. (2020)). Let
+K(N)
+θ
+= ∇φ(N)
+θ
+where φ(N)
+θ
+is the output of an optimization
+algorithm that solves the minimization objective J(f) with
+ϵ optimality gap. Then
+∥K(N)
+θ
+− K∥2
+L2ρ ≤ 2ϵ,
+where K = ∇φ is the exact gain function.
+The
+optimization
+gap
+ϵ
+depends
+on
+the
+selected
+parametrization Fθ, number of particles N, and the it-
+eration number of the employed optimization algorithm.
+Its characterization and analysis is open and the subject
+of ongoing work. In general, such analysis falls under the
+framework of statistical learning theory (Anthony et al.,
+1999; Shalev-Shwartz and Ben-David, 2014).
+The numerical results using this approach are depicted
+in Fig. 5. These results are for the bimodal example in-
+troduced in Fig. 3.
+The gain function is parameterized
+using a two-layer residual NN with 32 neurons per layer.
+The Adam algorithm is used to learn the parameters of
+the NN. Additional details on the numerics can be found
+in (Olmez et al., 2020).
+5. Optimal transport theory
+In this section, we describe a systematic procedure to
+construct the exact mean-field process ¯X introduced as
+step 1 in (10). The first aspect to note is that while the
+FPF (11) provides an explicit formula for u and K, the
+formula is not unique: One can interpret (10) as trans-
+porting the prior density p0 at time t = 0 to the posterior
+density pt at time t.
+Clearly, there are infinitely many
+maps that transport one density into another. This sug-
+gests that there are infinitely many choices of control laws
+that all lead to exact filters. This is not surprising: The
+exactness condition specifies only the marginal density at
+times t, which is not enough to uniquely identify a stochas-
+tic process, e.g., the joint density at two time instants has
+not been specified.
+In the following, we first discuss the non-uniqueness
+issue for the simpler linear Gaussian model.
+The non-
+uniqueness naturally motivates optimal transport ideas to
+uniquely solve for u and K. This is the subject of the re-
+mainder of this section to derive the feedback control law
+for the FPF (11).
+5.1. Non-uniqueness issue in linear-Gaussian setting
+Consider the linear Gaussian FPF (14) for the mean-
+field process { ¯Xt}t≥0. The conditional mean and variance
+of ¯Xt are denoted by ¯mt and ¯Σt, respectively. The condi-
+tional mean evolves according to
+d ¯mt = A ¯mtdt + ¯Kt(dZt − H ¯mtdt),
+where ¯Kt := ¯ΣtH T. Define an error process ξt := ¯Xt − ¯mt.
+Its equation is given by
+dξt = (A − 1
+2
+¯ΣtH
+TH)ξt + σBd ¯Bt.
+This is a linear system and therefore the variance of ξt,
+which equals ¯Σt (by definition), evolves according to the
+Lyapunov equation
+d
+dt
+¯Σt = (A − 1
+2
+¯ΣtH
+TH)¯Σt + ¯Σt(A − 1
+2
+¯ΣtH
+TH)
+T + ΣB
+= Ricc(¯Σt).
+The derivation helps show that the equations for the
+mean and variance are identical to the Kalman filter equa-
+tions, (8a) and (8b), respectively, and thus proves the ex-
+actness property of the linear FPF (14).
+These arguments suggest the following general proce-
+dure to construct an exact ¯X process: Express ¯Xt as a
+sum of two terms:
+¯Xt = ¯mt + ξt,
+t ≥ 0,
+15
+
+where ¯mt evolves according to (8a) and the evolution of ξt
+is defined by the SDE:
+dξt = Gtξtdt + σtd ¯Bt + σ′
+td ¯Wt,
+where { ¯W}t≥0 and { ¯B}t≥0 are independent copies of the
+measurement noise {W}t≥0 and the process noise {B}t≥0,
+respectively, and Gt, σt, and σ′
+t satisfy the matrix equation
+(for each time)
+Gt ¯Σt + ¯ΣtGT
+t + σtσ
+T
+t + σ′
+t(σ′
+t)
+T = Ricc(¯Σt),
+t ≥ 0. (30)
+By construction, the equation for the variance is given by
+the Riccati equation (8b). The result is summarized in the
+following Proposition:
+Proposition 5.1 (Prop.
+1 in Taghvaei et al. (2022)).
+Consider the linear-Gaussian filtering problem (7) and the
+following family of the mean-field processes
+d ¯Xt = A ¯mtdt + ¯Kt(dZt − H ¯mtdt)
++ Gt( ¯Xt − ¯mt)dt + σtd ¯Bt + σ′
+td ¯Wt, ¯X0 ∼ N(m0, Σ0),
+where Gt, σt, and σ′
+t satisfy the consistency condition (30).
+Then,
+¯Xt is exact, i.e.
+the density of
+¯Xt is Gaussian
+N( ¯mt, ¯Σt) where ¯mt and ¯Σt solve the Kalman filter equa-
+tions, (8a) and (8b), respectively.
+In general, with different choices of σt and σ′
+t, there are
+infinitely many solutions for (30). Below, we describe three
+solutions that lead to three established form of EnKF and
+linear FPF:
+1. EnKF with perturbed observation (Reich, 2011, Eq.
+(27)):
+Gt = A − ¯ΣtH
+TH,
+σt = σB,
+σ′
+t = ¯ΣtH
+T.
+2. Stochastic linear FPF (Yang et al., 2016, Eq. (26)) or
+square-root form of the EnKF (Bergemann and Reich,
+2012, Eq (3.3)) :
+Gt = A − 1
+2
+¯ΣtH
+TH,
+σt = σB,
+σ′
+t = 0.
+3. Deterministic linear FPF (Taghvaei and Mehta, 2016,
+Eq. (15)) (de Wiljes et al., 2018, Eq. (82)):
+Gt = A − 1
+2
+¯ΣtH
+TH + 1
+2ΣB ¯Σ−1
+t ,
+σt = 0,
+σ′
+t = 0.
+Fix σt, σ′
+t. Then given any particular solution Gt of (30),
+one can construct a family of solutions Gt + ¯Σ−1
+t Ωt, where
+Ωt is any arbitrary skew-symmetric matrix (Taghvaei and
+Mehta, 2020, Sec. III-B). For the linear Gaussian problem,
+the non-uniqueness issue is well known in literature: The
+two forms of EnKF, the perturbed observation form (Re-
+ich, 2011) and the square-root form (Bergemann and Re-
+ich, 2012) are standard. A homotopy of exact determinis-
+tic and stochastic EnKFs is given in (Kim et al., 2018). An
+explanation for the non-uniqueness in terms of the Gauge
+transformation appears in (Abedi and Surace, 2019). An
+extension to the case with correlated noise appears in Kang
+et al. (2022).
+Given the non-uniqueness issue, a natural question is
+how to identify a unique ¯X process? For this purpose, opti-
+mal transport theory is described in the following Sec. 5.2.
+For the linear Gaussian case, the theory is used to derive
+the following optimal transport form of the linear FPF
+(see (Taghvaei and Mehta, 2016, 2020) for details):
+d ¯Xt =A ¯Xtdt + 1
+2ΣB ¯Σ−1
+t
+( ¯Xt − ¯mt)dt
++ 1
+2
+¯Kt(dZt − H ¯Xt + H ¯mt
+2
+dt) + Ωt ¯Σ−1
+t ( ¯Xt − ¯mt)dt,
+(31)
+where Ωt = ΩOPT
+t
+is a specific skew-symmetric matrix.
+The optimal transport FPF (31) is exact and has two dif-
+ferences compared to the linear FPF (14):
+1. The stochastic term σBd ¯Bt is replaced with the de-
+terministic term 1
+2ΣB ¯Σ−1
+t ( ¯Xt − ¯mt)dt. Given a Gaus-
+sian prior, the two terms yield the same posterior.
+However, in a finite-N implementation, the stochastic
+term serves to introduce an additional error of order
+O(
+1
+√
+N ) (Taghvaei and Mehta, 2018, Prop. 4).
+2. The SDE (31) has an extra term involving the skew-
+symmetric matrix Ωt. The extra term does not effect
+the posterior, i.e., ¯X is exact for all skew-symmetric
+choices of Ωt. The specific optimal choice Ωt = ΩOPT
+t
+serves to pick the symmetric solution Gt of the consis-
+tency equation (30). For the scalar (d = 1) case, the
+skew-symmetric term is zero. Therefore, in the scalar
+case, the update formula in the linear FPF (14) is op-
+timal. In the vector case, it is optimal iff ΩOPT
+t
+≡ 0.
+5.2. FPF formula
+In this section, we provide a justification for the feedback
+control formula in the FPF (11). It is helpful to begin with
+the simpler deterministic case.
+5.2.1. Deterministic path
+Let P2(Rd) be the space of everywhere positive proba-
+bility densities on Rd with finite second moment. Given
+a smooth path {pt ∈ P2(Rd) : t ≥ 0} the problem is to
+construct a stochastic process { ¯Xt}t≥0 such that the prob-
+ability density of ¯Xt, denoted as ¯pt, equals pt for all t ≥ 0.
+The exactness condition is expressed as
+¯pt = pt,
+∀ t ≥ 0.
+(32)
+As has already been noted, there are infinitely many
+stochastic processes that satisfy the exactness condition.
+A unique choice is made by prescribing an additional op-
+timality criterion based on the optimal transport theory.
+To make these considerations concrete, assume that the
+given path {pt}t≥0 evolves according to the PDE
+∂pt
+∂t = V(pt),
+t > 0,
+16
+
+where V(·) is an operator (e.g., the Laplacian) that acts
+on probability densities. (This necessarily restricts the op-
+erator V, e.g.,
+�
+V(ρ)(x)dx = 0 for all ρ ∈ P2(Rd).) The
+following model is assumed for the process { ¯Xt}t≥0:
+d
+dt
+¯Xt = ut( ¯Xt),
+¯X0 ∼ p0,
+(33)
+where ut(·) is a control law that needs to be designed.
+From the continuity equation, the exactness condition (32)
+is satisfied if
+− ∇ · (¯ptut) = V(¯pt),
+∀ t > 0.
+(34)
+The non-uniqueness issue is now readily seen: The first-
+order PDE (34) admits infinitely many solutions. A unique
+solution ut(·) is picked by minimizing the transportation
+cost from ¯Xt to ¯Xt+∆t in the limit as ∆t → 0. The L2-
+Wasserstein cost is particularly convenient because
+lim
+∆t→0
+1
+∆t2 E[|Xt+∆t − Xt|2] =
+�
+Rd |ut(x)|2¯pt(x)dx.
+Therefore, for each fixed t, the control law ut(·) is obtained
+by solving the constrained optimization problem
+min
+ut(·)
+�
+Rd |ut(x)|2¯pt(x)dx,
+s.t
+− ∇ · (¯ptut) = V(¯pt).
+By a standard calculus of variation argument, the opti-
+mal solution is obtained as u∗
+t = ∇φt where φt solves the
+Poisson equation −∇ · (¯pt∇φt) = V(¯pt).
+The resulting
+stochastic process ¯X is defined by
+d ¯Xt
+dt
+= ∇φt( ¯Xt),
+¯X0 ∼ p0,
+φt solves the PDE − ∇ · (¯pt∇φt) = V(¯pt).
+The process is exact by construction.
+Example 5.2. Suppose the given path is a solution of
+the heat equation
+∂pt
+∂t
+= ∆pt (V(·) is the Laplacian).
+The solution of the Poisson equation is easily obtained as
+φt = log(¯pt). The optimal transport process then evolves
+according to
+d
+dt
+¯Xt = −∇ log(¯pt( ¯Xt)),
+¯X0 ∼ p0.
+(35a)
+This process should be compared to the well known example
+dXt = dBt,
+X0 ∼ p0,
+(35b)
+where {Bt}t≥0 is a W.P.. The density for Xt also solves
+the heat equation. In the language of optimal transporta-
+tion theory, the coupling defining (35a) is deterministic
+while it is stochastic in (35b).
+5.2.2. Stochastic path
+In the filtering problem, the path of the posterior prob-
+ability density is stochastic (because it depends upon the
+random observations {Zt}t≥0). Therefore, the preceding
+discussion is not directly applicable. Suppose the stochas-
+tic path {pt}t≥0 is governed by a stochastic PDE
+dpt = H(pt)dIt,
+where H(·) is an operator that acts on probability densities
+and {It : t ≥ 0} is a W.P..
+Consider the following SDE model:
+d ¯Xt = ut( ¯Xt)dt + Kt( ¯Xt)dIt,
+¯X0 ∼ p0
+where, compared to the deterministic model (33), an addi-
+tional stochastic term is now included. The problem is to
+identify control laws ut(·) and Kt(·) such that the condi-
+tional density of ¯Xt equals pt. Upon writing the evolution
+equation for the conditional density of ¯Xt (Yang et al.,
+2016, Prop. 1), the exactness condition is formally satis-
+fied by all such ut(·) and Kt(·) that solve
+− ∇ · (¯ptKt) = H(¯pt),
+(36a)
+− ∇ · (¯ptut) + 1
+2(∇ · (¯ptKt)Kt + ¯ptKt∇Kt) = 0.
+(36b)
+These equations are the stochastic counterpart of (34), and
+as with (34), their solution is not unique.
+The unique solution is obtained by requiring that the
+coupling from ¯Xt and ¯Xt+∆t is optimal in the limit as
+∆t → 0. In contrast to the deterministic setting, the lead-
+ing term in the transportation cost E[| ¯Xt+∆t − ¯Xt|2] is
+O(∆t) whereby
+lim
+∆t→0
+1
+∆tE[| ¯Xt+∆t − ¯Xt|2] =
+�
+Rd |Kt(x)|2¯pt(x)dx.
+Therefore, for each fixed t, the control law Kt(·) is obtained
+by solving the constrained optimization problem
+min
+Kt(·)
+�
+Rd |Kt(x)|2¯pt(x)dx,
+s.t
+− ∇ · (¯ptKt) = Ht(¯pt).
+As before, the optimal solution is given by K∗
+t = ∇φt where
+φt solves the second-order PDE −∇ · (¯pt∇φt) = H(¯pt).
+It remains to identify the control law ut(·). For this pur-
+pose, the second-order term in the infinitesimal Wasser-
+stein cost is used:
+lim
+∆t→0
+1
+∆t2
+�
+E[| ¯Xt+∆t − ¯Xt|2] − ∆t
+�
+Rd |K∗
+t (x)|2¯pt(x)dx
+�
+=
+�
+Rd |ut(x)|2¯pt(x)dx.
+The righthand-side is minimized subject to the con-
+straint (36b).
+Remarkably, the optimal solution is ob-
+tained in closed form as
+u∗
+t = − 1
+2¯pt
+H(¯pt)∇φt + 1
+2∇2φt∇φt + ξt,
+17
+
+where ξt is the (unique such) divergence free vector field
+(i.e., ∇ · (ptξt) = 0) such that u∗
+t is of a gradient form.
+That (36b) can be solved in an explicit manner was a major
+surprise at the time of its discovery (Yang et al., 2011b,
+2013b). The resulting optimal transport process is
+d ¯Xt = ∇φt( ¯Xt) ◦ (dIt − 1
+2¯pt
+H(¯pt)dt) + ξt( ¯Xt)dt, ¯X0 ∼ p0.
+(37)
+It is also readily shown that the process { ¯Xt}t≥0 is in
+fact exact for any choice of divergence free vector field
+{ξt}t≥0. The most convenient such choice is to simply set
+ξt ≡ 0. The resulting filter is exact and furthermore also
+(infinitesimally) optimal to the first-order.
+For the special case of the nonlinear filtering prob-
+lem, H(ρ) = (h − ¯h)ρ where ¯h =
+�
+h(x)ρ(x)dx and
+dIt = (dZt − ¯htdt) is the increment of the innovation pro-
+cess. For these choices, the optimal transport stochastic
+process (37) becomes
+d ¯Xt = ∇φt( ¯Xt) ◦ (dZt − 1
+2(h( ¯Xt) + ¯ht)dt) + ξt( ¯Xt)dt.
+The feedback control law in the FPF algorithm (11) repre-
+sents the particular sub-optimal choice ξt ≡ 0. The choice
+is optimal for d = 1.
+5.3. Optimal transport formula for the static example
+We now revisit the static example introduced in Sec. 3.1
+with the aim of deriving an explicit form of the control U
+and relating it to the FPF. As explained in Sec. 3.1, the
+problem is to find a control U such that E[f(X)|Y ] =
+E[f( ¯X1)|Y ] for all functions f ∈ Cb(Rd), where ¯X1 =
+¯X0 + U and ¯X0 is an independent copy of X. This con-
+dition is equivalently expressed as ( ¯X1, Y ) ∼ PXY , and
+the problem of finding U is formulated as the following
+optimal transportation problem:
+min
+U∈σ( ¯
+X0,Y ) E[|U|2],
+s.t
+¯X1 = ¯X0 + U,
+( ¯X1, Y ) ∼ PXY ,
+(38)
+where the notation U ∈ σ( ¯X0, Y ) means that U is allowed
+to be measurable with respect to ¯X0 and Y . This is an op-
+timal transportation problem between ( ¯X0, Y ) ∼ PX ⊗PY
+and (X, Y ) ∼ PXY where the transportation is constrained
+to be of the form ( ¯X0, Y ) → ( ¯X0 + U, Y ), i.e., the second
+argument Y remains fixed. Its solution is obtained as an
+extension of the celebrated Brenier’s result (Brenier, 1991)
+as follows:
+Theorem 5.3 (Thm. 1 in Taghvaei and Hosseini (2022)).
+Consider the optimal transportation problem (38). Sup-
+pose PX admits a density with respect to the Lebesgue mea-
+sure. Then the optimal control is
+U = ∇¯Φ( ¯X0; Y ) − ¯X0,
+where ¯Φ is the minimizer of the dual Kantorovich problem
+min
+Φ∈CVXx E[Φ( ¯X0; Y ) + Φ⋆(X; Y )],
+(39)
+where Φ ∈ CVXx means x �→ Φ(x; y) is convex in x for all
+y and Φ⋆(x; y) := supz zTx − Φ(z; y) is the convex conju-
+gate of Φ with respect to x.
+Remark 5.4 (Relationship to the update formula for
+FPF). In the continuous-time limit, the dual Kantorovich
+problem (39) is related to the variational form (25) of
+the Poisson equation (12).
+In particular, with ∆Zt =
+h(Xt)∆t + ∆Wt, the solution to the problem (39) is as
+follows (Taghvaei and Hosseini, 2022, Prop. 2):
+¯Φ( ¯Xt; ∆Zt) = 1
+2| ¯Xt|2 + φ( ¯Xt)∆Zt + ψ( ¯Xt)∆t + O(∆t2)
+where φ is the solution to the Poisson equation (12) with
+ρ taken as the density of PX, and ψ is the unique such
+function such that ∇ψ = − h+¯h
+2 ∇φ + 1
+4∇|∇φ|2 + ξ where ξ
+is divergence free. Therefore, the optimal transformation
+¯Xt �→ ¯Xt+∆t is given by,
+¯Xt+∆t = ∇x ¯Φ( ¯Xt; ∆Zt)
+= ¯Xt + ∇φ( ¯Xt)(∆Zt − h( ¯Xt) + ¯ht
+2
+∆t)
++ 1
+4∇|∇φ( ¯Xt)|2∆t + ξ( ¯Xt)∆t + O(∆t2)
+which in the limit as ∆t → 0 is the SDE for the optimal
+transport FPF (37).
+Remark 5.5 (Stochastic optimization and DNNs). The
+variational problem (39) is a stochastic optimization prob-
+lem which allows for application of machine learning tools
+to approximate its solution. In particular, deep neural net-
+works (DNNs) can be used to parameterize the function Φ
+and stochastic optimization algorithms employed to learn
+the parameters. Preliminary results in this direction are
+presented in (Taghvaei and Hosseini, 2022) with a com-
+prehensive development the subject of ongoing work.
+PART II
+6. CIPS for optimal control
+In order to elucidate the ideas as clearly as possible, our
+focus in this paper is entirely on the linear quadratic (LQ)
+problem. Its extension to the nonlinear optimal control
+problem (4) can be found in (Joshi et al., 2022).
+6.1. Problem statement and background
+The finite-horizon linear quadratic (LQ) optimal control
+problem is a special case of (4) as follows:
+18
+
+min
+u
+J(u) =
+� T
+0
+1
+2
+�
+|Cxt|2 + u
+T
+t Rut
+�
+dt + x
+T
+T PT xT
+(40a)
+subject to:
+˙xt = Axt + But,
+x0 = x
+(40b)
+It is assumed that (A, B) is controllable, (A, C) is observ-
+able, and matrices PT , R ≻ 0. The [T = ∞] limit is re-
+ferred to as the linear quadratic regulator (LQR) problem.
+It is well known that the optimal control ut = φt(xt)
+where the optimal policy is linear
+φt(x) = Ktx
+where
+Kt = −R−1B
+TPt,
+0 ≤ t ≤ T
+is the optimal gain matrix and {Pt : 0 ≤ t ≤ T} is a
+solution of the backward (in time) DRE
+− d
+dtPt = A
+TPt+PtA+C
+TC−PtBR−1B
+TPt,
+PT (given)
+(41)
+The ARE is obtained by setting the left-hand side to 0.
+As T → ∞, for each fixed time t, Pt → P ∞, expo-
+nentially fast (Kwakernaak and Sivan, 1972, Thm. 3.7),
+where P ∞ ≻ 0 is the unique such positive-definite solu-
+tion of the ARE, and therefore the optimal gain converges,
+Kt → K∞ := −R−1BTP ∞. Approximation of the gain
+K∞ is a goal in recent work on model-based RL for the
+LQR problem (Fazel et al., 2018; Mohammadi et al., 2022).
+6.2. Objectives and assumptions
+For the reasons noted in Sec. 1, we are interested in a
+simulation-based solution that does not rely on an explicit
+solution of the DRE (41).
+To clarify what is meant by
+a simulation-based solution in the context of model-based
+RL, we make a formal assumption as follows:
+Assumption 1.
+1. Functions f(x, α) = Ax + Bα and
+c(x) = Cx are available in the form of an oracle
+(which allows function evaluation at any state action
+pair (x, α) ∈ Rd × Rm).
+2. Matrices R and PT are available. Both of these ma-
+trices are strictly positive-definite.
+3. Simulator is available to simulate (40b).
+4. Simulator provides for an ability to add additional in-
+puts outside the control channel (e.g., see (5a)).
+This assumption is motivated from the data assimila-
+tion literature where it is entirely standard and widely
+used in applications, such as weather prediction, involving
+EnKF. Part 1 of the assumption means that the matri-
+ces A, B, C are not available explicitly. Rather, for any
+given (x, α) ∈ Rd × Rm, the vectors f(x, α) and c(x) can
+be evaluated. Function evaluation forms for the dynamics
+and the cost function is also a standard assumption for
+any model-based RL algorithm. Part 2 of the assumption
+is not too restrictive for the following two reasons:
+1. In physical systems, one is typically able to assess
+relative costs for different control inputs (actuators).
+This knowledge can be used to select R.
+2. For the LQR problem, under mild technical condi-
+tions, the optimal policy is stationary and does not
+depend upon the choice of PT .
+If these matrices are not available, one possibility is to
+take R and PT to be identity matrices of appropriate di-
+mensions. The main restriction comes from part 3 of the
+assumption. However, as the widespread use of EnKF am-
+ply demonstrates, it is not un-realistic to assume it for a
+simulation-based solution. Of course, it will not be possi-
+ble with a physical experiment.
+6.3. Dual EnKF
+The dual EnKF algorithm is obtained from making use
+of duality between optimal control and filtering. For this
+purpose, we need to first dualize the DRE (41). Under the
+assumptions of this paper, Pt ≻ 0 for 0 ≤ t ≤ T whenever
+PT ≻ 0 (Brockett, 2015, Sec. 24). Set St = P −1
+t
+. It is
+readily verified that {St : 0 ≤ t ≤ T} also solves a DRE
+(which represents the dual of (41))
+d
+dtSt = ASt + StA
+T − BR−1B
+T + StC
+TCSt,
+ST = P −1
+T
+(42)
+The strategy is to approximate {St : 0 ≤ t ≤ T} using
+simulations. As before, the construction proceeds in two
+steps: (i) definition of an exact mean-field process; and (ii)
+its finite-N approximation.
+Step 1. Mean-field process: Define a stochastic process
+¯Y = { ¯Yt ∈ Rd : 0 ≤ t ≤ T} as a solution of the following
+backward (in time) SDE:
+d ¯Yt = A ¯Ytdt + Bd
+�ηt + 1
+2 ¯StC
+T(C ¯Yt + C¯nt)dt, 0 ≤ t < T
+¯YT ∼ N(0, ST )
+(43)
+where η = {ηt ∈ Rm : 0 ≤ t ≤ T} is a W.P. with covariance
+matrix R−1, and
+¯nt := E[ ¯Yt],
+¯St := E[( ¯Yt − ¯nt)( ¯Yt − ¯nt)
+T], 0 ≤ t < T
+(44)
+The meaning of the backward arrow on d
+�η in (43) is that
+the SDE is simulated backward in time starting from the
+terminal condition specified at time t = T. The reader is
+referred to (Nualart and Pardoux, 1988, Sec. 4.2) for the
+definition of the backward Itˆo-integral.
+The mean-field
+process is useful because of the following proposition.
+Proposition 6.1 (Prop. 1 in Joshi et al. (2022)). The
+solution to the SDE (43) is a Gaussian stochastic process,
+in which the mean and covariance of ¯Yt are given by
+¯nt = 0,
+¯St = St,
+0 ≤ t ≤ T
+Consequently, ¯Xt := ¯S−1
+t
+( ¯Yt − ¯nt) is also a Gaussian ran-
+dom variable with
+E[ ¯Xt] = 0,
+E[ ¯Xt ¯X
+T
+t ] = Pt,
+0 ≤ t ≤ T
+19
+
+The significance of Prop. 6.1 is that the optimal control
+policy φt(·) can now be obtained in terms of the statis-
+tics of the random variable ¯Xt. Specifically, we have the
+following two cases:
+1. If the matrix B is explicitly known then the optimal
+gain matrix
+Kt = −R−1B
+TE[ ¯Xt ¯X
+T
+t ]
+2. If
+B
+is
+unknown,
+define
+the
+Hamiltonian
+(the
+continuous-time
+counterpart
+of
+the
+Q-
+function (Mehta and Meyn, 2009)):
+H(x, α, t)
+:= 1
+2|Cx|2 + 1
+2α
+TRα
+�
+��
+�
+cost function
++x
+TE[ ¯Xt ¯X
+T
+t ] (Ax + Bα)
+�
+��
+�
+model (40b)
+from which the optimal control law is obtained as
+φt(x) = arg min
+α∈Rm H(x, α, t)
+by recalling the minimum principle, which states
+that the optimal control is the unique minimizer of
+the Hamiltonian.
+It is noted that the Hamiltonian
+H(x, α, t) is in the form of an oracle because (Ax+Bα)
+is the right-hand side of the simulation model (40b).
+Step 2. Finite-N approximation: The particles {Y i
+t ∈
+Rd : 0 ≤ t ≤ T, i = 1, . . . , N} evolve according to the
+backward SDE:
+dY i
+t =
+AY i
+t dt + Bd
+�η
+i
+t
+�
+��
+�
+i-th copy of model (40b)
++ S(N)
+t
+C
+T
+�
+CY i
+t + Cn(N)
+t
+2
+�
+�
+��
+�
+coupling
+dt,
+(45)
+Y i
+T
+i.i.d
+∼ N(0, P −1
+T ),
+1 ≤ i ≤ N
+ηi := {ηi
+t : 0 ≤ t ≤ T} is an i.i.d copy of η and
+n(N)
+t
+= 1
+N
+N
+�
+i=1
+Y i
+t
+S(N)
+t
+=
+1
+N − 1
+N
+�
+i=1
+(Y i
+t − n(N)
+t
+)(Y i
+t − n(N)
+t
+)
+T
+The CIPS (45) is referred to as the dual EnKF.
+Optimal control: Set Xi
+t = (S(N)
+t
+)−1(Y i
+t − n(N)
+t
+). There
+are two cases as before:
+1. If the matrix B is explicitly known then
+K(N)
+t
+= −
+1
+N − 1
+N
+�
+i=1
+R−1(B
+TXi
+t)(Xi
+t)
+T
+(46)
+2. If B is unknown, define the Hamiltonian
+H(N)(x, α, t) := 1
+2|Cx|2 + 1
+2α
+TRα
+�
+��
+�
+cost function
++
+1
+N − 1
+N
+�
+i=1
+(x
+TXi
+t)(Xi
+t)
+T (Ax + Bα)
+�
+��
+�
+model (40b)
+from which the optimal control policy is approximated
+as
+φ(N)
+t
+(x) = arg min
+a∈Rm H(N)(x, a, t)
+There are several zeroth-order approaches to solve the
+minimization problem, e.g., by constructing 2-point
+estimators for the gradient. Since the objective func-
+tion is quadratic and the matrix R is known, m queries
+of H(N)(x, ·, t) are sufficient to compute φ(N)
+t
+(x).
+The overall dual EnKF algorithm can be found in (Joshi
+et al., 2022, Algorithm 1 and 2).
+6.4. Relating dual EnKF to model-based RL
+The following remarks are included to help provide an
+intuitive explanation of the various aspects of the dual
+EnKF and relate these to the model-based RL:
+1. Representation.
+In designing any RL algorithm,
+the first issue is the representation of the unknown
+value function (Pt in the linear case). Our novel idea
+is to represent Pt is in terms of statistics (variance) of
+the particles. Such a representation is distinct from
+representing the value function, or its proxies, such
+as the Q function, within a parameterized class of
+functions.
+2. Value iteration. The algorithm is entirely simula-
+tion based: N copies of the model (40b) are simulated
+in parallel where the terms on the right hand-side
+of (45) have the following intuitive interpretations:
+(a) Dynamics: The first term “AY i
+t dt” on the right-
+hand side of (45) is simply a copy of uncontrolled
+dynamics in the model (40b).
+(b) Control: The second term “Bd
+�η
+i
+t” is the con-
+trol input for the i-th particle.
+It is specified
+as a W.P. with covariance R−1. One may inter-
+pret this as an approach to exploration whereby
+cheaper control directions are explored more.
+(c) Coupling: The third term, referred to as the cou-
+pling, effectively implements the value iteration
+step. Coupling has a “gain times error” structure
+where S(N)
+t
+CT is the gain and 1
+2(CY i
+t + Cn(N)
+t
+)
+is the counterpart of the error in the linear
+FPF (14).
+3. Arrow
+of
+time.
+The particles are simulated
+backward—from terminal time t = T to initial time
+t = 0. This is different from most model-based RL
+but consistent with the dynamic programming (DP)
+equation which also proceeds backward in time.
+20
+
+0
+5
+10
+(i)
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+P11
+ARE
+EnKF
+DRE
+0
+5
+10
+(ii)
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+P12
+0
+5
+10
+(iii)
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+P21
+0
+5
+10
+(iv)
+1.0
+1.2
+1.4
+1.6
+P22
+T − t
+(a) d = 2
+0
+5
+10
+0
+5
+10
+15
+20
+ARE
+EnKF
+DRE
+T − t →
+0
+5
+10
+2
+4
+6
+8
+10
+ARE
+EnKF
+DRE
+(b) d = 10
+Figure 6: Comparison of the numerical solution obtained from the EnKF, the DRE, and the ARE. Note the x-axis for these plots is T − t for
+0 ≤ t ≤ T. d is the state-dimension.
+6.5. Convergence and error analysis
+In (Joshi et al., 2022, Prop. 3), under certain additional
+assumptions on system matrices, the following error bound
+is derived:
+E[∥S(N)
+t
+− ¯St∥F ] ≤ C1
+√
+N
++ C2e−2λ(T −t)E[∥S(N)
+T
+− ¯ST ∥F ],
+(47)
+where C1, C2, λ are positive constants and || · ||F denotes
+Frobenius norm for matrices.
+The significance of the
+bound (47) is as follows: The constant λ is same as the
+rate that governs the convergence of the solution of the
+DRE (41) to the stationary solution (of the infinite-horizon
+LQR problem). This means that the dual EnKF learns the
+optimal LQR gain exponentially fast with a rate that is as
+good as one would obtain from directly solving the DRE.
+Convergence
+is
+numerically
+illustrated
+for
+a
+d-
+dimensional system expressed in its controllable canonical
+form
+A =
+�
+����
+0
+1
+0
+0
+. . .
+0
+0
+0
+1
+0
+. . .
+0
+...
+...
+a1
+a2
+a3
+a4
+. . .
+ad
+�
+���� ,
+B =
+�
+����
+0
+0
+...
+1
+�
+����
+where the entries (a1, . . . , ad) ∈ Rd are i.i.d. samples from
+N(0, 1). The matrices C, R, PT are identity matrices of
+appropriate dimension. For numerics, we fix T = 10, chose
+the time-discretization step as 0.02, and use N = 1000
+particles to simulate the dual EnKF.
+Fig. 6(a) depicts the convergence of the four entries of
+the matrix P (N)
+t
+for the case where d = 2. Fig. 6(b) depicts
+the analogous results for d = 10. Fig. 7(a) and Fig. 7(b)
+depict the open-loop poles (eigenvalues of the matrix A)
+and the closed-loop poles (eigenvalues of the matrix (A +
+BK(N)
+0
+)), for d = 2 and d = 10, respectively. Note that
+the closed-loop poles are stable, whereas some open-loop
+poles have positive real parts.
+6.6. Comparison to literature
+We present a comparison of the dual EnKF with policy
+gradient algorithms in Mohammadi et al. (2022) (denoted
+as [M21]) and Fazel et al. (2018) (denoted as [F18]). In
+these prior works, by restricting the control policies to the
+linear form ut = Kxt, the LQR problem reduces to the
+finite-dimensional static optimization problem:
+K⋆ = arg min
+K
+J(K) = E
+�� ∞
+0
+x
+T
+t Qxt + u
+T
+t Rut dt
+�
+(48)
+where the expectation is over the initial condition. The
+authors apply a pure-actor method using “zeroth order”
+methods to approximate gradient descent, much like the
+early REINFORCE algorithm for RL (Sutton and Barto,
+2018).
+A qualitative comparison of the dual EnKF with these
+prior algorithms is given in Table 1.
+Choosing t = 0
+in (47), the error is smaller than ε if the number of particles
+N > O( 1
+ε2 ) and the simulation time T > O(log( 1
+ε)), while
+the iteration number is one. This is compared with pol-
+icy optimization approach in Fazel et al. (2018) where the
+21
+
+−0.6
+−0.4
+−0.2
+0.0
+−0.5
+0.0
+0.5
+Im
+Re
+OL
+CL
+(a) d = 2
+−1
+0
+1
+2
+−1.0
+−0.5
+0.0
+0.5
+1.0
+Re
+Im
+OL
+CL
+(b) d = 10
+Figure 7: Open and closed-loop poles for the two plots (parts (a) and (b)) depicted in Fig. 6.
+number of particles and the simulation time scales poly-
+nomially with ε, while the number of iterations scale as
+O(log( 1
+ε)). This result is later refined in Mohammadi et al.
+(2022) where the required number of particles and the sim-
+ulation time are shown to be O(1) and O(log( 1
+ε)) respec-
+tively (although this result is valid with probability that
+approaches zero as the number of iterations grow (Moham-
+madi et al., 2022, Thm. 3).).
+A numerical comparison is made on the benchmark
+spring mass damper example borrowed from (Mohammadi
+et al., 2019, Sec. VI). Fig. 8 depicts the relative mean-
+squared error, defined as
+MSE := 1
+T E
+�� T
+0
+∥Pt − P (N)
+t
+∥2
+F
+∥Pt∥2
+F
+dt
+�
+Two trends are depicted in the figure: the O( 1
+N ) decay of
+the MSE as N increases (for d fixed), which is a numerical
+illustration of the error bound (47), and a plot of the MSE
+as a function of dimension d (for N fixed).
+A side-by-side comparison with [F18] and [M21] is de-
+picted in Fig. 9. The comparison is for the following met-
+rics (taken from Mohammadi et al. (2022)):
+errorgain = ∥Kest − K∞∥F
+∥K∞∥F
+,
+errorvalue = cest − c∞
+c(N)
+init − c∞
+where the LQR optimal gain K∞ and the optimal value
+c∞ are computed from solving the ARE. The value c(N)
+init
+is approximated using the initial gain K = 0 (Note such
+a gain is not necessary for EnKF). Because [F18] is for
+discrete-time system, an Euler approximation is used to
+obtain a discrete-time model.
+In the numerical experiments, the dual EnKF is found
+to be significantly more computationally efficient—by two
+orders of magnitude or more.
+The main reason for the
+order of magnitude improvement in computational time is
+as follows: An EnKF requires only a single iteration over
+a fixed time-horizon In contrast, [F18] and [M21] require
+several steps of gradient descent, with each step requiring
+an evaluation of the LQR cost, and because these opera-
+tions must be done serially, these computations are slower.
+In carrying out these comparisons, the same time-
+horizon [0, T] and discretization time-step ∆t was used for
+all the algorithms. It is certainly possible that some of
+these parameters can be optimized to improve the perfor-
+mance of the other algorithms.
+In particular, one may
+consider shorter or longer time-horizon T or use paral-
+lelization to speed up the gradient calculation. Codes are
+made available on Github for interested parties to inde-
+pendently verify these comparisons1.
+7. Discussion and conclusion
+In this survey, we described CIPS to approximate the
+solution of the optimal filtering and the optimal control
+problems (in parts I and II, respectively). As explained
+in Sec. 1, there are close parallels with DA and RL. In
+this section, we expand on some of these parallels with the
+goal of highlighting some important points and directions
+for future work.
+1.
+Data assimilation, sampling, optimal transportation.
+CIPS may be viewed as a sampling algorithm. The FPF
+control law (coupling) is designed to sample from the pos-
+terior. Compared to the conventional particle filters, cou-
+pling is beneficial because the issue of particle degener-
+acy is avoided (as discussed in Sec. 3.4). To design the
+coupling, optimal transportation theory provides a use-
+ful framework (as described in Sec. 5). Variations of the
+1https://github.com/anantjoshi97/EnKF-RL
+22
+
+Algorithm
+particles/samples
+simulation time
+iterations
+dual EnKF
+O( 1
+ε2 )
+O(log( 1
+ε))
+1
+Fazel et al. (2018)
+poly
+� 1
+ε
+�
+poly
+� 1
+ε
+�
+O(log( 1
+ε))
+Mohammadi et al. (2022)
+O(1)
+O(log( 1
+ε))
+O(log( 1
+ε))
+Table 1: Computational complexity comparison of the algorithms to achieve ε error in approximating the infinite-horizon LQR optimal gain.
+102
+103
+Number of particles (N)
+10−2
+100
+102
+104
+MSE
+d = 2
+d = 10
+d = 20
+d = 50
+d = 80
+0
+20
+40
+60
+80
+Dimensions (d)
+10−2
+10−1
+100
+101
+102
+MSE
+O(1/N)
+N = 100
+N = 300
+N = 1000
+Figure 8: Performance of the dual EnKF algorithm: MSE as a function of the number of particles N and system dimension d.
+basic approach described here have been used in construc-
+tion of a class of filtering algorithms (Halder and Geor-
+giou, 2017, 2018, 2019; Garbuno-Inigo et al., 2020; Luo,
+2019). The optimal transport formulation has also been
+extended to the Schr¨odinger bridge setting by considering
+a cost with respect to the (prior) dynamics, or considering
+an entropic regularization (Chen et al., 2016; Reich, 2019).
+In related works, the coupling viewpoint along with geo-
+metric notions from optimal transportation theory, have
+enabled application of optimization algorithms to design
+sampling schemes (Liu and Wang, 2016; Richemond and
+Maginnis, 2017; Zhang et al., 2018b; Frogner and Poggio,
+2018; Chizat and Bach, 2018; Chen et al., 2018; Liu et al.,
+2018; Zhang et al., 2019; Taghvaei and Mehta, 2019).
+Part II of this paper is motivated by the enormous suc-
+cess of the CIPS (EnKF) in DA.
+2.
+Reinforcement learning and optimal control. Com-
+pared to typical RL approaches, there are two key innova-
+tions/differences:
+1. Representation of the unknown value function in
+terms of the statistics (variance) of a suitably designed
+process; and
+2. Design of interactions (coupling) between simulations
+for the purposes of policy optimization.
+We fully believe that the two key innovations may be useful
+for many other types of models including MDPs and par-
+tially observed problems. In the LQ setting of the problem,
+doing so is beneficial because of the learning rate: Since
+the [N = ∞] limit is exact tor the LQ problem, the dual
+EnKF algorithm yields a learning rate that closely approx-
+imates the exponential rate of convergence of the solution
+of the DRE. This is rigorously established with the aid of
+error bound (47). In numerical examples, this property is
+shown to lead to an order of magnitude better performance
+than the state-of-the-art algorithms.
+Apart from RL, model predictive control (MPC) is an-
+other area where a model in the form of a simulator
+is assumed to design optimal control for problems such
+as (4) (Rawlings et al., 2017). Using duality, MPC meth-
+ods have been adapted to design the moving horizon es-
+timator (MHE). A big selling point of MPC is its ability
+to handle constraints which has not been a major theme
+in the DA literature. Another notable distinction is that
+while MPC aims to find a single (optimal) trajectory, CIPS
+simulate multiple stochastic trajectories in a Monte Carlo
+manner. Notably, the solution of the deterministic optimal
+control problem (4) is based on simulating (5a) which is an
+SDE. For the stochastic MPC problems, multiple simula-
+tions have been considered in the scenario-based approach
+(Campi and Garatti, 2018).
+Some perspectives on future research. In basic sciences,
+there are a number of important examples of interacting
+particle systems. This paper presents results on the theme
+of “CIPS as an algorithm”. The most historical of such
+23
+
+0.05
+0.10
+0.15
+101
+103
+105
+Comp. Time (s)
+EnKF
+[M21]
+[F18]
+0.0000
+0.0025
+0.0050
+0.0075
+101
+103
+Error in gain
+Error in cost
+Figure 9: Comparison with algorithms in Fazel et al. (2018) (labeled [F18]) and Mohammadi et al. (2022) (labeled [M21]). The comparisons
+depict the computation time (in Python) as a function of the relative error in approximating the LQR gain and cost.
+algorithms is the EnKF which is used to solve the problem
+of data assimilation. It is hoped that this survey convinces
+the reader that the paradigm is also useful for solving other
+problems in estimation and control. A major selling point
+of CIPS, and also the reason for widespread use of the
+EnKF, is that it is able to work directly with a simulator.
+Therefore, it is amenable as a solution method for complex
+systems where models typically exist only in the form of a
+simulator. Apart from the open problems described in the
+main body of the paper, a few themes for future research
+are as follows:
+• MPC offers a useful benchmark for CIPS. With the
+exception of the geometric approaches, e.g., FPF on
+Riemannian manifolds (Zhang et al., 2017b), con-
+straints has not been an important theme in design
+of CIPS. It is an important problem to extend the
+design of mean-field process to handle general types
+of constraints in inputs and states. One possible next
+step is to extend the dual EnKF to the inequality-
+constrained LQR problems.
+• RL could be an important application for CIPS. A key
+difference is that CIPS-based solution does not rely
+on function approximation. Instead, the value func-
+tion is approximated in terms of the distribution of
+the particles. This has some advantages, e.g., avoids
+the need to select basis functions, and some disadvan-
+tages, e.g., availability of computational resources. It
+will be useful to understand some of these trade-offs.
+• Relationship to mean-field games and optimal control
+should be further developed. CIPS represent simple
+examples of mean-field type control laws. However,
+derivation of these control laws is, more often than
+not, rooted in methods from optimal transportation
+theory (Sec. 5). It remains an open problem to de-
+rive the FPF control law starting from a mean-field
+optimal control type objective (some partial results in
+this direction appear in (Zhang et al., 2018a)).
+• Extensions to partially observed optimal control prob-
+lems. For the linear Gaussian model, algorithms de-
+scribed in parts I and II are easily combined to obtain
+a CIPS for the partially observed problem. The solu-
+tion is based on the separation principle: A forward
+(in time) EnKF is run to solve the optimal filtering
+problem; and a completely independent backward (in
+time) dual EnKF is run to solve the optimal control
+problem. For the nonlinear problem, there may be
+benefit to couple the forward and backward CIPS.
+• Distributionally robust FPF. In order to handle un-
+certainty in signal and observation models, it may be
+useful to explore methods from distributionally ro-
+bust optimization framework (Rahimian and Mehro-
+tra, 2019). The framework has been used to develop
+the Wasserstein robust Kalman filter for the linear
+Gaussian model (Shafieezadeh Abadeh et al., 2018).
+Its extension to the nonlinear filtering model (1) is
+open and may be possible based on the optimal trans-
+port formulation of the FPF.
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+page_content='A Survey of Feedback Particle Filter and related Controlled Interacting Particle Systems (CIPS) Amirhossein Taghvaei, Prashant G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Mehta aWilliam E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Boeing Department of Aeronautics & Astronautics, University of Washington, Seattle, 98195, WA, USA bCoordinated Science Laboratory, University of Illinois, Urbana-Champaign, 61801, IL, USA Abstract In this survey, we describe controlled interacting particle systems (CIPS) to approximate the solution of the optimal filtering and the optimal control problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Part I of the survey is focussed on the feedback particle filter (FPF) algorithm, its derivation based on optimal transportation theory, and its relationship to the ensemble Kalman filter (EnKF) and the conventional sequential importance sampling-resampling (SIR) particle filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The central numerical problem of FPF—to approximate the solution of the Poisson equation—is described together with the main solution approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' An analytical and numerical comparison with the SIR particle filter is given to illustrate the advantages of the CIPS approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Part II of the survey is focussed on adapting these algorithms for the problem of reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The survey includes several remarks that describe extensions as well as open problems in this subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Introduction In many applications, dynamic models exist only in the form of a simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Our aim is to provide a survey of a class of algorithms, that use only a model simulator, to solve the two canonical problems of Control Theory: Design of optimal filter (in the sense of estimation);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Design of optimal control law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In this survey, such simulation-based algorithms are broadly referred to as controlled interacting particle sys- tems (CIPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Our research group’s most well known con- tribution to CIPS is the feedback particle filter (FPF), which is also the main focus of this survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The FPF algorithm is useful to approximate the optimal (nonlin- ear) filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' By making use of the duality between optimal control and filtering, the FPF algorithm is extended to approximate the solution of an optimal control problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' We begin by describing the high-level idea for the two problems of optimal filtering and optimal control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' CIPS in optimal filtering Mathematical problem: In continuous-time and continuous-space settings of the problem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' the standard model of nonlinear (or stochastic) filtering is the follow- ing Itˆo stochastic differential equations (SDEs): State: dXt = a(Xt)dt + σB(Xt)dBt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' X0 ∼ p0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (1a) Observation: dZt = h(Xt)dt + dWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (1b) where Xt ∈ Rd and Zt ∈ Rm are the state and observation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' at time t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' p0 is the probability density func- tion (PDF) at the initial time t = 0 (p0 is referred to as the prior density),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' and {Bt}t≥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' {Wt}t≥0 are mutually inde- pendent standard Wiener processes (W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=') taking values in Rq and Rm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The mappings a(·), h(·), σB(·), and the density p0(·) are smooth (continuously differen- tiable) functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The linear Gaussian model is obtained when the drift terms a(·), and h(·) are linear functions, σB(·) is a constant matrix, and p0 is a Gaussian density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The filtering problem is to compute the conditional PDF of the state Xt given the time-history (filtration) of obser- vations up to time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The conditional PDF is denoted by pt and is referred to as the posterior density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' CIPS algorithm: involves construction of N stochastic processes {Xi t ∈ Rd : t ≥ 0, 1 ≤ i ≤ N} where the i-th process (particle) evolves according to the SDE: dXi t = a(Xi t)dt + σB(Xi t)dBi t � �� � i−th copy of model (1a) + dU i t, Xi 0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' ∼ p0, (2) where U := {U i t : t ≥ 0, 1 ≤ i ≤ N} is referred to as the coupling (with U = 0, the N processes are un-coupled).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The goal is to design the coupling U so that the empirical distribution of the N particles at any time t approximates the posterior pt: 1 N N � i=1 f(Xi t) ≈ � Rd f(x)pt(x)dx, ∀ f ∈ Cb(Rd), (3) where “≈” means that the approximation error goes to zero (in a suitable sense) as N → ∞ (Cb(Rd) is the space of continuous and bounded functions on Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' A key breakthrough, that appeared around 2010, is that U can be realized as a mean-field type feedback control law (“mean-field type” means that the control law depends Preprint submitted to Annual Reviews in Control January 4, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='00935v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='SY] 3 Jan 2023 also on the statistics of the stochastic process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Feedback particle filter (FPF) is one such example of a mean-field type control law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In this paper, we describe the FPF, relate it to its historical precursor, the ensemble Kalman filter (EnKF) algorithm, and summarize the important de- velopments in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For the filtering model (1), the idea of controlling the particles to approximate the posterior appears in the work of three groups working independently: the first example of such a control law appears in (Crisan and Xiong, 2010) using a certain smoothed form of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The FPF formula appears in (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2011b,a) and its special case for the linear Gaussian model is described in (Reich, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Bergemann and Reich, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' A comparison of these three early works can be found in (Pathiraja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For the discrete-time filtering models, closely related ideas and algorithms were proposed, also around the same time- frame, by (Daum and Huang, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' El Moselhy and Mar- zouk, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Reich, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2014) (see (Spantini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2022) for a recent review of this literature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Our early work on FPF was closely inspired by the pi- oneering developments in mean-field games (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2007, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The topic of mean-field games and mean- field type optimal control is concerned with control and decision problems arising in interacting particle systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Over the past decade, this topic has grown in significance with theory and applications described in several mono- graphs (Bensoussan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Carmona et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Gomes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In the Physics literature, the study of interacting particle systems is a classical subject (Liggett, 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' A canonical example of an interacting particle sys- tem is the coupled oscillators model of Kuramoto (Ku- ramoto, 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Strogatz, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' D¨orfler and Bullo, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Extensions of the classical Kuramoto model to mean-field games appears in (Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Carmona and Graves, 2020) and to FPF is given in (Tilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Design of CIPS to approximate the optimal control law is a more recent development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The idea is described next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' CIPS in optimal control Mathematical problem: Consider a finite-horizon de- terministic optimal control problem: min u J(u) = � T 0 � 1 2|c(xt)|2 + 1 2u T t Rut � dt + g(xT ), (4a) subject to: ˙xt = a(xt) + b(xt)ut, x0 = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (4b) where xt ∈ Rd is the state at time t and u := {ut ∈ Rm : 0 ≤ t ≤ T} is the control input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The mappings a(·), b(·), c(·), g(·) are smooth functions and R is a strictly positive- definite matrix (henceforth denoted as R ≻ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The lin- ear quadratic (LQ) model is obtained when a(x) = Ax, b(x) = B, c(x) = Cx, and g(x) = xTPT x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The infinite- time horizon (T = ∞) case is referred to as the linear quadratic regulator (LQR) problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' CIPS algorithm: involves construction of N stochastic processes {Y i t ∈ Rd : 0 ≤ t ≤ T, 1 ≤ i ≤ N} where the i-th particle evolves according to an SDE dY i t = a(Y i t )dt + b(Y i t )dvi t � �� � i−th copy of model (4b) + U i tdt, 0 ≤ t ≤ T, (5a) where the input v := {vi t ∈ Rm : 0 ≤ t ≤ T} and the coupling U := {U i t ∈ Rd : 0 ≤ t ≤ T} are obtained as part of the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The goal is to design v and U so that the empirical distribution of the N particles at time t approx- imates a smooth density pt encoding the optimal control law ut = φ∗ t (xt) where φ∗ t (x) = R−1b T(x)∇ log pt(x), 0 ≤ t ≤ T, (5b) and ∇ denotes the gradient operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In the infinite- horizon case, a stationary policy is obtained by letting T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The righthand-side of the formula (5b) is a consequence of the log transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The transformation relates the value function of an optimal control problem to the pos- terior density of the dual optimal filtering problem (Flem- ing and Mitter, 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Mitter and Newton, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This manner of converting an optimal control problem into an optimal filtering problem (and vice-versa) is referred to as the minimum energy duality (Hijab, 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Mortensen, 1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The use of this duality to express and solve an estimation problem as an optimal control problem is a standard approach in model predictive control (Rawlings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2017, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The CIPS (5a) comes about from the use of duality in the opposite direction whereby an op- timal control problem (4) is solved using a filtering-type algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Related constructions, based on somewhat dif- ferent algorithmic approaches, is an important theme in the Robotics literature (Todorov, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Kappen, 2005a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Vijayakumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Toussaint, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Hoffmann and Rostalski, 2017) (see (Levine, 2018) for a recent review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Both (2) and (5) are examples of a “simulation-based” algorithm because multiple copies—of the model (1a) and (4b), respectively—are run in a Monte-Carlo manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The main message of our paper is that through a suit- able design of interactions between simulations—referred to as coupling—yields powerful algorithms for solving op- timal filtering and optimal control problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Relationship to other simulation-based algorithms For the two problems of filtering and control, related simulation-based solution approaches are considered in the data assimilation (DA) and reinforcement learning (RL) communities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' These relationships are dis- cussed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Data assimilation (DA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The term “Data Assimila- tion” means assimilating real-time observations (“data”) into models—which typically exist only as a software code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The term is used by a community of researchers working in 2 geophysical and atmospheric sciences (Van Leeuwen and Evensen, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Evensen, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Houtekamer and Mitchell, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Reich and Cotter, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The most celebrated appli- cation is weather prediction and forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For the abstract mathematical model, the nonlinear filter gives the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In practice, the filter must be approximated in a computationally tractable form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For this purpose, the EnKF algorithm was first introduced in (Evensen, 1994) as an alternative to the extended Kalman filter (EKF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In geophysical applications, there are two issues that ad- versely affect the implementation of an extended Kalman filter: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In high-dimensions, it is a challenge to compute the Kalman gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This is because the formula for the Kalman gain is based on the solution of a certain dif- ferential Riccati equation (DRE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The matrix-valued nature of the DRE means that any algorithm is O(d2) in the dimension d of the state-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The model parameters are not explicitly available to write down the DRE let alone solve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This is a con- cern whenever the model exists only in the form of a black-box numerical simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In an EnKF implementation, N processes are simulated (same as (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In order to compute the Kalman gain, the solution of the DRE at time t is approximated by the em- pirical covariance of the ensemble {Xi t}N i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Because an explicit solution of the DRE is avoided, an EnKF can be implemented using only a model simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This property has historically proved to be an important factor in ap- plications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Notably, the EnKF algorithm is a workhorse for the weather prediction application (Evensen, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Houtekamer and Zhang, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The computational com- plexity of the EnKF is O(Nd) and in high-dimensions, N is chosen to be much smaller than d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The historical significance of the FPF is that it repre- sents a simulation-based solution of the nonlinear filtering problem (1), for arbitrary types of non-Gaussian posterior density pt (under some mild technical conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' More- over, the EnKF was shown to arise as a special case in the linear Gaussian setting of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Like the Kalman filter, the FPF formula has a “gain times error” feedback structure which is useful in several ways, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', to handle additional uncertainty in signal and measurement models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For these reasons, FPF can be viewed as a modern exten- sion to the Kalman filter, a viewpoint stressed in a prior review paper (Taghvaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For the nonlinear filtering problem (1), the FPF rep- resents an alternative solution approach to the sequential importance sampling-resampling (SIR) particle filters and its many variants (Gordon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Bain and Crisan, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Del Moral, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Doucet, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In an SIR filter, the posterior is approximated as (compare with (3)) � Rd f(x)pt(x)dx ≈ N � i=1 W i t f(Xi t), ∀ f ∈ Cb(Rd), where Xi t is a copy of the hidden state Xt and {W i t }N i=1 are the importance weights obtained from the Bayes’ for- mula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In practice, all but a few weights can become very small—an issue known as particle degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This is- sue is ameliorated using a re-sampling procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The salient feature of the FPF, compared to the conventional particle filters, is that the weights are uniform (= 1 N ) by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Because of this difference, FPF does not suf- fer from the particle degeneracy issue and does not require re-sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In several independent numerical evaluations and comparisons, it has been observed that FPF exhibits smaller simulation variance (Berntorp, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Tilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2013b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Stano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2014) and better scaling properties with the problem dimension compared to particle filters (Surace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Some of these analytical and numerical comparisons are highlighted in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Reinforcement learning (RL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' RL is concerned with solving optimal control problems, such as (4) and its exten- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' All of the standard choices are treated in the litera- ture: continuous and discrete state-space and time, deter- ministic and stochastic dynamics, discounted and average cost structures, and finite and infinite time-horizon (Bert- sekas and Tsitsiklis, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Meyn, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' What makes the RL paradigm so different from optimal control as formal- ized by Bellman and Pontryagin in the 1950s is that in RL the system identification step is usually avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Instead, the optimal policy is approximated (“learned”) based on input-output measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In popular media, RL is described as an “agent” that learns an approximately optimal policy based on interac- tions with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Important examples of this idea include advertising, where there is no scarcity of real- time data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In the vast majority of applications we are not so fortunate, which is why successful implementation usu- ally requires simulation of the physical system for the pur- poses of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For example, DeepMind’s success story with Go and Chess required weeks of simulation for train- ing on a massive collection of super-computers (Schrit- twieser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' These success stories are largely empirical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In order to better understand the theoretical foundations of RL, there has been a concerted recent interest, in the Control com- munity, to revisit the classical linear quadratic (LQ) op- timal control problem (Fazel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Tu and Recht, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Dean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Malik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Mohammadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The two issues discussed as part of DA are relevant also to this problem: In high-dimensions, it is a challenge to solve the Riccati equation, and typically the model parameters are not explicitly available in RL set- tings of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' An outgrowth of this recent work is a class of simulation- based algorithms where multiple copies of the simulator are run in parallel to learn and iteratively improve the solution of the DRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The CIPS algorithm (5a) has the same structure where the important distinction is that the 3 simulations are now coupled with a coupling term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' We include comparisons on a benchmark problem to show how coupling helps improve performance over state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Structure of the paper and outline This paper is divided into two parts as follows: Part I on CIPS for the optimal filtering problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' It comprises Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2 - Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Part II on CIPS for the optimal control problem (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' It comprises Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The paper is written so that the key ideas are easily ac- cessible together with an understanding of the main com- putational problems and algorithms for the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For ex- ample, a reader should to be able to implement the FPF and EnKF algorithms after reading Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3 and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The more theoretical aspects related to optimal transportation theory appear in a self-contained manner in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The other significant aspect of this survey is analytical and nu- merical comparison against competing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' These appear in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='4 for part I where a comparison with the SIR filter is discussed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' and in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='6 for part II where a comparison with RL algorithms for the LQR problem is described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' We make note of two additional points: (i) While the paper presents some relatively novel ideas that are closely inspired by and connected to the work in mean-field mod- eling and control, and therefore of interest to the Control community, these algorithms have older roots (EnKF) in the DA community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Along with the discussion in the In- troduction, several remarks are included to highlight these roots and connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (ii) While the CIPS algorithms solve some problems (such as particle degeneracy), they also create new ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This informs the structure of the paper with a dedicated Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 4 on the central numerical problem of FPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In particular, the discussion of the bias- variance trade-off in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='3 is helpful to understand some of the key limitations in high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' PART I 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Background on optimal filtering Consider the filtering problem for the model (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The sigma-algebra (ot the time-history) of observations up to time t is denoted by Zt := σ(Zs : 0 ≤ s ≤ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The posterior density pt is defined as follows: � Rd f(x)pt(x)dx := E[f(Xt)|Zt], ∀ f ∈ Cb(Rd), where the conditional expectation on the righthand-side is referred to as the nonlinear filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The integral on the lefthand-side is denoted by ⟨pt, f⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The posterior pt is optimal in the sense that, among all Zt-measurable random variables, ⟨pt, f⟩ represents the best mean-squared error (MSE) estimate of the random variable f(Xt): ⟨pt, f⟩ = arg min S∈Zt E[|f(Xt) − S|2], (6) where the notation “S ∈ Zt” means S is allowed to be Zt-measurable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', an arbitrary measurable function of observations up to time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For the model (1), the evolution of the posterior pt is given by the Kushner-Stratonovich stochastic partial dif- ferential equation (Xiong, 2008, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In the special lin- ear Gaussian setting of the problem, the equation admits a finite-dimensional representation given by the Kalman- Bucy filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Linear Gaussian model and the Kalman-Bucy filter The linear Gaussian model is a special case of (1a)-(1b) and takes the following form: dXt = AXt + σBdBt, X0 ∼ N(m0, Σ0), (7a) dZt = HXtdt + dWt, (7b) where A, H, σB are matrices of appropriate dimensions and the prior is a Gaussian density with mean m0 and variance Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' It is denoted by N(m0, Σ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For the linear Gaussian model (7), it can be shown that the posterior pt is a Gaussian density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' It is denoted by N(mt, Σt), where mt and Σt are conditional mean and covariance, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Their evolution is described by the Kalman-Bucy filter (Kalman and Bucy, 1961): dmt = Amt + Kt(dZt − Hmtdt), m0 (given) (8a) d dtΣt = Ricc(Σt), Σ0 (given) (8b) where Kt := ΣtH T is referred to as the Kalman gain, and the Riccati function Ricc(Σ) := AΣ + ΣA T + ΣB − ΣH THΣ with ΣB := σBσT B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Apart from the linear Gaussian model, there are very few examples where the equation for the posterior pt ad- mits a finite-dimensional representation (Beneˇs, 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In the general setting of the nonlinear model (1) with a non- Gaussian posterior, pt is numerically approximated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Feedback particle filter Feedback particle filter (FPF) is a numerical algorithm to approximate the posterior pt for the filtering model (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Before describing the FPF, it is helpful to consider a sim- pler static problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Intuitive explanation with a simpler example Suppose the state X and the observation Y are vector- valued random variables of dimension d and m, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The probability distribution (prior) of X is denoted by PX and the joint distribution of (X, Y ) is denoted by PXY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For any given function f ∈ Cb(Rd), the problem is to obtain an MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' estimate of the unknown f(X) from a single observation of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Adapting (6) to the simple case, S∗ f(Y ) = arg min Sf (·) E[|f(X) − Sf(Y )|2], (9) where on the righthand-side Sf : Rm → R is allowed to be an arbitrary function of the Rm-valued observation (the sub-script means that the function may depend also upon f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The optimal estimator gives the conditional expecta- tion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', E[f(X)|Y ] = S∗ f(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='1 (Linear estimation and the update formula for Kalman filter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Consider the case where f is linear, f(x) = aTx, and Sf(·) is restricted to be an affine function of its argument: Sf(y) = u Ty + b, where u ∈ Rm and b ∈ R parametrize the estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' With such a choice, the optimization problem (9) is finite- dimensional whose solution is readily obtained as S∗ f(Y ) = a T(E[X] + K(Y − E[Y ])), where K = ΣXY Σ−1 Y , ΣXY = E[(X − E[X])(Y − E[Y ])T], ΣY = E[(Y − E[Y ])(Y − E[Y ])T], and it is assumed that ΣY is invertible with inverse Σ−1 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Because the vector a is arbitrary, this also shows that the optimal linear esti- mate of X is E[X] + K(Y − E[Y ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Under the stronger assumption that X and Y are jointly Gaussian, it can be shown that this is in fact the optimal estimate of X among all functions Sf(·) (not necessarily affine) (Hajek, 2015, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Therefore, in the Gaussian case E[X|Y ] = E[X] + K(Y − E[Y ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The righthand-side is the update formula for the discrete- time Kalman filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Note that the interpretation of the formula as the conditional expectation works only in the Gaussian case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In general, the formula gives only the best linear estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The example above illustrates the special and important case of obtaining optimal linear estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The question is how to extend the procedure to the nonlinear setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', the setting where both the function f(·) and the esti- mator Sf(·) are allowed to be nonlinear functions of their arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This is achieved through the concept of CIPS whose construction proceeds in two steps: Step 1: Let ¯X0 be an independent copy of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Design a control U such that, upon setting ¯X1 = ¯X0 + U, S∗ f(Y ) = E[f( ¯X1)|Y ], ∀ f ∈ Cb(Rd), Note that the control is not allowed to depend on the func- tion f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' It is designed to give the best estimate for any choice of function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' It is not yet clear that such a con- trol exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' But for now, let us assume that it exists and moreover takes the form U = u( ¯X0, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (Typically, the mapping u(·, ·) is designed to be a deterministic function but may in general also be random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=') Step 2: Generate N independent samples (particles) {X1 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' , XN 0 } from PX, update each particle according to Xi 1 = Xi 0 + u(Xi 0, Y ), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' , N, and form a Monte-Carlo approximation of the estimate: S∗ f(Y ) ≈ 1 N N � i=1 f(Xi 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='2 (CIPS and the update formula for EnKF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Continuing with Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='1 where PXY is assumed to be Gaus- sian, two formulae are described for the transformation ¯X0 �→ ¯X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The first of these formulae is based on optimal transportation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The second formula is based on the perturbed form of the discrete-time EnKF algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Optimal transport formula is given by a deterministic affine mapping ¯X1 = A( ¯X0 − E[ ¯X0]) + K(Y − E[Y ]) + E[ ¯X0], where A is the unique such symmetric positive-definite solution to a Lyapunov equation AΣXA = ΣX − ΣXY Σ−1 Y ΣY X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Perturbed EnKF formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Let ( ¯X0, ¯Y0) be an indepen- dent copy of (X, Y ) then ¯X1 = ¯X0 + K(Y − ¯Y0), where the formula for K is same as in Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' It is readily verified that, in either case, ¯X1 is a Gaus- sian random variable whose conditional mean and variance equals the conditional mean and variance of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' We defer the details on how these formulae came about to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='3 instead remarking here on several features which apply also to more general settings: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The transformation ¯X0 �→ ¯X1 is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Both the transformations are of “mean-field type” whereby the transformation depends also on statistics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', E[X] and E[Y ], of (X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In the optimal transport formula, u(·, ·) is a deter- ministic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In the EnKF formula, u(x, y) = K(y − ¯Y0) is a random map because Y0 is a random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' These considerations provide the background for the feedback particle filter algorithm which is described next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Feedback particle filter Just like the static example, the construction of FPF proceeds in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Step 1: Construct a stochastic process, denoted by ¯X = { ¯Xt}t≥0, according to a controlled SDE: d ¯Xt = a( ¯Xt)dt + σB( ¯Xt)dBt + utdt + KtdZt, ¯X0 ∼ p0, (10) where the controls ut and Kt are designed so that the con- ditional density of ¯Xt equals the posterior density pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Step 2: Simulate N stochastic processes, denoted by Xi = {Xi t}t≥0 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' , N, according to (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The two steps are summarized below: ⟨pt, f⟩ Step 1 = E[f( ¯Xt)|Zt] � �� � exactness condition Step 2 ≈ 1 N N � i=1 f(Xi t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The exactness condition refers to the fact that ¯Xt has the same conditional density as Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The N processes {Xi}N i=1 are referred to as particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' At this point, the first of these two steps appears to be aspirational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Even in the case of the static example, it is not at all clear that the function u(·, ·) exists in the general non-Gaussian case, and even if it does, it can be computed in a tractable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The case of the stochastic process where ut and Kt are allowed to be measurable with respect to the past values of observations Z and state ¯X appears, at the first glance, to be entirely hopeless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The surprising (at least at the time of its discovery) breakthrough of the FPF is that the control terms ut and Kt are given by a simple feedback control law where the computation reduces to solving a linear Poisson equation at each time-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' FPF: The process ¯X is defined according to the SDE d ¯Xt = a( ¯Xt)dt + σB( ¯Xt)d ¯Bt � �� � copy of model (1a) + Kt( ¯Xt) ◦ (dZt − h( ¯Xt) + ¯ht 2 dt) � �� � FPF feedback control law , ¯X0 ∼ p0 (11) where { ¯Bt}t≥0 is a copy of the process noise {Bt}t≥0, and ¯ht := E[h( ¯Xt)|Zt].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The ◦ indicates that the SDE is ex- pressed in its Stratonovich form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' At any fixed time t, the gain Kt(·) is a d × m matrix-valued function obtained by solving m partial differential equations: for j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' m, the j-th column K(j) t := ∇φ(j) where φ(j) is the solution of the Poisson equation: − 1 ρ(x)∇·(ρ(x)∇φ(j)(x)) = (h(j)(x)−¯h(j)), x ∈ Rd (12) where the density ρ = ¯pt (the conditional density of ¯Xt at time t), h(j) is the j-th component of the observation func- tion h, ¯h(j) = � h(j)(x)ρ(x)dx, and ∇ and ∇· denote the gradient and the divergence operators, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For a succinct presentation, the functions {φ(j)}m j=1 are collected to form the vector-valued function φ = [φ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' , φ(m)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' With such a notation, the gain function Kt is the Jacobian ∇φ = [∇φ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' , ∇φ(m)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The process ¯X is an example of a mean-field process because its evolution depends upon its own statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' An SDE of this type is called a McKean-Vlasov SDE or a mean-field SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Accordingly, (11) is referred to as the mean-field FPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The main result, first proved in Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (2013b), is that the mean-field process thus defined is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='3 (Thm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='3, Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (2013b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Consider the filtering model (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Suppose {pt}t≥0 denotes the condi- tional density of the process {Xt}t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Suppose the mean- field process { ¯Xt}t≥0 defined by (11)-(12) is well-posed with conditional density denoted by {¯pt}t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Then, pro- vided ¯p0 = p0, ¯pt = pt, ∀ t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='4 (Well-posedness and Poincar´e inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The well-posedness of (11)-(12) means that a strong solu- tion ¯X exists with a well-defined density {¯pt}t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' To show well-posedness, apart from the standard Lipschitz condi- tion on the drift terms a(·) and σB(·), the main technical condition is that the posterior density pt (of Xt) satis- fies the Poincar´e inequality (PI), and � |h(x)|2pt(x)dx < ∞ (Laugesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2015, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (A probability density ρ = e−V satisfies the PI if xT∇V (x) ≥ α|x| for |x| ≥ R where α and R are positive constants (Bakry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2008, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This condition is true, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', whenever ρ has a Gaussian tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=') An explanation of the relevance of the PI for the well-posedness (existence, uniqueness, and regularity) of the solution φ of the Poisson equation (12) is deferred to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 4, where algorithms for its approximation are also described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Once a solution φ of the Poisson equa- tion is obtained together with necessary apriori estimates, well posedness of ¯X follows from the standard theory of mean-field SDEs (Carmona et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Although the general case remains open, it has been possible to prove the PI under certain additional conditions on the filtering model (1) (Pathiraja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2021, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='1), (Laugesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2015, Prop 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' We next describe the finite-N algorithm which is how the FPF is implemented in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' CIPS: The particles {Xi t : t ≥ 0, 1 ≤ i ≤ N} evolve according to: dXi t = a(Xi t)dt + σ(Xi t)dBi t + K(N) t (Xi t) ◦ (dZt − h(Xi t) + h(N) t 2 dt), Xi 0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='d ∼ p0, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' N, (13) 6 where {Bi t}t≥, for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' , N, are mutually indepen- dent W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', h(N) t := N −1 �N i=1 h(Xi t), and K(N) t is the out- put of an algorithm that is used to approximates the so- lution to the Poisson equation (12): K(N) t := Algorithm({Xi t}N i=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The notation is suggestive of the fact that algorithm is adapted to the ensemble {Xi t}N i=1 and the function h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' the density ¯pt is not known in an explicit form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Before de- scribing the algorithms for gain function approximation in (the following) Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 4, we discuss the linear Gaussian case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The main computational challenge to simulate the finite- N FPF (13) is the computation of the gain function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The difficulty arises because, for a general nonlinear observa- tion function h and a non-Gaussian density ρ, there are no known closed-form solutions of the Poisson equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In the linear Gaussian special case, with a linear obser- vation function h(x) = Hx and a Gaussian density, the Poisson equation admits an explicit solution whereby the gain function is given by the Kalman gain: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='5 (Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='4, Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (2013b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Con- sider the Poisson equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Suppose ρ is a Gaussian density N(m, Σ) and h(x) = Hx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Then its unique solution is given by: φ(x) = (HΣ)(x − m), x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Consequently, the gain function ∇φ(x) = ΣH T is the Kalman gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Using the Kalman gain, the FPF algorithm simplifies to a square-root form of the ensemble Kalman filter (EnKF) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This is described next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Ensemble Kalman filter In the linear Gaussian case, upon replacing the gain function with the Kalman gain, the mean-field FPF (11) is the Itˆo-SDE d ¯Xt = A ¯Xtdt + σBd ¯Bt + ¯ΣtH T(dZt − H ¯Xt + H ¯mt 2 dt), (14) where ¯mt = E[ ¯Xt|Zt], ¯Σt = E[( ¯Xt − ¯mt)( ¯Xt − ¯mt) T|Zt].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' As a corollary of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='3, the mean-field process ¯X is exact which, in the linear Gaussian case, means that the conditional density of ¯Xt is Gaussian whose mean ¯mt and the covariance matrix ¯Σt evolve according to the Kalman filter (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' A direct proof showing (14) is exact appears in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The finite-N FPF is obtained as follows: dXi t = AXi tdt+σBdBi t+Σ(N) t H T(dZt − HXi t + Hm(N) t 2 dt), (15a) where the mean-field terms in (14) are approximated em- pirically as follows: m(N) t := 1 N N � j=1 Xi t, (15b) Σ(N) t := 1 N − 1 N � j=1 (Xi t − m(N) t )(Xi t − m(N) t ) T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (15c) The linear Gaussian FPF (15) is identical to the square- root form of the ensemble Kalman filter (Bergemann and Reich, 2012, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='6 (Historical context for EnKF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The EnKF algorithm was first introduced in Evensen (1994), in the discrete-time setting of the filtering problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' At the time, the algorithm was introduced as an alternative to the extended Kalman filter (EKF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' As already mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 1, a major reason for using an EnKF is that, un- like EKF, it does not require an explicit solution of the DRE (Van Leeuwen and Evensen, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Burgers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Houtekamer and Mitchell, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Since its intro- duction, a number of distinct types of EnKF algorithms have appeared in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Amongst these, the most well-known types are as follows: (i) EnKF based on per- turbed observation (Evensen, 2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' and (ii) The square root EnKF (Anderson, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Whitaker and Hamill, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Bishop et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The details for these algorithms can be found in (Reich and Cotter, 2015, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 6-7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The two aforementioned types of the EnKF algorithm have also been extended to the continuous-time setting (Bergemann and Reich, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In these settings, the EnKF is usually referred to as the ensemble Kalman-Bucy filter (EnKBF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' A review of the EnKBF algorithm and its connection to the FPF algorithm can be found in (Taghvaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The EnKBF algorithm and the linear FPF admits several extensions: (i) EnKBF with perturbed observation (Berge- mann and Reich, 2012) (Del Moral and Tugaut, 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (ii) Stochastic linear FPF (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2016, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (26)) which is same as the square root EnKBF (Bergemann and Reich, 2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='(iii) Deterministic linear FPF (Taghvaei and Mehta, 2016, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (15)) (de Wiljes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' EnKF was recently extended to the case with correlated observa- tion noise (Ertel and Stannat, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' An excellent recent survey on this topic appears in Calvello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='7 (Current research on EnKF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Error analysis of the EnKF algorithm remains an active area of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For the discrete-time EnKF algorithm, these results ap- pear in (Le Gland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Mandel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Tong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Kelly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Kwiatkowski and Mandel, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The analysis for continuous-time EnKF is more re- cent (Del Moral and Tugaut, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Bishop and Del Moral, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Taghvaei and Mehta, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Del Moral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' de Wiljes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Bishop and Del Moral, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Typically, one is interested in obtaining a 7 uniform error bound as follows: E[∥m(N) t − mt∥2] + E[∥Σ(N) t − Σt∥2] ≤ C √ N , (16) where (mt, Σt) are the solutions of the Kalman filter (8) and (m(N) t , Σ(N) t ) are obtained from simulating an EnKF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' and C > 0 is a time-independent constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In the most recent result (Bishop and Del Moral, 2020), (16) is shown under the assumption that H TH is a positive-definite ma- trix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' It is expected that (16) also holds under the weaker condition of the pair (A, H) being detectable, which is the condition for the stability of the Kalman filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' However, a complete resolution is still open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' A comprehensive review of recent developments in this area can be found in Bishop and Del Moral (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Comparison In this section, we provide an analytical comparison of the FPF with the importance sampling-based particle fil- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For this purpose, consider a parameter estimation example with a fully observed model as follows: dXt = 0, X0 ∼ N(0, σ2 0Id) = p0, dZt = Xtdt + σwdWt, (17) where the time t ∈ [0, 1], σW , σ0 > 0, and Id is the d × d identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The posterior p1 at time t = 1 is a Gaus- sian N(m1, Σ1) with m1 = σ2 0 σ2 0+σ2 W Z1 and Σ1 = σ2 0σ2 w σ2 0+σ2w Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Let {Xi 0}N i=1 be N i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='d samples from the prior p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The importance sampling-based particle filter yields an empir- ical approximation of the posterior p1 as follows: π(N) PF (f) := N � i=1 W i 1f(Xi 0), W i 1 = e − |Z1−Xi 0|2 2σ2w �N i=1 e − |Z1−Xi 0|2 2σ2w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (18) In contrast, given the initial samples {Xi 0}N i=1, the FPF approximates the posterior by implementing a feedback control law as follows: π(N) FPF(f) := 1 N N � i=1 f(Xi 1), dXi t = Σ(N) t σ2w (dZt−Xi t + m(N) t 2 dt), (19) where the mean m(N) t and covariance Σ(N) t are empirically approximated using (15b) and (15c), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The MSE in estimating the conditional expectation of a given function f is defined as follows: MSE∗(f) := E[|π(N) ∗ (f) − ⟨p1, f⟩|2], where the subscript ∗ is either the PF or the FPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For f(x) = 1 √ d1Tx, a numerically computed plot of the level-sets of MSE, as a function of N and d, is depicted in Figure 1-(a)-(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The expectation is approximated by averaging over M = 1000 independent simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' It is observed that, in order to have the same error, the im- portance sampling-based approach requires the number of samples N to grow exponentially with the dimension d, whereas the growth using the FPF for this numerical ex- ample is O(d 1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This conclusion is consistent with other numerical studies reported in the literature (Surace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Stano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Berntorp, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For the purposes of the analysis, a modified form of the particle filter is considered whereby the denominator is replaced by its exact form: π(N) PF (f) := N � i=1 ¯W i 1f(Xi 0), ¯W i 1 = e − |Z1−Xi 0|2 2σ2w NE[e − |Z1−X0|2 2σ2w |Z1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (20) Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='8 (Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 4 in (Taghvaei and Mehta, 2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Consider the filtering problem (17) with state di- mension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Suppose σ0 = σw = σ > 0 and f(x) = aTx where a ∈ Rd with |a| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Then: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' for the modified importance sampling esti- mator (20) is given by MSEPF(f) = σ2 N � 3(2d) − 1 2 � ≥ σ2 N 2d+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The MSE for the FPF estimator (19) is bounded as MSEFPF(f) ≤ σ2 N (3d2 + 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (21) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='9 (Curse of Dimensionality (CoD)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In the limit as d → ∞, the performance of the importance sampling-based particle filters is studied in the litera- ture (Bickel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Bengtsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Snyder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Rebeschini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The main focus of these studies is on the particle degeneracy (or the weight collapse) issue: it is shown that if log N log d d → 0 then the largest weight max1≤i≤N W i t → 1 in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Conse- quently, in order to prevent the weight collapse, the number of particles must grow exponentially with the dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This phenomenon is referred to as the curse of dimension- ality for the particle filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In contrast, the weights in an FPF are uniform by design (see (19)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Therefore, the FPF does not suffer from the weight collapse issue and, in particular, does not require resampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' A complete com- parison of the two types of particle filters remains open (see (Abedi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2022) for recent progress on this topic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='10 (Scaling with the dimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The scal- ing with dimension depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 1 (b) suggests that the O(d2) bound in (21) is loose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This is the case because, in deriving the bound, the inequality ∥ · ∥2 ≤ ∥ · ∥F is used (Taghvaei and Mehta, 2020, Appendix E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The in- equality is loose particularly so as the dimension grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Also, it is observed that the MSE for the particle filter 8 1 2 3 4 5 6 7 8 9 10 5 6 7 8 9 log(N) d N 2d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='10 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='e (PF) (a) 1 2 3 4 5 6 7 8 9 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='5 log(N) d N d 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='010 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='e (FPF) (b) Figure 1: Numerical comparison for the filtering model (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Level sets of the MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' using: (a) importance sampling-based algorithm (18) and (b) the FPF (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' As the state dimension d grows, in order to have same performance (MSE), the number of particles N must increase as 2d for (18) while they increase as d 1 2 for (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' grows slightly slower than the lower-bound 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This is be- cause the lower-bound is obtained for the modified particle filter (20), while the MSE is numerically evaluated for the standard particle filter (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The correlation between the numerator and denominator in (18) reduces the MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Extensions of FPF In deriving the FPF, the main modeling assumption is the nature of observation model (1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (Such a model is referred to as the white noise observation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=') In sev- eral follow on works, the basic FPF is extended to handle more general types of models for the state process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' These extensions are briefly described next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 1) FPF on Riemannian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The feedback control form of the FPF formula (11) holds not only for the Euclidean state-space but also for the cases where the state {Xt}t≥0 evolves on a Riemannian manifold, such as the matrix Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' These extensions are described in (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2016b,a, 2017a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In these papers, the FPF is shown to provide an intrinsic description of the fil- ter that automatically satisfies the geometric constraints of the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The gain is expressed as grad φ and ob- tained as a solution of the Poisson equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' It is shown that the gain is also intrinsic that furthermore does not depend upon the choice of the Riemannian metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For the special case when the manifold is a matrix Lie group, explicit formulae for the filter are derived, using the ma- trix coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Filters for two example problems are presented: the attitude estimation problem on SO(3) and the robot localization problem in SE(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Comparisons are also provided between the FPF and popular algorithms for attitude estimation, namely the multiplicative EKF, the invariant EKF, the unscented quaternion estimator, the invariant ensemble Kalman filter, and the bootstrap particle filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Specifically, under a certain assumption of a “concentrated distribution”, the evolution equations for the mean and the covariance are shown to be identical to the left invariant EKF algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2) FPF on discrete state-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (2015), FPF is extended to the filtering problem where the hidden state {Xt}t≥0 is a continuous-time Markov process that evolves on a finite state-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (For this model, the optimal non- linear filter is called the Wonham filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=') A standard algo- rithm to simulate a Markov process is based on the use of Poisson counters to simulate transitions between discrete states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In order to define the process ¯X, a control process U is introduced that serves to modulate the rates of these counters based on causal observations of data Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' An ex- plicit formula for the FPF feedback control law is derived and shown to be exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Similar to (11), the formula is in the form of “gain times error” where the gain is now obtained by solving a certain linear matrix problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The linear matrix problem is the finite state-space counterpart of the Poisson equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3) FPF with data association and model uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In applications such as multiple target tracking, the filter- ing problem often involves additional uncertainties in the state model (1a) and the observation model (1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In the classical linear Gaussian settings, algorithms based on the Kalman filter have been developed to provide a solution to these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' These algorithms are referred to as the interacting multiple model (IMM) filter (Blom, 2013) and the probabilistic data association (PDA) filter (Bar- Shalom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In the PDA filter, the Kalman gain is allowed to vary based on an estimate of the instantaneous uncertainty in the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In the IMM filter, mul- tiple Kalman filters are run in parallel and their outputs combined to form an estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 9 Like the Kalman filter, the FPF is easily extended to handle additional uncertainties in the observation and sig- nal models: These extensions, namely, the probabilistic data association (PDA)-FPF and the interacting multiple model (IMM)-FPF are derived in our prior works (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2012, 2013a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Yang and Mehta, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Structurally, the FPF based implementations are similar to the classical algorithms based on the Kalman filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In the linear Gaus- sian settings, the equations for the mean and the variance of the FPF-based filters evolve according the classical PDA and IMM filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 4) Collective inference FPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The term “collective infer- ence” is used to describe filtering problems with a large number of aggregate and anonymized data (Sheldon and Dietterich, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Some of these problems have gained in importance recently because of COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Indeed, the spread of COVID-19 involves dy- namically evolving hidden processes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', number of in- fected, number of asymptomatic etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='.) that must be de- duced from noisy and partially observed data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', num- ber of tested positive, number of deaths, number of hos- pitalized etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In carrying out data assimilation for such problems, one typically only has aggregate observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For example, while the number of daily tested positives is available, the information on the disease status of any particular agent in the population is not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (2021), the FPF algorithm is extended for a model with M agents and M observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The M observations are non-agent specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Therefore, in its ba- sic form, the problem is characterized by data association uncertainty whereby the association between the observa- tions and agents must be deduced in addition to the agent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (2021), the large-M limit is interpreted as a problem of collective inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This viewpoint is used to derive the equation for the empirical distribution of the hidden agent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' An FPF algorithm for this problem is presented and illustrated via numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For- mulae are described for both the Euclidean and the finite state-space case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The classical FPF algorithm is shown to be the special case (with M = 1) of these more gen- eral results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The simulations help show that the algorithm well approximates the empirical distribution of the hidden states for large M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Before closing this section, we remark on the Stratonovich form of the mean-field FPF SDE (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The FPF is expressed in this form because of two reasons: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The feedback control law is “gain times error” which is appealing to control engineers, and structurally sim- ilar to the update formula in a Kalman filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' More- over, for the linear Gaussian model, the gain is the Kalman gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Expressed in its Stratonovich form, the gain times er- ror formula carries over to the Riemannian manifolds settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This is because of the intrinsic nature of the Stratonovich form (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2017b, Remark 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Notably, for the linear Gaussian model, the gain function is a constant (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', does not depend upon x) and therefore the Stratonovich form and the Itˆo form are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For the general case, the Itˆo form involves a Wong-Zakai correction term as described in the following remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='11 (Itˆo form of FPF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In its Itˆo form, the mean-field FPF (11) is expressed as d ¯Xt =a( ¯Xt)dt + σ( ¯Xt)d ¯Bt + Kt( ¯Xt)(dZt − h( ¯Xt) + ¯ht 2 dt) + 1 4 m � j=1 ∇|K(j) t ( ¯Xt)|2dt, where 1 4 �m j=1 ∇|K(j) t ( ¯Xt)|2 is the Wong-Zakai correction term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The Itˆo-Stratonovich relationship discussed here is based on interpreting Kt(x) as a function of space x and time t, and interpreting the ◦ in the Stratonovich form only with respect to the space x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In a recent paper (Pathiraja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2021, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3), the gain function is defined and in- terpreted as a function of space x and the density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This is natural because the dependence upon time t comes because of the changes in density (¯pt) as the time evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Because the density is a stochastic process, it is argued that the appropriate interpretation of ◦ in the Stratonovich form should involve both space x and the density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Using such an interpretation, the Stratonovich form involves extra-terms that are solutions to accompanying Poisson equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Algorithms for gain function approximation The exact gain K is a d × m matrix-valued function, where the j-th column of K is the solution of the Poisson equation (12) for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For the ease of presen- tation, the exposition in this section is restricted to the scalar-valued observation setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' m = 1, so that K becomes a d-dimensional vector-valued function and the superscript j is dropped from the Poisson equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In practice, the Poisson equation must be solved numeri- cally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The numerical gain function approximation problem is as follows: input: samples {Xi : 1 ≤ i ≤ N} i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' ∼ ρ, h(·) output: gain function {Ki : 1 ≤ i ≤ N} where ρ is the (posterior) density and Ki := K(Xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The explicit dependence on time t is suppressed in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' An illustration of the gain function approximation problem appears in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Motivation and overview of approaches The Poisson equation is a linear PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In order to mo- tivate the various solution approaches, it is useful to first consider a finite-dimensional counterpart Ax = b, (22) 10 Figure 2: Gain function approximation problem in the feedback par- ticle filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The exact gain function K(x) = ∇φ(x) where φ solves the Poisson equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The numerical problem is to approximate Ki = ∇φ(x)|x=Xi using only the particles {Xi : 1 ≤ i ≤ N} sam- pled from density ρ (depicted as shaded region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The dashed line indicates the constant gain approximation, where the gain function is approximated by its expected value according to (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' where A is a n × n (strictly) positive-definite symmetric matrix and the righthand-side b is a given n × 1 vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The problem is to compute the unknown n × 1 vector x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For this purpose, the following equivalent formulations of the finite-dimensional problem are first introduced: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' x is the solution of the weak form y TAx = y Tb, ∀ y ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For some chosen positive ϵ, x is the solution to the fixed-point equation x = e−ϵAx + � ϵ 0 e−sAb ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' x is the solution of an optimization problem x = arg min z∈Rn 1 2z TAz − z Tb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' When n is large, these formulations are useful to numeri- cally approximate the solution of (22): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For each fixed y ∈ Rn, the weak form is a single equa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' By restricting y to a suitable low-dimensional subspace S ⊂ Rn, the number of linear equations is reduced for the purposes of obtaining an approximate solution (possibly also in S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The fixed-point equation is useful because e−ϵA is a strict contraction for ϵ > 0 (because A is strictly positive-definite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' So, a good initial guess for x can readily be improved by using the Banach iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The optimization form is useful to develop alternate (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', search type) algorithms to obtain the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' With this background, we turn our attention to the Pois- son equation (12) expressed succinctly as −∆ρφ = (h − ¯h), where ¯h := � h(x)ρ(x)dx and ∆ρ := 1 ρ∇ · (ρ∇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The lin- ear operator ∆ρ is referred to as the probability weighted Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Functional analytic considerations require in- troduction of the function spaces: L2(ρ) is the space of square integrable functions with respect to ρ with inner product ⟨f, g⟩ := � f(x)g(x)ρ(x)dx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' H1(ρ) is the Hilbert space of functions in L2(ρ) whose first derivative, defined in the weak sense, is the also in L2(ρ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' and H1 0(ρ) = {ψ ∈ H1(ρ)| � ψ(x)ρ(x)dx = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' These definitions are important because H1 0(ρ) is the natural space for the solution φ of the Poisson equa- tion (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The operator −∆ρ is symmetric (self-adjoint) and positive definite because −⟨f, ∆ρg⟩ = ⟨∇f, ∇g⟩ = −⟨∆ρf, g⟩, ∀f, g ∈ H1 0(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In the infinite-dimensional settings, one requires an addi- tional technical condition—the Poincar´e inequality (PI)— to conclude that the operator is in fact strictly positive- definite (Taghvaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2020, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Assuming the PI holds, it is also readily shown that ∆−1 ρ is well de- fined, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', a unique solution φ ∈ H1 0(ρ) exists for any given h ∈ L2(ρ) (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2016, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For the purposes of numerical approximation, entirely analogous to the finite-dimensional case, the following equivalent formulations of the Poisson equation are intro- duced: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' φ is a solution of the weak form ⟨∇ψ, ∇φ⟩ = ⟨ψ, h − ¯h⟩ ∀ ψ ∈ H1 0(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (23) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For some chosen positive ϵ, φ is a solution of the fixed- point equation φ = eϵ∆ρφ + � ϵ 0 es∆ρ(h − ¯h)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (24) The notation eϵ∆ρ is used to denote the semigroup as- sociated with ∆ρ (Bakry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The semigroup is readily shown to be a Markov operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' φ is the solution of an optimization problem φ = arg min f∈H1 0(ρ) 1 2⟨∇f, ∇f⟩ + ⟨f, h − ¯h⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (25) Each of the three formulations has been used to develop numerical algorithms for gain function approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' A review of the resulting constructions appears in the follow- ing three subsections: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Galerkin and constant gain approximation The starting point is the weak form (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' A relaxation is considered whereby ψ ∈ S = span{ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' , ψM}, a finite- dimensional subspace of H1 0(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The functions ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' , ψM need to be picked and are referred to as the basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The resulting algorithm is referred to as the Galerkin algo- rithm (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2016, Sec 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The algorithm is given in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 11 Algorithm 1 Synthesis of the gain function: Galerkin approximation Input: {Xi}N i=1, {h(Xi)}N i=1, basis functions {ψl(x)}L l=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Output: {Ki}N i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 1: Calculate h(N) = 1 N �N i=1 h(Xi t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2: Calculate bk = 1 N �N i=1(h(Xi t) − h(N))ψk(Xi t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3: Calculate Akl = 1 N �N i=1 ∇ψl(Xi t)T∇ψk(Xi t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 4: Solve the linear matrix equation Aκ = b for κ, where A = [Akl] and b = [bk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 5: Ki = �L l=1 κl∇ψl(Xi t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Algorithm 2 Synthesis of the gain function: constant gain approximation Input: {Xi}N i=1, {h(Xi)}N i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Output: {Ki}N i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 1: Calculate ˆh(N) = 1 N �N i=1 h(Xi t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2: Ki = 1 N �N j=1 Xj t � h(Xj t ) − ˆh(N)� The most important special case of the Galerkin al- gorithm is obtained upon picking S to be the subspace spanned by the d coordinate functions {x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' , xd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The special case yields the constant gain approximation of the gain K as its expected value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Remarkably, the ex- pected value admits a closed-form expression which is then readily approximated empirically using the particles: K(cnst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' apprx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=') := � ∇φ(x)ρ(x)dx = � (h(x) − ¯h)xρ(x)dx ≈ 1 N N � i=1 (h(Xi) − h(N))Xi, (26) where h(N) := N −1 � i h(Xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2 for an illus- tration of the constant gain approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=') With the constant gain approximation, the FPF algorithm is a non- linear EnKF algorithm (Taghvaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' While its derivation starting from an FPF is novel, the for- mula (26) has been used as a heuristic in the EnKF liter- ature (Evensen, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Bergemann and Reich, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The main issue with the Galerkin approximation is that it is in general very difficult to pick the basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' There have been a number of studies to refine and improve upon this formula (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2016, 2013b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Berntorp and Grover, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Matsuura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Radhakrishnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Radhakrishnan and Meyn, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Berntorp, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In the following two subsections, we describe two approxima- tions which appear to be more promising approaches in general settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Diffusion map-based algorithm The starting point is the fixed-point equation (24) based on the Markov semigroup eϵ∆ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For small values of ϵ, there is a well known approximation of eϵ∆ρ in terms of the so- called diffusion map (which too is a Markov operator): (Tϵf)(x) := 1 nϵ(x) � Rd gϵ(|x − y|) �� gϵ(|y − z|)ρ(z)dz f(y)ρ(y)dy, (27) where gϵ(z) := e− z2 4ϵ is the Gaussian kernel in R and nϵ(x) is the normalization factor chosen so that � (Tϵ1)(x)dx = 1 (Coifman and Lafon, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' A representative approxi- mation result is as follows: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='1 (Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='4 in (Taghvaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Let n ∈ N, t0 < ∞, and t ∈ (0, t0) with ϵ = t n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Then, for all functions f such that f, ∇f ∈ L4(ρ): ∥(T n t n − et∆ρ)f∥L2(ρ) ≤ t √ t n C(∥f∥L4(ρ) + ∥∇f∥L4(ρ)), where the constant C depends only on t0 and ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Because the diffusion map (27) is defined using Gaussian kernels, its empirical approximation is straightforward: (T (N) ϵ f)(x) = 1 n(N) ϵ (x) N � i=1 gϵ(|x − Xi|) ��N j=1 gϵ(|Xi − Xj|) f(Xi), where n(N) ϵ (x) is the normalization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The nature of the approximation is as follows: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='2 (Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='5 in Taghvaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Consider the diffusion map kernel Tϵ and its empirical ap- proximation {T (N) ϵ }N∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Then for any bounded continu- ous function f ∈ Cb(Rd): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (Almost sure convergence) For all x ∈ Rd lim N→∞(T (N) ϵ f)(x) = (Tϵf)(x), a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (Convergence rate) For any δ ∈ (0, 1), in the asymp- totic limit as N → ∞, � |(T (N) ϵ f)(x) − (Tϵf)(x)|2ρ(x)dx ≤ O(log( N δ ) Nϵd ), with probability higher than 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' With these approximations, the fixed-point equa- tion (24) is approximated in two steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The semigroup eϵ∆ρ is approximated by the diffusion map Tϵ: (step 1) φϵ = Tϵφϵ + ϵ(h − ¯hϵ), (28a) where ¯hϵ = � h(x)ρ(ϵ)(x)dx with ρ(ϵ)(x) = nϵ(x)ρ(x) � nϵ(x)ρ(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Tϵ is approximated by its empirical approximation T (N) ϵ : (step 2) φ(N) ϵ = T (N) ϵ φ(N) ϵ + ϵ(h − ¯h(N) ϵ ), (28b) where ¯h(N) ϵ = � h(x)ρ(N) ϵ (x)dx with ρ(N) ϵ (x) = �N i=1 nϵ(x)δXi �N i=1 nϵ(Xi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 12 3 2 1 0 1 2 3 x 0 2 4 6 8 10 K const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' gain exact = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='02 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='10 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='50 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='00 (a) variance dominates bias dominates diffusion map constant gain (b) Figure 3: Bias variance trade-off in the diffusion map-based gain function approximation algorithm: (a) Gain function computed for different values of ϵ with N = 200 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The dashed line is the constant gain solution (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' As ϵ gets larger, the diffusion map gain converges to the constant gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (b) Plot of the MSE as a function of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The shaded area in the background of part (a) is the density ρ which is taken as sum of two Gaussians N(−1, σ2) and N(+1, σ2) with σ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The exact gain function K(x) is computed for h(x) = x by using an (exact) integral formula forr the solution (Taghvaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2020, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In part (b), the MSE is computed as an empirical approximation of the lefthand-side of (29) by averaging over 1000 simulation runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Algorithm 3 Synthesis of the gain function: diffusion map-based algorithm Input: {Xi}N i=1, {h(Xi)}N i=1, Φprev, ϵ, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Output: {Ki}N i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 1: Calculate gij := e− |Xi−Xj |2 4ϵ for i, j = 1 to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2: Calculate kij := gij √� l gil√� l gjl for i, j = 1 to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3: Calculate di = � j kij for i = 1 to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 4: Calculate Tij := kij di for i, j = 1 to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 5: Calculate πi = di � j dj for i = 1 to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 6: Calculate ˆh = �N i=1 πjh(Xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 7: Initialize Φ = Φprev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 8: for t = 1 to L do 9: Φi = �N j=1 TijΦj + ϵ(h − ˆh) for i = 1 to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 10: end for 11: Calculate ri = Φi + ϵhi for i = 1 to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 12: Calculate sij = 1 2ϵTij(rj − �N k=1 Tikrk) for i, j = 1 to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 13: Calculate Ki = � j sijXj for i = 1 to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Based on the finite-dimensional fixed-point equa- tion (28b), an algorithm for gain function approximation is given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Error analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The error in diffusion map approximation comes from two sources: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The bias error due to the diffusion map approximation of the semigroup (step 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The variance error due to empirical approximation in terms of particles (step 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The error is analyzed in (Taghvaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2020) where the following result is proved: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='3 (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='4 in (Taghvaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2020) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Consider the fixed-point formulation of the Pois- son equation (24), its diffusion-map approximation (28a), and its empirical approximation (28b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For each fixed ϵ > 0, there exists a unique solu- tion to (28a) with a uniform bound ∥φϵ∥L2(ρϵ) ≤ C∥h∥L2(ρϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In the asymptotic limit as ϵ → 0 ∥φϵ − φ∥L2(ρϵ) ≤ O(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The operator T (N) ϵ is a strict contraction on L2 0(ρ(N) ϵ ) and the fixed-point equation (28b) admits a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The approximate solution φ(N) ϵ converges to the kernel solution φϵ lim N→∞ ∥φ(N) ϵ − φϵ∥L∞(Ω) = 0, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The following diagram illustrates the convergence and the respective types of errors: φ(N) ϵ N↑∞ −→ (variance) φϵ ϵ↓0 −→ (bias) φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' A quantitative bound on the mean-squared error (MSE) is obtained in the asymptotic limit as ϵ ↓ 0 and N → ∞ as follows: � E[ 1 N N � i=1 |Ki − ∇φ(Xi)|2] � � �� � MSE ≤ O(ϵ2) � �� � bias + O( 1 ϵ(2+d)N ) � �� � variance , (29) where {Ki}N i=1 is computed from the Algorithm (Table 3) and ∇φ is the exact gain function from solving the Poisson 13 const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' gain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='16d - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='3 o( ) m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='e (a) const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' gain m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='e (b) Figure 4: Bias-variance trade-off as a function of (a) the state dimension d ∈ {1, 2, 5, 10} (for a fixed N = 1000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' and (b) the number of particles N ∈ {100, 200, 500, 1000} (for a fixed d = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In the vector case, ρ(x) = ρb(x1) �d n=2 ρg(xn) where ρb is the bimodal density (same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3) and ρg is the Gaussian density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The error due to bias converges to zero as ϵ → 0 and the error due to variance converges to zero as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' There is trade-off between the two errors: To reduce bias, one must reduce ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' However, for any fixed value of N, one can reduce ϵ only up to a point where the variance starts increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The bais-variance trade-off is illustrated with the aid of a scalar (d = 1) example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3: If ϵ is large, the error due to bias dominates, while if ϵ is small, the error due to variance dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' An numerical illustration of scalings with N and d appears in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Additional details on both these examples can be found in (Taghvaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2020, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='4 (Relationship to the constant gain for- mula (26)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' There is a remarkable and somewhat unex- pected relationship between the diffusion map and the con- stant gain approximation (Taghvaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2020, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In particular, in the limit as ϵ → ∞, the diffusion map gain converges to the constant gain (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This sug- gests a systematic procedure to improve upon the constant gain by de-tuning the value of ϵ away from the [ϵ = ∞] limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For any fixed N, a finite value of ϵ is chosen to minimize the MSE according to the bias variance trade- off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Based on this, a rule of thumb for choosing the ϵ value appears in (Taghvaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2020, Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='5 (Analysis of FPF with diffusion map approx- imation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' An analysis of the finite-N FPF using the diffu- sion map approximation appears in (Pathiraja and Stan- nat, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Under mild technical conditions on the drift a(·), σ(·), h(·), it is shown that the finite-N FPF is well- posed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', a strong solution exists for all time t (Pathiraja and Stannat, 2021, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Based on a propagation of chaos type analysis, convergence estimates are derived to relate the finite-N system to its mean-field limit (Pathiraja and Stannat, 2021, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' These estimates are shown to hold up to a certain stopping time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For arbitrary time t, well-posedness and convergence remains an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Variational approximation The starting point is the variational form (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The objective function is denoted by J(f) with its empirical approximation is obtained as J(N)(f) := 1 N N � i=1 1 2|∇f(Xi)|2 − f(Xi)(h(Xi) − h(N)) The problem of minimizing the empirical approximation over all functions is ill-posed: the minimum is unbounded and minimizer does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (Abstractly, this is because the empirical probability distribution does not satisfy the Poincar´e inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=') Therefore, we consider min fθ∈FΘ J(N)(fθ) where FΘ is a parameterized class of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' A function in the class FΘ is denoted by fθ(x) or f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' θ) where θ ∈ Θ is the parameter, and Θ is the parameter set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The two main examples are as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' FΘ = {�M j=1 θjψj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' ψj ∈ H1 0, θj ∈ R for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' , M} is a linear combination of selected basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' With a linear parametrization, the solution of the empirical optimization problem is given by the Galerkin algorithm (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2016, Remark 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' FΘ is a neural network where the parameters θ are the weights in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In practice, it is not possible to solve the optimization problem exactly, but up to some optimization gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In par- ticular, let φ(N) θ be the output of an optimization algorithm that solves the problem up to ϵ error, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', J(φ(N) θ ) ≤ min f∈H1 0 J(f) + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The good news is that it is possible to upper-bound the error in approximating the gain function in terms of this optimization gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 14 3 2 1 0 1 2 3 x 0 1 2 3 4 5 6 7 exact Iter = 100 Iter = 200 Iter = 300 Iter = 1000 0 200 400 600 800 1000 iterations 10 1 100 101 optimization gap Figure 5: Results of the variational gain function approximation using a neural network parameterization: Plot of (a) the gain function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' and (b) the optimization gap as the number of iterations of the Adam algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The problem setup is the same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='6 (Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 1 in Olmez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Let K(N) θ = ∇φ(N) θ where φ(N) θ is the output of an optimization algorithm that solves the minimization objective J(f) with ϵ optimality gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Then ∥K(N) θ − K∥2 L2ρ ≤ 2ϵ, where K = ∇φ is the exact gain function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The optimization gap ϵ depends on the selected parametrization Fθ, number of particles N, and the it- eration number of the employed optimization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Its characterization and analysis is open and the subject of ongoing work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In general, such analysis falls under the framework of statistical learning theory (Anthony et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Shalev-Shwartz and Ben-David, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The numerical results using this approach are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' These results are for the bimodal example in- troduced in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The gain function is parameterized using a two-layer residual NN with 32 neurons per layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The Adam algorithm is used to learn the parameters of the NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Additional details on the numerics can be found in (Olmez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Optimal transport theory In this section, we describe a systematic procedure to construct the exact mean-field process ¯X introduced as step 1 in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The first aspect to note is that while the FPF (11) provides an explicit formula for u and K, the formula is not unique: One can interpret (10) as trans- porting the prior density p0 at time t = 0 to the posterior density pt at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Clearly, there are infinitely many maps that transport one density into another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This sug- gests that there are infinitely many choices of control laws that all lead to exact filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This is not surprising: The exactness condition specifies only the marginal density at times t, which is not enough to uniquely identify a stochas- tic process, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', the joint density at two time instants has not been specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In the following, we first discuss the non-uniqueness issue for the simpler linear Gaussian model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The non- uniqueness naturally motivates optimal transport ideas to uniquely solve for u and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This is the subject of the re- mainder of this section to derive the feedback control law for the FPF (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Non-uniqueness issue in linear-Gaussian setting Consider the linear Gaussian FPF (14) for the mean- field process { ¯Xt}t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The conditional mean and variance of ¯Xt are denoted by ¯mt and ¯Σt, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The condi- tional mean evolves according to d ¯mt = A ¯mtdt + ¯Kt(dZt − H ¯mtdt), where ¯Kt := ¯ΣtH T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Define an error process ξt := ¯Xt − ¯mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Its equation is given by dξt = (A − 1 2 ¯ΣtH TH)ξt + σBd ¯Bt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This is a linear system and therefore the variance of ξt, which equals ¯Σt (by definition), evolves according to the Lyapunov equation d dt ¯Σt = (A − 1 2 ¯ΣtH TH)¯Σt + ¯Σt(A − 1 2 ¯ΣtH TH) T + ΣB = Ricc(¯Σt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The derivation helps show that the equations for the mean and variance are identical to the Kalman filter equa- tions, (8a) and (8b), respectively, and thus proves the ex- actness property of the linear FPF (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' These arguments suggest the following general proce- dure to construct an exact ¯X process: Express ¯Xt as a sum of two terms: ¯Xt = ¯mt + ξt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' t ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 15 where ¯mt evolves according to (8a) and the evolution of ξt is defined by the SDE: dξt = Gtξtdt + σtd ¯Bt + σ′ td ¯Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' where { ¯W}t≥0 and { ¯B}t≥0 are independent copies of the measurement noise {W}t≥0 and the process noise {B}t≥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' and Gt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' σt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' and σ′ t satisfy the matrix equation (for each time) Gt ¯Σt + ¯ΣtGT t + σtσ T t + σ′ t(σ′ t) T = Ricc(¯Σt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (30) By construction, the equation for the variance is given by the Riccati equation (8b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The result is summarized in the following Proposition: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='1 (Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 1 in Taghvaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Consider the linear-Gaussian filtering problem (7) and the following family of the mean-field processes d ¯Xt = A ¯mtdt + ¯Kt(dZt − H ¯mtdt) + Gt( ¯Xt − ¯mt)dt + σtd ¯Bt + σ′ td ¯Wt, ¯X0 ∼ N(m0, Σ0), where Gt, σt, and σ′ t satisfy the consistency condition (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Then, ¯Xt is exact, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' the density of ¯Xt is Gaussian N( ¯mt, ¯Σt) where ¯mt and ¯Σt solve the Kalman filter equa- tions, (8a) and (8b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In general, with different choices of σt and σ′ t, there are infinitely many solutions for (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Below, we describe three solutions that lead to three established form of EnKF and linear FPF: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' EnKF with perturbed observation (Reich, 2011, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (27)): Gt = A − ¯ΣtH TH, σt = σB, σ′ t = ¯ΣtH T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Stochastic linear FPF (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2016, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (26)) or square-root form of the EnKF (Bergemann and Reich, 2012, Eq (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='3)) : Gt = A − 1 2 ¯ΣtH TH, σt = σB, σ′ t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Deterministic linear FPF (Taghvaei and Mehta, 2016, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (15)) (de Wiljes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2018, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (82)): Gt = A − 1 2 ¯ΣtH TH + 1 2ΣB ¯Σ−1 t , σt = 0, σ′ t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Fix σt, σ′ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Then given any particular solution Gt of (30), one can construct a family of solutions Gt + ¯Σ−1 t Ωt, where Ωt is any arbitrary skew-symmetric matrix (Taghvaei and Mehta, 2020, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' III-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For the linear Gaussian problem, the non-uniqueness issue is well known in literature: The two forms of EnKF, the perturbed observation form (Re- ich, 2011) and the square-root form (Bergemann and Re- ich, 2012) are standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' A homotopy of exact determinis- tic and stochastic EnKFs is given in (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' An explanation for the non-uniqueness in terms of the Gauge transformation appears in (Abedi and Surace, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' An extension to the case with correlated noise appears in Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Given the non-uniqueness issue, a natural question is how to identify a unique ¯X process?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For this purpose, opti- mal transport theory is described in the following Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For the linear Gaussian case, the theory is used to derive the following optimal transport form of the linear FPF (see (Taghvaei and Mehta, 2016, 2020) for details): d ¯Xt =A ¯Xtdt + 1 2ΣB ¯Σ−1 t ( ¯Xt − ¯mt)dt + 1 2 ¯Kt(dZt − H ¯Xt + H ¯mt 2 dt) + Ωt ¯Σ−1 t ( ¯Xt − ¯mt)dt, (31) where Ωt = ΩOPT t is a specific skew-symmetric matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The optimal transport FPF (31) is exact and has two dif- ferences compared to the linear FPF (14): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The stochastic term σBd ¯Bt is replaced with the de- terministic term 1 2ΣB ¯Σ−1 t ( ¯Xt − ¯mt)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Given a Gaus- sian prior, the two terms yield the same posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' However, in a finite-N implementation, the stochastic term serves to introduce an additional error of order O( 1 √ N ) (Taghvaei and Mehta, 2018, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The SDE (31) has an extra term involving the skew- symmetric matrix Ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The extra term does not effect the posterior, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', ¯X is exact for all skew-symmetric choices of Ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The specific optimal choice Ωt = ΩOPT t serves to pick the symmetric solution Gt of the consis- tency equation (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For the scalar (d = 1) case, the skew-symmetric term is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Therefore, in the scalar case, the update formula in the linear FPF (14) is op- timal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In the vector case, it is optimal iff ΩOPT t ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' FPF formula In this section, we provide a justification for the feedback control formula in the FPF (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' It is helpful to begin with the simpler deterministic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Deterministic path Let P2(Rd) be the space of everywhere positive proba- bility densities on Rd with finite second moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Given a smooth path {pt ∈ P2(Rd) : t ≥ 0} the problem is to construct a stochastic process { ¯Xt}t≥0 such that the prob- ability density of ¯Xt, denoted as ¯pt, equals pt for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The exactness condition is expressed as ¯pt = pt, ∀ t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (32) As has already been noted, there are infinitely many stochastic processes that satisfy the exactness condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' A unique choice is made by prescribing an additional op- timality criterion based on the optimal transport theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' To make these considerations concrete, assume that the given path {pt}t≥0 evolves according to the PDE ∂pt ∂t = V(pt), t > 0, 16 where V(·) is an operator (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', the Laplacian) that acts on probability densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (This necessarily restricts the op- erator V, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', � V(ρ)(x)dx = 0 for all ρ ∈ P2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=') The following model is assumed for the process { ¯Xt}t≥0: d dt ¯Xt = ut( ¯Xt), ¯X0 ∼ p0, (33) where ut(·) is a control law that needs to be designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' From the continuity equation, the exactness condition (32) is satisfied if − ∇ · (¯ptut) = V(¯pt), ∀ t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (34) The non-uniqueness issue is now readily seen: The first- order PDE (34) admits infinitely many solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' A unique solution ut(·) is picked by minimizing the transportation cost from ¯Xt to ¯Xt+∆t in the limit as ∆t → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The L2- Wasserstein cost is particularly convenient because lim ∆t→0 1 ∆t2 E[|Xt+∆t − Xt|2] = � Rd |ut(x)|2¯pt(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Therefore, for each fixed t, the control law ut(·) is obtained by solving the constrained optimization problem min ut(·) � Rd |ut(x)|2¯pt(x)dx, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='t − ∇ · (¯ptut) = V(¯pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' By a standard calculus of variation argument, the opti- mal solution is obtained as u∗ t = ∇φt where φt solves the Poisson equation −∇ · (¯pt∇φt) = V(¯pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The resulting stochastic process ¯X is defined by d ¯Xt dt = ∇φt( ¯Xt), ¯X0 ∼ p0, φt solves the PDE − ∇ · (¯pt∇φt) = V(¯pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The process is exact by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Suppose the given path is a solution of the heat equation ∂pt ∂t = ∆pt (V(·) is the Laplacian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The solution of the Poisson equation is easily obtained as φt = log(¯pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The optimal transport process then evolves according to d dt ¯Xt = −∇ log(¯pt( ¯Xt)), ¯X0 ∼ p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (35a) This process should be compared to the well known example dXt = dBt, X0 ∼ p0, (35b) where {Bt}t≥0 is a W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='. The density for Xt also solves the heat equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In the language of optimal transporta- tion theory, the coupling defining (35a) is deterministic while it is stochastic in (35b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Stochastic path In the filtering problem, the path of the posterior prob- ability density is stochastic (because it depends upon the random observations {Zt}t≥0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Therefore, the preceding discussion is not directly applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Suppose the stochas- tic path {pt}t≥0 is governed by a stochastic PDE dpt = H(pt)dIt, where H(·) is an operator that acts on probability densities and {It : t ≥ 0} is a W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='. Consider the following SDE model: d ¯Xt = ut( ¯Xt)dt + Kt( ¯Xt)dIt, ¯X0 ∼ p0 where, compared to the deterministic model (33), an addi- tional stochastic term is now included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The problem is to identify control laws ut(·) and Kt(·) such that the condi- tional density of ¯Xt equals pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Upon writing the evolution equation for the conditional density of ¯Xt (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2016, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 1), the exactness condition is formally satis- fied by all such ut(·) and Kt(·) that solve − ∇ · (¯ptKt) = H(¯pt), (36a) − ∇ · (¯ptut) + 1 2(∇ · (¯ptKt)Kt + ¯ptKt∇Kt) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (36b) These equations are the stochastic counterpart of (34), and as with (34), their solution is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The unique solution is obtained by requiring that the coupling from ¯Xt and ¯Xt+∆t is optimal in the limit as ∆t → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In contrast to the deterministic setting, the lead- ing term in the transportation cost E[| ¯Xt+∆t − ¯Xt|2] is O(∆t) whereby lim ∆t→0 1 ∆tE[| ¯Xt+∆t − ¯Xt|2] = � Rd |Kt(x)|2¯pt(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Therefore, for each fixed t, the control law Kt(·) is obtained by solving the constrained optimization problem min Kt(·) � Rd |Kt(x)|2¯pt(x)dx, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='t − ∇ · (¯ptKt) = Ht(¯pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' As before, the optimal solution is given by K∗ t = ∇φt where φt solves the second-order PDE −∇ · (¯pt∇φt) = H(¯pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' It remains to identify the control law ut(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For this pur- pose, the second-order term in the infinitesimal Wasser- stein cost is used: lim ∆t→0 1 ∆t2 � E[| ¯Xt+∆t − ¯Xt|2] − ∆t � Rd |K∗ t (x)|2¯pt(x)dx � = � Rd |ut(x)|2¯pt(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The righthand-side is minimized subject to the con- straint (36b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Remarkably, the optimal solution is ob- tained in closed form as u∗ t = − 1 2¯pt H(¯pt)∇φt + 1 2∇2φt∇φt + ξt, 17 where ξt is the (unique such) divergence free vector field (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', ∇ · (ptξt) = 0) such that u∗ t is of a gradient form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' That (36b) can be solved in an explicit manner was a major surprise at the time of its discovery (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2011b, 2013b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The resulting optimal transport process is d ¯Xt = ∇φt( ¯Xt) ◦ (dIt − 1 2¯pt H(¯pt)dt) + ξt( ¯Xt)dt, ¯X0 ∼ p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (37) It is also readily shown that the process { ¯Xt}t≥0 is in fact exact for any choice of divergence free vector field {ξt}t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The most convenient such choice is to simply set ξt ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The resulting filter is exact and furthermore also (infinitesimally) optimal to the first-order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For the special case of the nonlinear filtering prob- lem, H(ρ) = (h − ¯h)ρ where ¯h = � h(x)ρ(x)dx and dIt = (dZt − ¯htdt) is the increment of the innovation pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For these choices, the optimal transport stochastic process (37) becomes d ¯Xt = ∇φt( ¯Xt) ◦ (dZt − 1 2(h( ¯Xt) + ¯ht)dt) + ξt( ¯Xt)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The feedback control law in the FPF algorithm (11) repre- sents the particular sub-optimal choice ξt ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The choice is optimal for d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Optimal transport formula for the static example We now revisit the static example introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='1 with the aim of deriving an explicit form of the control U and relating it to the FPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' As explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='1, the problem is to find a control U such that E[f(X)|Y ] = E[f( ¯X1)|Y ] for all functions f ∈ Cb(Rd), where ¯X1 = ¯X0 + U and ¯X0 is an independent copy of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This con- dition is equivalently expressed as ( ¯X1, Y ) ∼ PXY , and the problem of finding U is formulated as the following optimal transportation problem: min U∈σ( ¯ X0,Y ) E[|U|2], s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='t ¯X1 = ¯X0 + U, ( ¯X1, Y ) ∼ PXY , (38) where the notation U ∈ σ( ¯X0, Y ) means that U is allowed to be measurable with respect to ¯X0 and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This is an op- timal transportation problem between ( ¯X0, Y ) ∼ PX ⊗PY and (X, Y ) ∼ PXY where the transportation is constrained to be of the form ( ¯X0, Y ) → ( ¯X0 + U, Y ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', the second argument Y remains fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Its solution is obtained as an extension of the celebrated Brenier’s result (Brenier, 1991) as follows: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='3 (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 1 in Taghvaei and Hosseini (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Consider the optimal transportation problem (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Sup- pose PX admits a density with respect to the Lebesgue mea- sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Then the optimal control is U = ∇¯Φ( ¯X0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Y ) − ¯X0, where ¯Φ is the minimizer of the dual Kantorovich problem min Φ∈CVXx E[Φ( ¯X0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Y ) + Φ⋆(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Y )], (39) where Φ ∈ CVXx means x �→ Φ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' y) is convex in x for all y and Φ⋆(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' y) := supz zTx − Φ(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' y) is the convex conju- gate of Φ with respect to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='4 (Relationship to the update formula for FPF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In the continuous-time limit, the dual Kantorovich problem (39) is related to the variational form (25) of the Poisson equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In particular, with ∆Zt = h(Xt)∆t + ∆Wt, the solution to the problem (39) is as follows (Taghvaei and Hosseini, 2022, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2): ¯Φ( ¯Xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' ∆Zt) = 1 2| ¯Xt|2 + φ( ¯Xt)∆Zt + ψ( ¯Xt)∆t + O(∆t2) where φ is the solution to the Poisson equation (12) with ρ taken as the density of PX, and ψ is the unique such function such that ∇ψ = − h+¯h 2 ∇φ + 1 4∇|∇φ|2 + ξ where ξ is divergence free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Therefore, the optimal transformation ¯Xt �→ ¯Xt+∆t is given by, ¯Xt+∆t = ∇x ¯Φ( ¯Xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' ∆Zt) = ¯Xt + ∇φ( ¯Xt)(∆Zt − h( ¯Xt) + ¯ht 2 ∆t) + 1 4∇|∇φ( ¯Xt)|2∆t + ξ( ¯Xt)∆t + O(∆t2) which in the limit as ∆t → 0 is the SDE for the optimal transport FPF (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='5 (Stochastic optimization and DNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The variational problem (39) is a stochastic optimization prob- lem which allows for application of machine learning tools to approximate its solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In particular, deep neural net- works (DNNs) can be used to parameterize the function Φ and stochastic optimization algorithms employed to learn the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Preliminary results in this direction are presented in (Taghvaei and Hosseini, 2022) with a com- prehensive development the subject of ongoing work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' PART II 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' CIPS for optimal control In order to elucidate the ideas as clearly as possible, our focus in this paper is entirely on the linear quadratic (LQ) problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Its extension to the nonlinear optimal control problem (4) can be found in (Joshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Problem statement and background The finite-horizon linear quadratic (LQ) optimal control problem is a special case of (4) as follows: 18 min u J(u) = � T 0 1 2 � |Cxt|2 + u T t Rut � dt + x T T PT xT (40a) subject to: ˙xt = Axt + But, x0 = x (40b) It is assumed that (A, B) is controllable, (A, C) is observ- able, and matrices PT , R ≻ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The [T = ∞] limit is re- ferred to as the linear quadratic regulator (LQR) problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' It is well known that the optimal control ut = φt(xt) where the optimal policy is linear φt(x) = Ktx where Kt = −R−1B TPt, 0 ≤ t ≤ T is the optimal gain matrix and {Pt : 0 ≤ t ≤ T} is a solution of the backward (in time) DRE − d dtPt = A TPt+PtA+C TC−PtBR−1B TPt, PT (given) (41) The ARE is obtained by setting the left-hand side to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' As T → ∞, for each fixed time t, Pt → P ∞, expo- nentially fast (Kwakernaak and Sivan, 1972, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='7), where P ∞ ≻ 0 is the unique such positive-definite solu- tion of the ARE, and therefore the optimal gain converges, Kt → K∞ := −R−1BTP ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Approximation of the gain K∞ is a goal in recent work on model-based RL for the LQR problem (Fazel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Mohammadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Objectives and assumptions For the reasons noted in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 1, we are interested in a simulation-based solution that does not rely on an explicit solution of the DRE (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' To clarify what is meant by a simulation-based solution in the context of model-based RL, we make a formal assumption as follows: Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Functions f(x, α) = Ax + Bα and c(x) = Cx are available in the form of an oracle (which allows function evaluation at any state action pair (x, α) ∈ Rd × Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Matrices R and PT are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Both of these ma- trices are strictly positive-definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Simulator is available to simulate (40b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Simulator provides for an ability to add additional in- puts outside the control channel (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', see (5a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This assumption is motivated from the data assimila- tion literature where it is entirely standard and widely used in applications, such as weather prediction, involving EnKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Part 1 of the assumption means that the matri- ces A, B, C are not available explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Rather, for any given (x, α) ∈ Rd × Rm, the vectors f(x, α) and c(x) can be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Function evaluation forms for the dynamics and the cost function is also a standard assumption for any model-based RL algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Part 2 of the assumption is not too restrictive for the following two reasons: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In physical systems, one is typically able to assess relative costs for different control inputs (actuators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This knowledge can be used to select R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For the LQR problem, under mild technical condi- tions, the optimal policy is stationary and does not depend upon the choice of PT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' If these matrices are not available, one possibility is to take R and PT to be identity matrices of appropriate di- mensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The main restriction comes from part 3 of the assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' However, as the widespread use of EnKF am- ply demonstrates, it is not un-realistic to assume it for a simulation-based solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Of course, it will not be possi- ble with a physical experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Dual EnKF The dual EnKF algorithm is obtained from making use of duality between optimal control and filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For this purpose, we need to first dualize the DRE (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Under the assumptions of this paper, Pt ≻ 0 for 0 ≤ t ≤ T whenever PT ≻ 0 (Brockett, 2015, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Set St = P −1 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' It is readily verified that {St : 0 ≤ t ≤ T} also solves a DRE (which represents the dual of (41)) d dtSt = ASt + StA T − BR−1B T + StC TCSt, ST = P −1 T (42) The strategy is to approximate {St : 0 ≤ t ≤ T} using simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' As before, the construction proceeds in two steps: (i) definition of an exact mean-field process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' and (ii) its finite-N approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Mean-field process: Define a stochastic process ¯Y = { ¯Yt ∈ Rd : 0 ≤ t ≤ T} as a solution of the following backward (in time) SDE: d ¯Yt = A ¯Ytdt + Bd �ηt + 1 2 ¯StC T(C ¯Yt + C¯nt)dt, 0 ≤ t < T ¯YT ∼ N(0, ST ) (43) where η = {ηt ∈ Rm : 0 ≤ t ≤ T} is a W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' with covariance matrix R−1, and ¯nt := E[ ¯Yt], ¯St := E[( ¯Yt − ¯nt)( ¯Yt − ¯nt) T], 0 ≤ t < T (44) The meaning of the backward arrow on d �η in (43) is that the SDE is simulated backward in time starting from the terminal condition specified at time t = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The reader is referred to (Nualart and Pardoux, 1988, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='2) for the definition of the backward Itˆo-integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The mean-field process is useful because of the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='1 (Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 1 in Joshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The solution to the SDE (43) is a Gaussian stochastic process, in which the mean and covariance of ¯Yt are given by ¯nt = 0, ¯St = St, 0 ≤ t ≤ T Consequently, ¯Xt := ¯S−1 t ( ¯Yt − ¯nt) is also a Gaussian ran- dom variable with E[ ¯Xt] = 0, E[ ¯Xt ¯X T t ] = Pt, 0 ≤ t ≤ T 19 The significance of Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='1 is that the optimal control policy φt(·) can now be obtained in terms of the statis- tics of the random variable ¯Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Specifically, we have the following two cases: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' If the matrix B is explicitly known then the optimal gain matrix Kt = −R−1B TE[ ¯Xt ¯X T t ] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' If B is unknown, define the Hamiltonian (the continuous-time counterpart of the Q- function (Mehta and Meyn, 2009)): H(x, α, t) := 1 2|Cx|2 + 1 2α TRα � �� � cost function +x TE[ ¯Xt ¯X T t ] (Ax + Bα) � �� � model (40b) from which the optimal control law is obtained as φt(x) = arg min α∈Rm H(x, α, t) by recalling the minimum principle, which states that the optimal control is the unique minimizer of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' It is noted that the Hamiltonian H(x, α, t) is in the form of an oracle because (Ax+Bα) is the right-hand side of the simulation model (40b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Finite-N approximation: The particles {Y i t ∈ Rd : 0 ≤ t ≤ T, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' , N} evolve according to the backward SDE: dY i t = AY i t dt + Bd �η i t � �� � i-th copy of model (40b) + S(N) t C T � CY i t + Cn(N) t 2 � � �� � coupling dt, (45) Y i T i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='d ∼ N(0, P −1 T ), 1 ≤ i ≤ N ηi := {ηi t : 0 ≤ t ≤ T} is an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='d copy of η and n(N) t = 1 N N � i=1 Y i t S(N) t = 1 N − 1 N � i=1 (Y i t − n(N) t )(Y i t − n(N) t ) T The CIPS (45) is referred to as the dual EnKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Optimal control: Set Xi t = (S(N) t )−1(Y i t − n(N) t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' There are two cases as before: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' If the matrix B is explicitly known then K(N) t = − 1 N − 1 N � i=1 R−1(B TXi t)(Xi t) T (46) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' If B is unknown, define the Hamiltonian H(N)(x, α, t) := 1 2|Cx|2 + 1 2α TRα � �� � cost function + 1 N − 1 N � i=1 (x TXi t)(Xi t) T (Ax + Bα) � �� � model (40b) from which the optimal control policy is approximated as φ(N) t (x) = arg min a∈Rm H(N)(x, a, t) There are several zeroth-order approaches to solve the minimization problem, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', by constructing 2-point estimators for the gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Since the objective func- tion is quadratic and the matrix R is known, m queries of H(N)(x, ·, t) are sufficient to compute φ(N) t (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The overall dual EnKF algorithm can be found in (Joshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2022, Algorithm 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Relating dual EnKF to model-based RL The following remarks are included to help provide an intuitive explanation of the various aspects of the dual EnKF and relate these to the model-based RL: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In designing any RL algorithm, the first issue is the representation of the unknown value function (Pt in the linear case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Our novel idea is to represent Pt is in terms of statistics (variance) of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Such a representation is distinct from representing the value function, or its proxies, such as the Q function, within a parameterized class of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Value iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The algorithm is entirely simula- tion based: N copies of the model (40b) are simulated in parallel where the terms on the right hand-side of (45) have the following intuitive interpretations: (a) Dynamics: The first term “AY i t dt” on the right- hand side of (45) is simply a copy of uncontrolled dynamics in the model (40b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (b) Control: The second term “Bd �η i t” is the con- trol input for the i-th particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' It is specified as a W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' with covariance R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' One may inter- pret this as an approach to exploration whereby cheaper control directions are explored more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (c) Coupling: The third term, referred to as the cou- pling, effectively implements the value iteration step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Coupling has a “gain times error” structure where S(N) t CT is the gain and 1 2(CY i t + Cn(N) t ) is the counterpart of the error in the linear FPF (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Arrow of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The particles are simulated backward—from terminal time t = T to initial time t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This is different from most model-based RL but consistent with the dynamic programming (DP) equation which also proceeds backward in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 20 0 5 10 (i) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='0 P11 ARE EnKF DRE 0 5 10 (ii) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='5 P12 0 5 10 (iii) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='5 P21 0 5 10 (iv) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='6 P22 T − t (a) d = 2 0 5 10 0 5 10 15 20 ARE EnKF DRE T − t → 0 5 10 2 4 6 8 10 ARE EnKF DRE (b) d = 10 Figure 6: Comparison of the numerical solution obtained from the EnKF, the DRE, and the ARE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Note the x-axis for these plots is T − t for 0 ≤ t ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' d is the state-dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Convergence and error analysis In (Joshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2022, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3), under certain additional assumptions on system matrices, the following error bound is derived: E[∥S(N) t − ¯St∥F ] ≤ C1 √ N + C2e−2λ(T −t)E[∥S(N) T − ¯ST ∥F ], (47) where C1, C2, λ are positive constants and || · ||F denotes Frobenius norm for matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The significance of the bound (47) is as follows: The constant λ is same as the rate that governs the convergence of the solution of the DRE (41) to the stationary solution (of the infinite-horizon LQR problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This means that the dual EnKF learns the optimal LQR gain exponentially fast with a rate that is as good as one would obtain from directly solving the DRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Convergence is numerically illustrated for a d- dimensional system expressed in its controllable canonical form A = � ���� 0 1 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 0 0 0 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' a1 a2 a3 a4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' ad � ���� , B = � ���� 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 1 � ���� where the entries (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' , ad) ∈ Rd are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' samples from N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The matrices C, R, PT are identity matrices of appropriate dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For numerics, we fix T = 10, chose the time-discretization step as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='02, and use N = 1000 particles to simulate the dual EnKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 6(a) depicts the convergence of the four entries of the matrix P (N) t for the case where d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 6(b) depicts the analogous results for d = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 7(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 7(b) depict the open-loop poles (eigenvalues of the matrix A) and the closed-loop poles (eigenvalues of the matrix (A + BK(N) 0 )), for d = 2 and d = 10, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Note that the closed-loop poles are stable, whereas some open-loop poles have positive real parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Comparison to literature We present a comparison of the dual EnKF with policy gradient algorithms in Mohammadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (2022) (denoted as [M21]) and Fazel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (2018) (denoted as [F18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In these prior works, by restricting the control policies to the linear form ut = Kxt, the LQR problem reduces to the finite-dimensional static optimization problem: K⋆ = arg min K J(K) = E �� ∞ 0 x T t Qxt + u T t Rut dt � (48) where the expectation is over the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The authors apply a pure-actor method using “zeroth order” methods to approximate gradient descent, much like the early REINFORCE algorithm for RL (Sutton and Barto, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' A qualitative comparison of the dual EnKF with these prior algorithms is given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Choosing t = 0 in (47), the error is smaller than ε if the number of particles N > O( 1 ε2 ) and the simulation time T > O(log( 1 ε)), while the iteration number is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This is compared with pol- icy optimization approach in Fazel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (2018) where the 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='5 Im Re OL CL (a) d = 2 −1 0 1 2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='0 Re Im OL CL (b) d = 10 Figure 7: Open and closed-loop poles for the two plots (parts (a) and (b)) depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' number of particles and the simulation time scales poly- nomially with ε, while the number of iterations scale as O(log( 1 ε)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This result is later refined in Mohammadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (2022) where the required number of particles and the sim- ulation time are shown to be O(1) and O(log( 1 ε)) respec- tively (although this result is valid with probability that approaches zero as the number of iterations grow (Moham- madi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2022, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' A numerical comparison is made on the benchmark spring mass damper example borrowed from (Mohammadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2019, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' VI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 8 depicts the relative mean- squared error, defined as MSE := 1 T E �� T 0 ∥Pt − P (N) t ∥2 F ∥Pt∥2 F dt � Two trends are depicted in the figure: the O( 1 N ) decay of the MSE as N increases (for d fixed), which is a numerical illustration of the error bound (47), and a plot of the MSE as a function of dimension d (for N fixed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' A side-by-side comparison with [F18] and [M21] is de- picted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The comparison is for the following met- rics (taken from Mohammadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (2022)): errorgain = ∥Kest − K∞∥F ∥K∞∥F , errorvalue = cest − c∞ c(N) init − c∞ where the LQR optimal gain K∞ and the optimal value c∞ are computed from solving the ARE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The value c(N) init is approximated using the initial gain K = 0 (Note such a gain is not necessary for EnKF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Because [F18] is for discrete-time system, an Euler approximation is used to obtain a discrete-time model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In the numerical experiments, the dual EnKF is found to be significantly more computationally efficient—by two orders of magnitude or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The main reason for the order of magnitude improvement in computational time is as follows: An EnKF requires only a single iteration over a fixed time-horizon In contrast, [F18] and [M21] require several steps of gradient descent, with each step requiring an evaluation of the LQR cost, and because these opera- tions must be done serially, these computations are slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In carrying out these comparisons, the same time- horizon [0, T] and discretization time-step ∆t was used for all the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' It is certainly possible that some of these parameters can be optimized to improve the perfor- mance of the other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In particular, one may consider shorter or longer time-horizon T or use paral- lelization to speed up the gradient calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Codes are made available on Github for interested parties to inde- pendently verify these comparisons1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Discussion and conclusion In this survey, we described CIPS to approximate the solution of the optimal filtering and the optimal control problems (in parts I and II, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' As explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 1, there are close parallels with DA and RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In this section, we expand on some of these parallels with the goal of highlighting some important points and directions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Data assimilation, sampling, optimal transportation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' CIPS may be viewed as a sampling algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The FPF control law (coupling) is designed to sample from the pos- terior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Compared to the conventional particle filters, cou- pling is beneficial because the issue of particle degener- acy is avoided (as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' To design the coupling, optimal transportation theory provides a use- ful framework (as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Variations of the 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='com/anantjoshi97/EnKF-RL 22 Algorithm particles/samples simulation time iterations dual EnKF O( 1 ε2 ) O(log( 1 ε)) 1 Fazel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (2018) poly � 1 ε � poly � 1 ε � O(log( 1 ε)) Mohammadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (2022) O(1) O(log( 1 ε)) O(log( 1 ε)) Table 1: Computational complexity comparison of the algorithms to achieve ε error in approximating the infinite-horizon LQR optimal gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 102 103 Number of particles (N) 10−2 100 102 104 MSE d = 2 d = 10 d = 20 d = 50 d = 80 0 20 40 60 80 Dimensions (d) 10−2 10−1 100 101 102 MSE O(1/N) N = 100 N = 300 N = 1000 Figure 8: Performance of the dual EnKF algorithm: MSE as a function of the number of particles N and system dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' basic approach described here have been used in construc- tion of a class of filtering algorithms (Halder and Geor- giou, 2017, 2018, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Garbuno-Inigo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Luo, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The optimal transport formulation has also been extended to the Schr¨odinger bridge setting by considering a cost with respect to the (prior) dynamics, or considering an entropic regularization (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Reich, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In related works, the coupling viewpoint along with geo- metric notions from optimal transportation theory, have enabled application of optimization algorithms to design sampling schemes (Liu and Wang, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Richemond and Maginnis, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Frogner and Poggio, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Chizat and Bach, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Taghvaei and Mehta, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Part II of this paper is motivated by the enormous suc- cess of the CIPS (EnKF) in DA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Reinforcement learning and optimal control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Com- pared to typical RL approaches, there are two key innova- tions/differences: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Representation of the unknown value function in terms of the statistics (variance) of a suitably designed process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Design of interactions (coupling) between simulations for the purposes of policy optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' We fully believe that the two key innovations may be useful for many other types of models including MDPs and par- tially observed problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In the LQ setting of the problem, doing so is beneficial because of the learning rate: Since the [N = ∞] limit is exact tor the LQ problem, the dual EnKF algorithm yields a learning rate that closely approx- imates the exponential rate of convergence of the solution of the DRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This is rigorously established with the aid of error bound (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In numerical examples, this property is shown to lead to an order of magnitude better performance than the state-of-the-art algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Apart from RL, model predictive control (MPC) is an- other area where a model in the form of a simulator is assumed to design optimal control for problems such as (4) (Rawlings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Using duality, MPC meth- ods have been adapted to design the moving horizon es- timator (MHE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' A big selling point of MPC is its ability to handle constraints which has not been a major theme in the DA literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Another notable distinction is that while MPC aims to find a single (optimal) trajectory, CIPS simulate multiple stochastic trajectories in a Monte Carlo manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Notably, the solution of the deterministic optimal control problem (4) is based on simulating (5a) which is an SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For the stochastic MPC problems, multiple simula- tions have been considered in the scenario-based approach (Campi and Garatti, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Some perspectives on future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In basic sciences, there are a number of important examples of interacting particle systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This paper presents results on the theme of “CIPS as an algorithm”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The most historical of such 23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='15 101 103 105 Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Time (s) EnKF [M21] [F18] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='0075 101 103 Error in gain Error in cost Figure 9: Comparison with algorithms in Fazel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (2018) (labeled [F18]) and Mohammadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' (2022) (labeled [M21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The comparisons depict the computation time (in Python) as a function of the relative error in approximating the LQR gain and cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' algorithms is the EnKF which is used to solve the problem of data assimilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' It is hoped that this survey convinces the reader that the paradigm is also useful for solving other problems in estimation and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' A major selling point of CIPS, and also the reason for widespread use of the EnKF, is that it is able to work directly with a simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Therefore, it is amenable as a solution method for complex systems where models typically exist only in the form of a simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Apart from the open problems described in the main body of the paper, a few themes for future research are as follows: MPC offers a useful benchmark for CIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' With the exception of the geometric approaches, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', FPF on Riemannian manifolds (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2017b), con- straints has not been an important theme in design of CIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' It is an important problem to extend the design of mean-field process to handle general types of constraints in inputs and states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' One possible next step is to extend the dual EnKF to the inequality- constrained LQR problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' RL could be an important application for CIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' A key difference is that CIPS-based solution does not rely on function approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Instead, the value func- tion is approximated in terms of the distribution of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' This has some advantages, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', avoids the need to select basis functions, and some disadvan- tages, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', availability of computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' It will be useful to understand some of these trade-offs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Relationship to mean-field games and optimal control should be further developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' CIPS represent simple examples of mean-field type control laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' However, derivation of these control laws is, more often than not, rooted in methods from optimal transportation theory (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' It remains an open problem to de- rive the FPF control law starting from a mean-field optimal control type objective (some partial results in this direction appear in (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2018a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Extensions to partially observed optimal control prob- lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For the linear Gaussian model, algorithms de- scribed in parts I and II are easily combined to obtain a CIPS for the partially observed problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The solu- tion is based on the separation principle: A forward (in time) EnKF is run to solve the optimal filtering problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' and a completely independent backward (in time) dual EnKF is run to solve the optimal control problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' For the nonlinear problem, there may be benefit to couple the forward and backward CIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Distributionally robust FPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' In order to handle un- certainty in signal and observation models, it may be useful to explore methods from distributionally ro- bust optimization framework (Rahimian and Mehro- tra, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' The framework has been used to develop the Wasserstein robust Kalman filter for the linear Gaussian model (Shafieezadeh Abadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' Its extension to the nonlinear filtering model (1) is open and may be possible based on the optimal trans- port formulation of the FPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
+page_content=' References Abedi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfA_pb/content/2301.00935v1.pdf'}
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+Federated Fog Computing for Remote Industry 4.0 Applications
+Razin Farhan Hussain
+A Dissertation presented to the Graduate Faculty
+in Partial Fulfillment of the Requirements for the Degree
+Doctor of Philosophy
+University of Louisiana at Lafayette
+Fall 2022
+APPROVED:
+Mohsen Amini Salehi, Chair
+The Center for Advanced Computer Studies
+Sheng Chen
+The Center for Advanced Computer Studies
+Xu Yuan
+The Center for Advanced Computer Studies
+Li Chen
+The Center for Advanced Computer Studies
+Mary Farmer-Kaiser
+Dean of the Graduate School
+arXiv:2301.00484v1 [cs.DC] 1 Jan 2023
+
+© Razin Farhan Hussain
+2022
+All Rights Reserved
+
+Abstract
+Industry 4.0 operates based on IoT devices, sensors, and actuators,
+transforming the use of computing resources and software solutions in diverse
+sectors. Various Industry 4.0 latency-sensitive applications function based on
+machine learning and utilize the generated sensor data for automation and other
+industrial activities. Sending sensor data to cloud systems is time consuming and
+detrimental to the latency constraints of the applications. In this circumstance, fog
+computing can be used to support latency-sensitive applications. Executing these
+applications across heterogeneous fog systems demonstrates stochastic execution
+time behaviour that affects the task completion time. Hence, we investigate and
+model various Industry 4.0 ML-based applications’ stochastic executions and
+introduce real-world execution time traces of Industry 4.0 applications.
+Remote offshore industries like oil and gas are prone to disasters requiring
+the coordination of various latency-sensitive activities. Accordingly, their procured
+fog computing resources can get oversubscribed due to the surge in the computing
+demands during a disaster. Hence, in this dissertation, we propose federating nearby
+fog computing systems and forming a fog federation to make remote Industry 4.0
+sites resilient against the surge in computing demands. We propose a statistical
+resource allocation method across fog federation for latency-sensitive tasks.
+Many of the modern Industry 4.0 applications operate based on a workflow
+of micro-services that are used alone within an industrial site. As such, industry 4.0
+solutions need to be aware of applications’ architecture, particularly monolithic vs.
+iii
+
+micro-service. Therefore, we propose a probability-based resource allocation method
+that can partition micro-service workflows across fog federation to meet their
+latency constraints. Another concern in Industry 4.0 is the data privacy of the
+federated fog. As such, we propose a solution based on federated learning to train
+industrial ML applications across federated fog systems without compromising the
+data confidentiality.
+iv
+
+To my wife, Rezwana Mahjabeen, my daughter, Ruzainah Shehzeen Hussain, my
+parents, H.K.M. Altaf Hussain and Fariha Akhter, my Sister, Sabrina Shahreen
+Sarna, my brother, Shoumin Rafsun Hussain and to all my friends and loved ones.
+
+Acknowledgments
+I sincerely thank my supervisor, Dr. Mohsen Amini Salehi, for his constant
+encouragement and passion for computer science and, especially, for his guidance,
+support, and cooperation. Thanks to my dissertation committee, Dr. Xu Yuan, Dr.
+Li Chen, and Dr. Sheng Chen. Thanks to Sm Zobaed, Ali Mokhtari, Davood
+Ghatreh Samani, and Chavit Denninart for their assistance in the work of this
+dissertation. Finally, thanks go to the Center for Advanced Computer Studies and
+the Graduate School at the University of Louisiana at Lafayette for their support
+and guidance.
+vi
+
+Table of Contents
+Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
+Dedication
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
+Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
+List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
+List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
+Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
+1.1
+Research Problem and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . 7
+1.2
+Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
+1.3
+Dissertation Organisation
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
+Chapter 2: Background and Literature Study . . . . . . . . . . . . . . . . . . . . . . 13
+2.1
+Computing as a Prominent Aspect of Industry 4.0 . . . . . . . . . . 13
+2.2
+Distributed Computing Systems in Industry 4.0
+. . . . . . . . . . . . 14
+2.2.1
+Cloud Computing
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
+2.2.2
+Edge and Fog Computing for Remote Industry 4.0 . . . . 16
+2.2.3
+Edge-to-Cloud Continuum
+. . . . . . . . . . . . . . . . . . . . . . . . . 18
+2.2.4
+Use Case of Edge-to-Cloud Continuum in Smart
+O&G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
+2.2.5
+Landscape of Computing in O&G . . . . . . . . . . . . . . . . . . . 21
+2.3
+Smart O&G: Data and Software Aspects
+. . . . . . . . . . . . . . . . . . 22
+2.3.1
+Big Data in the O&G industry
+. . . . . . . . . . . . . . . . . . . . . 22
+2.3.2
+Machine Learning as a Data-driven applications in
+O&G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
+2.3.3
+Digital Twin: Another Data-driven Applications in
+O&G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
+2.4
+Edge-to-Cloud for AI and other Data-driven Applications
+in Smart O&G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
+2.5
+Federated Fog and It’s Challenges in Remote Industry 4.0 . . . . 27
+2.5.1
+Real-time Services of Industry 4.0 . . . . . . . . . . . . . . . . . . 28
+2.5.2
+Heterogeneous Fog Systems in Remote Industry
+4.0
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
+2.5.3
+Uncertainty of Task Completion in Fog Systems
+. . . . . 30
+2.5.4
+Software Architecture of Industry 4.0
+Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
+vii
+
+2.6
+The Scope of Fog Federation in Industry 4.0
+. . . . . . . . . . . . . . . 33
+2.7
+Data Privacy Aspects of a Federated Fog Computing
+System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
+2.7.1
+Major Challenges of FL in Fog Federation
+. . . . . . . . . . 35
+2.8
+Downside of Smart Solutions in Industry 4.0 . . . . . . . . . . . . . . . 36
+2.9
+Summary and Positioning of this Dissertation . . . . . . . . . . . . . . 38
+Chapter 3: Performance Analysis of DNN-based Application in
+Cloud and Fog Systems
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
+3.1
+Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
+3.2
+DNN-Based Applications in Industry 4.0
+. . . . . . . . . . . . . . . . . . 41
+3.2.1
+Fire Detection
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
+3.2.2
+Human Activity Recognition
+. . . . . . . . . . . . . . . . . . . . . . . 42
+3.2.3
+Oil Spill Detection
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
+3.2.4
+Acoustic Impedance Estimation
+. . . . . . . . . . . . . . . . . . . . 44
+3.3
+Computing Platforms for Industry 4.0 . . . . . . . . . . . . . . . . . . . . . 44
+3.3.1
+Amazon Cloud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
+3.3.2
+Chameleon as Fog Computing System . . . . . . . . . . . . . . . 46
+3.4
+Environmental Setup for Performance Modeling . . . . . . . . . . . . 46
+3.5
+Application-Centric Analysis of Inference Time . . . . . . . . . . . . . 47
+3.5.1
+Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
+3.5.2
+Statistical Distribution of Inference Execution
+Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
+3.5.3
+Analysis of Central Tendency and Dispersion
+Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
+3.6
+Resource-Centric Analysis of Inference Time . . . . . . . . . . . . . . . 53
+3.6.1
+Estimating Confidence Interval using Jackknife
+Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
+3.6.2
+Estimating Confidence Interval using Bootstrap
+Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
+3.7
+Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
+Chapter 4: The Benefits of Federated Fog to Manage Monolithic
+Workload in Remote Industrial Sites
+. . . . . . . . . . . . . . . . . . . . . . . . . . 60
+4.1
+Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
+4.2
+End-to-End Latency in Federated Fog Systems
+. . . . . . . . . . . . . 61
+4.2.1
+Estimating Communication Latency
+. . . . . . . . . . . . . . . . 61
+4.2.2
+Estimating Computational Latency
+. . . . . . . . . . . . . . . . . 63
+4.2.3
+Estimating End-to-End Latency
+. . . . . . . . . . . . . . . . . . . . 64
+viii
+
+4.3
+Robust Resource Allocation in the Federated Fog
+Computing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
+4.4
+Performance Evaluation of Federated Fog . . . . . . . . . . . . . . . . . . 69
+4.4.1
+Baseline Task Assignment Heuristics for Load
+Balancer
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
+4.4.2
+Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
+4.5
+Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
+Chapter 5: Adapting Remote Industry 4.0 Smart Micro-Service
+Applications to Federated Fog Computing Systems . . . . . . . . . . . . . . 76
+5.1
+Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
+5.1.1
+Smart Micro-Service Applications for Industry
+4.0
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
+5.1.2
+Federated Fog Systems for Industry 4.0
+Micro-service Applications . . . . . . . . . . . . . . . . . . . . . . . . . 78
+5.2
+Partitioning Method for Micro-service Application
+Workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
+5.3
+Resource Allocation Method for Partitioned Micro-service
+Applications Across Fog Federation
+. . . . . . . . . . . . . . . . . . . . . . . 85
+5.4
+Performance Evaluation of Software Architecture-Aware
+Federated Fog Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
+5.4.1
+Comparison of Micro-service Workflow
+Partitioning Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
+5.4.2
+Comparison of Resource Allocation Methods
+. . . . . . . . 91
+5.4.3
+Fog Federation Scaling Impact . . . . . . . . . . . . . . . . . . . . . . 93
+5.5
+Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
+Chapter 6: Data Security & Privacy Aspects in Federated Fog
+Computing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
+6.1
+Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
+6.2
+Problem Formulation for Federated Learning . . . . . . . . . . . . . . 100
+6.3
+Federated Learning to Mitigate the Class Imbalance
+. . . . . . . 102
+6.4
+Experimental Setup
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
+6.5
+Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
+6.5.1
+Tuning Loss Function
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
+6.5.2
+The Impact of Using IID Data Distribution
+. . . . . . . . 107
+6.5.3
+The Impact of Using non-IID Data Distribution . . . . 108
+6.5.4
+The Impact of Using non-IID and Unbalanced Data
+Distribution
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
+6.5.5
+The Impact of Class Imbalance Intensity . . . . . . . . . . . 111
+ix
+
+6.5.6
+The Impact of Number of Workers on the Global
+Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
+6.5.7
+Verifying the Impact of Aggregation Scheme on
+Global Model
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
+6.6
+Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
+Chapter 7: Threats and Side-Effects of Smart Solutions in Industry
+4.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
+7.1
+Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
+7.2
+Taxonomy of Cyber-Threats and Side-Effects in the Smart
+O&G Industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
+7.3
+Vulnerabilities caused by the Interplay of Informational and
+Operational technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
+7.4
+Cyber Threats in Smart Oil and Gas Industry . . . . . . . . . . . . . 124
+7.4.1
+Vulnerabilities of Sensitive Data
+. . . . . . . . . . . . . . . . . . 125
+7.4.2
+Vulnerabilities of Smart Systems
+. . . . . . . . . . . . . . . . . . 126
+7.4.3
+Malware and Vulnerability of Information
+Technology (IT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
+7.5
+Incompatible IoT Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
+7.5.1
+Hardware-level Incompatibility
+. . . . . . . . . . . . . . . . . . . . 132
+7.5.2
+Software-level Incompatibility
+. . . . . . . . . . . . . . . . . . . . . 133
+7.6
+Blockchain to Overcome Cyber-Threats in Smart O&G . . . . . 134
+7.6.1
+Blockchain-based Control Systems (SCADA)
+. . . . . . . 134
+7.6.2
+Blockchain to Enable Trust Across Industrial IoT
+. . . 137
+7.7
+Risks of Smart Solutions in industrial IoT
+. . . . . . . . . . . . . . . . 138
+7.7.1
+Human-Machine Interaction Issues . . . . . . . . . . . . . . . . 138
+7.7.2
+Machine-to-Machine Interaction Issues
+. . . . . . . . . . . . 140
+7.8
+Bias in Smart Industry
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
+7.8.1
+Biases Caused by the Artificial Intelligence (AI)
+Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
+7.8.2
+Automation Bias in Smart Solutions of Industry
+4.0
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
+7.8.3
+Gender Bias in O&G Industry
+. . . . . . . . . . . . . . . . . . . . 145
+7.8.4
+Cognitive Bias in Smart O&G Solutions
+. . . . . . . . . . . 145
+7.9
+Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
+Chapter 8: Conclusion and Future Research Directions . . . . . . . . . . . . 148
+8.1
+Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
+8.2
+Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
+x
+
+8.2.1
+Resource Allocation Using Reinforcement Learning
+for Industry 4.0 Applications across Federated Fog
+System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
+8.2.2
+Data Locality-Aware Resource Allocation Across
+Federated Fog System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
+8.2.3
+Dynamic Fault-Tolerant Federated Fog Systems for
+Industry 4.0 Operation
+. . . . . . . . . . . . . . . . . . . . . . . . . . . 156
+8.2.4
+The Cognitive Aspects of Human-Machine
+Interaction for Smart Industry 4.0 Solutions
+. . . . . . . 158
+8.2.5
+Fog Computing and Advanced Analytics for
+Human-Machine Interaction in Industrial Sector
+. . . 160
+Bibliography
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
+Biographical Sketch
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
+xi
+
+List of Tables
+Table 3.1.
+DNN-based applications used in O&G Industry 4.0 along with
+their network model, input data type, usage scope, and code base. . . . . . . . 42
+Table 3.2.
+Heterogeneous machine types and VM configurations in
+Amazon EC2 that are considered for performance modeling of
+DNN-based applications. In this table, ML Optimized represents
+Inferentia machine type offered by AWS.
+. . . . . . . . . . . . . . . . . . . . . . . 45
+Table 3.3.
+Various VM flavors in Chameleon cloud are configured to
+represent a consistently heterogeneous system. . . . . . . . . . . . . . . . . . . . 46
+Table 3.4.
+The execution time distributions of DNN-based applications in
+AWS clouds machines using Shapiro-Wilk test. . . . . . . . . . . . . . . . . . . . . . 48
+Table 3.5.
+The execution time distributions of DNN applications in
+Chameleon cloud using Shapiro-Wilk test.
+. . . . . . . . . . . . . . . . . . . . . . . . 49
+Table 3.6.
+Inference time distributions of DNN-based applications in AWS
+cloud machines using Kolmogorov-Smirnov test.
+. . . . . . . . . . . . . . . . . . . . 50
+Table 3.7.
+Inference time distributions of DNN-based applications in
+Chameleon’s machines using the K-S test.
+. . . . . . . . . . . . . . . . . . . . . . . . 50
+Table 3.8.
+The measurement of central tendency metric (µ), and data
+dispersion metric (σ) on the observed inference times in AWS. . . . . . . . . . . 52
+Table 3.9.
+Central tendency metric (µ), and data dispersion metric (σ) of
+the inference times in the Chameleon cloud.
+. . . . . . . . . . . . . . . . . . . . . . . 52
+Table 3.10. MIPS values of heterogeneous machines in AWS for each
+DNN-based application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
+Table 3.11. MIPS vales for heterogeneous machines on Chameleon cloud for
+each DNN-based application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
+Table 3.12. The confidence intervals of MIPS values for DNN-based
+applications in AWS machines, resulted from Jackknife re-sampling
+method.
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
+xii
+
+Table 3.13. Confidence intervals of MIPS values for different DNN-based
+applications in Chameleon machines, resulted from Jackknife re-sampling
+method.
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
+Table 6.1.
+Pixel distribution for each of the class in oil spill detection
+data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
+xiii
+
+List of Figures
+Figure 1.1.
+Advanced computing systems in various smart industries
+(e.g., oil and gas, healthcare, transportation) for real-time
+latency-sensitive tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
+Figure 1.2.
+A remote offshore smart oil field consists of multiple oil rigs
+(oil extraction sites). In this scenario, the oil rigs, drill ships, or even
+rescue ships have fog computing systems in the form of mobile data
+centers to support the oil extraction computing demands along with
+any unpredictable emergencies (e.g., oil spill detection, toxic gas
+detection)
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
+Figure 1.3.
+A typical micro-service application, “fire safety” execution
+scenario in edge-fog-cloud paradigm.
+. . . . . . . . . . . . . . . . . . . . . . . . . . . 6
+Figure 2.1.
+Various cloud services (e.g., simulation, analytics,
+visualization, compute, machine learning, reporting) can be
+employed to store, process, and analyze sensor-generated data and
+to control industrial equipment in a smart oil and gas industry. . . . . . . 15
+Figure 2.2.
+Edge-to-Cloud continuum for oil and gas industry as an
+example of Industry 4.0. The continuum is mainly divided into four
+tiers, namely end devices, edge, fog, and cloud data centers. The
+bottom of the triangle has end devices that are energy limited,
+whereas traversing to the top, we find more energy-consuming
+systems.
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
+Figure 2.3.
+Drone-based inspection scenario where drone captures
+images and real-time analysis can be performed in edge computing
+resource whereas long term analysis is performed in distant cloud
+computing facility.
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
+Figure 2.4.
+Various software architecture for Industry 4.0 smart
+applications. Seismic analysis is represented as a monolithic
+application, whereas fire safety application exhibit micro-service
+architecture.
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
+xiv
+
+Figure 2.5.
+A typical federated learning scenario that consists of FL
+workers and a central server having the global model. At the
+beginning of the training, the global model is broadcast to the
+participating workers to train with their corresponding local training
+data. After a period of training in FL workers, the updated model is
+sent back to the server for integration with the global model.
+. . . . . . . 35
+Figure 3.1.
+The FCN-8 model is presented in block diagram that consist of
+5 fully convolutional network blocks, and 2 up-sampling blocks. The
+model receives input as a SAR image and perform pixel-wise classification
+to output a labeled image.
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
+Figure 3.2.
+Schematic view of Temporal Convolutional Network (TCN)
+model that consists of six temporal blocks, the input data, and the
+output in form of the predicted AI.
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
+Figure 3.3.
+The stochastic nature of inference execution time of oil spill
+application while running on heterogeneous VMs in the AWS. For every
+VM instance, the oil spill detection application is executed 30 times and
+those executions are plotted as number of attempts along x-axis. The
+y-axis represents the inference time for every attempts. . . . . . . . . . . . . . . . 47
+Figure 3.4.
+Comparative analysis of the MIPS values of AWS and
+Chameleon machines for various DNN-based applications. For the sake of
+presentation, the MIPS values are normalized between [0,1].
+. . . . . . . . . . . 57
+Figure 4.1.
+A Fog system with load balancer module that facilitates fog
+federation. Task requests generated by sensors are received by the
+load balancer module and are assigned to the fog system that
+maximizes the likelihood of success for the task.
+. . . . . . . . . . . . . . . . . 66
+Figure 4.2.
+The impact of increasing oversubscription level (number of
+arriving tasks) on deadline miss rate using different task assignment
+heuristics in the load balancer.
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
+Figure 4.3.
+Mean communication latency overhead introduced to each
+task in fog federation by different heuristics.
+. . . . . . . . . . . . . . . . . . . . 73
+Figure 4.4.
+Average makespan time(seconds) of tasks using various
+task assignment heuristics.
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
+xv
+
+Figure 5.1.
+The structure of a microservice-based workflow is presented
+in a block diagram. Every microservice need to be processed to
+complete the fire safety application.
+. . . . . . . . . . . . . . . . . . . . . . . . . . . 77
+Figure 5.2.
+Offshore oil and gas industry has the fog federation
+infrastructure that can support smart microservice-based
+applications.
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
+Figure 5.3.
+The flowchart of the workflow partitioning method. The
+partitioned workflow is sent to the resource allocation module, which
+is denoted as the end box for this flow chart.
+. . . . . . . . . . . . . . . . . . . . 82
+Figure 5.4.
+Comparison of the partitioning techniques in terms of
+workflow deadline meet rate while utilizing proposed probabilistic
+partitioning technique. The x-axis represents the increasing number
+of arriving workflow execution requests, whereas the y-axis
+represents the workflow deadline meet rate.
+. . . . . . . . . . . . . . . . . . . . . 90
+Figure 5.5.
+Comparison of resource allocation techniques while
+utilizing proposed workflow partitioning technique for
+microservice-based workflow applications. . . . . . . . . . . . . . . . . . . . . . . . 92
+Figure 5.6.
+Comparison of resource allocation techniques for monolithic
+applications. The proposed resource allocation technique MR
+outperforms other baselines in every application arrival trial. . . . . . . . . 93
+Figure 5.7.
+Impact of scaling the fog federation for proposed
+partitioning and resource allocation techniques in increasing
+oversubscribed situations considering microservice applications. The
+degree represents the number of neighbors each fog system has for
+executing the Industry 4.0 applications.
+. . . . . . . . . . . . . . . . . . . . . . . . 94
+Figure 5.8.
+Impact of scaling the fog federation for proposed resource
+allocation techniques on monolithic applications. The degree
+represents the number of neighbors each fog system has for
+executing the Industry 4.0 applications.
+. . . . . . . . . . . . . . . . . . . . . . . . 95
+Figure 6.1.
+A federated learning setup in fog federation. Multiple
+company share their fog systems to train oil spill detection DNN
+model where data security is preserved by federated learning.
+. . . . . . . 99
+xvi
+
+Figure 6.2.
+Federated learning training considering class imbalance and
+global convergence. Tversky loss is used in the training considering
+class imbalance. After training of each epoch, mean intersection over
+union (mIoU) is checked with a dynamic threshold for global
+convergence.
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
+Figure 6.3.
+Comparison of FedAvg and FedBal training loss utilizing
+tversky loss function. The alpha parameter of tversky index is
+changed from 0.6 to 0.8 (left to right) and the loss per epoch is
+captured for both FedAvg and FedBal algorithm. . . . . . . . . . . . . . . . . 107
+Figure 6.4.
+Comparison of FedBal with FedAvg, FedSGD, and FedProx
+method’s global model performance in IID setup. . . . . . . . . . . . . . . . . 107
+Figure 6.5.
+The performance comparison of global models in terms of
+mIoU using FedAvg and FedBal methods. The data distribution is
+non-IID, the number of workers are 6, and in each fed round 50
+epochs of training has been performed. . . . . . . . . . . . . . . . . . . . . . . . . 109
+Figure 6.6.
+Comparison of FedBal with FedAvg, FedProx, and FedSGD
+method’s global model performance in non-IID and unbalanced data
+distribution.
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
+Figure 6.7.
+Comparison of FedAvg, and FedBal method’s global model
+performance in non-IID data distribution from high intensity(only 1
+class per worker) to low intensity(3 classes per worker). The
+difference of mIoU of FedBal, and FedAvg is plotted as barchart for
+3 case scenarios (1 class, 2 class, and 3 class).
+. . . . . . . . . . . . . . . . . . 112
+Figure 6.8.
+The influence of federated worker on global models
+performance (mIoU) for FedBal, and FedAvg is measured by
+increasing the number of federated worker from 6 worker to 25
+worker. For each case of worker pool 20 federated rounds of training
+are performed for both FedBal, and FedAvg method, and for each
+case maximum mIoU of both methods are considered for plotting as
+a barchart.
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
+Figure 6.9.
+The impact of workers weight (averaged on each of the
+federated round) on global model’s mIoU.
+. . . . . . . . . . . . . . . . . . . . . 114
+xvii
+
+Figure 7.1.
+A taxonomy reflecting the downsides of smart solutions
+implemented with advanced technology is organized using box
+flow-chart form. The main three levels are colored in orange, blue,
+and yellow. The white boxes represent different types (examples) of
+its parent box. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
+Figure 7.2.
+Information technology (IT) and operational technology
+(OT) platforms of a smart oil and gas company that operates using
+different networks to run the entire operation of smart O&G
+industry. The IT platform is significantly related to business
+applications and the financial side of O&G, whereas the OT
+platform directly involves with oil or gas extraction and production
+operations. Both IT and OT platform is connected at some point
+which creates the sweet spot for cyber-attackers to penetrate into
+the whole system.
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
+Figure 7.3.
+The anatomy of ransomware from start to end. The
+ransomware client enters the IT platform through malicious email or
+other external mediums. The client then communicate with hacker’s
+command and control server to download the encryption key. The
+user’s data encrypted by the ransomware client, and finally extortion
+notice is sent.
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
+Figure 7.4.
+Blockchain based data transmission within end-to-end
+SCADA system of an oil and gas company. Blockchain enable
+encryption while transmitting the data for processing that increase
+the data security even data is hijacked while transmitting.
+. . . . . . . . 136
+Figure 7.5.
+Human-machine interaction workflow from sensing to
+control operation.
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
+Figure 7.6.
+A small fire breakout accident occurs in a closed oil
+production area in a compressor unit. The fire alarm generates, and
+water sprinkler starts to sprinkle water that causes power failure in
+power generator that made the electric door locked. Unfortunately,
+workers were working on pipeline maintenance, and were trapped
+inside the facility due to door closure. Here, machine to machine
+interaction cause the safety issue of the onsite worker. . . . . . . . . . . . . 141
+xviii
+
+Chapter 1:
+Introduction
+Industry 4.0 is revolutionizing the utilization of computing resources across
+various industries [1]. With the emergence of the Internet of Things (IoT) and
+modern computing systems (e.g., edge computing, fog computing, and serverless
+computing), industries are becoming more intelligent with smart sensors and
+actuators that create a large quantity of data [2] every day. However, the
+computational resources required to store and analyze sensor-generated data are
+expensive and particularly scarce in remote areas [3]. In the industrial sector,
+various sorts of applications (e.g., machine learning (ML), reporting, alarm
+generators, and surveillance) employ sensor-generated data to automate or conduct
+complex operations. Sometimes these data require real-time feedback to conduct
+fault-intolerant latency-sensitive activities (e.g., drilling operation in an oil rig,
+workers’ safety operation, manufacturing products). Alternatively, certain tasks
+need large computing capacity and are delay tolerance, necessitating cloud data
+center assistance. For instance, the “Fire safety” application [4] utilizes a deep
+neural network (DNN) model that needs extensive training in highly configured
+cloud data centers. Similarly, “reservoir simulation,” widely used in the petroleum
+industry to anticipate the field performance under varies producing strategies,
+requires a large quantity of seismic data and high-performance computing systems
+[5].
+In remote or distant locations of the industrial sectors (e.g., offshore oil
+extraction sites, solar fields), transferring the sensor-generated data to a cloud data
+1
+
+Figure 1.1. Advanced computing systems in various smart industries (e.g., oil and
+gas, healthcare, transportation) for real-time latency-sensitive tasks
+Fog systems
+Sensors
+Fog systems
+Cloud Computing
+Smart healthcare
+Smart oil and gas
+industry
+Smart transport system using vehicle to infrastructure (V2I)
+Satellite
+Edge system
+Edge system
+Edge system
+center is expensive and latency intensive, influencing the use of computing near the
+data sources [6]. Additionally, real-time applications require a faster response time,
+which is usually not feasible with cloud computing resources. Hence, bringing
+computational resources to the data sources near the end clients is an essential
+requirement for remote industries.
+One of the solutions for computing near data sources is edge computing [7]
+as depicted in figure 1.1, which brings computational resources closer to the end
+devices, and data generation sources. As such, edge computing can be defined as a
+distributed computing platform that puts industrial applications closer to data
+sources like IoT devices or local computer servers. This closeness to data at its
+2
+
+Base Station
+Task
+Processing
+Task Request
+nse
+Road
+vehicle moving left to right
+米
+Road
+vehicle moving right to leftsource can result in significant business benefits such as faster insights, faster
+reaction times, and increased bandwidth availability. Although edge computing
+supports real-time latency-sensitive applications, edge devices are resource
+constraints that need efficient resource allocation [8] mechanism. Hence, another
+solution for computing platforms near end users is fog computing systems [9] that
+complement edge computing by having more computing resources, having more
+comprehensive middleware for managing workload efficiently.
+The main driving force of Industry 4.0 revolution is machine learning (ML)
+or deep neural network (DNN) applications [10, 11, 12] that ensure efficient
+industrial operations and workplace safety. Hence the ML or DNN-based
+applications encompass both the training and inference stages [13]. The training
+stage is generally carried out offline due to time and computing resource constraints.
+Whereas the inference execution can be completed utilizing general-purpose
+computing systems. The DNN-based applications are mainly trained on cloud data
+centers or computing servers with high configuration hardware (e.g., GPU, TPU,
+FPGA) support. In contrast, the inference operations are performed on the fog
+computing systems near the end users. As such, it is essential for a system engineer
+or system administrator to understand the performance of these DNN-based
+applications in fog computing systems [14]. Especially for fault-intolerance
+latency-sensitive critical DNN-based applications, the forecast of inference execution
+time within a computing resource can be significantly vital that sometimes save lives
+in a disaster situation. As such, we perform a statistical analysis of the inference
+3
+
+execution time of various Industry 4.0 applications on the cloud and fog systems.
+Consequently, we introduce an execution time workload trace that can help system
+architects to develop a load-balancing solution robust against stochastic execution
+times of Industry 4.0 smart applications. Therefore, efficiency, productivity, and
+industrial safety can be ensured by utilizing these robust solutions.
+In remote offshore industry, at times of emergencies (e.g., disasters,
+accidents), the demand for task processing in the edge or even fog computing
+systems can be significantly high, leading to a drop in some tasks due to not meeting
+their latency constraints (a.k.a deadlines). As such, we propose to federate nearby
+privately owned computing resources by forming a federation of fog systems to
+support the surge of task processing requests in times of emergencies. For instance,
+in a remote offshore smart oil field, as depicted in Figure 1.2, multiple oil extraction
+sites can be built by the respective companies that typically contain privately owned
+fog computing systems to support their regular computing demands. At a disaster
+time or other emergencies such as an explosion, the computing demands surge to
+support multiple recovery procedures to be coordinated. For instance, in a fire
+breakout event, various activities such as drone-based inspection, fire detection, and
+alert generation with precise fire locations need real-time coordination to neutralize
+the emergency. In this scenario, some latency-sensitive tasks can be offloaded [15] to
+other fog systems that may have more computational resources or less busy. Hence,
+the federated fog systems’ resilience depends on supporting the surge in computing
+demands by efficient resource allocation across the federation. Therefore, we
+4
+
+propose a probabilistic resource allocation method across fog federation for latency
+sensitive monolithic tasks to support computing demands in emergency situations.
+Figure 1.2. A remote offshore smart oil field consists of multiple oil rigs (oil ex-
+traction sites). In this scenario, the oil rigs, drill ships, or even rescue ships have
+fog computing systems in the form of mobile data centers to support the oil extrac-
+tion computing demands along with any unpredictable emergencies (e.g., oil spill
+detection, toxic gas detection)
+Smart applications in Industry 4.0 can have various software architectures
+(e.g., monolithic, micro-service [16]) that serve different purposes of industrial
+operations. Hence, micro-service architecture is one of the widely used software
+architectures that comprise various micro-level services having immense benefits on
+development and deployment [17]. For instance, as depicted in Figure 1.3, the “fire
+safety” micro-service-based application comprises video pre-processing, noise
+removal, feature extraction, fire detection, location mapping, and alert generation
+micro-services performing different activities. In a typical industrial scenario, these
+micro-services are supported by various execution platforms that can have
+5
+
+'mmWave Wireless Link
+between Rigs
+区
+Edge
+Computing
+()
+Edge
+Computing
+Cooperative
+Monitoring
+Edge
+Computing
+Oil rig
+Sensors
+Smart Oil Field
+Sensors
+Smart
+Smart
+Well
+WellFigure 1.3. A typical micro-service application, “fire safety” execution scenario in
+edge-fog-cloud paradigm.
+video
+preprocessing
+noise
+removal
+input video
+feature
+extraction
+fire
+detection
+location
+mapping
+alert
+generation
+cloud data
+center
+fog system
+edge system
+stochastic execution latencies. In contrast, a monolithic architecture is the
+conventional unified paradigm for constructing a software application. Monolithic
+software is intended to be self-contained, with firmly connected rather than loosely
+coupled components or services, as in a micro-service architecture. Although the
+industrial revolution influenced the utilization of micro-service applications, various
+industries have monolithic legacy applications that are still in operation and need to
+be supported by existing execution platforms. In an emergency or disaster, various
+application requests with different latency requirements are generated in the
+proximity of the disastrous area that needs distinct computational support. Hence,
+6
+
+the nearby execution platform gets oversubscribed with the surge in demand for
+executing numerous applications on time, that can degrade the execution platform’s
+performance. In this case, to support the high computation demands utilizing the
+proposed federated fog system need an efficient resource allocation method that is
+aware of receiving applications’ internal structure, computation, and communication
+latencies. Therefore, the reliability of an execution platform in an oversubscribed
+situation depends on accommodating various computational demands that ensure
+industrial safety.
+Federating computing resources in remote industrial areas imposes security
+concerns for each participant fog system of the federation, that is owned by private
+companies. In addition, individual fog systems can have sensitive data that imposes
+privacy issues for the company owning the computing systems. Hence, considering
+ML application training across the fog federation suffers from data scarcity, that is
+an obstacle to building accurate ML models. As such, a secure and
+privacy-preserving distributed ML training method should be in place to build an
+accurate ML model that can be crucial in emergency situations in Industry 4.0.
+1.1 Research Problem and Objectives
+The fundamental purpose of this research is to identify, evaluate, and
+manage robust execution of Industry 4.0 applications in remote areas (e.g., offshore
+oil fields) across modern edge and fog computing systems. Hence, we develop
+solutions that use fog systems in emergency and oversubscribed circumstances to
+satisfy the computational demand in remote industrial sectors. This dissertation
+7
+
+address the following research challenges to enable a robust and QoS-efficient
+federated fog system for industry 4.0 applications:
+1. How to utilize modern distributed computing systems in the context of remote
+smart industries (e.g., oil and gas, energy) considering the industrial
+revolution, Industry 4.0?
+2. What are the statistical execution behaviors of Industry 4.0 applications in fog
+systems?
+3. How to enable a robust federated fog computing system that can efficiently
+procure computing demands during a workload surge?
+4. How to support Industry 4.0 applications with modern micro-service
+architecture along with monolithic legacy applications and maintain the
+Quality of Service (QoS) of a fog federation?
+5. How to utilize federated fog securely to improve the performance of Industry
+4.0 applications?
+1.2 Contributions
+We identified various obstacles as we investigated many facets of federated
+fog computing systems in the industrial sector. Therefore, in addition to addressing
+significant challenges, we present state-of-the-art solutions and give exhaustive
+performance assessments for recommended methodologies. In light of the research
+topics outlined in the preceding section, the considerable contributions of this
+dissertation are as follows:
+8
+
+• Identifying the connection of industry 4.0 and modern distributed computing
+systems (e.g., edge, fog) and addressing the scope of utilizing advanced
+analytics (e.g., artificial intelligence, machine learning, deep learning) in the
+context of the remote smart oil field.
+• Performance analysis of ML-based Industry 4.0 applications across fog and
+cloud computing systems?
+• Proposing a real-world workload benchmark of inference execution times for
+four different ML-based Industry 4.0 applications.
+• Enabling the notion of federated fog via resource allocation methods operating
+based on Bayesian probability utilizing fog systems for latency-sensitive tasks
+in an oversubscribed system that tries to recover from a disaster.
+• Proposing a statistical resource allocation solution across federated fog
+systems that is aware of internal software architecture and stochastic latency
+requirements of Industry 4.0 micro-service workflow applications.
+• Proposing a data privacy preserving ML-based Industry 4.0 application
+training solution across federated fog system in remote industrial sites.
+1.3 Dissertation Organisation
+• Chapter 2 explores the related research works and provides background for
+edge & fog computing, fog federation systems, load balancing & task
+allocation techniques, and data privacy aspects of a federated fog system.
+9
+
+Hence, the scope of utilizing the modern distributed systems in the remote
+smart oil fields is addressed with various use case scenarios.
+– Razin Farhan Hussain, Ali Mokhtari, Mohsen Amini Salehi, and Ali
+Ghalambor IoT for Smart Operations in the Oil and Gas Industry
+published as a book by Elsevier (ISBN:9780323998444).
+• Chapter 3 studies the performance of ML-based Industry 4.0 applications in
+heterogeneous cloud computing resources. The statistical analysis of
+ML-based applications helped to generate a real-world Industry 4.0
+application inference execution time workload that can be beneficial for the
+system architect to develop robust load-balancing solutions.
+– Razin Farhan Hussain, Alireza Pakravan, and Mohsen Amini Salehi
+Analyzing the Performance of Smart Industry 4.0 Applications on Cloud
+Computing Systems published in Proceedings of the 22nd IEEE
+International Conference on High-Performance Computing and
+Communications (HPCC-2020)
+• Chapter 4 explores the possible advantages and practicality of building a fog
+federation in a distant offshore smart oil field in the event of a disaster. Using
+probabilistic load balancing heuristics across the fog federation for resource
+allocation can efficiently assure the system’s resiliency. Moreover, compared to
+baseline approaches, the advantage of employing probabilistic methods is
+backed by a synthetic workload created in EdgeCloudSim simulation[18].
+10
+
+– Razin Farhan Hussain, Mohsen Amini Salehi, Anna Kovalenko, and
+Omid Semiari Federated Edge Computing for Disaster Management in
+Remote Smart Oil Fields published in Proceedings of the 21st IEEE
+International Conference on High Performance Computing and
+Communications (HPCC-2019)
+– Razin Farhan Hussain, Omid Semiari, and Mohsen Amini Salehi
+Robust Resource Allocation Using Edge Computing for Vehicle to
+Infrastructure (V2I) Networks published in Proceedings of the 3rd IEEE
+International Conference on Fog and Edge Computing (ICFEC’19)
+– Razin Farhan Hussain, Mohsen Amini Salehi, and Omid Semiari
+Serverless Edge Computing for Green Oil and Gas Industry published in
+Proceedings of IEEE Green Technologies Conference(GreenTech) - 2019
+• Chapter 5 explores the advanced micro-service software architecture for
+Industry 4.0 applications to enhance the robustness of remote federated fog
+systems. Hence, the load balancer should be aware of the software architecture
+of the receiving applications as well as the uncertainties of the execution
+platform. As a result, the distribution of receiving applications across the fog
+federation enhances the possibility of applications being completed on time.
+– Razin Farhan Hussain, Mohsen Amini Salehi Adapting Remote
+Industry 4.0 Applications to Federated Fog Computing Systems prepared
+for submission to Future Generation Computing System journal in 2022
+11
+
+• Chapter 6 explores the data privacy aspects of ML-based application training
+across the federated fog computing systems in Industry 4.0.
+• Chapter 7 explores the downsides and side effects of smart solutions for
+Industry 4.0. This chapter identifies and proposes various cutting-edge
+solutions for security issues of different industrial sectors.
+– Razin Farhan Hussain, Ali Mokhtari, Mohsen Amini Salehi, and Ali
+Ghalambor IoT for Smart Operations in the Oil and Gas Industry
+published as a book by Elsevier (ISBN:9780323998444).
+• Chapter 8 concludes the dissertation with a discussion of our main findings
+and future research directions in the area of efficient utilization of fog
+computing platforms for Industry 4.0.
+12
+
+Chapter 2:
+Background and Literature Study
+2.1 Computing as a Prominent Aspect of Industry 4.0
+Industrial systems are quickly transitioning from human-controlled processes
+to closed-loop control services supporting their operations autonomously using
+extensive sensor and computing infrastructure. This revolutionary change is critical
+for supporting growing data-intensive and time-sensitive Industry 4.0 applications,
+particularly at remote locations such as offshore Oil and Gas (O&G) fields where
+computer infrastructure is restricted and human resources are limited. Realizing
+these systems necessitates interdisciplinary research and study at the interplay of
+Industry 4.0 in remote industry, modern computing infrastructure (such as an Edge
+and Cloud), and advanced analytics (e.g., ML, DNN).
+Consequently, this chapter aims to illustrate the challenges, prospects, and
+solutions for establishing a smart and robust remote industry based on the
+fundamentals of the Industry 4.0 paradigm. As a result of this study, researchers
+and practitioners can be more effective in making the remote industry safer, more
+sustainable, greener, automated, and, subsequently, more cost-efficient. This
+chapter investigates several computer technologies that support the computing
+needs of distant industries. Furthermore, it explains how the synergy of
+cutting-edge computing solutions, such as the Internet of Things (IoT), Machine
+Learning methodologies, and distributed computing platforms, can be employed to
+improve industrial processes. As an ideal example of a remote offshore industry, we
+consider Oil and Gas that has been transforming significantly with the industrial
+13
+
+revolution Industry 4.0. On the other hand, the remote offshore O&G industry has
+been facing various disasters and catastrophes that raise concerns about production
+efficiency and safety measures. For instance, the deepwater horizon (2010)[19],
+usumacinta jack-up disaster (2007) [20], mumbai high north disaster (2005) [21],
+and the ocean ranger disaster (1982) incidents are significantly connected with
+safety failures in the industrial sites. Hence these incidents motivated us to improve
+the computing support in remote industries to ensure safety and productivity.
+Therefore, this chapter explores various distributed computing technologies,
+federation-friendly execution platforms, software architecture, and security aspects
+of Industry 4.0, focusing on the O&G industry.
+2.2 Distributed Computing Systems in Industry 4.0
+2.2.1 Cloud Computing
+Cloud computing is a concept that enables resources (e.g., computing,
+storage, services) to be available as a service, on-demand, configurable, and also
+shareable [22]. Modern cloud systems provide diverse services in different levels,
+such as infrastructure as a service (IaaS), platform as a service (PaaS), software as a
+service (SaaS), and function as a service (FaaS).
+As presented in Figure 2.1, smart O&G industry increasingly relies on
+cloud-based services that are hosted on remote Internet servers (a.k.a. cloud data
+centers). These data centers are utilized to store and process their data. According
+to Figure 2.1, various sensor-generated data are sent to cloud providers to avail of
+different kinds of cloud services. Among these services, some of them send insightful
+14
+
+Figure 2.1. Various cloud services (e.g., simulation, analytics, visualization, com-
+pute, machine learning, reporting) can be employed to store, process, and analyze
+sensor-generated data and to control industrial equipment in a smart oil and gas
+industry.
+Sensors
+Sensor
+Generated Data
+Actuators
+response
+Smart Oil Field
+Cloud services
+Storage
+Analytics
+Machine
+Learning
+Compute
+Simulation
+Service
+Visualization
+Reports
+decisions to actuators to close the automation loop in the smart oil field. Cloud
+technology enables O&G companies to utilize various data-related and
+computational services (e.g., machine learning and visualization) without the need
+to maintain any computing infrastructure. However, data privacy and security have
+remained a concern for such companies to fully embrace the cloud services. These
+security concerns have caused a small pause and hesitation in adoption cloud
+services, particularly by major players in this industry. An alternative and more
+secure approach is to store the data on an on-premise computing facility that is
+known as a private cloud (more recently called fog computing).
+On the positive side, cloud systems’ performance and ease-of-use are
+tempting for the O&G industry. For instance, one of the main users of data-driven
+cloud services is the North American shale industry that drills thousands of wells
+15
+
+every year [23]. The scalability feature of cloud services helped the growing amount
+of data from these wells to be utilized efficiently, allowing the industry to expand
+remarkably. As such, various modern cloud-based data analytics services have
+emerged to help O&G companies to improve their operational workflows and make
+profitable decisions.
+2.2.2 Edge and Fog Computing for Remote Industry 4.0
+Due to the increasing importance of latency-sensitive applications, real-time
+operations close to the end user in remote offshore industries, the interest in the
+notion of edge computing has begun to increase. Additionally, the proliferation of
+the Internet of Things (IoT) devices and smart sensors in the industrial sector
+results in a massive amount of data that need to be processed locally. In a typical
+scenario, the data is transported to cloud data centers [24], and responses or results
+are transmitted back to clients through the internet, both of which take time and
+money. As a result, a distributed computing paradigm has been introduced, which
+is located close to the end client and processes client data at the network’s edge [25].
+Researchers call this type of computing “Edge Computing” since it operates at the
+network’s periphery.
+The conventional definition of edge computing is difficult to come by.
+Different organizations or sources have different definitions, heavily impacted by
+context. The general perception of edge computing is to provide various computing
+services (e.g., application execution, data pre-processing) through distributed
+computer systems instead of centralized cloud data centers. Hence, edge computing
+16
+
+enables analysis and knowledge collection at the point of information source. In
+network design, an “edge computer” is located directly next to or even on top of
+network endpoints (such as controllers and sensors). The data is then partially or
+fully processed before being transmitted to the cloud for storage or further
+processing. Edge computing, on the other hand, may result in the direct transfer of
+huge volumes of data to the cloud. This might have an impact on system capacity,
+efficiency, and security. Fog computing [26] solves this problem by inserting a
+processing layer between the edge and the cloud. As a result, ‘fog computing’
+collects and analyses data at the edge before it reaches the cloud. Hence, the place
+from the data source where computing service is offered can be a defining element in
+distinguishing Fog/Edge computing from cloud computing. For instance, a
+renewable energy company geared with numerous sensors utilizes fog computing for
+sensor-data analysis in their operational fields. Accordingly, company‘s production
+efficiency improved by 15% by reducing data analysis latency from 10 minutes to
+few seconds. Therefore, fog computing placed near data source in remote industries
+can enhance efficiency in production.
+The emergence of edge and fog computing does not substitute the cloud
+computing services; instead, it brings some portion of those services (e.g.,
+computing, storage, analytical services) near the end clients. Especially with the
+ever-growing Internet of Things (IoT) devices, a considerable amount of data is
+generated [2] that is significantly valuable for Industry 4.0 ML applications. The
+generated data sometimes need immediate processing (i.e., edge computing
+17
+
+support), and alternatively, sometimes need complex processing (i.e., cloud
+computing support). Therefore, a continuous computing platform (i.e.,
+Edge-to-Cloud Continuum [27]) is required to support both real-time nature and
+complex analytical tasks.
+Figure 2.2.
+Edge-to-Cloud continuum for oil and gas industry as an example of
+Industry 4.0. The continuum is mainly divided into four tiers, namely end devices,
+edge, fog, and cloud data centers. The bottom of the triangle has end devices that
+are energy limited, whereas traversing to the top, we find more energy-consuming
+systems.
+ Latency, Elasticity,
+Computing Power,
+Centrality, Un-
+trustworthy
+Edge
+Fog
+(Cloudlet)
+Cloud Data
+Centers
+Edge-to-Cloud Continuum
+ Devices
+Sensors
+Worker Equipment
+Actuators
+PDA
+Smart
+Gateway
+Smart
+Phone
+Laptop
+ASCI
+Device
+Drone
+Pressure
+Temp. Smart
+Helmet Safety Vest
+Smart
+Watch Camera
+Robot
+Energy Limited
+Medium Energy
+Energy Hungry
+2.2.3 Edge-to-Cloud Continuum
+Although edge and cloud computing has a difference in terms of distance and
+resources, they can be utilized as a complement to each other. For a massive
+industry such as O&G, diverse operations and services are needed that require
+various underlying computing platforms. Hence, the integration of edge or fog
+computing with cloud computing is a need of time that reflects the usability of the
+18
+
+edge-to-cloud continuum[28]. Accordingly, the Edge-to-Cloud continuum is a service
+platform that provides various computational resources and infrastructures for
+supporting different types of services essential for O&G industry.
+Figure 2.2 demonstrates the Edge-to-Cloud continuum as a triangle where
+edge devices reside close to end devices (bottom of the triangle) and cloud data
+centers are the furthest computing entity from end devices. Accordingly, this is a
+hierarchical arrangement that is distributed vertically. Hence cloud computing has
+high latency than edge and fog computing. Alternatively, cloud computing has high
+availability in terms of elasticity and computing power, whereas edge and fog
+devices are highly secure and privacy-preserving than cloud computing. Therefore,
+various computing platforms within the continuum serve different purposes for
+industrial operations.
+2.2.4 Use Case of Edge-to-Cloud Continuum in Smart O&G
+We investigate drone-based pipeline inspection scenarios in the oil and gas
+industry to understand how the edge-to-cloud continuum supports computing
+demands in the industrial sector. Let’s consider a scenario where 4K drone-mounted
+cameras can collect hundreds of gigabytes of data per hour. The current method of
+analyzing data is to transfer the massive data to the cloud data center, which is
+cost-prohibitive and impractical, especially if the analysis is real-time. Hence one
+critical question Is it feasible or scalable for the future to have any cloud vendor
+send their container truck with petabytes of storage?
+An example of the same scenario (presented in figure 2.3) from the oil and
+19
+
+Figure 2.3. Drone-based inspection scenario where drone captures images and real-
+time analysis can be performed in edge computing resource whereas long term analysis
+is performed in distant cloud computing facility.
+Drone capture
+images and pre-
+process them
+Edge computing:
+light weight
+processing
+Cloud computing:
+complex analysis
+1
+2
+3
+Pipeline Fracture
+gas industry perspective is that the drone-based inspection system could use
+multi-stage value extraction using an edge-to-cloud continuum. The O&G pipelines
+can be thousands of miles long and pass through an immense landscape. Pipe
+sections are generally fitted with analog gauges and smart sensors to measure
+pressure, flow, and other metrics. By employing an edge AI-enabled surveillance
+drone to capture these analog gauge images presented in step 1 of figure 2.3, it is
+possible to separate (step 2) the gauge images and transfer only that critical
+information to the next compute layer. Here, data pre-processing (image
+separation) is real-time nature that is performed in the edge computing system.
+Therefore only the localized necessary data is processed for an accurate reading in
+the cloud data center (step 3). Then the output of actionable intelligence is sent to
+20
+
+the on-site maintenance team to resolve pipeline fracture. Therefore, data
+pre-processing, lightweight processing, and complex analysis are performed across
+the edge-to-cloud continuum to conduct efficient drone-based pipeline surveillance.
+2.2.5 Landscape of Computing in O&G
+Modern computing systems, such as edge, fog, or cloud enable the smooth
+operation of different fault-intolerant processes across different sectors of the O&G
+industry. As a cyber-physical system, the computing technology stack of the O&G
+industry is composed of the following components:
+• Sensors: Numerous sensors of different types (e.g., to gauge pressure, emission
+of toxic gases, security cameras, etc.) continuously procure multi-modal data
+in the form of signal, text, images, video, and audio. The data is stored or
+communicated for offline or online processing to monitor the operation of the
+oil field or to make management decisions.
+• Computer networks: In a smart oil field, short- and long-range wireless and
+wired computer networks (e.g., Bluetooth, CBRS, satellite, etc.) have to be
+configured for low-latency and high data-rate communication of devices (e.g.,
+sensors, servers, and actuators) both for onsite and offsite communication.
+• Computing systems and middleware: All the collected data have to be
+eventually processed to be useful. That is why, in the back-end, smart oil
+fields are reliant on different forms of computing systems (e.g., HPC, cloud,
+fog, edge, and real-time systems) to perform batch or online data processing
+21
+
+for purposes like monitoring, visualization, and human-based or automatic
+decision making.
+• Data processing and software technologies: The rule of thumb in a smart oil
+field is that “the more data can be processed, the more informed decisions can
+be made”. The large amount of multi-modal data (text, images, video, and
+signals) continuously generated in a smart oil filed form what is known as big
+data. Such diverse formats of big data have to be processed using various
+algorithmic techniques, particularly Machine Learning, to provide an insight
+from the data or to make informed decisions upon them.
+• Actuators: Once a decision is made, it is communicated to an actuator (e.g.,
+drilling head and pressure valve) to enact the decision (e.g., increase or
+decrease the pressure).
+2.3 Smart O&G: Data and Software Aspects
+2.3.1 Big Data in the O&G industry
+The oil and gas industry generates a large volume of data on a daily basis,
+necessitating the need for large-scale computing resources and the cloud. The three
+key sources of such considerable data in the O&G industry are as follows::
+• Hydrocarbon reservoirs are commonly found between 5,000 and 35,000 feet
+below the Earth’s surface. High-resolution images and expensive well logs are
+the main options for finding and characterizing reservoirs (after the wells are
+dug).
+22
+
+• Fluids must pass through complex rock to reach the wellbore, and the fluids
+themselves are complex, having many different physical properties. Therefore,
+learning about the unique characteristics of each oil well and evaluating the
+extracted fluid to treat it properly necessitates collecting vast amounts of data
+via sensors installed in the oil well and on the drill-head.
+• Oil production entails environmental and human safety hazards, and
+preventing it requires significant sensor deployment across a large geographical
+region to gather data regularly and therefore be able to respond rapidly to any
+ecologically polluting discharge.
+Big data analytics aids in the automation of critical oil and gas operations,
+such as exploration, drilling, production, and delivery. The upstream sector, which
+consists of exploration and drilling, is the most dominant data source among all
+other sectors, owing to the increasing use of big data analytics for detecting
+non-conventional shale gas. Furthermore, the oil and gas industry is becoming more
+volatile due to fluctuating oil prices. As a result, in addition to the engineering
+team, business teams are increasingly adopting a data-driven strategy to forecast
+the market and mitigate risks.
+2.3.2 Machine Learning as a Data-driven applications in O&G
+The smart O&G industry is a subset of the Industry 4.0 revolution,
+supported primarily by artificial intelligence (AI), IoT, and cutting-edge computing
+systems (e.g., edge, fog, and cloud computing). An extensive range and volume of
+23
+
+relevant data are acquired from many sectors of the O&G industry due to the
+widespread adoption of smart sensors and actuators. These data may be evaluated
+using machine learning models to derive valuable insights and knowledge for the
+industry and the environment. As a result, in a broad sense, AI is a vital tool for
+transforming sensor-generated data into new and valuable information and
+knowledge via Edge-to-Cloud computing.
+The term “data-driven approaches” refers to an arsenal of techniques that
+can be used to combine different kinds of data, evaluate uncertainties, spot trends,
+and recover useful facts. Data-dominated software applications running on ML and
+Deep Neural Network (DNN) models, such as oil production control and emergency
+surveillance systems [29, 30, 31, 32, 33], have emerged as the fundamental pillars of
+the Industry 4.0 revolution [34, 35, 36, 37]. Especially in remote areas where there is
+a need for real-time closed-loop automated processes of these applications. The
+ML-based solutions often take the shape of micro-service processes, each of which
+may have one or more critical paths that together determine the latency of the
+whole application [29]. These applications require:
+• A large amount of data to be collected in real-time
+• Seamless communications of sensing data despite wireless link uncertainties
+• Dependable execution of ML applications with latency constraints in the face
+of unexpected load surges (for example, during emergencies)
+• Transparent deployment and provisioning of applications and resources (also
+24
+
+known as “serverless”).
+Tackling these needs may be difficult, particularly in out-of-the-way places
+(such as offshore oil fields) with inconsistent connectivity and unstable access to
+cloud services. These communication and computation constraints become crucial
+when a remote system must handle massive volumes of data in real-time to manage
+several facets of an emergency circumstance (for example, an oil spill). While micro
+datacenters (also known as fog systems) are employed to meet the computing
+demands of such distant systems, their capabilities are sometimes inadequate to
+deal with the real-time data transport, and processing demands of the load spike
+[38]. In the following subsection (ref:edgeAi), we revisit the difficulty posed by
+limited resources for processing surge in computing demands.
+2.3.3 Digital Twin: Another Data-driven Applications in O&G
+The term “digital twin” (DT) refers to a computer simulation of an existing
+system. Input to the twin may be set from the sensors collecting data from
+real-world imitation. The twin may then offer real-time feedback to the
+management about the predicted performance or other repercussions by stimulating
+the physical object. DT is a data-dominant application that operates based on
+Machine Learning and the scalability of cloud computing to bring the goal of data
+integration closer to actuality [39]. The importance of data in a DT system cannot
+be overstated since it is required for many different types of analysis, prediction,
+and automation. High-quality, verified, and referenced data is required to produce a
+practical duplicate. Since the DT operates in real-time, all previously collected data
+25
+
+and models must remain accurate, and up-to-date [40]. By enabling operators and
+management in the O&G sector to transform enormous amounts of data into
+insights that might make asset failure predictable and hidden revenue opportunities
+revealed, DT systems can contribute to operational excellence.
+2.4 Edge-to-Cloud for AI and other Data-driven Applications in Smart
+O&G
+The wide variety of sensors that communicate through heterogeneous
+protocols like Modbus, CAN bus, PROFINET, and MQTT [41] makes it challenging
+to operationalize an Edge-to-Cloud continuity with local appliances linked to
+sensors. It is already difficult to implement, with hundreds of agencies and oil rigs
+involved. In addition, the next generation of cloud-native apps needs different
+machine learning (ML) frameworks, configurations, and requirements. Furthermore,
+applications need to be interoperable to function on a variety of devices with diverse
+processing capabilities (e.g., CPU, several kinds of GPU, ASICs, and FPGAs). In
+addition, the human aspect of IT operational technologies, developers, and data
+scientists all need to join together to manage the IoT application deployed in the
+edge-to-cloud continuum. Thus, the primary difficulties throughout the
+Edge-to-Cloud spectrum might be summed up as follows:
+1. Connecting a huge number of IoT devices, as well as the edge and the cloud.
+2. Costs associated with wireless communication technologies.
+3. Having access to high-quality computer resources on demand.
+26
+
+4. Wireless connections that are stagnant, inconsistent, or not operating.
+5. The need for real-time operation of ML-based and other data-driven
+applications (e.g., digital twin).
+6. Data integrity and privacy across Edge-to-Cloud systems.
+The Edge-to-Cloud continuum problems for the O&G industry are broad,
+complicated, and distinct from traditional solutions. As a result, petroleum
+professionals and technological specialists are the primary driving forces in
+developing lucrative eco-friendly solutions for the smart O&G industry. Meanwhile,
+academic publications, research papers, and books addressing the junction of
+petroleum and computer science domains are uncommon. Therefore narrowing the
+gap between knowing the issue space and providing efficient solutions can help the
+industry to be more productive and safe.
+2.5 Federated Fog and It’s Challenges in Remote Industry 4.0
+The earlier sections of this chapter demonstrate the edge-fog-cloud
+continuum in a hierarchical arrangement where computing resources are distributed
+vertically. Hence, multiple tiers of execution platforms can be conceptualize where
+higher tier (i.e., cloud) imposes significant latency that may not suitable for
+latency-sensitive tasks. Accordingly, we investigate the horizontal scalable execution
+platform, fog systems, in a peer-to-peer arrangement to reduce latency issues. In the
+industrial sector, fog computing systems are typically located in close proximity
+that sometimes potential candidates for forming a federation in a peer-to-peer
+27
+
+setting to support computing demands in emergencies. For instance, multiple oil
+rigs with drillships[42] can be deployed near an offshore hydrocarbon reservoir to
+extract oil having their private fog systems. Moreover, rescue ships with mobile
+data centers at disaster time comprise fog systems deployed near disastrous areas
+whose computing ability can be augmented by forming the fog federation. Hence, it
+is feasible to assume that some fog systems are underutilized and can support more
+task processing than their day-to-day requests. Thus, efficient resource allocation
+across federated computing systems can increase the federated system’s quality of
+Service (QoS). In a related study, [43], Xu et al.offer a resource allocation instance
+for edge computing platforms. It uses a decoupled architecture that separates
+infrastructure management at Edge Computing Infrastructures (ECIs) from service
+delivery and administration by service providers (SPs). In addition, the authors
+offer an auction-based resource contract mechanism and a latency-aware scheduling
+approach that optimizes edge computing systems and service providers’ utility.
+Hence, federating edge computing systems with efficient resource allocation may be
+used in an emergency to accommodate a spike in task requests. However, several
+other problems should be addressed to establish a robust and efficient edge
+federation in an emergency or disaster. The main challenges can be addressed in the
+following subsections.
+2.5.1 Real-time Services of Industry 4.0
+To improve the response time of latency-intolerant services (e.g., sensor data
+analysis, production monitoring), fog computing systems have been exploited in the
+28
+
+literature from the network latency perspective. Lorenzo et al.[44] proposed a
+resource allocation model for wireless edge systems that harvest unused resources of
+mobile devices to mitigate network congestion. The proposed model utilizes
+solutions at the physical, access, networking, application, and business layers to
+reinforce network robustness. This work solely considers networking latency and not
+end-to-end latency. In [45], Chang et al.proposed an optimized resource migration
+scheme from mobile IoT devices to a heterogeneous Cloud-Fog-Edge computing
+environment that is aware of the resource-constrained nature of edge devices. It
+focuses on the performance gain of process migration and assigns tasks based on
+their run time expectations on the participating systems.
+2.5.2 Heterogeneous Fog Systems in Remote Industry 4.0
+Fog systems in remote industries can be heterogeneous, and the research
+community addresses two forms of heterogeneity: consistent and inconsistent
+heterogeneity [46], respectively. Consistent heterogeneity occurs when the same kind
+of machine has different computational resources. Inconsistent heterogeneity occurs
+when various types of machines have disparate computational capacities. The
+requested job may have different execution times depending on the heterogeneity,
+which substantially impacts the task completion time in an edge system. The
+problem of heterogeneous data acquisition from sensors in various sectors (e.g.,
+upstream, midstream, downstream) of smart oil fields is addressed in [47] where
+khan et al.proposed an IoT-based architecture to enable the data acquisition process
+more simple, secure, robust, reliable and quick. There are several other works (e.g.,
+29
+
+[48, 49, 50]) that either do not consider the emergency (oversubscription) or ignore
+the uncertainties that exist in federated fog environments. In another related work
+[51] by the same author, the main focus was on optimizing the wireless network
+while no resource allocation was performed at the fog system. Therefore,
+considering both computing and communication latencies is critical to maintaining
+the QoS of a fog federation.
+2.5.3 Uncertainty of Task Completion in Fog Systems
+The primary uncertainty of task completion in a fog federation is influenced
+by execution and communication latencies. Hence, execution uncertainty mainly
+refers to the computational resources that execute the assigned task, whereas
+communication uncertainty is primarily rooted in network systems, especially the
+upload and download time uncertainty. Both of these uncertainly significantly
+influence task completion within a fog federation. Xu et al.addressed uncertainty in
+a similar study [52], mentioning that the execution uncertainty caused by
+performance degradation, service failure, and new service additions remains a
+significant barrier to the user’s service experience. To overcome the uncertainty, this
+study proposes a software-defined network (SDN)-based fog computing architecture
+and a dynamic resource provisioning mechanism. Furthermore, the nondominated
+sorting genetic algorithm-III is used to maximize two objectives, namely energy
+consumption and completion time, to produce balanced scheduling strategies. In
+another related paper [53] on resource allocation and uncertainty, Li et al.suggested
+a multi-objective optimization problem. Three parallel methods have been
+30
+
+developed to increase latency, performance, and resource management. First, a
+queuing model was investigated in conjunction with task buffering, offloading, and
+resource allocation algorithms. The authors designed the resource allocation
+strategy using Lyapunov drift [54]. An exchange between latency and throughput is
+found in outcomes for improved system performance.
+Figure 2.4. Various software architecture for Industry 4.0 smart applications. Seis-
+mic analysis is represented as a monolithic application, whereas fire safety application
+exhibit micro-service architecture.
+user
+user
+data pre-
+processing
+fire
+detection
+alert
+generation
+seismic analysis
+monolithic
+micro-service
+fire safety
+MS-1
+MS-2
+MS-3
+seismic
+modeling
+Vs
+2.5.4 Software Architecture of Industry 4.0 Applications
+The industrial revolution has created the demand for emerging smart
+applications with different software architectures, as depicted in figure 2.4. Hence,
+smart applications consist of various micro-services that can be separately deployed
+with the least amount of administration. For instance, a “fire safety” application
+based on micro-service architecture comprised of data pre-processing, fire detection,
+and alert generation can be deployed in remote industries to ensure the safety of
+31
+
+Thnsmission of
+Data centre
+earthquake signal
+(speed of light)
+Sesmigeations
+Transmission of earthguake
+early warning message
+(speed of light)
+S-waves (approx.3.5km/s)
+Blind Zone
+Hypocenter/
+P-waves(approx.6km/s)
+seismicfocusonsite workers from fire hazards. However, the rise of micro-service architecture was
+mainly introduced to reduce the complexity of large monolithic applications with
+huge code-base [17]. Hence, the research community suggests maintaining the size of
+micro-service applications optimal [55, 56, 57], not too large, that can impose
+complexity in administrating application workflow. In contrast, considering old
+operational systems, the centralized cloud can only support legacy applications
+[58, 59] with huge latency-tolerant nature. In contrast, modern Industry 4.0
+applications are latency sensitive that need a dynamic execution platform to enable
+smartness and support swift response time. As such, Rao et al.in [60] proposed a
+dynamic runtime for smart industrial applications that utilize 5G technology with
+edge-cloud architecture. This work uses application-specific knowledge to map the
+micro-services into the execution platform. Hence, authors consider only the critical
+path’s latency ignoring various generic micro-services that could play an important
+role in completing the smart solutions. Additionally, this work considers utilizing
+cloud data centers to ignore emergency and oversubscribed situations. Similarly,
+Faticanti et al.in a related study [61], analyzes the throughput needs of
+micro-service applications while offloading to various fog systems. The authors in
+this work addressed resource allocation challenges for the fog-native application
+architectures built on containerized micro-service modules. Two cascading
+algorithms make up the entirety of the answer. The first one separates fog
+application components according to throughput, whereas the second governs
+application orchestration across geographically distributed data centers.
+32
+
+2.6 The Scope of Fog Federation in Industry 4.0
+The smart industry’s numerous sensors create massive volumes of data that
+are often not analyzed due to a lack of storage and processing capabilities [62].
+Alternatively, only some of the data is relevant to any analytical findings. As a
+result, data pre-processing and filtering of noises and anomalies may be performed
+in the fog federation [63], leading to effective training of ML models in cloud data
+centers.
+Augmented reality (AR) and real-time video analytics need a quick response
+and efficient, secure storage systems that fog federation can support [64]. For
+instance, a significant processing delay may confuse a process engineer to perform
+fault-intolerant work, leading to an accident. Hence AR systems supported by fog
+computing can maximize throughput and reduce latency in both processing and
+transmission. Accordingly, K. Ha, et al.in [65] design and implement a wearable
+cognitive assistance spanning backed up by Google Glass and Cloudlet that assists
+the user by providing hints for social interaction via real-time scene analysis.
+To ensure security and safety, an immense amount of camera sensors are
+deployed in smart industries (e.g., oil and gas, transportation, manufacturing) that
+perform surveillance 24/7 to detect any anomaly and monitor the hazardous area.
+Therefore, the captured video needs storage and computational services that can be
+supported by fog federation. In addition, videos’ live streams, transcoding, and ML
+processing (e.g., object detection, classification, object tracking) are more frequent
+in Industry 4.0 applications. After completing the required services with captured
+33
+
+videos, the response can be sent to users in the form of notification, events,
+description, or video summary. Hence fog federation can be useful for achieving
+real-time processing (inference) and feedback on a huge amount of video streaming.
+In addition, scalability can be ensured on low-bandwidth output data. Furthermore,
+privacy-preserving techniques can also be applied at the fog side to ease the concern
+of personal privacy leakage in public surveillance systems.
+2.7 Data Privacy Aspects of a Federated Fog Computing System
+The technological advancement in smart IoT devices and smartphones has
+increased the possibility of using end devices for various complex ML applications,
+especially training ML models. The ever-growing power of end devices (e.g., mobile
+phones, PDAs, laptops, wearable) in computing and communication makes the
+complex Ml model training possible in fog devices. Hence, considering the fog
+federation, training with various fog systems’ local data in a distributed manner can
+enrich the ML model’s accuracy.
+Although, data security and privacy are the major challenges in this
+scenario. Hence, federated learning [66] is one solution that shares the ML model
+rather than data that does not leave the owner’s fog system. In federated learning,
+a global model is sent (global model broadcast) to the participating workers’ system
+to train with their local data as presented in figure 2.5. After a certain training
+period, the updates are sent back to the central server to incorporate the updates
+into the global model. Then the updated global model is again sent back to the
+participating FL workers. The process continues until the global model achieves a
+34
+
+Figure 2.5. A typical federated learning scenario that consists of FL workers and a
+central server having the global model. At the beginning of the training, the global
+model is broadcast to the participating workers to train with their corresponding local
+training data. After a period of training in FL workers, the updated model is sent
+back to the server for integration with the global model.
+Central server
+Model Updating
+FL workers
+Global model
+broadcasting
+certain accuracy. Different techniques (e.g., fedAvg, fedSGD, fedProx) can be
+utilized considering the global model’s accuracy to incorporate the updates from FL
+workers. Furthermore, considering the heterogeneity of FL workers’ computation
+and communication capability, the updates can be generated at different times.
+Accordingly, two different types of FL techniques are considered in the literature:
+asynchronous and synchronous FL, respectively [67]. Considering the various time
+to generate updates by the federated worker, some stragglers need to catch up to
+the certain period of sending the updates to the server. Therefore, asynchronous FL
+tries to incorporate as many updates as possible, whereas synchronous updates
+discard the updates that lag behind.
+2.7.1 Major Challenges of FL in Fog Federation
+The federated learning technique in fog federation ensures the ML services
+35
+
+while preserving the privacy of the data owners to the end clients. However, due to
+heterogeneous fog devices and data anomaly, some major challenges need to be
+addressed that are as following:
+• Class Imbalance Issue in Training Data: In FL technique, various FL
+workers’ local data are utilized for ML network model training. Hence, it is
+possible to have class imbalance issues within some participating workers’
+local data that can impact the global models’ robustness.
+• Communication Cost for Aggregating Updates into Global
+Model: To perform FL training, the global model needs to be transferred to
+participating workers via the internet. After training, the updates are sent
+back to the server for synchronization with the global model, and finally
+updated global model is sent to the FL workers. All the transfer operations
+utilize internet protocol which can incur a huge amount of communication
+cost.
+• Efficient Management of FL Workers: The number of participating FL
+workers can be huge where unexpected network connectivity and
+heterogeneous communication protocol make the management scenario nearly
+impossible[68].
+2.8 Downside of Smart Solutions in Industry 4.0
+Advances in hardware and software technology have evolved the oil and gas
+sector into a completely automated and machine-dependent industry [69]. Although
+36
+
+this digital revolution enhances production efficiency, it may produce numerous
+types of vulnerability and side effects that can lead to catastrophic incidents such as
+hazardous gas emissions, fire dangers, and oil spills [70]. Furthermore, the constant
+advancement of technology opens the potential to hack into information technology
+(IT) platforms [71] that deals with diverse industrial data and communication with
+the outside network. Another critical technology stack is the operational technology
+(OT) platform [72], primarily concerned with direct oil and gas production and
+processing operations with limited external access. Hence, the bridge between the
+IT and operational technology (OT) platforms , in particular, raises cyber-threats to
+oil and gas operations. As a result, while creating smart technology for oil and gas,
+researchers must study or be cognizant of the drawbacks of smart solutions. As a
+result, new and current smart solutions should contain better security approaches to
+ensure the system’s reliability. Furthermore, the possible side effects of smart
+solutions might impede operational efficiencies and become counter-productive.
+Accordingly, it is necessary to explore and identify various vulnerable areas
+of IT and OT platform as well as their interplay aspects in structural categories.
+Hence, cyber-threats and device incompatibility should be addressed properly to
+identify various open doors for cyber criminals. One of the issues issue with the oil
+and gas industry is that it relies on systems that were not designed with network
+connectivity in consideration. Industrial plants, for example, were never designed to
+be connected to networks, but with the continuous digital revolution, they are
+today. This can lead to a risky scenario since a cyber-attack on such a system can
+37
+
+impair operations and cause the death of life.
+The industrial revolution has increased the utilization of various types of
+machines that robots or human workers operate. Moreover, these machines
+sometimes communicate with other machines to complete an industrial operation.
+Hence, machine-machine and human-machine interactions can go wrong and create
+opportunities for cyber criminals to sabotage industrial processes. As such,
+identification of industrial interaction challenges can help to build smart solutions
+that are safe and secure. Finally, developing any physical or software solution
+requires human and machine involvement that leads to the engagement of various
+biases (e.g., artificial intelligence, automation, and human-related biases) in smart
+solutions of Industry 4.0. These biases can lead to unwanted accidents or loopholes
+for cyber criminals. Therefore, addressing different forms of bias in industrial
+sectors can help build smart solutions that are resilient to cyber-threats and attacks.
+2.9 Summary and Positioning of this Dissertation
+This section introduced the Edge-to-Cloud continuum and federation of fog
+computing paradigms and their goals. First, we discuss various scopes to utilize the
+Edge-Fog-Cloud continuum for different Industry 4.0 applications, especially
+real-time nature and machine learning (ML) based applications. Then in chapter 3,
+we analyze the performance of various Industry 4.0 applications in widely used AWS
+cloud and Chameleon fog servers. After that, we investigate the challenges of
+federated fog systems and suggest a statistical resource allocation method across
+federated fog systems for monolithic workloads in remote industrial sites in chapter
+38
+
+4. Then in chapter 5, we explore the micro-service software architecture of the
+Industry 4.0 applications and propose a probabilistic partitioning and resource
+allocation method to improve the robustness of the fog federation. After that, we
+study the data security and privacy aspects of fog federation by addressing
+state-of-the-art challenges in privacy-preserving ML-application training for the oil
+and gas industry in chapter 6. Finally, in chapter 7, we identify the downsides of
+smart solutions and suggest state-of-the-art solutions for the remote oil and gas
+industry. In the end, we conclude the dissertation by disclosing a summary of our
+findings and future avenues to explore in chapter 8.
+39
+
+Chapter 3:
+Performance Analysis of DNN-based Application in Cloud
+and Fog Systems
+3.1 Overview
+This chapter analyzes the performances of Deep Neural Network
+(DNN)-based Industry 4.0 applications to study the inference execution times on
+cloud and fog computing resources. Being an indispensable part of Industry 4.0,
+DNN-based smart applications make the latency-sensitive inference that needs to be
+accurate and execute certain application constraints with a specific deadline. The
+quality of service(QoS) could be compromised due to missing each application’s
+deadline even if the inference accuracy is high. Due to the multi-tenancy and
+resource heterogeneity inherent to the cloud and fog computing environments, the
+inference time of DNN-based applications is stochastic. Such stochasticity, if not
+captured, can potentially lead to a disaster in critical sectors, such as Oil and Gas
+industry. To make Industry 4.0 robust, solution architects and researchers need to
+understand the behavior of DNN-based applications and capture the stochasticity
+that exists in their inference times. Accordingly, in this study, we provide a
+descriptive analysis of the inference time in the popular cloud platform, Amazon,
+and in Chameleon as Fog system.
+We employ two statistical methodologies to evaluate DNN-based
+applications: application-centric and resource-centric. First, we begin with an
+application-centric analysis in which we statistically model the inference execution
+time of four categorically unique DNN applications executing on both Amazon and
+40
+
+Chameleon. Second, we examine a rate-based indicator known as Million
+Instruction Per Second (MIPS) for heterogeneous cloud and fog systems using a
+resource-centric approach. The confidence interval of MIPS for heterogeneous cloud
+and fog systems is then estimated using non-parametric modeling approaches such
+as Jackknife and Bootstrap re-sampling. The findings of this work might help
+academics and cloud solution architects build robust solutions against the stochastic
+nature of inference time in the cloud, allowing them to deliver higher QoS to their
+users while avoiding unanticipated repercussions. Furthermore, we provide a
+DNN-based applications benchmark a for system architects to employ in building
+effective resource allocation solutions.
+3.2 DNN-Based Applications in Industry 4.0
+Among various DNN-based applications utilized in Industry 4.0, we consider
+four different applications used in the smart O&G industry. The summary of the
+chosen applications is demonstrated in table 3.1, which presents the abbreviated
+name for each application, its DNN (network) model, the type of its input data, the
+scope of deployment in O&G Industry [73], and the code base to build the model.
+The applications’ code base, input data, and analysis results are publicly available
+for reproducibility purposes in the GitHub repository mentioned earlier. In the rest
+of this section, we explore the characteristics of each application type.
+3.2.1 Fire Detection
+The fire detection application is an essential component of monitoring
+ahttps://github.com/hpcclab/Benchmarking-DNN-applications-industry4.0
+41
+
+Table 3.1. DNN-based applications used in O&G Industry 4.0 along with their network
+model, input data type, usage scope, and code base.
+Application Type
+DNN Model
+Input Type
+Scope
+Code Base
+Fire Detection (Fire)
+Customized Alexnet
+Video Segment
+Control &
+Monitoring
+Tensorflow
+(tflearn)
+Human Activity
+Recognition (HAR)
+Customized Sequential
+Neural Network
+Motion sensors
+Workers
+Safety
+keras
+Oil Spill Detec. (Oil)
+FCN-8
+SAR Images
+Disaster
+Management
+keras
+Acoustic Impedance
+Estimation (AIE)
+Temporal Convolutional
+Network
+Seismic Data
+Seismic
+Exploration
+PyTorch
+systems designed to provide safety and resilience in Industry 4.0. We used a
+convolutional neural network (CNN) to investigate a fire detection DNN-based
+application proposed by Dunnings and Breckon [74]. It identifies fire areas (pixels)
+in real-time in the frames of a monitored video. We use the FireNet model, which
+correctly identifies and locates fire in each frame of a given video segment, among
+the several fire detection models offered by the authors. FireNet is a simplified
+version of the AlexNet model [75], with three convolutional layers of sizes 64, 128,
+and 256. To obtain high-frequency features with a significant response from the
+preceding layer, each convolutional layer in this model is enhanced with a
+max-pooling layer and a local response normalization. We created a benchmarking
+dataset of 240 videos with varied backgrounds to examine the inference time of the
+fire detection application. All videos are regarded as two seconds long for a fair and
+accurate appraisal.
+3.2.2 Human Activity Recognition
+Human Activity Recognition (HAR) systems are widely used in Industry 4.0
+to ensure workers’ safety in hazardous zones. In the HAR system, various sensor
+42
+
+data are analyzed that are generated from different sensors used by human workers
+while performing any physical movement. In this case, motion sensors, such as
+accelerometers and gyroscopes, that are widely available on handheld PDA devices
+are utilized to capture human activity-related sensor data. The HAR system we use
+operates based on the sequential neural network model with four layers to identify
+the worker’s activities (walking, walking upstairs, walking downstairs, sitting). For
+analysis, we use a dataset of the UCI machine learning repository, known as Human
+Activity Recognition Using Smartphones [76].
+Figure 3.1.
+The FCN-8 model is presented in block diagram that consist of 5 fully
+convolutional network blocks, and 2 up-sampling blocks. The model receives input as a
+SAR image and perform pixel-wise classification to output a labeled image.
+FCN-Block 1
+FCN-Block 2
+FCN-Block 3
+FCN-Block 4
+FCN-Block 5
+MP-1
+MP-2
+MP-3
+MP-4
+MP-5
+Convolution
+Convolution
+Upsample
+Upsample
+U*U*U
+Input SAR Image
+captured from
+Satellite
+Output Labeled
+Image
+Look-alike
+Oil Spill
+Sea surface
+3.2.3 Oil Spill Detection
+Detecting the oil spill is of paramount importance to have a safe and clean
+O&G Industry 4.0. The accuracy of DNN-based oil spill detection systems has been
+promising [77]. We utilize a detection system that operates based on the FCN-8
+43
+
+model [78], which is depicted in Figure 3.1. As we can see, the model contains five
+Fully Convolutional Network (FCN) blocks and two up-sampling blocks that
+collectively perform semantic segmentation (i.e., classifying every pixel) of an input
+image and output a labeled image. The FCN-8 model functions based on the
+satellite (a.k.a. SAR) [79] images. We configure the analysis to obtain the inference
+time of 110 SAR images collected by MKLab [77].
+Figure 3.2.
+Schematic view of Temporal Convolutional Network (TCN) model that
+consists of six temporal blocks, the input data, and the output in form of the predicted AI.
+Input Seismic Traces from
+Marmousi Model
+Temporal
+Block
+(1,3)
+Temporal
+Block
+(3,5)
+Temporal
+Block
+(5,5)
+Temporal
+Block
+(5,5)
+Temporal
+Block
+(5,5)
+Temporal
+Block
+(5,6)
+Concatenation
+Linear Layer
+Temporal Convolutional Network
+Output Predicted Acoustic Impedance (AI)
+3.2.4 Acoustic Impedance Estimation
+Estimating acoustic impedance (AI) from seismic data is an important step
+in O&G exploration. To estimate AI from seismic data, we utilize a solution
+functions based on the temporal convolutional network [80], shown in Figure 3.2.
+The solution outperforms others in terms of gradient vanishing and overfitting.
+Marmousi 2 dataset [81] is used to estimate AI.
+3.3 Computing Platforms for Industry 4.0
+3.3.1 Amazon Cloud
+AWS is a pioneer in the Cloud computing industry and provides more than
+175 services, including Amazon EC2 [82], across a large set of distributed data
+44
+
+centers. Amazon EC2 provides inconsistently heterogeneous machines (e.g., CPU,
+GPU, and Inferentia) in form of various VM instance types (e.g., general purpose,
+compute-optimized, and machine learning (ML)). Within each VM type, a range of
+VM configurations (e.g., large, xlarge, 2xlarge) are offered that reflect the
+consistent heterogeneity within that VM type. To realize the impact of machine
+heterogeneity on the inference time of various applications, we choose one
+representative VM type of each offered machine type. Table 3.2 represents the type
+of machines and their specification in terms of number of cores and memory. We
+note that all the machine types use SSD storage units. Although General Purpose
+machines are not considered suitable for latency-sensitive DNN-based applications,
+the reason we study them is their similarity to the specifications of machine types
+often used in the fog computing platforms. As such, considering these types of
+machines (and similarly m1.small in the Chameleon cloud) makes the results of this
+study applicable to cases where fog computing is employed for latency-sensitive
+applications [83].
+Table 3.2. Heterogeneous machine types and VM configurations in Amazon EC2
+that are considered for performance modeling of DNN-based applications. In this
+table, ML Optimized represents Inferentia machine type offered by AWS.
+Machine Type
+VM Config.
+vCPU
+GPU
+Mem. (GB)
+Mem. Optimized
+r5d.xlarge
+4
+0
+32
+ML Optimized
+inf1.xlarge
+4
+0
+8
+GPU
+g4dn.xlarge
+4
+1
+16
+General Purpose
+m5ad.xlarge
+4
+0
+16
+Comp. Optimized
+c5d.xlarge
+4
+0
+8
+45
+
+Table 3.3. Various VM flavors in Chameleon cloud are configured to represent a
+consistently heterogeneous system.
+VM Config.
+vCPU
+Mem. (GB)
+m1.xlarge
+8
+16
+m1.large
+4
+8
+m1.medium
+2
+4
+m1.small
+1
+2
+3.3.2 Chameleon as Fog Computing System
+Chameleon [84] is a large-scale public computing platform maintained by
+National Science Foundation (NSF) that usually utilized for academic research
+purposes. Due to Chameleon’s maintenance issues (e.g., transient failures,
+unexpected downtime, resource scarcity), less large scale VM flavors, and
+distributed nature, we consider Chameleon as Fog computing system. Chameleon
+supports VM-based heterogeneous computing. It offers the flexibility to manage the
+compute, memory, and storage capacity of the VM instances. In this study, we use
+the Chameleon to configure a set of consistently heterogeneous machines (Fog
+Systems). We configure four VM flavors, namely m1.xlarge, m1.large,
+m1.medium, and m1.small, as detailed in Table 3.3. We note that VMs offered by
+Chameleon uses HDD unit as storage.
+3.4 Environmental Setup for Performance Modeling
+The focus of this study is on latency-sensitive DNN-based applications that
+are widely used in Industry 4.0. Accordingly, we consider a dynamic (online) system
+that is already loaded with pre-trained DNN-based applications, explained in the
+previous section, and executes arriving requests on the pertinent application. This
+46
+
+means that we measure the hot start inference time [85] in the considered
+applications. The DNN-based applications mostly use TensorFlow, and the VMs
+both in AWS and Chameleon are configured to use the framework on top of Ubuntu
+18.04.
+Figure 3.3. The stochastic nature of inference execution time of oil spill application while
+running on heterogeneous VMs in the AWS. For every VM instance, the oil spill detection
+application is executed 30 times and those executions are plotted as number of attempts
+along x-axis. The y-axis represents the inference time for every attempts.
+1
+10
+20
+30
+16.5
+16.6
+Inference Time(s)
+Compute Optimized
+1
+10
+20
+30
+132.5
+135.0
+137.5
+General Purpose
+1
+10
+20
+30
+Number of Attempts
+8.0
+8.2
+GPU Instance
+1
+10
+20
+30
+15.8
+16.0
+ML Optimized
+1
+10
+20
+30
+16
+17
+Memory Optimized
+Our initial evaluations in AWS (shown in Figure 3.3) demonstrate that, in
+different attempts, the inference execution time of an application (Oil Spill) on the
+same machine type can be highly stochastic. Similar stochasticity is found for
+chameleon cloud while we run the oil spill detection application 30 times within
+same VM instance. Hence to capture this randomness (aka consistent heterogeneity)
+that is caused by several factors, such as transient failures or multi-tenancy [86, 87],
+we base our analysis on 30 times execution of the same request within same VM.
+3.5 Application-Centric Analysis of Inference Time
+3.5.1 Overview
+In this part, we capture the inference time of the four DNN applications and
+try to identify their underlying statistical distributions using various statistical
+methods. Then, to describe the behavior of inference execution time using a single
+metric, we explore the central tendency of the distributions.
+47
+
+3.5.2 Statistical Distribution of Inference Execution Time
+Among various statistical methods, normality tests are widely employed to
+understand the distribution of the collected samples. Hence, we first use the
+Shapiro-Wilk test [88] to verify the normality of the inference time distribution on
+each machine type. Next, we employ the Kolmogorov-Smirnov test [89] to find the
+best fit distribution based on the sampled inference execution times.
+3.5.2.1 Shapiro-Wilk test to verify normality of the sampled data.
+The null hypothesis is that the inference execution times are normally distributed.
+To understand whether a random sample comes from a normal distribution, we
+perform the Shapiro-Wilk test. The result of this test is considered as W, whose low
+value (lower than wα threshold) indicates that the sampled data are not normally
+distributed and vice versa.
+The value of W is used to perform the significant testing (i.e., calculating
+P-value). The higher P-value, especially greater than a threshold value (typically
+0.05), supports the null hypothesis that the sampled data are normally distributed.
+Table 3.4. The execution time distributions of DNN-based applications in AWS clouds
+machines using Shapiro-Wilk test.
+Execution Time Distribution with Shapiro-Wilk Test in AWS Cloud
+App. Type
+Mem.
+Opt.
+ML Opt.
+GPU
+Gen.
+Pur.
+Compt.
+Opt.
+Fire
+Not Gaussian
+(P=2.73e−16)
+Not Gaussian
+(P=5.42e−16)
+Not Gaussian
+(P=6.59e−16)
+Not Gaussian
+(P=2.06e−13)
+Not Gaussian
+(P=3.9e−16)
+HAR
+Not Gaussian
+(P=7.12e−8)
+Not Gaussian
+(P=1.04e−8)
+Gaussian
+(P=0.19)
+Not Gaussian
+(P=1.76e−8)
+Not Gaussian
+(P=0.4.62e−5)
+Oil
+Not Gaussian
+(P=8e−4)
+Not Gaussian
+(P=2.9e−16)
+Not Gaussian
+(P=0.012)
+Not Gaussian
+(P=1.27e−16)
+Not Gaussian
+(P=5.86e−14)
+AIE
+Gaussian
+(P=0.46)
+Gaussian
+(P=0.23)
+Gaussian
+(P=0.08)
+Not Gaussian
+(P=1.99e−10)
+Gaussian
+(P=0.96)
+The results of Shapiro-Wilk test on the collected inference times for AWS are
+48
+
+Table 3.5.
+The execution time distributions of DNN applications in Chameleon cloud
+using Shapiro-Wilk test.
+Execution Time Distribution with Shapiro-Wilk Test in Chemeleon
+App. Type
+m1.xlarge
+m1.large
+m1.medium
+m1.small
+Fire
+Not Gaussian
+(P=4.05e−5)
+Not Gaussian
+(P=1.e−4)
+Not Gaussian
+(P=7.58e−6)
+Not Gaussian
+(P=1.32e−7)
+HAR
+Gaussian
+(P=0.74)
+Not Gaussian
+(P=0.02)
+Gaussian
+(P=0.18)
+Gaussian
+(P=0.84)
+Oil
+Not Gaussian
+(P=0.01)
+Not Gaussian
+(P=5.5e−7)
+Not Gaussian
+(P=0.01)
+N/A
+AIE
+Not Gaussian
+(P=2.77e−10)
+Not Gaussian
+(P= 3.46e−6)
+Not Gaussian
+(P= 1.4e−4)
+Not Gaussian
+(P=2.46e−6)
+presented in Table 3.4, where columns present the various machine types and rows
+define the application types. The table reflects that our initial assumption is not
+totally valid. The Shapiro-Wilk test result for the Chameleon cloud, depicted in
+Table 3.5, shows that for only three of the total cases, the normality assumption
+holds. Considering the lack of normality in several cases, in the next section, we
+utilize Kolmogorov-Smirnov test to more granularly explore the best fitting
+distribution for the inference time of each application and also cross validate the
+prior results we obtained using another statistical method.
+3.5.2.2 Kolmogorov-Smirnov test to identify the execution time
+distribution. The null hypothesis for the Kolmogorov-Smirnov test is that the
+inference times of a certain application type on a given machine type follows a
+statistical distribution. The Kolmogorov-Smirnov Goodness of Fit test (a.k.a. K-S
+test) identifies whether a set of samples derived from a population fits to a specific
+distribution. Precisely, the test measures the largest vertical distance (called test
+statistic D) between a known hypothetical probability distribution and the
+distribution generated by the observed inference times (a.k.a. empirical distribution
+49
+
+function (EDF)). Then, if D is greater than the critical value obtained from the K-S
+test P-Value table, then the null hypothesis is rejected.
+Table 3.6. Inference time distributions of DNN-based applications in AWS cloud machines
+using Kolmogorov-Smirnov test.
+Execution Time Distribution with Kolmogorov-Smirnov Test in AWS Cloud
+App. Type
+Mem.
+Opt.
+ML Opt.
+GPU
+Gen.
+Pur.
+Compt.
+Opt.
+Fire
+No Distr.
+No Distr.
+No Distr.
+No Distr.
+No Distr.
+HAR
+Student’s t
+(P=0.08)
+Student’s t
+(P=0.77)
+Student’s t
+(P=0.99)
+Student’s t
+(P=0.57)
+Student’s t
+(P=0.95)
+Oil
+Student’s t
+(P=0.44)
+Student’s t
+(P=0.96)
+Student’s t
+(P=0.5)
+Student’s t
+(P=0.20)
+Exponential
+(P=0.21)
+AIE
+Normal
+(P=0.99)
+Normal
+(P=0.54)
+Normal
+(P=0.47)
+Exponential
+(P=0.16)
+Normal
+(P=0.99)
+Table 3.7. Inference time distributions of DNN-based applications in Chameleon’s ma-
+chines using the K-S test.
+Execution Time Distribution with Kolmogorov-Smirnov test in Chameleon
+App. Type
+m1.xlarge
+m1.large
+m1.medium
+m1.small
+Fire
+No Distr
+No Distr
+No Distr
+Log-normal
+HAR
+Normal
+(P=0.98)
+Student’s t
+(P=0.88)
+Normal
+(P=0.66)
+Normal
+(P=0.96)
+Oil
+Log-normal
+(P=0.36)
+Log-normal
+(P=0.99)
+Log-normal
+(P=0.81)
+N/A
+AIE
+Student’s t
+(P= 0.47)
+Student’s t
+(P=0.12)
+Student’s t
+(P=0.25)
+Student’s t
+(P=0.83)
+The results of the K-S test on the observed inference times in AWS and
+Chameleon clouds are depicted in Table 3.6 and 3.7, respectively. From Table 3.6,
+we find that, in AWS, majority of the entries either represent Normal distribution or
+Student’s t-distribution that exposes similar properties. However, we observe that
+the inference time of Fire Detection application does not follow any particular
+distribution with an acceptable P-Value. We also observe that the inference times of
+both Oil Spill application on Compute Optimized machine and AIE application on
+General Purpose machine follow Exponential distribution. However, low P-Value in
+both of these cases indicate a weak acceptance of the null hypothesis.
+50
+
+On the contrary, Table 3.7 reflects the dominance of Log-normal (i.e., the
+logarithm of the random variable is normally distributed) and Student’s
+t-distribution over other distributions in the Chameleon cloud. Analyzing the
+execution traces shows us that the inference times in Chameleon are predominantly
+larger than the ones in AWS that causes right-skewed property. Hence, the
+distribution tends to be Log-normal. Consistent with AWS observations, we see
+that the Fire Detection application does not follow any distribution in most cases.
+Our further analysis showed that the lack of distribution is due to the input videos’
+variety (e.g., frame rate and resolution). When we reduced the degree of freedom in
+the input videos and limited them to those with the same configuration (frame
+rate), we noticed the inference time followed a Log-normal distribution. The
+observation shows that the characteristics and variation of input data can be
+decisive in the statistical behavior of inference times (mentioned in highlighted
+insight). Finally, we note that the Oil Spill application cannot be run on m1.small
+machine owing to its limited memory.
+Insights: The key insights of the analysis we conducted on identifying the
+distribution of inference time are as follows:
+• Shapiro-Wilk test for AWS and Chameleon rejects the null hypothesis that
+the inference times of DNN-based applications follow the Normal distribu-
+tion.
+• The K-S test reflects the dominance of Student’s t-distribution of inference
+time in both AWS (Table 3.6), and Chameleon (Table 3.7).
+• Various configurations of input data is decisive on the statistical distribution
+of the inference time.
+51
+
+Table 3.8. The measurement of central tendency metric (µ), and data dispersion metric
+(σ) on the observed inference times in AWS.
+Mean and Standarad Deviation of Inference Execution Times (ms) in AWS
+App. Type
+Mem.
+Opt.
+ML Opt.
+GPU
+Gen.
+Pur.
+Compt.
+Opt.
+Fire
+µ=1461.8
+σ=457.3
+µ=1281.7
+σ=387.93
+µ=1349.5
+σ=418.9
+µ =1534.8
+σ=494.7
+µ=1421.4
+σ=441.8
+HAR
+µ=1.27
+σ=0.082
+µ=0.66
+σ=0.006
+µ=0.51
+σ=0.006
+µ =1.17
+σ=0.042
+µ=0.66
+σ=0.003
+Oil
+µ=269.9
+σ=1.01
+µ=218.8
+σ=0.66
+µ=65.98
+σ=0.47
+µ=667.1
+σ=2.26
+µ=242.9
+σ=0.68
+AIE
+µ=7.02
+σ=0.02
+µ=6.41
+σ=0.03
+µ=7.55
+σ=0.04
+µ=9.35
+σ=0.06
+µ=7.95
+σ=0.02
+Table 3.9. Central tendency metric (µ), and data dispersion metric (σ) of the inference
+times in the Chameleon cloud.
+Mean and Std. of Inference Execution Times (ms) in Chameleon
+App. Type
+m1.xlarge
+m1.large
+m1.medium
+m1.small
+Fire
+µ=2155.20
+σ=725.48
+µ=2213.30
+σ=731.50
+µ=2330.80
+σ=742.20
+µ=3184.80
+σ=1033.30
+HAR
+µ=22.14
+σ=0.76
+µ=47.69
+σ=1.26
+µ=49.24
+σ=0.57
+µ=52.69
+σ=0.78
+Oil
+µ=147.16
+σ=5.23
+µ=222.22
+σ=2.89
+µ=412.78
+σ=4.99
+N/A
+AIE
+µ=6.23
+σ=0.25
+µ=6.23
+σ=0.15
+µ=6.40
+σ=0.13
+µ=7.72
+σ=0.24
+3.5.3 Analysis of Central Tendency and Dispersion Measures
+Leveraging the statistical distributions of inference times, in this part, we
+explore their central tendency metric that summarizes the behavior of collected
+observations in a single value. In addition, to analyze the statistical dispersion of
+the observations, we measure the standard deviation of the inference times. In
+particular, we estimate the arithmetic mean and standard deviation of the inference
+times. The central tendency metric of inference times for AWS and Chameleon
+systems are shown in Tables 3.8 and 3.9, respectively. The key insights are as
+follows:
+52
+
+• Machine Learning Optimized and GPU instances often outperform other
+AWS machine types.
+• In both clouds, the inference times of Fire and Oil experience a higher stan-
+dard deviation in compare with other applications. The high uncertainty
+is attributed to the characteristics of their input data; Oil Spill input im-
+ages suffer from class imbalance [77], whereas, Fire input videos are highly
+variant.
+• In Chameleon VMs with a consistent heterogeneity, DNN applications with
+dense network models (e.g., Oil and Fire) can take advantage of powerful
+machines (e.g., m1.xlarge) to significantly reduce their inference times.
+• Overall, AWS offers a lower inference time than Chameleon. The reason is
+utilizing SSD units in AWS rather than HDD in Chameleon. In addition, we
+noticed that Chameleon experiences more transient failures that can slow
+down the applications.
+3.6 Resource-Centric Analysis of Inference Time
+In performance analysis of computing systems, a rate-based metric
+[90] is defined as the normalization of number of computer instructions executed to
+a standard time unit. MIPS is a popular rate-based metric that allows comparison
+of computing speed across two or more computing systems. Given that computing
+systems (e.g., AWS ML Optimized and GPU) increasingly use instruction-level
+facilities for ML applications, our objective in this part is to analyze
+the performance of different machine types in processing DNN-based applications.
+The results of this analysis can be of particular interest to researchers and
+cloud solution architects whose endeavor is to develop tailored resource allocation
+solutions for Industry 4.0 use cases. As for rate-based metrics we do not assume
+any distribution [91], we conduct a non-parametric approach. In addition to MIPS,
+we provide the range of MIPS in form of Confidence Intervals (CI) for each case.
+53
+
+Table 3.10. MIPS values of heterogeneous machines in AWS for each DNN-based appli-
+cation.
+The MIPS for DNN Applications in AWS Cloud
+App. Type
+Mem.
+Opt.
+ML Opt.
+GPU
+Gen.
+Pur.
+Compt.
+Opt.
+Fire
+1938.63
+2196.35
+2092.72
+1862.04
+1989.56
+HAR
+838640.65
+1595874.34
+2040057.33
+891754.48
+1581709.12
+Oil
+164.54
+168.58
+331.98
+20.46
+162.01
+AIE
+145.58
+180.28
+150.25
+131.25
+160.32
+Table 3.11. MIPS vales for heterogeneous machines on Chameleon cloud for each DNN-
+based application.
+The MIPS for DNN Applications in Chameleon
+App. Types
+m1.xlarge
+m1.large
+m1.medium
+m1.small
+Fire
+1327.81
+1282.33
+1249.63
+871.36
+HAR
+91.78
+102.51
+124.76
+136.62
+Oil
+18267.35
+11233.41
+6243.94
+N/A
+AIE
+246366.52
+249551.29
+236300.93
+201807.49
+Let application i with ni instructions have tim inference time
+on machine m. Then, MIPS of machine m to execute the application is defined as
+MIPSmi = ni/(tim × 106). Hence, before calculating MIPS for any machine, we need
+to estimate the number of instructions (n) of each DNN-based application. For that
+purpose, we execute each task ti on a machine whose MIPS is known and estimated
+ni. Then, for each machine m, we measure tim and subsequently calculate MIPSmi.
+Tables 3.10 and 3.11 show the MIPS values for AWS and Chameleon, respectively.
+To measure the confidence intervals (CI) of MIPS for each application type
+in each machine type, we use the non-parametric statistical methods [91] that
+perform prediction based on the sample data without making any assumption about
+their underlying distributions. As we deal with a rate-based metric, we use
+harmonic mean that offers a precise analysis for this type of metric rather than the
+arithmetic mean. We utilize Jackknife [91] re-sampling method and validate it using
+54
+
+Bootstrap [91], which is another well-known re-sampling method. Both of these
+methods employ harmonic mean to measure the confidence intervals of MIPS.
+Table 3.12. The confidence intervals of MIPS values for DNN-based applications in AWS
+machines, resulted from Jackknife re-sampling method.
+CI of MIPS using Jackknife Method in AWS cloud
+App. Type
+Mem.
+Opt.
+ML Opt.
+GPU
+Gen.
+Pur.
+Compt.
+Opt.
+Fire
+[1549.42,
+1975.65]
+[1770.81,
+2243.04]
+[1671.78,
+2131.66]
+[1465.31,
+1889.77]
+[1594.78,
+2028.36]
+HAR
+[812040.26,
+856355.96]
+[1592214.75,
+1599426.64]
+[2033084.47,
+2046727.57]
+[880417.69,
+901345.49]
+[1580275.10,
+1585598.85]
+Oil
+[163.55,
+165.47]
+[168.36,
+168.81]
+[330.68,
+333.22]
+[20.35,
+20.57]
+[161.86,
+162.17]
+AIE
+[139.02,
+141.04]
+[155.56,
+156.01]
+[141.57,
+142.03]
+[118.06,
+119.82]
+[148.35,
+149.00]
+Table 3.13. Confidence intervals of MIPS values for different DNN-based applications in
+Chameleon machines, resulted from Jackknife re-sampling method.
+CI of MIPS using Jackknife Method in Chameleon Cloud
+App. Type
+m1.xlarge
+m1.large
+m1.medium
+m1.small
+Fire
+[1032.11,
+1341.75]
+[1010.62,
+1303.02]
+[964.76,
+1259.68]
+[670.82,
+872.85]
+HAR
+[88.27,
+94.20]
+[99.84,
+104.49]
+[122.33,
+126.67]
+[135.13,
+137.92]
+Oil
+[18083.59,
+18628.64]
+[11159.71,
+11662.41]
+[6139.59,
+6262.15]
+N/A
+AIE
+[237710.12,
+252686.82]
+[247166.73,
+251673.68]
+[168804.58,
+268273.11]
+[199676.71,
+203681.17]
+3.6.1 Estimating Confidence Interval using Jackknife Method
+Let p be the number of observed inference times. The Jackknife method
+calculates the harmonic mean in p iterations, each time by eliminating one sample.
+That is, each time it creates a new sample (re-sample) with size p − 1. Let xj be the
+jth observed inference time. Then, the harmonic mean of re-sample i is called the
+pseudo-harmonic value (denoted as yi) and is calculated based on Equation 3.1.
+yi =
+p − 1
+p�
+j=1,j̸=i
+1
+xj
+(3.1)
+55
+
+Next, the arithmetic mean (denoted ¯y) of the p pseudo-harmonic values is
+computed, and is used to estimate the standard deviation. Finally, the
+t-distribution table is used to calculate the CI boundaries with a 95% confidence
+level. The result of the Jackknife method for AWS machines is shown in Table 3.12
+that conforms with the MIPS calculation in Table 3.10. Similarly, the results of
+analysis for Chameleon cloud using Jackknife method, shown in Table 3.13, validate
+the prior MIPS calculations in Table 3.11. However, in the next part, we
+cross-validate these results using Bootstrap method.
+3.6.2 Estimating Confidence Interval using Bootstrap Method
+Bootstrap repeatedly performs random sampling with a replacement technique
+[91] on the observed inference times. The random sampling refers to the selection of a
+sample with the chance of non-zero probability and the number (represented as k) of
+re-sample data depends on the user’s consideration. After re-sampling, the harmonic
+means of k number of samples are calculated and sorted in ascending order to estimate
+the confidence intervals. Finally, for a specific confidence level, the (α/2 × k)th and
+((1 − α/2) × k)th values are selected from the sorted samples as the lower and upper
+bounds of the CI. We set the k value to 100 and α to 0.05 for 95% confidence level.
+For both AWS and Chameleon, the results of CI analysis using the Bootstrap
+method are similar to, thus validate, the ranges estimated by the Jackknife method.
+Therefore, due to the shortage of space, we do not report the table of MIPS values
+for the Bootstrap method. However, we note that the CI ranges provided by the
+Bootstrap method are shorter (i.e., have less uncertainty), regardless of the
+56
+
+application type and the cloud platform. The reason for the shorter range is that
+Bootstrap performs re-sampling with a user-defined number of samples that can be
+larger than the original sample size.
+Figure 3.4. Comparative analysis of the MIPS values of AWS and Chameleon machines
+for various DNN-based applications. For the sake of presentation, the MIPS values are
+normalized between [0,1].
+Fire
+HAR
+Oil
+AIE
+DNN-based Applications
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Normalized MIPS
+AWS Comp. Optimized
+Chameleon m1.large
+To perform a cross-platform analysis of the MIPS values, in Figure 3.4, we
+compare the range of MIPS values for AWS Compute Optimized against m1.large
+that is a compatible machine type in Chameleon (see Tables 3.2 and 3.3). The
+horizontal axis of this figure shows different application types and the vertical axis
+shows the MIPS values, normalized based on MinMax Scaling in the range of [0,1],
+for the sake of better presentation. Due to high variation in the input videos, we
+observe a broad CI range for Fire detection across both cloud platforms. However,
+for HAR, Oil Spill, and AIE applications, we observe that the first and third
+57
+
+quartiles of the CI range in Chameleon (whose machines are prone to more transient
+failures [92]) is larger than those in AWS. This wide range indicates that, apart
+from variations in the input data, the reliability of underlying resources is also
+decisive on the stochasticity of the inference times.
+3.7 Summary and Discussion
+Accurately estimating the inference time of latency-sensitive DNN-based
+applications plays a critical role in robustness and safety of Industry 4.0. Such
+accurate estimations enable cloud providers and solution architects to devise
+resource allocation and load balancing solutions that are robust against uncertainty
+exists in the execution time of DNN-based applications. In this work, we provide
+application- and resource-centric analyses on the uncertainty exists in the inference
+times of several DNN-based applications deployed on heterogeneous machines of two
+computing platforms, namely AWS and Chameleon. In the first part, we utilized
+the Shapiro-Wilk test to verify if the assumption of Normal distribution for the
+inference time holds. We observed that the inference times often do not follow a
+Normal distribution. Therefore, in the second part, we broaden our distribution
+testing investigation and utilized the Kolmogorov-Smirnov test to verify the
+underlying distributions in each case. The analysis showed that inference times
+across the two computing platforms often follow Student’s t-distribution. However,
+in several cases in Chameleon system we observed the Log-normal distribution that
+we attribute it to the uncertain performance of VMs in this platform. Next, to
+conduct a resource-centric analysis, we modeled MIPS (as a rate-based performance
+58
+
+metric) of the heterogeneous machines for each application type. In the analysis, we
+took a non-parametric approach, which is suitable for rate-based metrics, and
+utilized the Jackknife and Bootstrap re-sampling methods with harmonic mean to
+determine the range of confidence intervals of the MIPS values in each case. The
+calculated MIPS values and their CI ranges reflect the behavior of different
+DNN-based applications under various machine types of cloud and fog systems. A
+comparative analysis of the CI ranges across AWS and Chameleon demonstrate that
+the uncertainty in the inference time is because of variations in the input data and
+unreliability of the underlying platforms. In the future, we plan to incorporate the
+findings of this research to devise accurate resource allocation methods in IoT and
+edge computing systems. In addition, we plan to develop a predictive analysis to
+determine the execution of each inference task upon arrival.
+59
+
+Chapter 4:
+The Benefits of Federated Fog to Manage Monolithic
+Workload in Remote Industrial Sites
+4.1 Overview
+In the previous chapters, our preliminary research found that fog federation
+can be a potential computational platform for remote smart industries with
+stochastic execution behaviors for Industry 4.0 applications. Hence, the stochastic
+execution of Industry 4.0 applications has an influence on task completion times. In
+this case, an efficient resource allocation and load balancing technique that is aware
+of stochastic execution behaviors of Industry 4.0 applications can ensure the
+system’s robustness by enabling the on-time completion of receiving tasks.
+Accordingly, in this chapter, we first strategically develop a load-balancing method
+for allocating arriving tasks to a fog federation. Hence, our primary goal is to ensure
+the system’s robustness (fog federation) in terms of meeting the deadlines of
+arriving tasks. To achieve the goal, we estimate the end-to-end latency of a
+receiving task in a fog system and utilize the latency to predict the task completion
+time across the fog federation. Hence, we propose a probabilistic task allocation
+method in the load balancer of each fog system that is aware of the latency
+constraints of the receiving tasks. Then, in the second part, we evaluate our
+proposed load balancing method using the synthetic workload (customized to
+industrial tasks workload) of EdgeCloudSim [18] simulator.
+60
+
+4.2 End-to-End Latency in Federated Fog Systems
+When a task request arrives at a fog system’s load balancer, communication
+and computational latencies combine to generate the end-to-end latency.
+Furthermore, several factors impact each of these latencies, causing them to behave
+stochastically. For these reasons, calculating end-to-end latency and capturing its
+stochastic character in fog computing systems is difficult. In the following sections,
+we go over the elements that influence communication and processing latencies. In
+addition, we present a model for estimating end-to-end latency while accounting for
+its stochastic character.
+4.2.1 Estimating Communication Latency
+The time it takes to process and return a response to a task request is the
+communication latency. More specifically, communication latency is caused by
+transmission latency and propagation latency. The transmission latency between
+any two points m and n (e.g., two fog systems in the fog federation) for task t of
+type i, denoted Θi(m, n), is defined as the sum of uplink transmission latency,
+denoted τu(m, n, i), and downlink transmission latency, denoted τd(m, n, i). That is,
+we have Θi(m, n) = τu(m, n, i) + τd(m, n, i). Let Iu(i) be the size of data payload (in
+bits), originally captured by a sensor, serving as input for task type i. Note that, for
+some sensors (e.g., cameras), there can be randomness in the size of captured data,
+in every sensor reading. Also, let Ru(m, n) represent the uplink bandwidth, through
+which the data is transmitted. T is the time required to transmit each data packet
+to the uplink channel (known as Transmission Time Intervals (TTI)). Then, the
+61
+
+uplink latency is calculated based on Equation 4.1.
+τu(m, n, i) = ⌈
+Iu(i)
+Ru(m, n)· T ⌉
+(4.1)
+Similarly, the downlink latency is defined as Equation 4.2.
+τd(m, n, i) = ⌈
+Id(i)
+Rd(m, n)· T ⌉
+(4.2)
+An orthogonal frequency-division multiplexing (OFDM) with total bandwidth W is
+divided equally into a set of k sub-channels (where k ∈ K) each with bandwidth w.
+Accordingly, the downlink bandwidth is defined based on Equation 4.3.
+Rd(m, n) = w·
+�
+k∈K
+ymnk log2(1 + γd(m, n, k))
+(4.3)
+where ymnk = 1, if sub-channel k is allocated, otherwise ymnk = 0. As the wireless
+communication is prone to noise and interference from other fog systems in the
+federation, the value of Rd(m, n) also depends on downlink signal to noise plus
+interference ratio (also known as SINR [93]). SINR is defined as the power of a
+particular signal divided by the sum of the interference power (from all the other
+interfering signals) along with the power of background noise. We note that, details
+of calculating uplink transmission latency (τu(m, n, i)) is similar to those for
+downlink.
+In fog federation, due to the vicinity, the propagation latency between fog
+systems is negligible. In contrast, the communication between fog systems and
+cloud datacenters is commonly achieved via satellite that introduces a substantial
+propagation latency [94]. The propagation latency, denoted τp, is calculated based
+62
+
+on Equation 4.4.
+τp = 2· d(n, st)
+Sl
+(4.4)
+In the Equation 4.4, d(n, st) is the distance between fog n to satellite st and Sl is
+the propagation speed in medium or link. To calculate propagation latency in the
+round trip time, the fraction value should be doubled. Once we know propagation
+latency, the overall communication latency, denoted dcomm, to access cloud
+datacenter is calculated based on Equation 4.5.
+dcomm = Θi(m, n) + τp
+(4.5)
+As we noticed, there are several factors that collectively form the communication
+latency with stochastic behavior. To capture this stochastic behavior, we treat
+communication latency as a random variable and model it using statistical
+distribution. That is, we represent the communication latency between any two
+points (e.g., two fog systems in the federation) using a probability density function
+(PDF), built upon historical communication information [46]. Based on the central
+limit theorem, communication latency can be modeled using Normal distribution.
+4.2.2 Estimating Computational Latency
+Once the load balancer assigns arriving task request t to a fog system, the
+task has to wait in the scheduling queues of the fog system before its execution. For
+a given task t of type i, denoted ti, its completion time (i.e., computational latency)
+is influenced by the waiting time in the queue (queuing latency), plus the task’s
+execution time (execution latency) on the machines of the assigned fog system.
+63
+
+Importantly, both of these factors are stochastic, as a result, the task completion
+time exhibits a stochastic behavior.
+The queuing latency of task ti is dependent on the number and execution
+times of tasks ahead of it in the fog system. The stochasticity in execution time can
+be due to different task types and characteristics of machines in different fog
+systems. Even the execution time of tasks from the same type on homogeneous
+machines of the same fog system is stochastic. This can be because of variations in
+the size of data to be processed and multi-tenancy of tasks in the fog system [95].
+Other factors, such as machine failure, can also be reasons for stochastic task
+execution time.
+To capture the stochasticity in computational latency, we consider the task
+completion time of each task type on each fog system as a random variable. Then,
+we model the computational latency using statistical distribution. That is, the
+computational latency is modeled using PDF, built upon historical completion time
+information of each task type on each fog system. Based on the central limit
+theorem, the computational latency of each task type on each fog system can be
+modeled using Normal distribution.
+4.2.3 Estimating End-to-End Latency
+Once we estimate the communication and computational latencies, their
+compound latency forms the end-to-end latency. More specifically, the compound
+latency can be obtained by convolving the PDF of communication latency with the
+PDF of the computational latency. For an arriving task ti to a load balancer, let Ni
+64
+
+be PDF of its communication latency to another fog system in the federation. Also,
+let Mi be PDF of the computational latency of ti on the other fog system. Then,
+the end-to-end latency for ti, denoted Ei, is calculated as Ei = Ni ⊛ Mi.
+4.3 Robust Resource Allocation in the Federated Fog Computing System
+The synopsis of the proposed resource allocation model in the federated fog
+computing system is demonstrated in Figure 4.1. The resource allocation model
+utilizes a load balancer module that is the main enabler of fog federation. Every fog
+system is equipped with a load balancer that, for each arriving task, it determines
+the appropriate fog system (either the receiving fog or to a neighboring one) where
+the task has the highest likelihood of completion before its deadline.
+The functionality of load balancer is particularly prominent to cope with the
+uncertainty exists in task arrivals (e.g., during disaster time) and make the fog
+system robust against it. The load balancer operates in immediate mode [96] and
+assigns arriving tasks to the appropriate fog system, immediately upon task arrival.
+The appropriateness is characterized based on the fog system that maximizes the
+probability of the task meeting its deadline (known as the probability of success).
+The probability of success for task ti with deadline δi can be calculated for each
+neighboring fog system, by leveraging the end-to-end latency distribution of
+executing task ti on that system. To avoid repetitive task reassignment and
+compound latency, we determine that once a task assignment decision is made, the
+task cannot be re-allocated.
+The resource allocation of each fog system leverages the historical
+65
+
+Figure 4.1. A Fog system with load balancer module that facilitates fog federation.
+Task requests generated by sensors are received by the load balancer module and are
+assigned to the fog system that maximizes the likelihood of success for the task.
+Batch
+Queue
+Scheduler
+Fog
+system 1
+Load
+Balancer
+Pressure
+Sensor
+Flow rate monitor
+Sensor
+H2S Gas Sensor
+Load
+Balancer
+Actuators
+Actuators
+ETT Matrix
+ETC Matrix
+ETC Matrix
+ETT Matrix
+Batch
+Queue
+Scheduler
+Fog
+system2
+information of computational and communication latencies to build PDF of their
+distributions. For that purpose, each load balancer maintains two matrices, namely
+Estimated Task Completion (ETC) [97] and Estimated Task Transfer (ETT), to
+keep track of computational and communication latencies for each task type on each
+neighboring fog system. Entry ETC(i, j) keeps the PDF of computational latency
+for task type i on fog system j. Similarly, entry ETT(i, j) keeps the PDF of
+communication latency for task type i to reach fog system j. The entries of ETC
+and ETT matrices are periodically updated in an offline manner and they do not
+interfere with the real-time operation of the load balancer.
+66
+
+Upon arriving task ti, load balancer of the receiving fog can calculate the
+end-to-end latency distribution of ti on any neighboring fog j, using ETC(i, j) and
+ETT(i, j). The end-to-end distribution can be used to obtain the probability of
+completing ti before its deadline, denoted pj(ti), on any of those fog systems. We
+have: pj(ti) = P(Ei ≤ δi). We note that the probability calculation for task ti on the
+receiving fog does not imply further communication latency. As such, for the
+receiving fog r we have: pr(ti) = P(Mi ≤ δi). In the next step, the fog system that
+provides the highest probability of success is chosen as a suitable destination to
+assign task ti. This implies that task ti is assigned to a neighboring fog system, only
+if even after considering the communication latency, the neighboring fog provides a
+higher probability of success.
+It is noteworthy that the probability of success on a neighboring fog can be
+higher than the receiving fog by a non-significant amount. In practice, a task should
+be assigned to a neighboring fog, only if the neighboring fog system offers a
+substantially higher probability of success. To understand if the difference between
+the probabilities is substantial, we leverage confidence intervals (CI) of the
+underlying end-to-end distributions, from which the probability of success for
+receiving and remote fogs are calculated. More specifically, we determine a
+neighboring fog offers a significantly higher probability of success for a given task,
+only if CI of end-to-end distribution of the neighboring fog does not overlap with
+the CI of end-to-end distribution of the receiving fog.
+The pseudo-code provided in Algorithm 1 expresses the robust task
+67
+
+Algorithm 1: Task assignment algorithm for load balancer.
+Input
+: Task ti; ETC and ETT matrices; G (set of neighboring fog
+systems)
+Output: Chosen fog j ∈ G to assign ti
+1 pr(ti) ← Probability of success on receiving fog r
+2 foreach fog system j ∈ G do
+3
+pj(ti) ← Probability of success on neighbor fog j
+4
+if pj(ti) > pr(ti) then
+5
+Add pj(ti) to P, as a potential fog for assignment
+6
+end
+7 end
+8 Sort elements of set P in descending order
+9 Consider receiving fog r as default assignment for ti
+10 foreach pj ∈ P do
+11
+if CI of Ej does not overlap with CI of Nr then
+12
+Choose fog j as destination and assign ti to it
+13
+Exit the loop
+14
+end
+15 end
+assignment heuristic that load balancer utilizes to take advantage of federated fog
+system and increase the robustness of the system. The heuristic is called Maximum
+Robustness (MR) and invoked upon arrival of a new task ti to the load balancer of a
+fog system. Based on the deadline of the arriving task (δi), the algorithm first
+calculates the probability of success for ti on the receiving fog and on its
+neighboring fog systems (Step 1-7 in Algorithm 1). Then, in Step 8, the calculated
+probabilities are sorted in the descending order. If the probability of success on the
+receiving fog is higher, then the task is allocated to the receiving fog system (Step
+9). Otherwise, CI of the end-to-end latency distribution for the neighbor with the
+highest probability of success is compared against receiving fog CI. If the CIs do not
+overlap, then task ti is assigned to the neighboring fog (Step 12). Otherwise, the
+68
+
+same procedure is performed for the rest of the neighbors of the receiving fog
+system. If there is no no-overlap neighbor found then, task ti is assigned to the
+receiving fog system (default assignment in Step 9).
+4.4 Performance Evaluation of Federated Fog
+We have used EdgeCloudSim [18], which is a discrete event simulator for
+performance evaluation. We simulate five fog systems (micro-datacenters) each one
+with eight cores and [1500, 2500] Million Instructions Per seconds (MIPs)
+computational capacity. Cores of each fog system are homogeneous: however,
+different fog systems have different MIPs that represents the heterogeneity across
+the fog systems. We also consider a cloud datacenter with 40,000 MIPs to process
+non-urgent tasks. Task within each fog is mapped in the first come first serve
+manner. The bandwidth to access cloud is based on satellite communication and set
+to 200 Mbps, and the propagation delay is 0.57 seconds [98].
+In each workload trial, generated to simulate load of a smart oil field, we
+consider half of the tasks represent urgent and the other half represent non-urgent
+tasks. Each task is of a certain type that represents its service type. In each
+workload trial, urgent tasks are instantiated from two different task types and
+non-urgent tasks are instantiated from two other task types. The execution time of
+each task instantiated from a certain type is sampled from a normal distribution,
+representing that particular task type. Each task is considered to be sequential
+(requires one core) and its execution time is simulated in the form of MIPs. Poisson
+distribution (with different means for different task types) is used to generate the
+69
+
+inter-arrival rate of the tasks and simulate task arrival during oversubscription
+periods. The number of tasks in each workload trial is varied to represent different
+oversubscription levels.
+Deadline for task i in a workload trial is generated as: δi =
+arri + β· avgi
+comp + α· avgi
+comm + ϵ, where arri is the task arrival time, avgi
+comp is
+average computational latency of the task type across fog systems, and avgi
+comm is
+average communication latency. β and α are coefficients, respectively, represent
+computation and communication uncertainties, and ϵ is the slack of other
+uncertainties exist in the system. We consider maintaining ETC and ETT matrices
+in every fog system and update them in every 10% of the workload execution. The
+entries of these matrices are considered as normal distribution as mentioned in the
+system model. For accuracy, each experiment was conducted 30 times and the mean
+and 95% confidence interval of the results are reported.
+4.4.1 Baseline Task Assignment Heuristics for Load Balancer
+Minimum Expected Completion Time (MECT): This heuristic [46] uses the
+ETC matrix to calculate the average expected completion time for the arriving task
+on each fog system and selects the fog system with the minimum expected
+completion time.
+Maximum Computation Certainty (MCC): This heuristic (used in [99])
+utilizes ETC matrix to calculate the difference between the task’s deadline and
+average completion time (called certainty). Then, the task is assigned to the fog
+that offers the highest certainty.
+70
+
+Edge Cloud (EC): This heuristic operates based on conventional fog
+computing model where no federation is recognized. Specifically, urgent tasks are
+assigned to the receiving fog and non-urgent tasks are assigned to the cloud
+datacenter.
+4.4.2 Experimental Results
+4.4.2.1 Analyzing the impact of oversubscription. The main metric to
+measure the robustness of an oversubscribed fog system in a smart oil field is the
+deadline miss rate of tasks. In this experiment, we study the performance of our
+system by increasing the number of tasks sensors generate (i.e., oversubscription
+level). Figure 4.2 shows the results of varying the number of arriving tasks (from
+1,500 to 7,500 in the horizontal axis) on deadline miss rate (vertical axis) when
+different task assignment heuristics is applied.
+Figure 4.2.
+The impact of increasing oversubscription level (number of arriving
+tasks) on deadline miss rate using different task assignment heuristics in the load
+balancer.
+1500
+3000
+4500
+6000
+7500
+Number of arriving tasks
+0
+20
+40
+60
+80
+100
+Tasks deadline miss rate (%)
+MR
+MECT
+MCC
+EC
+71
+
+In Figure 4.2, it is visible that as the number of tasks increases, the deadline
+miss rate grows for all of the heuristics. Under low oversubscription level (1,500
+tasks), MR, MECT, and MCC perform similarly. However, as the system gets more
+oversubscribed (4,500 tasks) the difference becomes substantial. With 7,500 tasks,
+MR offers around 16% lower deadline miss rate than MECT and MCC and
+approximately 21% better than EC. The reason is that MR captures end-to-end
+latency and proactively utilizes federation, only if it has a remarkable impact on the
+probability of success. Nonetheless, EC does not consider federation, and other
+baseline heuristics only consider the computational latency. We can conclude that
+considering end-to-end latency and capturing its underlying uncertainties can
+remarkably improve the robustness, particularly, when the system is oversubscribed
+(e.g., at a disaster time).
+4.4.2.2 Analyzing communication overhead of fog federation.
+Although we showed in the previous experiment that using federation improves
+system robustness, we are unaware of the communication overhead of task
+assignment in the federated environment. Therefore, in this experiment, we evaluate
+the communication latency imposed as a result of applying different task assignment
+heuristics. Specifically, we measure the mean communication latency overhead
+(vertical axis in Figure 4.3) induced to each task, for the various number of arriving
+tasks (horizontal axis in Figure 4.3).
+Figure 4.3 shows that MECT and MCC cause higher average communication
+latency. The reason is that these heuristics do not consider the communication
+72
+
+Figure 4.3. Mean communication latency overhead introduced to each task in fog
+federation by different heuristics.
+2000
+3000
+4000
+5000
+6000
+7000
+Number of arriving tasks
+3.05
+3.10
+3.15
+3.20
+3.25
+3.30
+3.35
+3.40
+Mean network latency overhead (s)
+1e
+2
+MECT
+MR
+MCC
+latency and aggressively redirect tasks to the same fog system, making that
+particular network link (between receiving fog and redirected fog system) congested.
+In contrast, MR that considers communication latency and redirect tasks more
+conservatively, only if the improvement in the probability of success is substantial.
+4.4.2.3 Analyzing average makespan of tasks. Different task
+assignment heuristics cause various computational latencies for the tasks. To
+understand the computational latency, we measure the average makespan of tasks,
+resulted by applying various task assignment heuristics.
+Figure 4.4 demonstrates that EC leads to the maximum average makespan
+time. The reason is that EC does not utilize federation, making the receiving fog
+system highly oversubscribed while other neighboring fog systems are underutilized.
+73
+
+Figure 4.4. Average makespan time(seconds) of tasks using various task assignment
+heuristics.
+2000
+3000
+4000
+5000
+6000
+7000
+Number of arriving tasks
+0
+10
+20
+30
+40
+50
+60
+70
+Average makespan time of tasks (s)
+MECT
+MR
+EC
+MCC
+Hence, average makespan time rapidly rises after the receiving fog is saturated with
+3,000 tasks. MECT and MCC do not consider the stochastic nature of task
+completion time; hence, they can potentially assign arriving tasks to one fog and
+oversubscribe that. As a result, the average makespan of tasks rises. In contrast,
+MR considers stochastic nature of end-to-end latency and calculates the probability
+of success on neighboring fog systems. Besides, it assigns tasks to a neighboring fog
+system, only if it offers a sufficiently higher probability of success. Hence, MR offers
+the lowest average makespan time than other heuristics.
+4.5 Summary
+In this chapter, we explored the usability of a fog federation for a smart
+Industry (Oil and Gas) in a disastrous situation. To support the computational
+demands in an emergency situation allocating various tasks in suitable fog system is
+74
+
+challenging due to heterogeneity across fog systems. Hence, maintaining the
+robustness of the system in terms of every real-time urgent tasks deadline can be
+difficult unless any efficient load balancing technique adopted by the system. To
+achieve that, we presented dynamic federation of fog computing systems, exist in
+nearby industries. Within the federated environment, we captured two sources of
+uncertainty, namely communication and computation, that are otherwise
+detrimental to the real-time services. The federation is achieved by a load-balancer
+module in each fog system that is aware of the end-to-end latency between fog
+systems and can capture the stochasticity in it. The load balancer leverages this
+awareness to find the fog system that can substantially improve the probability of
+success for each arriving task. Experimental results demonstrate that our proposed
+federated system can enhance the robustness of fog computing systems against
+uncertainties in arrival time, communication, and computational latencies. We
+concluded that the load balancer could be particularly useful (by up to 27%) for
+higher levels of oversubscription. Even for na¨ıve load balancing methods (MCC and
+MECT) in the federation, the performance improvement is approximately 13%.
+75
+
+Chapter 5:
+Adapting Remote Industry 4.0 Smart Micro-Service
+Applications to Federated Fog Computing Systems
+5.1 Overview
+The advancement of IoT technologies with smart applications drives the
+wheel of Industry 4.0 [71] revolution. Various smart sensors, actuators, and smart
+devices are deployed in different industries (e.g., manufacturing, food processing, oil
+& gas) to control the operational technology platform [100, 101]. Accordingly,
+sensors utilized in industrial operations frequently produce tons of data every day
+[102]. The oil and gas industry is an example of generating enormous amounts of
+sensor data and the necessity for processing close to the data source. For instance, a
+typical offshore oil rig produces 1 to 2 TB of data daily [103]. The majority of this
+data is fed to advanced computing applications (e.g., machine learning, report
+generation, automation) that can make smart latency-sensitive decisions to improve
+energy efficiency, production, and safety measures. For example, applications like
+workplace air quality estimation [104] for workers’ safety utilize environmental
+sensors that measure the quality (i.e., the existence of harmful particles in the air)
+of breathable air in the surrounding of the workplace. Hence, the air quality
+estimation must be fast to avoid potential occurrences.
+In the remote offshore industry, several services (e.g., data acquisition, alert
+generation, object tracking) are critical for complex or safety-related operations that
+need to be performed synchronously. The situation can worsen when any unwanted
+emergency brings many more computational activities completed within limited
+76
+
+time frames. In this case, our motivation is the smart Oil and Gas industry that has
+been facing various disasters and catastrophes (e.g., the deepwater horizon
+(2010)[19], usumacinta jack-up disaster (2007) [20], mumbai high north disaster
+(2005) [21], the ocean ranger disaster (1982) [105] ) due to complex fault intolerant
+industrial processes in exploration, drilling, and production operations. Therefore,
+remote offshore industries need latency-aware support [106] that can not be feasible
+with typical cloud data centers due to the remote locations of the industrial
+operation sites. The current solution utilizes fluctuating satellite communication
+[107] for sending data to mainland cloud data centers reducing the quality of service
+(QoS) and increasing the industrial safety risk. Hence, the high-level challenge is
+the lack of computational resources to support over-subscribed situations in remote
+industries.
+Figure 5.1. The structure of a microservice-based workflow is presented in a block
+diagram. Every microservice need to be processed to complete the fire safety appli-
+cation.
+fire
+detection
+input video
+noise
+removal
+feature
+extraction
+alert
+generation
+location
+mapping
+video
+preprocessing
+expansion
+prediction
+5.1.1 Smart Micro-Service Applications for Industry 4.0
+Industry 4.0 smart applications typically follow modern software architecture
+[16, 108] where various micro-services [17] need to be executed in order. However,
+77
+
+micro-services can be separately deployed using an automated deployment process,
+require the least amount of administration, can be developed using a variety of
+programming languages and data storage techniques, and can each be independently
+updated, changed, and scaled. Thus, we concentrate on micro-services applications
+frequently used in remote industries. For instance, as depicted in figure 5.1, a “fire
+safety” application can include micro-services for capturing video surveillance data,
+pre-processing captured video, noise removal, feature extraction, fire detection,
+location mapping, alert generation, and expansion prediction. In contrast, many
+industries have previously deployed legacy applications [109] with inflexible software
+architecture. Hence, the execution platform should support both monolithic legacy
+applications and modern micro-services to ensure industrial safety and fault-tolerant
+operations. However, modern industry 4.0 applications are comprised of
+micro-services that pose new challenges for the execution platforms. Under this
+arrangement, an application‘s latency constraint is subject to the completion time of
+the micro-services defined by the underlying software architecture. Therefore, to
+develop a robust execution platform for industry 4.0, system architects need to
+understand the software architecture of the receiving applications.
+5.1.2 Federated Fog Systems for Industry 4.0 Micro-service
+Applications
+The emerging industrial IoT and advancements in communication technology
+have brought computational resources near the data sources and end devices. For
+instance, nowadays, fog computing systems [110] in remote industries typically
+78
+
+Figure 5.2. Offshore oil and gas industry has the fog federation infrastructure that
+can support smart microservice-based applications.
+G1
+G2
+G3
+Monolithic
+Seismic
+Analysis
+Fire
+Detection
+Fed. Fog Platform
+Wireless Fed. Fog
+Industry 4.0 Applications
+AP
+Fog
+Oil Spill
+Oil Spill
+Detection
+execute the industrial computational process to enable smooth production and
+workplace safety. However, as depicted in figure 5.2, a federated fog platform can be
+conceptualized from chapter 4 that can form by connecting through wireless
+gateways denoted as Gi. Hence, various applications with heterogeneous latency
+constraints require computational support from federated fog computing systems.
+Accordingly, the federation should be cognizant of communications and computing
+uncertainties, as well as the applications’ software structure and latency
+requirements. Thus, an application execution plan needs to perform for monolithic
+and micro-service software structures, considering the stochastic execution times
+and uncertainties that derive from the execution platform and communication
+technology.
+Consequently, in our previous work [38] presented in chapter 4, the resource
+79
+
+allocation methods are explored intensively for monolithic applications.
+Furthermore, considering a complex operational process performed by various
+micro-services, one of the main problems is ensuring the completion of the whole
+application workflow within the time limit known as the deadline. Hence, it is
+crucial to know the optimal point to partition the application workflow so that it
+can be completed on time. Accordingly, the question that needs to be addressed is
+“How to distribute Industry 4.0 applications (e.g., monolithic, micro-service) across
+fog federation so that the application workflow can be completed within the given
+time frame?”. Hence, from a system administration perspective executing the smart
+micro-service applications raises two more questions, and they are 1) How to
+partition the micro-service workflows so that its deadline constraint can be realized?
+2) How do we allocate partitioned micro-services across fog federation so that it has
+the highest likelihood of completing on time?
+Our prior work [38] suggests that federating nearby resources is one solution
+to the resource restrictions (i.e., oversubscribe) encountered by edge computing
+systems in distant sectors like Oil and Gas. Furthermore, we explore new challenges
+imposed by smart software architecture, a.k.a micro-service workflows. Therefore, to
+address the difficulties faced by the offshore O&G sector at large, we propose a
+resource provisioning method for Industry 4.0 applications across the federated fog
+system that is aware of both the software architecture and the underlying execution
+platform’s structure. More so, the solution maintains the deadline limitations of the
+micro-service workflow, which in turn makes the execution platform more reliable.
+80
+
+As a result, our approach consists of two stages: understanding the software
+architecture of the receiving applications and allocating computational resources for
+the successful completion of these applications. Therefore, the following are the
+contribution of this research:
+• Proposing a probabilistic partitioning method that is aware of the underlying
+software architecture of Industry 4.0 applications.
+• Proposing a statistical resource allocation heuristic considering the time
+constraints of the application.
+• Providing extensive evaluation of partitioning technique along with resource
+allocation across fog federation.
+The suggested solution can serve as a foundation upon which system
+architects or industry-focused research associates might construct more elaborate
+solutions referring to distant offshore sectors at peak demand. In addition, the
+solution is compatible with monolithic legacy applications, which may aid
+conventional industries in transitioning to and adapting to the changes brought
+about by Industry 4.0.
+5.2 Partitioning Method for Micro-service Application Workflow
+Maintaining latency constraints of a smart application comprising multiple
+micro-services depends on underlying software architecture and mapping of
+computational resources. For example, executing a micro-service application into a
+single fog system may not be possible or may not maintain its deadline constraint.
+81
+
+On the other hand, a monolithic application can not be partitioned and can be
+considered an application with a single micro-service. For micro-service software
+architecture, partitioning the application into multiple partitions and allocating
+them across fog federation can increase the likelihood of its completion within the
+latency constraint. Furthermore, allocating appropriate computational resources to
+the partitioned micro-services also ensures the completion of the whole application
+workflow.
+Figure 5.3. The flowchart of the workflow partitioning method. The partitioned
+workflow is sent to the resource allocation module, which is denoted as the end box
+for this flow chart.
+estimate the chance of on
+time completion for workflow
+on the local fog
+no
+yes
+submit to the
+resource allocation
+module
+partition workflow into
+two sub-graphs and
+estimate the chance of
+sucess for each partition ,
+across fog federation
+rollback to
+yes
+no
+yes
+no
+
+OR
+OR
+The main goal of the partitioning method is to partition the micro-service
+application in a way such that the application can meet its deadline. Hence, we
+82
+
+considered an application having micro-service architecture as a set of micro-services
+that are connected together in some manner to form a graph G = (V, E), where the
+set of vertices V = (m1, m2, ..mn) denotes the micro-services and edge
+e(mi, mj) ∈ E represents the communication between micro-service mi and mj. As
+the first step of the partitioning method, we consider executing the whole
+micro-service workflow into the local fog system without partitioning. As such, the
+partitioning method estimates the chance of on-time completion for workflow w on
+the local fog, which is the first processing box of flowchart 5.3. To estimate the
+deadline for the whole application workflow w, we perform a summation of the
+deadlines for the micro-services that can be defined as δw = mδ
+1 + mδ
+2 + ... + mδ
+n.
+Hence, each micro-service has a deadline (mδ
+i) known in advance to the load
+balancer. Furthermore, for each micro-service type, we have computational latency
+distribution (md
+i ) that represents the execution times across fog federation. Hence,
+to estimate the probability of success for the entire w, we convolve the
+computational latency distributions of the application’s micro-services that can be
+defined as
+Dw = md
+1 ⊛ md
+2 ⊛ ....md
+n
+(5.1)
+Finally, using the convolved distribution DA, we measure the probability of success
+as follows,
+P(w) = P(DA ≤ δA)
+(5.2)
+The output of equation 5.2 is compared with a conditional variable α as depicted in
+first condition of flowchart 5.3. We choose an average success rate (i.e., 50%) for α
+83
+
+as our experimental evaluation scenario. When the likelihood of completing the
+workflow is less than α, the partitioning service takes place using the min-cut [111]
+graph partitioning algorithm, which is the partition workflow w into two sub-graphs
+i and j process box in figure 5.3. Hence, considering the flow of actions within the
+application, we employ one of the widely utilized graph theorems, max-flow
+min-cute [112] in our proposed solution.
+Due to finding the minimum number of partitions which is an np-hard
+problem, we developed our customized solution for Industry 4.0 micro-service
+applications. Thus, the partitions resulting from the min-cut are estimated for the
+chance of success across fog federation using equation 5.1 and 5.2 in the third
+process box of flowchart 5.3. As we utilize probability to determine the partitions,
+we named the proposed partitioning method as Probabilistic Paritioning (ProPart).
+If the new sub-graphs completion success is less than the prior success rate (2nd
+condition of the flowchart), we consider earlier partitions as optimal (rollback to w
+process box). Accordingly, the resource allocation methods for those partitions are
+started, which is the end process box of the flowchart.
+On the other hand, if the latest partition’s chance of on-time completion is
+greater than the prior success rate, then we evaluate each partition’s micro-service
+architecture, which is the third condition of the flowchart. If the condition fails
+(“no” line from the third condition), the partitioning process is halted for partitions
+with only one micro-service, and the resource allocation service takes place. In
+contrast, for partitions with more than one micro-service (“yes” line from the third
+84
+
+condition), the partitioning process is repeated until each has a single micro-service.
+Therefore, the partitioning method is a repeated process where the output is the
+optimal number of partitions that are submitted to the resource allocation module.
+5.3 Resource Allocation Method for Partitioned Micro-service Applica-
+tions Across Fog Federation
+Resource allocation occurs when the partitioning is completed with an
+optimum number of partitions. The partitioning method returns the whole
+application as one part to the resource allocation module for a monolithic
+application that is considered a single micro-service workflow. The efficacy of the
+resource allocation approach is significant in dealing with the unpredictability that
+occurs in applications’ arrival (e.g., during disaster time) and making the fog
+system resilient. The resource allocation module runs in immediate mode [96] and
+quickly allocates incoming applications or micro-service partitions to the relevant
+fog system. The relevance is defined by the fog system, which increases the
+likelihood of the micro-services achieving their deadlines (a.k.a the probability of
+success). Hence, the likelihood of on-time completion for a micro-service mi on a
+particular fog system can be estimated using the historical end-to-end latency
+distribution. Furthermore, to minimize frequent application reassignment and
+compound delay, we have decided that the micro-service cannot be relocated once
+an assignment choice is made.
+Each fog system’s resource allocation module uses prior data on
+computational and communication latencies of various micro-service types across
+85
+
+fog federation to generate PDFs of their distributions. To that end, each load
+balancer keeps track of computational and communication latencies for each
+micro-service type on each nearby fog using two matrices: Estimated Task
+Completion (ETC) [97] and Estimated Task Transfer (ETT). The PDF of
+computational delay for micro-service type i on fog system j is stored in entry
+ETC(i, j) that is previously used in the partitioning method. Similarly, the item
+ETT(i, j) maintains the PDF of communication delay for micro-service type i to
+reach fog system j. Hence, the resource allocation module is aware of
+communication latencies as well, whereas the partitioning method is only aware of
+computation latencies. The entries of the ETC and ETT matrices are regularly
+updated offline and do not interfere with the load balancer’s real-time functionality.
+The resource allocation module can compute the end-to-end latency
+distribution across fog federation upon the arrival of a partition of micro-services
+using convolution of ETC(i, j) and ETT(i, j). On any fog system j, the end-to-end
+distribution can be used to calculate the probability of completing each
+micro-service partition mpi before its deadline, denoted pj(mpi). Hence, we estimate
+the deadline δi for the given partition mpi by adding each micro-service’s deadline
+within that partition. Then we convolve each micro-service’s computational latency
+distribution dcomp with communication distribution dcomm to measure the
+completion time ei in a particular fog system. To estimate the completion time of
+the partition mpi denoted as Ei, we convolve the completion time distribution for
+each micro-service within a given partition. We have: pj(mpi) = P(Ei ≤ δi). We see
+86
+
+that the probability of mpi on the receiving fog does not entail any additional
+communication delay. Consequently, for receiving fog system, we don’t convolve
+communication latency distribution to completion time estimation. In the
+subsequent stage, the fog system with the greatest likelihood of completion is
+selected as a viable destination to allocate mpi. This assignment entails that the
+micro-service partition mpi is only given to an adjacent fog system if the
+surrounding fog offers a greater chance of on-time completion after accounting for
+communication delay.
+It‘s important to note that the success rate on a neighboring fog could be
+greater than on the receiving fog. This is because assigning a micro-service partition
+to a fog system in close proximity should only be done if doing so significantly
+increases the likelihood of the partition being completed successfully. Hence, we use
+confidence intervals (CI) of the underlying end-to-end distributions, from which we
+derive the likelihood of success for receiving and distant fogs, to assess the
+significance of the discrepancy. In particular, we find that the CI of the end-to-end
+distribution of the nearby fog does not overlap with the CI of the receiving fog only
+if the neighboring fog gives a much better likelihood of success for a given
+micro-service partition.
+The pseudo-code provided in Algorithm 2 expresses the resource allocation
+method that the load balancer utilizes to take advantage of the federated fog system
+and increase the system’s robustness. The method is called Maximum Robustness
+(MR) and is invoked when the partitioning method sends micro-service partitions
+87
+
+Algorithm 2: Resource allocation algorithm
+Input
+: Micro-service partition set M; ETC and ETT matrices; G (set of
+neighboring fog systems)
+Output: Chosen fog f ∈ G to assign micro-service partitions mpn ∈ M
+1 foreach micro-service partition mpi ∈ M do
+2
+pr(mpi) ← Probability of success on receiving fog r
+3
+foreach fog system f ∈ G do
+4
+pf(mpi) ← Probability of success on neighbor fog f
+5
+if pf(mpi) > pr(mpi) then
+6
+Add pf(mpi) to P, as a potential fog for assignment
+7
+end
+8
+end
+9
+Sort elements of set P in descending order
+10
+Consider receiving fog r as default assignment for mpi
+11
+foreach pf ∈ P do
+12
+if CI of Ef does not overlap with CI of Nr then
+13
+Choose fog f as destination and assign mpi to it
+14
+Exit the loop
+15
+end
+16
+end
+17 end
+M for resource allocation. At first, the micro-service partitions are separated for
+further processing. Then based on the deadline (δi) of each micro-service partition
+mpi, the algorithm calculates the probability of success on the receiving fog and on
+its neighboring fog systems (Step 2-8 in Algorithm 2). Next, step 9 sorts the
+calculated probabilities in descending order. If the probability of success on the
+receiving fog is higher, then the micro-service partition mpi is considered for
+allocation to the receiving fog system (Step 10). Otherwise, the CI of the end-to-end
+latency distribution for the neighbor with the highest probability of success is
+compared against receiving fog’s CI. If the CIs do not overlap, then partition mpi is
+assigned to the neighboring fog (Step 13). Otherwise, the same procedure is
+88
+
+performed for the rest of the neighbors of the receiving fog system. If no
+non-overlap neighbor is found, then partition mpi is assigned to the receiving fog
+system (default assignment in Step 10).
+5.4 Performance Evaluation of Software Architecture-Aware Federated
+Fog Systems
+The partitioning and resource allocation components of the proposed
+technique occur one after the other within the load balancer module. As a
+consequence, we evaluate each component separately in various experiments. The
+recommended partitioning approach is compared to different baselines in the first
+experiment to examine how the deadline constraints for workflow applications based
+on microservices have improved. Following partitioning, the allocation of resources
+to those partitions must be evaluated. We execute the second category of trials to
+assess the system’s efficacy, which compares our proposed resource allocation
+techniques to alternative baselines. The third experiment is then performed to
+determine how scaling the fog federation impacts the suggested solution. Finally, for
+microservice and monolithic applications, we examine the computational latencies
+resulting from partitioning and resource allocation approaches. The experiments are
+thoroughly described in the subsections that follow.
+5.4.1 Comparison of Micro-service Workflow Partitioning Methods
+In this experiment, we use the suggested partitioning technique
+(Probabilistic Partitioning, defined as ProPart) for accepting microservice-based
+workflow applications and compare it to the other two baselines (Min-cut, Least
+89
+
+data transfer, for example). In this experiment, we increase the number of
+microservice applications submitted to the system to generate oversubscribed
+conditions and record the applications’ deadline meet rate in each round of request
+submission, shown as a bar chart in figure 5.4. The figure’s x-axis indicates the
+number of microservice-based applications received by the system, while the y-axis
+reflects the rate at which application deadlines have been met.
+Figure 5.4. Comparison of the partitioning techniques in terms of workflow deadline
+meet rate while utilizing proposed probabilistic partitioning technique. The x-axis
+represents the increasing number of arriving workflow execution requests, whereas the
+y-axis represents the workflow deadline meet rate.
+100
+200
+300
+400
+Number of Receiving Microservice-based Workflows
+0
+20
+40
+60
+80
+100
+Workflow deadline meet rate (%)
+ProPart
+Min-Cut
+Least data transfer
+The results of this experiment, shown in figure 5.4, indicate that the deadline
+meet rate decreases as the number of workflow requests to the system increases for
+all partitioning techniques. However, in every round of submissions, ProPart
+surpasses other baselines. For less overloaded scenarios (e.g., 100 & 200 requests),
+the performance gap between the least efficient strategy (least data transfer) and
+the suggested technique ProPart is greater than for completely oversubscribed
+90
+
+conditions (e.g., 300 & 400 requests). The primary reason for ProPart’s superior
+performance is its statistical assessment of each partition’s success likelihood. For
+up to 200 application requests, the min-cut strategy performed better than the least
+data transfer. In contrast, in totally overloaded scenarios, the least data transfer
+performed marginally better than the min-cut because it considers the connection
+that generates the least output data for splitting. Min-cut, in contrast, examines
+the smallest communication channel when partitioning. Finally, due to the repeated
+probabilistic calculation of deadline fulfillment for all microservices, ProPart
+performed better in totally oversubscribed conditions.
+5.4.2 Comparison of Resource Allocation Methods
+The load balancer in every fog system utilizes a resource allocation technique
+after the partitioning steps for microservice-based workflow applications. In
+contrast, for monolithic applications, as soon as load balancer receives a request, it
+performs resource allocation using probabilistic estimation across fog federation. As
+such, to compare the proposed resource allocation technique, we performed the
+following experiments with three different resource allocation methods for
+microservice and monolithic applications respectively.
+Microservice-based Workflow Applications:
+Similar to the previous
+experiment, the number of receiving microservice-based application is incremented
+to create more oversubscribed situations(i.e., the x-axis of the graph). To visualize
+the performance of the resource allocation techniques, the deadline meet rates of
+receiving applications are captured and plotted in figure 5.5.
+91
+
+Figure 5.5. Comparison of resource allocation techniques while utilizing proposed
+workflow partitioning technique for microservice-based workflow applications.
+100
+200
+300
+400
+Number of Receiving Microservice-based Workflows
+0
+20
+40
+60
+80
+100
+Workflow deadline meet rate (%)
+MR
+MECT
+MCC
+The result represents a downward trend for all the resource allocation
+techniques with increasing oversubscribed situations. Hence, it is visible that the
+proposed resource allocation technique, MR outperforms other baselines in every
+oversubscribed situation. This is because MR is aware of uncertainty in
+computation and communication of receiving microservices. In contrast, MECT is
+only aware of computation, and Certainty utilizes deadlines in its resource
+allocation technique which lacks communication information.
+Monolithic Independent Applications:
+In this experiment, we investigate the
+performance of our system by increasing the number of monolithic applications
+generated by sensors (i.e., the oversubscription level). Figure 5.6 shows the effects
+of altering the number of incoming applications (from 400 to 1000 on the horizontal
+axis) on the deadline meet rate (vertical axis) when various resource allocation
+heuristics are employed.
+92
+
+Figure 5.6. Comparison of resource allocation techniques for monolithic applica-
+tions. The proposed resource allocation technique MR outperforms other baselines
+in every application arrival trial.
+400
+600
+800
+1000
+Number of Receiving Monolithic Tasks
+0
+20
+40
+60
+80
+100
+120
+Tasks deadline meet rate (%)
+MR
+MECT
+MCC
+In figure 5.6, it is visible that as the number of applications increases, the
+deadline meets rate decreases for all of the heuristics. Under low oversubscription
+levels (400 tasks), MR, MECT, and MCC perform similarly. However, the difference
+becomes substantial as the system gets more oversubscribed (800 applications).
+With 1000 applications, MR offers around 18-20% higher deadline-meeting rates
+than MECT and MCC. The reason is that MR captures end-to-end latency and
+proactively utilizes federation only if it remarkably impacts the probability of
+success. Nonetheless, other baseline heuristics only consider computational latency.
+Therefore, we can conclude that for monolithic applications considering end-to-end
+latency and capturing its underlying uncertainties can remarkably improve the
+robustness, particularly when the system is oversubscribed (e.g., at a disaster time).
+5.4.3 Fog Federation Scaling Impact
+Fog federation in remote offshore areas can be scaled up in times of
+93
+
+emergencies by utilizing mobile fog systems mounted on a boat or other vehicles. In
+contrast, a scaled-down fog federation can decrease the system’s robustness. Hence,
+to understand the impact of federation scaling over the proposed solution, we
+increase the fog federation degree that represents the number of neighbors and
+captures the deadline meet rates of the received applications within the increasing
+oversubscribed situations. The result of this experiment is presented in figure 5.7.
+In addition, we performed a similar experiment for monolithic applications, where
+we fixed the number of receiving tasks and incremented the fog federation degree.
+The result for monolithic applications is presented in figure 5.8.
+Figure 5.7.
+Impact of scaling the fog federation for proposed partitioning and
+resource allocation techniques in increasing oversubscribed situations considering mi-
+croservice applications.
+The degree represents the number of neighbors each fog
+system has for executing the Industry 4.0 applications.
+100
+200
+300
+400
+Number of Receiving Microservice-based Workflows
+0
+20
+40
+60
+80
+100
+Workflow deadline meet rate (%)
+Fog Fed. Degree: 2
+Fog Fed. Degree: 3
+Fog Fed. Degree: 4
+Fog Fed. Degree: 1
+Microservice-based Application:
+The result shown in 5.7 demonstrates the
+advantages of scaling up the fog federation. As a result, in any overcrowded
+circumstance, the federation with the greatest number of neighbors (i.e., fog fed.
+degree 4) excels. Despite this, considerable performance improvements are seen in
+94
+
+most oversubscribed circumstances (i.e., a system processing 400 microservice-based
+workflows). For less overloaded scenarios (for example, a system with 100 - 200
+receiving microservice-based workflows), the performance difference for minor
+scale-up fog federation is negligible. This is due to the suggested method,
+particularly the partitioning technique, attempting to put the whole application into
+a fog system rather than partitioning and distributing them around the federation
+in less oversubscribed conditions. As a result, the performance increase is
+substantial in the fully oversubscribed scenario with the most neighbors.
+Figure 5.8. Impact of scaling the fog federation for proposed resource allocation
+techniques on monolithic applications. The degree represents the number of neighbors
+each fog system has for executing the Industry 4.0 applications.
+Degree 1
+Degree 2
+Degree 3
+Degree 4
+Fog Federation Scaling
+0
+20
+40
+60
+80
+100
+120
+Tasks deadline meet rate (%)
+MR
+MECT
+MCC
+Monolithic Applications:
+In this experiment, we compare the resource
+allocation techniques for monolithic applications while scaling up the fog federation.
+Similar to microservice-based workflows, the monolithic applications positively
+impact federation scaling, which is visible from figure 5.8. The result reflected a
+significant performance improvement when the federation scaled up from degree 1 to
+95
+
+degree 2 for all heuristics. Hence, degree 1 defines only one neighbor, and the
+federation is formed with two fog systems. Therefore, none of the heuristics
+performed well. Even though the proposed method MR, performed better than
+baselines. Whereas for the highest degree of the federation, the proposed MR
+heuristic performed approximately 18-20% better than MECT and MCC. However,
+for all of the federation scales up, the proposed MR heuristic outperforms others.
+The main reason is that MR, efficiently utilizes fog federation resources, considering
+the communication and computation latencies to complete every monolithic
+application on time.
+5.5 Summary
+The advancement in software and hardware stack has brought the industrial
+revolution, Industry 4.0, that changed many legacy system architectures and
+imposed latency constraints. As such, complex industrial processes are adopting
+smart solutions every day. Hence, computation near the data source supports smart
+microservice-based solutions that significantly face resource scarcity and latency
+constraints challenges. Especially in remote offshore Industries (e.g., Oil and Gas,
+mining ), the latency issue can be critical for complex fault-intolerant industrial
+processes (e.g., hydrocarbon exploration, drilling). Moreover, in emergency
+situations, the computational execution platform gets oversubscribed with various
+types of microservices. To overcome challenges enforced by the smart
+microservice-based solutions, a robust task allocation scheme proposed in this
+research work that is aware of the software architecture of the solution as well as
+96
+
+uncertainties imposed by fog federation. Hence, the proposed solution works on two
+levels within a load balancer module that exists in every fog system of the
+federation. The first level considers the software architecture of the receiving
+application and performs partitioning if necessary, utilizing the probabilistic success
+rate to complete the applications. Then in the second level, the received
+applications (e.g., monolithic applications or partitioned microservices) are mapped
+across fog federation, considering the computation and communication constraints.
+The evaluation results reflect the benefits of using the proposed solution in
+oversubscribed situations that are approximately 15∼20% better than the baseline
+partitioning and resource allocation techniques. In the future, we plan to
+incorporate an ML-based resource provisioning method to improve the robustness of
+the federated fog system.
+97
+
+Chapter 6:
+Data Security & Privacy Aspects in Federated Fog
+Computing System
+6.1 Overview
+The rise of Industry 4.0 [113] elevates the utilization of IoT devices (e.g.,
+sensors, actuators) and fog computing for developing deep neural network (DNN)
+applications in various industrial sectors (e.g., smart oil field, smart farms, smart
+factory). The DNN-based applications mainly backed up by ML network models
+that are supposed to train with huge amount of data for achieving relatively high
+accuracy. Although the training data could be privacy preserving (i.e., sensitive to
+any company), and sometimes data acquisition (e.g., Satellite image data, high
+resolution camera data) is expensive, and time consuming. The expense of
+developing these DNN-based applications could also increase with data transfer to
+cloud datacenter using internet for training operation. Hence fog federation (formed
+by multiple private companies fog systems) can be a potential candidate for
+supporting computational demand of ML-model training where data security and
+privacy of the participant private fog systems in the federation need to be addressed
+to efficient ML training. In this case, federated Learning (FL) techniques [66] that
+brings ML-model to participant end user without leaving the data their source
+device, can be applied to overcome the privacy constrains of the fog systems owned
+by private companies. Although the privacy preservation constraints are mitigated
+by FL as depicted in Figure 6.1, it can impose some new challenges for the ML
+models training operation. Here, the problem is that data are coming from various
+98
+
+sources, and it is feasible to assume data distribution tends to be non-identical and
+independent distribution (non-IID). As such, lack of any priority class (i.e., consider
+oil spill class in oil spill detection problem) that is termed as class imbalance [114]
+can reduce the performance of the global DNN model in a FL setup as presented in
+Figure 6.1. Hence, ignoring the class imbalance issue, current federated learning
+methods [66] are providing less robust DNN model for oil spill detection. In this
+case, an object detection model can show misleading high accuracy for all other
+classes while providing low performance for the desired class (oil spill).
+Figure 6.1. A federated learning setup in fog federation. Multiple company share
+their fog systems to train oil spill detection DNN model where data security is pre-
+served by federated learning.
+Global DNN
+model
+Local DNN
+model 1
+Local DNN
+model 2
+Local data
+Fog computing
+(micro-data center)
+Aggregator
+Fog System
+5G
+communication
+Oil spill
+Oil spill
+Drone capturing
+images
+Federated learning is a special branch of distributed machine learning where
+the global model needs to be converged at a constraint rate. Hence, the convergence
+of FL mainly depends on the local workers’ aggregation that affects the global
+model’s performance. Among two types of federated learning (synchronize and
+99
+
+0asynchronous), we propose to utilize the synchronize FL method as it is a proven
+model, especially for class imbalance issue [115, 116]. As such to overcome the
+challenges of FL for oil spill detection, we have adopted an objective function (loss
+function) to train the local model considering the class imbalance problem.
+Considering the priority class (i.e., oil spill), we introduce a weight for each
+participating worker that intensifies or attenuates its influence over the global
+model. The relevant worker selection based on the worker weights is verified by
+empirical evaluation in section 6.5
+. Finally, a dynamic threshold mechanism has been
+proposed to select relevant workers efficiently considering the global model’s
+performance and fast convergence.
+6.2 Problem Formulation for Federated Learning
+The oil spill detection problem can be well defined in semantic segmentation
+domain of deep learning where various classes are identified in pixel level from
+original source image. In oil spill detection training various classes (e.g., oil-spill,
+look-alike, land, ship, sea-surface) are found in real world satellite image data set
+[77]. Here, each class is labeled as an individual color in ground truth image. For
+training a deep neural network (DNN) model (e.g., Unet) with federated learning
+settings a set of workers (i.e., fog systems of a fog federation) S = 1, 2, 3, ..., S are
+considered with its own local data set DL where L ∈ S with nL samples. Here, D =
+�
+L∈S
+DL is the full training data set. The total size of these workers’ data set for a
+random set of workers S′ is N(S′) =
+�
+L∈S′
+nL. The objective loss function over a
+model m and a sample z can be denoted as L(m, z).
+100
+
+Then in most prior FL work, the goal is to solve the following
+min
+w
+f(w) =
+M
+�
+m=1
+pmFm(w)
+where pm = nm
+D is the fraction of the total data worker, and thus,
+�
+m
+pm = 1. The
+local objective Fm is typically defined by the empirical loss over local data,
+Fm(w) = 1
+nm
+nm
+�
+j=1
+Lj(m, z). Here, w is the model parameter that used for predicting
+loss over a sample data, and the goal is to find the optimal w for which the loss
+should be minimized. Accordingly, we focus on utilizing a loss function that consider
+class imbalance problem in local data samples, and select a set of client worker’s
+(fog system) models to aggregate that have certain level of accuracy (i.e., mean
+intersection over union (mIoU) for semantic segmentation) to ensure the robustness
+of the global model. Hence, our new objective for this work would be as following:
+min
+w
+f(w) =
+M
+�
+m=1
+pmFm(w)
+s.t.
+mIoU(m) >= γ,
+θ > 1
+Where, γ is a dynamic threshold (initial value set to 50% or 0.5) for checking the
+local trained model’s mIoU with auxiliary test data, and θ is the user defined
+worker’s weight with respect to oil spill class. Both of this parameters are used to
+select the relevant worker’s model for aggregation into the global model that ensure
+the robustness, and consistency of the convergence for the aggregated model.
+101
+
+Table 6.1. Pixel distribution for each of the class in oil spill detection data set
+Class
+Pixels
+Sea Surface
+797.7 M
+Oil Spill
+9.1 M
+Look-alike
+50.4 M
+Ship
+0.3 M
+Land
+45.7 M
+6.3 Federated Learning to Mitigate the Class Imbalance
+In a typical federated learning setup, the server (e.g., fog device, cloud)
+stores a global ML model for training with local data of the participating workers
+(i.e., fog systems). We use one of the popular semantic segmentation DNN models
+named as Unet [117] model for oil spill detection in the FL setup. The figure 6.2
+represents a pictorial view of our solution. At first, some fog nodes agree to
+participate in the FL training, and they download the global model (i.e., Unet)
+from the fog server presented in step 1 of figure 6.2. Then, downloaded ML models
+are trained with their local data in step 2. Hence, we utilize tversky loss function for
+local training that work efficiently for class imbalance issue proven by the research
+community [118, 119, 120, 121, 122]. In step 3, ML models are checked for relevant
+worker model selection. Finally, in step 4, selected workers updated models are
+aggregated (new model), and the previous global model is updated accordingly.
+This whole process is considered a federated round. The updated model is again
+downloaded by participating worker for the next federated round, and the training
+continues. The proposed solution is presented in algorithm 3.
+Usually, the aggregator fog server provides the global model and aggregates
+102
+
+Figure 6.2.
+Federated learning training considering class imbalance and global
+convergence. Tversky loss is used in the training considering class imbalance. After
+training of each epoch, mean intersection over union (mIoU) is checked with a dynamic
+threshold for global convergence.
+GPU
+Global DNN model
+Aggregator fog server
+Model training
+Local data
+Download global model
+Check mIoU
+& worker
+weight
+Check mIoU
+& worker
+weight
+Check mIoU
+& worker
+weight
+Fog worker 1
+Fog worker 2
+Fog worker 3
+Download global model
+Updated model
+send
+Model aggregartion
+Update global model
+Tversky loss
+Tversky loss
+Tversky loss
+Relevant worker
+selection
+Local Level
+Global Level
+Step 1
+Step 2
+Step 3
+Step 4
+the updates sent by the worker fogs. The FedBal algorithm starts with initializing
+global model mg, relevant worker list, rf, and setting the threshold, th value to 0.50.
+After that, the federated round continues as a for loop that is presented with
+variable f. Then m number of workers are selected from K participating worker,
+and assigned to selected worker list, St for training (“ClientUpdate” function) with
+their local data in the second for loop of the algorithm 3. Finally, relevant workers
+are selected using function “selectionCriteria”, and aggregate into the new global
+model, mg using “averageModel” function. The “ClientUpdate” function performs
+the training with the tversky loss function and defined number of epochs to reduce
+the class imbalance at the local level. The proposed solution’s global level is
+103
+
+Algorithm 3: The K workers are indexed by k, C is the initial worker
+selection percentage
+1 Initialize global model, mg, test data, Dtest, relevant Worker List, rf;
+2 Set threshold, th = 0.50;
+3 for each federated round f = 1,2,... do
+4
+m ← max(C.K, 1);
+5
+St ← (random set of m workers);
+6
+for each worker k ∈ St in parallel do
+7
+ClientUpdate(k, mg);
+8
+rf = selectionCriteria(St, Dtest, th);
+9
+mg = averageModels(mg, rf);
+triggered in the “selectionCriteria” function, where trained worker models are
+evaluated according to their weight, θ, and mIoU value. The dynamic threshold
+mechanism also takes place in the “selectionCriteria” function to ensure the
+robustness of the global model. In this way, in every federated round,f, the global
+model updates and converges to a model robust against class imbalance with
+guaranteeing performance for our priority class, oil spill.
+6.4 Experimental Setup
+The Federated learning setup can be synthesized by PyTorch’s one of the
+popular library pysyft [123], and TensorFlow’s federated learning library named as
+tff [124]. Due to pysyft’s customization capability, we have selected pysyft as our
+development library. The oil spill detection is considered a semantic segmentation
+problem that typically uses real-world SAR image data sets (in this work, the data
+set is collected from MKLab [77], a research institute in Greece) for training a DNN
+model. To execute the DNN training operation, we used Google’s Colab [125]
+run-time environment that provides a GPU platform with a high-speed ram of size
+104
+
+24 GB with storage of 128 GB.
+The Colab provides Tesla P100, T4, or similar GPUs for the paid “pro”
+version. It also has the high-RAM option for faster execution while using GPU. We
+utilize pysyft’s virtual worker’s concept to synthesize fog devices. Our primary focus
+in this work is to reduce class imbalance issues and ensure a robust global model.
+Hence we concentrate on the computation part of FL and ignore the communication
+(i.e., network) of conventional FL setup. Our federated learning setup can be
+utilized for any aggregation algorithms (e.g., FedAvg, FedSGD, FedProx), and as
+such, we develop our codebase on top of these baseline algorithms. As our FL setup
+works on reducing the class imbalance, we named this setup “FedBal”. In most of
+our experiments, we use 20 federated rounds where each round consists of 50
+epochs. The reason behind choosing these values for the training parameters (e.g.,
+number of epoch, number of federated rounds) is to observe a significant difference
+among the aggregation algorithms. Finally, due to time constraints, we bound our
+experiments within 20 federated rounds of aggregation.
+6.5 Performance Evaluation
+The federated learning setup is always beneficial for fog devices where data
+tends to be generated frequently. Hence, to understand the advantage of utilizing
+federated learning, we perform an experiment capturing the loss found in each
+epoch of training using federated learning and single machine training. The
+federated learning setup (a) could use more data as there are four workers perform
+the DNN model(Unet) training. On the other hand, a non-federated learning setup
+105
+
+uses fewer data to train the model with a single fog device. Moreover, the
+uncertainty in federated learning setup is less severe than non-federated learning
+that we found in our initial experiment, (b). Considering the convergence of the
+training model, FL is also faster than non-fl. Although FL has better performance
+than typical machine learning, the class imbalance issue in the local data can make
+the global model’s performance degradation. Hence, our local worker level solution
+utilizes the tversky loss function where α for penalizing false negative and β for
+penalizing false positive parameters need to be tuned for better performance. The
+experiments with these parameters are provided in the following section.
+6.5.1 Tuning Loss Function
+To find the optimal Tversky loss function, we change the alpha parameter
+value from 0.6 to 0.8 and capture each training epoch’s loss. The main goal is to
+find the optimum alpha value for which the loss will be minimal. The results of
+these experiments are demonstrated in figure 6.3.
+The figure 6.3 represents the training loss (i.e., y-axis of the figure) for each
+epoch (i.e., the x-axis of the figure) while using alpha values 0.6, 0.7, and 0.8
+respectively within FedAvg, and FedBal FL setup. From figure 6.3, we find that
+FedBal performed similar in comparison with FedAvg. It is also visible that for
+alpha value 0f 0.7, FedBal has minimum training loss. When we increase the alpha
+value to 0.8, the training loss does not decrease, which means we can penalize false
+negatives up to a certain point (i.e., α = 0.7). The reason is that while we are
+penalizing false negatives, the false positive predictions are ignored (i.e., α + β = 1)
+106
+
+Figure 6.3. Comparison of FedAvg and FedBal training loss utilizing tversky loss
+function. The alpha parameter of tversky index is changed from 0.6 to 0.8 (left to
+right) and the loss per epoch is captured for both FedAvg and FedBal algorithm.
+FedBal
+FedAvg
+Alpha = 0.6
+training loss
+training loss
+Alpha = 0.7
+Alpha = 0.8
+epochs
+as well. Hence for a higher value of alpha, we get less benefit by penalizing
+false-negative predictions. Therefore, we use α = 0.7 for the rest of our experiments
+throughout this work.
+Figure 6.4. Comparison of FedBal with FedAvg, FedSGD, and FedProx method’s
+global model performance in IID setup.
+Global model mIoU comparison in IID setup
+(a) Comparison with FedAvg
+(b) Comparison with FedSGD
+(c) Comparison with FedProx
+6.5.2 The Impact of Using IID Data Distribution
+The benefit of a federated learning setup is reflected in the performance (i.e.,
+accuracy (mIoU)) of the global model after the aggregation step of FL is completed.
+107
+
+0.0008
+0.0006
+0.0004
+0.0002
+0000'0
+0
+20
+40
+60
+80
+1000.0008
+0.0006
+0.0004
+0.0002
+0.0000
+0
+20
+40
+60
+80
+1000.0008
+0.0006
+0.0004
+0.0002
+0.0000
+0
+20
+60
+80
+1000.0008
+0.0006
+0.0004
+0.0002
+0.0000
+0
+20
+40
+60
+80
+1000.0008
+0.0006
+0.0004
+0.0002
+0.0000
+0
+20
+40
+60
+80
+1000.0008
+0.0006
+0.0004
+0.0002
+0.0000
+0
+20
+40
+60
+80
+100FedProx
+★-
+FedBalFedSGD
+★
+FedBalFedAvg
+★
+FedBal
+NNHence, we measure the mIoU of the global model after every communication or
+federated round of FedBal with FedAvg, FedSGD, and FedProx, respectively. Then,
+we plot the result as a line graph in figure 6.4. For this experiment, we consider the
+data distribution among the FL workers is identical and independent distribution
+(a.k.a IID) which means every worker gets all the classes of images in their local
+data for training.
+The x-axis of the figure 6.4 represents the federated rounds, whereas the
+y-axis presents the mIoU of the global model. From the figure 6.4 (a), it is visible
+that FedBal has outperformed FedAvg in most of the fed rounds. Considering
+FedSGD, in figure 6.4 (b), FedBal performed significantly well in the last few
+rounds. Although, in the initial rounds, FedSGD performed better than FedBal.
+Finally, from figure 6.4 (c), we find that comparing FedProx, our FedBal method
+performed significantly well. This improvement mainly comes from the utilization of
+left-out workers in the global model. In addition, the worker selection in FedBal
+considers the class imbalance issue and the priority class (i.e., oil spill class) for
+aggregation into the global model. In contrast, other methods randomly select
+active workers for aggregation, leading to a less robust global model than FedBal.
+6.5.3 The Impact of Using non-IID Data Distribution
+In a real-world scenario, data distribution among FL workers is typically
+non-IID. That means every worker will get some fixed number of classes (not all the
+classes) for local training. Hence, we consider providing two classes for each worker,
+and these classes are different for every worker. Similar to our previous experiment,
+108
+
+we measure the mIoU of the global models for FedAvg and FedBal algorithms in
+each federated round. The result is provided in figure 6.5 for 20 federated rounds
+with six federated fog workers.
+Figure 6.5. The performance comparison of global models in terms of mIoU using
+FedAvg and FedBal methods. The data distribution is non-IID, the number of workers
+are 6, and in each fed round 50 epochs of training has been performed.
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+Fed Rounds
+38
+40
+42
+44
+46
+mean intersection over union (mIoU) %
+Global Model Performance nonIID
+FedAvg
+FedBal
+The figure 6.5 reflects that FedBal has a consistent performance (mIoU) for
+20 federated rounds then FedAvg. The FedBal method has less uncertainty (fewer
+spikes in orange line of figure 6.5) across the federated rounds for selecting relevant
+workers in every federated round. Although FedBal has less significant performance
+improvement than FedAvg, the average mIoU of the global model of FedBal is
+higher than FedAvg. This consistent performance of FedBal represents the
+robustness of our method across the federated rounds.
+6.5.4 The Impact of Using non-IID and Unbalanced Data Distribution
+The non-IID and unbalance data distribution means each FL worker has a
+different number of classes. For instance, worker one can have two classes, whereas
+worker two can have only one class in its training data. Hence, we measure the
+109
+
+mIoU of the global model and compare our method (FedBal) with the other three
+baseline methods named FedAvg, FedProx, and FedSGD, respectively. As our
+method is considered an improvement of any federated learning aggregation method
+(e.g., FedAvg, FedProx, FedSGD), we compare the baselines separately in three
+different sub-figures. The result demonstrates as a line plot in the figure 6.6 where
+the x-axis represents the federated rounds, and the y-axis represents the mean
+intersection over union (mIoU) of the global model.
+Figure 6.6. Comparison of FedBal with FedAvg, FedProx, and FedSGD method’s
+global model performance in non-IID and unbalanced data distribution.
+Global model mIoU comparison in non-IID and unbalanced setup
+(a) Comparison with FedAvg
+(b) Comparison with FedProx
+(c) Comparison with FedSGD
+Figure 6.6 represents that FedBal outperforms FedAvg, FedProx, and
+FedSGD respectively in the final round. Although, in the 19th federated round,
+FedSGD and FedBal perform similarly. The performance improvement for FedBal is
+significant for FedProx, due to the utilization of the left out workers in the global
+model. It is also visible that baseline methods performance has severe uncertainty
+(more spikes than FedBal), whereas FedBal has comparatively consistent
+performance throughout the federated rounds. The main reason behind this
+consistency is the relevant worker selection in FedBal with a dynamic threshold
+mechanism that maintains a certain performance and provides a robust global
+110
+
+FedSGD
+★-FedBalFedProx
+★- FedBalFedAvg
+★
+FedBalmodel against class imbalance issues in the local data set.
+6.5.5 The Impact of Class Imbalance Intensity
+The class imbalance is a common phenomenon in the oil spill data set where
+the intensity of the imbalance can be severe within the FL setup. As such, we
+consider our solution, FedBal, to be performed consistently well than the baseline
+FedAvg algorithm in all levels of class imbalance intensity. To explore the class
+imbalance intensity, we distribute the classes from high imbalance to low imbalance
+using a non-IID setup and measure the mIoU of the global model for FedBal and
+FedAvg across the federated rounds of FL training. We estimate the difference of
+mIoU values for each federated round for three cases (one class, two classes, and
+three classes distribution) of imbalanced data distribution. Furthermore, then we
+plot a bar chart presented in figure 6.7 where positive values indicate FedBal’s
+improvement over FedAvg, and negative values represent the opposite.
+Figure 6.7 represents the advantage or disadvantage of FedBal over FedAvg
+algorithm across 20 federated rounds of training. For the first nine federated
+rounds, the improvement of FedBal over FedAvg is not significant. In the 10th
+federated round, the difference values are all positive, and in the final federated
+round (20th), we find the highest performance of FedBal over FedAvg. We also
+notice that for 2 class per worker distribution, FedBal constantly outperforms
+FedAvg. For high intense class distribution (only 1 class per worker), FedBal starts
+to perform well after the 10th round. In the final round, we find that for 3 class
+distribution FedBal has the most remarkable improvement. The main reason behind
+111
+
+Figure 6.7. Comparison of FedAvg, and FedBal method’s global model performance
+in non-IID data distribution from high intensity(only 1 class per worker) to low inten-
+sity(3 classes per worker). The difference of mIoU of FedBal, and FedAvg is plotted
+as barchart for 3 case scenarios (1 class, 2 class, and 3 class).
+1
+5
+10
+15
+20
+Fed Rounds
+3
+2
+1
+0
+1
+2
+3
+4
+5
+mIoU
+Class Imbalance Intensity - Advantage over FedAvg
+1 Class Per Worker
+2 class Per Worker
+3 class Per Worker
+the less significant performance could be the dynamic threshold mechanism that
+starts with a good mIoU value (50%) and dynamically change over federated rounds
+to increase the performance of the global model. After the 10th round, the
+threshold becomes stable with a sufficient number of relevant workers, and we see
+performance improvement for the last ten rounds of federated training.
+6.5.6 The Impact of Number of Workers on the Global Model
+To understand the influence of FL workers in our federated learning method,
+we measure the maximum mIoU of the global model for FedBal and FedAvg
+methods by gradually increasing workers from 6 to 25. The main focus of this
+experiment is to compare the performance of our method, FedBal over FedAvg,
+112
+
+Figure 6.8. The influence of federated worker on global models performance (mIoU)
+for FedBal, and FedAvg is measured by increasing the number of federated worker
+from 6 worker to 25 worker. For each case of worker pool 20 federated rounds of
+training are performed for both FedBal, and FedAvg method, and for each case
+maximum mIoU of both methods are considered for plotting as a barchart.
+6 worker
+10 worker
+15 worker
+20 worker
+25 worker
+Fed Rounds
+0
+10
+20
+30
+40
+50
+60
+mIoU
+Influence of workers on performance
+FedBal
+FedAvg
+while increasing the FL workers gradually. The result of this experiment is provided
+in the figure 6.8.
+The figure 6.8 presents that the performance improvement of FedBal is
+significant when we have an increased number of FL workers. It is visible that up to
+15 workers FedBal does not show performance improvement then FedAvg. The
+reason is that FedBal selects relevant workers from the active workers’ pool, and
+sometimes the relevant workers are very few, leading to a less improved global
+model. On the contrary, FedAvg always selects the same number of FL workers
+throughout the federated round, thus having better performance than the low
+number of FL workers pool. Therefore, an increased number of workers can
+significantly improve the global model’s performance using the FedBal method.
+6.5.7 Verifying the Impact of Aggregation Scheme on Global Model
+The selection technique of the FedBal algorithm considers the relevant
+113
+
+Figure 6.9. The impact of workers weight (averaged on each of the federated round)
+on global model’s mIoU.
+2
+4
+6
+8
+10
+Fed Rounds
+1.4
+1.5
+1.6
+1.7
+1.8
+1.9
+Workers Avg. weight
+weight
+37
+38
+39
+40
+41
+42
+43
+44
+45
+mIoU
+The influence of weights on Global Model mIoU
+mIoU
+worker by checking their weights defined by the priority class (oil spill class). Hence,
+workers with significant performance have a positive impact on the global model’s
+mIoU. To explore the impact of the workers’ weight in the global model, we measure
+the average workers’ weight in each federated round and estimate the global model’s
+mIoU after the aggregation. The result of this experiment is presented in figure 6.9,
+where the x-axis presents the federated round, the left y-axis presents the average
+workers’ weight, and the right y-axis presents the global model’s mIoU.
+The figure 6.9 presents that selected worker’s weight has a positive impact on
+the global model. It is visible that with the increase of the average weight, the
+mIoU of the global model goes up, and with the decrease, the mIoU goes down.
+Therefore, the FedBal method’s relevant workers are supported by the user-defined
+weight based on priority class, oil spill.
+114
+
+6.6 Summary
+The federated learning technique has revolutionized distributed machine
+learning, especially considering data privacy and computational flexibility. With the
+emergence of IoT, Edge, and Fog computing, the data generation is getting faster
+and expensive when needed to be transferred utilizing network bandwidth. Hence
+federated learning can bring the ML model to the data generation sources that is
+less expensive and secure than cloud data centers. This case is more applicable than
+conventional centralized DNN model training considering ML support in remote
+areas. In addition, the data captured or collected in remote oil fields are sensitive,
+and privacy preservation is of significant importance for oil and gas companies.
+Although federated learning can overcome these challenges, the class imbalance
+issue can degrade the DNN model’s performance. As such, we focus on reducing the
+effect of class imbalance at the local level while training the model, and the global
+level while aggregating the federated worker into the global model. At the local
+level, we use the tversky loss function with appropriate parameters (e.g., α, β)
+tuning to train each federated workers model considering the class imbalance issue.
+Then we assign each worker a weight, considering our priority class, oil spill.
+Finally, we check each federated worker’s model mIoU with a predefined widely
+accepted mIoU value (50% or 0.50) and dynamically change the threshold to ensure
+the robustness of the global model.
+In the empirical evaluation considering the global model’s mIoU we find that
+for IID setup, FedBal has around 3% performance improvement than FedAvg. For
+115
+
+non-IID setup, we find similar performance in the final federated round (20th)
+compared to FedAvg. Although, FedBal’s average performance for the non-IID
+setup is better than FedAvg. For non-IID and unbalanced setup, FedBal
+outperforms FedAvg, FedProx, and FedSGD respectively in the 20th federated
+round. Although, FedSGD has similar performance compared to FedBal, it has
+more uncertainty than FedBal (figure 6.6 (c)). In the class imbalance intensity, we
+find FedBal performs better than FedAvg in the final federated round (20th round)
+in three of the cases. Although, for high-class imbalance (only one class per worker)
+intensity FedBal has less significant improvement (0.25%) whereas for low-class
+imbalance (three class per worker) intensity FedBal shows significant performance
+improvement (more than 2%). The experiment with the increasing number of
+federated workers reflects that FedBal’s performance can be improved (up to 2%)
+with an increased number of federated workers. Due to time constraints and
+network vulnerability, we could not scale up the experiments, especially with the
+increased number of workers and class imbalance intensity. Finally, the impact of
+FL workers weight in FedBal method’s global model reflects a positive relation that
+verifies the selection methods acceptability. The ML training parameters (number of
+epochs per federated rounds, optimizer, batch size) can be tuned in a more granular
+way to explore the areas of improvement using the FedBal method that is
+considered as the future work of this research. Accordingly in future, we also plan
+to develop a custom loss function for the semantic segmentation field of deep
+learning and enhance our method (FedBal) as a service plugin that can be used on
+116
+
+top of any federated learning algorithm to improve the robustness of the ML model.
+117
+
+Chapter 7:
+Threats and Side-Effects of Smart Solutions in Industry 4.0
+7.1 Overview
+The convergence of new IoT technologies, cloud computing systems,
+improved wireless networks, and machine learning solutions have enabled smooth
+operations of large-scale cyber-physical industrial systems. Wireless connection, in
+particular, has altered operating paradigms to the point that most, if not all,
+production activity may now be managed remotely using a variety of sensors and
+actuators. Furthermore, these technologies have significantly increased the
+production and efficiency of various complex industrial operations. However, not
+everything about the digitization and smartness paradigm shift is positive! There
+are some disadvantages to consider as well—digital transformation and pervasive
+connection present weaknesses that criminals might use to launch cyber-attacks,
+thus jeopardizing industrial production, distribution, and even safety. As we have
+noticed in several recent instances, such as colonial pipeline [126],
+Amsterdam-Rotterdam-Antwerp (ARA) cyber-attack [127], and Norwegian energy
+company [128], malicious software systems (a.k.a. malware) have been able to take
+over the control of a system and block its regular operation until the intruders’
+demand has been fulfilled. Indeed, these recent cyber-attacks have proven that
+cyber-attacks can be as harmful as physical attacks in terms of both implications
+and severity.
+Smart sectors, such as O&G, have unique security challenges that can only be
+addressed through in-depth research and diagnostics of the entire system. However,
+118
+
+specific security solutions for smart industries have yet to be available due to their
+high implementation complexity. This security vacuum has allowed countless
+cyberattacks to flourish in recent years, endangering people and communities
+worldwide. Therefore, it is imperative that, as part of the Industry 4.0 revolution,
+all-encompassing security solutions be investigated for smart industries due to the
+crucial nature of these sectors and the managerial and technical gaps between them.
+As the oil and gas industry becomes increasingly complicated and digitized,
+we are considering researching key areas of smart O& G that pose a security risk.
+The upstream, midstream and downstream deployment of a vast network of linked
+“things” (IoT devices) presents a significant security risk for the oil and gas industry
+as a whole. Predictive maintenance and on-site worker safety are just two examples
+of the kinds of efficiency gains that may be made possible by processing the massive
+amounts of real-time, real-world data generated by smart sensors. There is a risk
+that the use of internet-connected devices might compromise the physical security
+and safety of O&G infrastructure. For instance, interconnected cameras equipped
+with object-tracking capabilities, geofencing perimeter protection solutions,
+third-party infiltration, and other access control systems can cause security breaches
+in operational sites. Therefore, this thesis section focuses on the adverse outcomes
+of smart solutions for Industry 4.0 and the strategies for minimizing those outcomes.
+119
+
+Figure 7.1. A taxonomy reflecting the downsides of smart solutions implemented
+with advanced technology is organized using box flow-chart form. The main three
+levels are colored in orange, blue, and yellow. The white boxes represent different
+types (examples) of its parent box.
+Cyber-Threats
+Device
+Incompatibilities
+Interaction
+Challenges
+Bias in Smart
+Solutions
+Unauthorized Data
+Exposure
+Information Technology
+(IT) Platform
+Operational Technology
+(OT) Platform
+Hardware
+Software
+Data Pipeline
+Human-to-Machine
+Interaction
+Machine-to-Machine
+Interaction
+Biased AI
+Automation Bias
+Gender, Age, Language,
+& Culture Bias
+Vulnerability of
+Smart Solutions
+Side-Effects
+Bridging IT & OT
+Platforms
+Malware
+Direct attack on actuators
+Ransomware
+SCADA attacks
+Man-in-the-middle
+(MIMT) attack
+Denial-of-service
+(DoS) attack
+Biased Data
+Biased Model
+Machine Bias
+Format
+Workflow
+Data Accuracy
+Downsides of Smartness in O&G
+Version
+Visual Interface
+Connection
+Lack of Conyrol
+7.2 Taxonomy of Cyber-Threats and Side-Effects in the Smart O&G In-
+dustry
+To categorically explore various drawbacks of smart solutions in the O&G
+industry, we develop a taxonomy that is presented in Figure 7.1. We separate the
+possible drawbacks of smart solutions into two groups in this taxonomy:
+vulnerabilities and side-effects. The vulnerability section investigates cyber-threats
+120
+
+and challenges caused by device incompatibilities in a smart O&G system, focusing
+on software, hardware, infrastructure, and data-related vulnerabilities in the oil and
+gas industry. On the other hand, the side-effect category focuses on difficulties that
+develop as a result of interactions with smart solutions (e.g., human-machine and
+machine-machine interactions) and biases in a smart system.
+Figure 7.1 categorizes various drawbacks of smart solutions implemented or
+will be implemented in the near future. This taxonomy serves as the blueprint for
+this chapter, enabling readers to keep track of sophisticated smart solutions and
+their accompanying outcomes. Therefore, we will traverse major taxonomy sections
+in the following parts to comprehend the magnitude of smart solutions’ drawbacks.
+7.3 Vulnerabilities caused by the Interplay of Informational and Opera-
+tional technologies
+A smart oil and gas industry’s technological operations are organized into
+two key technological platforms: information technology (IT) and operational
+technology (OT). Figure 7.2 depicts an overview of an oil and gas company’s IT and
+OT components. As seen in the diagram, the IT component is primarily concerned
+with the movement of data and information throughout the company. IT
+components frequently access outside networks due to their operational context,
+which is mainly business logic. In contrast, the OT component is involved with the
+operation of physical processes of oil and gas production and the machinery needed
+to carry them out. As a result, cyber thieves primarily target IT and OT platforms
+to meet their needs. Traditionally, the IT component has been more susceptible than
+121
+
+the OT platforms because IT has numerous open windows (e.g., operating systems,
+email servers, direct communication applications) that attackers may exploit.
+On the other hand, OT platforms mainly deal with direct oil and gas
+production and processing activities with limited external access. Notably, the
+junction of IT and OT platforms is frequently a target for cyber attacks that system
+architects must effectively handle. Furthermore, smart IoT solutions based on
+sophisticated computing technologies are opening up access to OT platforms with
+the rise of IoT. As a result, we explore the extent of the vulnerabilities in these two
+platforms as well as their overlap.
+Figure 7.2. Information technology (IT) and operational technology (OT) platforms
+of a smart oil and gas company that operates using different networks to run the entire
+operation of smart O&G industry. The IT platform is significantly related to business
+applications and the financial side of O&G, whereas the OT platform directly involves
+with oil or gas extraction and production operations. Both IT and OT platform is
+connected at some point which creates the sweet spot for cyber-attackers to penetrate
+into the whole system.
+Information Technology (IT)
+Operational Technology (OT)
+ERP Solution
+Refinery operation
+Pipeline management
+Hydrocarbon extraction
+Vulnerability
+Smart O&G
+Supervison
+system
+Control
+system
+Database
+management
+wireless network
+router
+Sensor
+users
+applications
+network
+users
+operations
+applications
+network
+The OT platform is comprised of technologies that are actively engaged in
+the production of petroleum end products. The activities include extraction,
+refining, pipelines, production, control, and monitoring systems. On the other hand,
+122
+
+兰
+$the oil and gas IT commodity primarily deals with finance, database administration,
+digital asset management, and other business operations using different computer
+platforms and communication protocols. In this case, the OT entity provides the
+petroleum end products, while the IT entity develops commercial prospects and
+financial policies by exploiting the OT entity’s output. Therefore, compared to the
+OT entity, the contact with the outside network from the O&G company’s internal
+network is substantially greater for the IT entity. Because of this relationship, a
+petroleum company might become a victim of ransomware and other cyber-attacks.
+OT was traditionally an “air-gapped” environment, which was not linked to
+public networks or other digital technologies. For decades, traditional OT has
+depended on computers to monitor or modify a system’s physical state, such as
+employing SCADA systems to monitor and control equipment to increase
+operational efficiency. Traditional OT security largely comprises simple physical
+tasks, such as ensuring that a machine performs the same operation correctly and
+that an assembly line continues to run. Nonetheless, the emergence of Industry 4.0
+in recent years has altered the conventional OT environment. Companies have
+started to deploy new digital solutions in their networks to boost automation via
+the addition of “smart devices” that can gather data more effectively and have
+network access. The IT and OT systems were integrated as a consequence of this
+connection and to process/analyze the OT data as it was generated. Although this
+technological paradigm change (referred to as IT-OT Convergence [129, 130]) has
+generated new possibilities and unlocked new use cases, it has also offered scope for
+123
+
+cybersecurity vulnerabilities. For example, Colonial Pipeline’s assault [131]
+demonstrates how poor password management may harm the country’s largest
+gasoline pipeline. The hackers found the password for an old but still working VPN
+account. In light of this threat, oil and gas companies should establish strict
+cybersecurity safeguards, including employee training. STUXNET [132] was the
+first specialized hack into industrial control system (ICS) to attract considerable
+attention, although not being the first cyberattack against an industrial
+environment. STUXNET is a computer worm that is accused of creating havoc on
+Iran’s nuclear programme, damaging more than 20% of the country’s nuclear
+centrifuges. Since then, cyber-attacks on industrial organizations have progressively
+risen, affecting a wide range of industries, including power grids (Industroyer),
+energy (Black Energy), petrochemicals (Havex), and oil and gas (Havex) (TRISIS).
+Hackers are hacking into industrial networks, among other things, to shut down
+machines, demand ransom, and steal data.
+7.4 Cyber Threats in Smart Oil and Gas Industry
+The challenge with the oil and gas industry is that its systems need to be
+designed with network connections in mind. For instance, plants were never
+designed to be network-connected. However, they are today as a result of the
+developing digital revolution. This may create a dangerous scenario since a
+cyberattack on such a system can damage operations and cause loss of life. In terms
+of cybersecurity, the O&G industry lags behind other industries. Even though
+cybersecurity is vital to the company’s sustainability, many companies still need to
+124
+
+spend more on robust systems. The remainder of this section discusses some
+security problems the O&G industry confronts.
+7.4.1 Vulnerabilities of Sensitive Data
+When stored on industrial IoT devices (sensors and actuators), sensitive
+information must be protected by rigorous security protocols. As a result, oil and
+gas companies now routinely examine private information gathered from a wide
+range of sources. Here are some examples of such data sources:
+• Historical oil & gas exploration, delivery, and pricing data
+• Demographic data
+• Response data from job postings
+• Web browsing patterns (on informational websites)
+• Social Media
+• Traditional enterprise data from operational systems
+• Data from sensors during oil and gas drilling exploration, production,
+transportation, and refining
+The aforementioned are examples of highly confidential information for any private
+corporation. Various confidential information belonging to one company might be
+precious to a company’s competitors in the oil and gas industry due to the intense
+rivalry in this sector. As a result, hackers with questionable ethics increasingly focus
+on gaining access to these sensitive records.
+125
+
+7.4.2 Vulnerabilities of Smart Systems
+In earlier chapters, we covered smart solutions that empower Industry 4.0.
+Although these are intriguing and future technologies that might assist the oil and
+gas sector as a whole, their weaknesses should also be acknowledged. The following
+are some of the ways a smart solution might fail or be compromised:
+Inherent bias in a machine learning method: The quality of the training
+dataset is crucial to the success of any machine learning model. A biased dataset is
+one that has been selected in such a way that some types of examples are given
+more weight than others. For example, suppose the photo dataset used to train the
+model for pipeline leakage detection by drones mostly covers bright weather settings.
+In that case, the trained model will perform badly in rainy or snowy weather.
+Predictive maintenance may also be used when a model is taught to work with a
+certain brand of equipment under certain conditions. Consequently, the model failed
+to generalize previously observed data correctly. Therefore, it would need to
+improve in accuracy before it could be used as a predictive maintenance model.
+Uncertainty exists in the machine learning model: Machine learning models
+are susceptible due to their inherent ambiguity. However, it is feasible that the
+model may provide false-positive or false-negative findings, which might have
+disastrous repercussions. For example, if a refinery’s smart fire detection system
+overlooks a fire, it might cause severe damage quite rapidly.
+Failure in the workflow of a smart application: Smart solutions are usually
+composed of many parts that work together to build a directed acyclic graph or
+126
+
+DAG. Face detection on an oil rig, for example, entails capturing videos, removing
+frames, and then analyzing each frame individually. Interrupting such a smart
+application cycle at any point might cause the whole application to fail, making it
+vulnerable. Similarly, if the command is not sent to the actuator, the whole
+workflow may be deactivated, resulting in a loss of control over the system.
+7.4.3 Malware and Vulnerability of Information Technology (IT)
+Malware, an abbreviation for “malicious software,” refers to any invasive
+program created by cyber criminals (also referred to as “hackers”) to steal data and
+damage or destroy computers and computer systems. Examples of malware include
+viruses, worms, trojans, spyware, adware, and ransomware. Recent malware attacks
+have resulted in massive data leaks. Therefore, the malicious actor(s) must be
+identified swiftly to remove malware. Among many forms of malwares, we discuss
+four major types in the following paragraphs.
+Virus: In order to infect other computers, viruses often attach themselves
+to files that can run macros. The virus will remain latent inside the downloaded file
+until it is opened. Viruses are malicious programmes that interfere with normal
+system functioning. This means that infections may interrupt operations and lead
+to lost data.
+Worm: A worm is a piece of malicious software that can quickly copy itself
+and infect any system on a network. In contrast to viruses, worms may spread
+without the help of any host software. For example, a worm may infect a device by
+a file download or a network connection, then rapidly replicate and spread. Worms,
+127
+
+like viruses, may drastically impair a device’s functionality and lead to data loss.
+Trojan: Often, Trojan malware may mask as seemingly valuable pieces of
+software. However, once downloaded, the Trojan virus may access the user’s private
+information and make changes, prevent access, or even erase it. The device’s
+functionality may suffer severely as a result. In contrast to common viruses and
+worms, Trojan viruses are not programmed to multiply.
+Spyware: Spyware is malicious software that works surreptitiously on a
+computer and feeds data back to an outside source. Spyware is especially hazardous
+since it affects device performance, targets sensitive data, and allows would-be
+attackers remote access. Spyware often targets financial or personal data. A
+key-logger, for example, is a kind of spyware that records users’ keystrokes in order
+to steal passwords and other confidential information.
+Ransomware: Ransomware is a kind of malicious software that infiltrates
+a system, encrypts its data so that the user cannot access it, and then demands
+payment in exchange for decrypting the data. The use of ransomware is often
+associated with a phishing scheme. Figure 7.3 depicts the stages of an actual
+ransomware attack. As can be seen, in these assaults, the victim downloads the
+ransomware by accidentally clicking on a spoofed link. The attacker then encrypts
+the targeted data using a cryptographic key that is known only to the attacker.
+Finally, in exchange for money, the hacker will release the information. We then go
+on to analyze this threat in further depth because of its rising prevalence over the
+last several years, especially in the oil and gas sector.
+128
+
+7.4.3.1 Ransomware attack incidents. During a targeted cyberattack, a
+single virus may be used for a variety of reasons, including data theft, spread, and
+penetration. The threat actor’s goal is to maintain persistence inside the victim’s
+network. Therefore, they have to constantly communicate with and update their
+virus. Using the DNS protocol, a process known as DNS tunneling [133] transmits
+information between malware and the controller. Additionally, email and cloud
+services have greatly expanded the scope of modern-day communication, which
+creates a wide door for ransomware criminals.
+Figure 7.3. The anatomy of ransomware from start to end. The ransomware client
+enters the IT platform through malicious email or other external mediums.
+The
+client then communicate with hacker’s command and control server to download the
+encryption key.
+The user’s data encrypted by the ransomware client, and finally
+extortion notice is sent.
+ransomware
+encrypts data
+and post extortion
+notice
+ransomware
+self install,
+contact
+attacker C2
+server, and
+download
+public key
+user must
+pay the
+ransom in
+exchange of
+dycription
+key
+malicous code from
+email, flash drive or
+other external medium
+The command and
+control (C2) server
+1
+public key
+hacker's
+control server
+that store
+encryption key
+2
+3
+4
+5
+129
+
+Historically, hackers used spam botnets to infiltrate as many systems as
+possible and propagate ransomware. reference mandal2020digital. Although
+ransomware has always been a huge problem for everyone with digital files, it has
+become an even bigger problem as criminals have begun to specifically target
+businesses in assaults that may have devastating effects on operations. The
+following are examples of some of the most notable ransomware attacks:
+BitPaymer19:
+BitPaymer19 [134] is a particularly deadly type of
+ransomware that recently attacked a U.S. firm providing oil well drilling services.
+BitPaymer actors often employ phishing emails to infect their victim with first
+malware before moving laterally throughout the network to compromise other
+sensitive data. When IT personnel are unavailable (e.g., on weekends and holidays),
+the ransomware attacks are
+APT33: One well-known actor group’s primary concentration is on the oil
+sector and its supply networks. Organizations in the energy sector with linkages to
+petrochemical manufacturing and the aviation industry, where APT33 is involved in
+both military and commercial capacities, have been targeted. APT33 has also hit
+energy companies in Europe and Asia. From October 2018 through December 2018
+and into 2019, a Powerton C&C server was hosted on the C&C timesync.com
+website and communicated with a small number of IP addresses belonging to oil
+corporations. Over the course of three weeks in late November and early December
+of 2019, a database server run by a European oil company in India spoke with a
+Powerton C&C server used by APT33. In the fall of 2018, it was discovered that a
+130
+
+significant UK-based corporation offering specialized services to oil refineries and
+petrochemical plants might have been penetrated by APT33.
+Email phishing was APT33’s most common method of infiltration. For many
+years, this scam has relied on the same bait: an email that seems legitimate but is a
+spear phishing attempt to offer a job. Other campaigns were directed against the
+recruiting procedure in the aviation and oil industries [135]. Additionally, a link to
+the malicious “.hta” file is provided in the email. To further entrench themselves in
+the target’s network, APT33 may use the PowerShell script downloaded with the
+“.hta” file to download further malware.
+7.5 Incompatible IoT Devices
+Among the smart O&G sector’s most common vulnerabilities is the use of
+incompatible Internet of Things (IoT) devices, as seen in Figure 7.1. In fact,
+automated systems that take data from a wide variety of Internet of Things (IoT)
+devices and sensors, a process that data using machine learning or statistical
+models, and then implement their decisions via a variety of actuation operations are
+the real engines behind a smart industry like oil and gas. In reality, the sensors and
+other linked IoT devices are created and purchased by a wide variety of companies,
+making them inherently heterogeneous. This diversity may cause incompatibilities
+and can be exploited by cyber-attackers or lead to gaps in service during times of
+crisis. For effective data transmission and offering real-time communication during
+emergency scenarios (for example, poisonous gas detection), it is crucial to configure
+linked and suitable IoT devices that can interact seamlessly.
+131
+
+Since acquiring uniform and completely compatible IoT equipment is
+difficult, if not impossible, researchers are looking at other solutions, such as the
+development of standard protocols that would enable effective communication
+across all industrial IoT devices—because of this, leading IT firms are collaborating
+to create a single protocol (called matter protocol [136]) that will be compatible
+with any and all Internet of Things (IoT) gadgets. The issue is a new protocol for
+inter-network communication between smart homes that are being backed by the
+Connectivity Standards Alliance, which includes tech giants like Apple, Google,
+Amazon, and others. The problem is the lack of a standard, IP-based
+communication protocol that is based on tried and true technologies to construct
+safe and secure IoT ecosystems.
+In smart O&G and other smart industrial contexts, we might examine three
+different kinds and degrees of incompatibility. In the following sections, we will
+discuss the incompatibilities that exist: those at the hardware, the software, and the
+data pipeline.
+7.5.1 Hardware-level Incompatibility
+Commonly available products are increasingly being utilized to replace
+specialized equipment in the oil and gas sector. They are more susceptible to
+security problems than traditional process control systems because of their
+adaptability. Because they are so widely used and deployed, the attack surface is
+widened significantly. There are several methods to assault an oil field. For
+instance, a smart real-time video camera may be employed to keep an eye on a
+132
+
+potentially dangerous region for anomaly detection. Still, an unauthorized user
+might be able to use the control system to open a valve that lets poisonous or
+explosive gas escape. Sensors, actuators, cameras, and their supporting hardware
+might be protected against this kind of assault if they all used the same protocol to
+check for vulnerabilities and flag any unusual activity.
+7.5.2 Software-level Incompatibility
+Compatibility issues at the software level might arise from the usage of
+outdated or unsupported software, which can lead to system failure. Furthermore,
+there is a risk that malicious viruses will be introduced into internal systems
+through third-party software. But antiquated software must be updated to work
+with modern hardware and applications. Systems are more likely to be attacked if
+they haven’t been kept up-to-date or are using enhancements that weren’t made for
+their operating system.
+Companies in the oil and gas industry often purchase digital items on the
+assumption that they are secure and can be integrated into the more extensive
+system. However, it is common practice for them to verify that everything else in
+the system is compatible with the new component. Also, oil and gas companies may
+not always have the resources available to verify incompatibility at the software
+level. That’s how cybercriminals get into oil and gas firms’ private networks via
+vulnerabilities in “smart” technology.
+As a result of the Internet of Things (IoT) smartness, business leaders in the
+petroleum supply chain must come up with novel approaches to preventing
+133
+
+cybersecurity concerns. In recent hacks, vulnerabilities in the software were used. In
+2017, a cyberattack known as NotPetya hit a variety of institutions, including a
+single electricity provider, banks, public transportation networks, and a large
+international container shipping firm. Interestingly, the virus propagated via
+Ukrainian companies’ updated accounting software. When the infection spread to
+other computers, it caused crashes; in this case, the cybercriminals had infected
+customer-ready, certified software with spyware known as “SolarWinds” (2021). In
+both cases, hackers used vulnerabilities in software to get into connected vendors’
+systems. In addition, they put in place loopholes that might be used to steal IP
+financial data or propagate malware among user machines.
+7.6 Blockchain to Overcome Cyber-Threats in Smart O&G
+Blockchain technology, which has recently risen to prominence as the
+foundation of cryptocurrencies such as BitCoin and Etherium, is an effective
+security method. As the data is stored, it is linked in a series of blocks, and the
+hash value of the preceding block is kept in each block. Since the hash value of a
+tampered data block would no longer be consistent with that of the succeeding
+block, the attack could be traced. Several subsystems of Industry 4.0 and smart
+O&G are now using blockchain technology.
+7.6.1 Blockchain-based Control Systems (SCADA)
+The Industry 4.0 movement has transformed the role of IT and OT in the
+modern industry. The SCADA systems that gather information from the smart IoT
+devices and send it to the servers where it is analyzed constitute the backbone of
+134
+
+most OT platforms. However, this kind of data collection is inherently unsafe and
+unreliable, providing an opening for hackers. For this reason, edge and fog
+computing-based blockchain security procedures have been suggested to safeguard
+SCADA systems’ data collection transactions. The gathered sensor data are
+encrypted in data blocks before being processed on a cloud-based SCADA system,
+and a high-level overview of this method is shown graphically in Figure 7.4. Data
+hashes from the previous and current blocks are stored in each block. Then, the
+Data Aggregator (DA) and all relay servers participate in the block verification
+procedure. The servers will answer many times for the purpose of verification. Upon
+consensus that the block is legitimate, the DA will forward the request to all
+participating servers. The DA adds the new block to the blockchain and then
+successfully sends the updated blockchain to the command center. Both the mining
+node selection technique and a more secure consensus process that is compatible
+with Industry 4.0 have been suggested in a recent paper [137], and these are
+discussed in the following sections.
+7.6.1.1 Consensus mechanism. Simply put, a consensus mechanism is a
+process by which validators/miners verify the authenticity of freshly released blocks
+before adding them to the blockchain, hence preserving the integrity of the network.
+Various blockchain networks have spent considerable time and energy developing
+various consensus techniques. Both public and private blockchains use some
+consensus mechanisms. Proof-of-Work (PoW), Proof-of-Stake (PoS), Delegated
+Proof-of-Stake (DPoS), Practical Byzantine Fault Tolerance (PBFT),
+135
+
+Figure 7.4. Blockchain based data transmission within end-to-end SCADA system
+of an oil and gas company. Blockchain enable encryption while transmitting the data
+for processing that increase the data security even data is hijacked while transmitting.
+Pressure
+sensor
+Flow
+Rate
+Sensor
+Satellite
+Sensor layer
+Edge systems
+Cellular,
+Radio
+Wireless long haul
+communication
+Tank
+Level
+Sensor
+Temparature
+sensor
+Human Machine
+Interface
+ Database
+Service
+Machine
+Learning
+Reporting
+Service
+Cloud-based
+SCADA system
+RTU
+Data Flow
+Elastic Load
+Balancing
+Analytic
+Service
+Corporate
+Lan
+Data
+Hash of
+current
+block
+Hash of
+previous
+block
+Blockchain based data acquisition with edge systems
+Data transmission
+Data processing
+Data
+Hash of
+current
+block
+Hash of
+previous
+block
+Blockchain block
+Fog-based data
+processin
+Fog systems
+Proof-of-Authority (PoA), and RAFT [138] are all examples of popular consensus
+techniques. Both the benefits and drawbacks of each consensus technique are
+distinct. PoW, for instance, is unjust to new entrants since it has a large processing
+expense and favors the richest validators. On the other hand, DPoS is less robust
+and decentralized. Due to its lack of anonymity, PBFT has restricted to permission
+(non-public) blockchains [139].
+7.6.1.2 Mining node selection. A machine that participates in a
+blockchain network by hosting blockchain software and facilitating data transfer is
+called a “node.” Nodes in a network might be anything from a laptop to a phone to
+a router. “mining nodes” are the nodes that participate in the processing and
+verification of blockchain transactions. Any participant in the blockchain network
+may choose to take part in the mining process. As a term, “mining” refers to the
+activity of adding new transactions to a blockchain. Figure 7.4 shows the Data
+136
+
+TEMPERATURE SENS
+Serial No.13 signal TypC
+AADIAAND二二Aggregator (DA) edge server collecting data, processing it, and coordinating mining
+node selection and verification. If you want to save as much time and processing
+power as possible, place the DA on the same private network as the relay servers.
+Because of this, the fog servers used in the pre-processing stage of the data shown in
+Figure 7.4 are hosted inside the DAs’ own private network
+In [137], the authors provide a specialized method for selecting mining nodes.
+To begin, the DA server initiates a data request to the relay servers. Once the DA
+collects all of the readings from the various relays, it will produce a random number
+and send it out across the network. To count how many times a random number
+appears, relay servers hash their data and compare the results. At this point, the
+DA’s server statistics are identical to every other server’s. Ultimately, every server
+casts a vote for the one that has made the most random appearances during the
+process. The DA server will choose the relay with the highest count as the mining
+node for the current cycle if all other relay servers agree. On the other hand, let’s
+pretend that a large number of relay hosts have the same highest count or that they
+all have 0. The DA here selects the mining node at random using a
+cryptographically sound process [140].
+7.6.2 Blockchain to Enable Trust Across Industrial IoT
+The problem of trust is one of the barriers to the security of the industrial
+Internet of Things (IIoT). The conventional Public Key Infrastructure (PKI) design,
+which is built on a single root of trust, does not operate well in this heterogeneous
+dispersed IoT environment, which may be subject to several administrative
+137
+
+domains. Therefore, a distributed trust model that can be constructed on top of
+current trust domains and produce end-to-end trust across IoT devices without
+depending on a single root of trust is necessary for this sort of scenario. As a result,
+establishing a credit-based Blockchain with an integrated reputation system might
+be beneficial [141].
+Another potential use of blockchain in the oil and gas sector is the storage of
+credentials required to operate safety-critical industrial machinery. For example,
+employee and contractor qualifications, such as H2S training, first aid, and welding,
+may be securely recorded and preserved on a company’s blockchain network. By
+storing such information in a blockchain network [142], all members may perform
+verification of credentials and standard operating procedures at any time.
+7.7 Risks of Smart Solutions in industrial IoT
+As technology improves and more industries and products are connected to
+the internet, it is important to understand the risks of industrial IoT installations.
+Any business that wants to use IoT in manufacturing or industry or connect
+existing technologies for automated and remote monitoring should consider the
+advantages and disadvantages. In the next section, we’ll discuss about the possible
+inadequate performances of smart solutions.
+7.7.1 Human-Machine Interaction Issues
+The industrial IoT has come a long way; machines can now process data
+from connected devices automatically. In addition, various automated sensors and
+actuators (like video cameras, smart glasses, and automatic valves with audio
+138
+
+input/output) are in place to help or replace the human worker in order to make
+sure that production runs smoothly and/or that workers are safe when using
+different machines to do their jobs.
+Figure 7.5. Human-machine interaction workflow from sensing to control operation.
+Sensing
+Information
+Processing
+Controlling
+Controls
+Operation
+Display
+Human
+Machine
+Figure 7.5 shows how the human and machine sides of human-machine
+interaction work together. As this picture shows, people use different senses (like
+sight, smell, and hearing) to look at the machine’s results. So, a human worker uses
+information processing to run or control the machines. Then, machines do their
+139
+
+.jobs, and the results are shown to people so they can figure out what they mean.
+The whole process of how people and machines work together is called the
+human-machine interaction process.
+Production and safety on the job site may be jeopardized if the
+interdependent machines fail to operate as intended or are not user-friendly. To
+perform vital industrial tasks or, more crucially, to ”emphasize overrule” the choice
+of a smart system, human interactions are often required beyond those with a
+computational interface through input/output devices. Take the case of a drone or
+ROV sent to a politically sensitive location (like a border region) to conduct
+autonomous oil and gas surveys. However, inefficiently or a glitch in its algorithm
+may cause it to survey regions beyond the designated zone and prevent the operator
+from navigating the survey route. Unforeseen repercussions on the political or
+military front may result from such a glitch in human-machine interaction.
+7.7.2 Machine-to-Machine Interaction Issues
+Machines communicate with one another in networked autonomous systems
+to complete various activities. In these systems, an automated sequence of actions is
+carried out using multiple devices; if anything goes wrong, it could be due to (A)
+the devices producing misleading output (for instance, automated valve shutdown
+with wrong anomaly detection or automatic door closing that traps onsite workers
+with false alarm), or (B) incompatibility across devices. Accidents or catastrophes
+may arise due to machine-to-machine interface issues in a production setting with
+fault-intolerant operations.
+140
+
+Figure 7.6. A small fire breakout accident occurs in a closed oil production area in
+a compressor unit. The fire alarm generates, and water sprinkler starts to sprinkle
+water that causes power failure in power generator that made the electric door locked.
+Unfortunately, workers were working on pipeline maintenance, and were trapped in-
+side the facility due to door closure. Here, machine to machine interaction cause the
+safety issue of the onsite worker.
+electric door locked
+trapped
+worker
+fire alarm
+water sprinkler
+power
+generator
+compressor
+unit
+fire
+1
+2
+3
+4
+5
+Figure 7.6 depicts one scenario illustrating the repercussions of the
+machine-to-machine interaction problem. Consider pipeline maintenance and
+communication with numerous pieces of equipment in a production scenario.
+Consider a pipeline linked to a machine (for example, a distillation unit) in an
+enclosed space that requires repair. As a result, maintenance staff works within the
+enclosed space when a fire danger occurs. As a precaution, the gas sensor detects
+smoke (Step 2 in Figure 7.6), activates the water sprinkler (Step 3), and sends an
+alert to the controller. The electricity generator shuts down due to sprinkler water
+141
+
+fimtgroup(step 4). To safeguard the safety of the outside employees, the controller instantly
+sends the automatic door to shut (Step 5) while disregarding the workers within the
+area. In this scenario, the controller cannot recognize personnel within the facility
+and executes a safety action for outside workers, putting the workers inside at risk.
+We can see that machine-to-machine interaction concerns may sometimes lead to
+scenarios that must be solved by evaluating a smart solution for the O&G industry.
+7.8 Bias in Smart Industry
+Human intervention at different stages of software development may
+introduce a wide range of biases that might undermine the quality of otherwise
+intelligent software solutions [143]. Accidents and potentially dangerous situations
+have occurred as a result of prejudices in the past. Many types of bias exist,
+including those based on age, race, sexual orientation, disability, and other
+demographics. In addition, many onsite team members (e.g., workers, engineers,
+and coordinators) rely on software tools and simulations that are prone to the above
+mentioned flaws. Our discussion here will center on the different kinds of bias and
+the damage they might do to the smart O&G business and beyond.
+7.8.1 Biases Caused by the Artificial Intelligence (AI) Solutions
+Even though AI systems have shown to be revolutionary in several contexts,
+they are prone to the following two types of bias:
+• There are gaps in the training dataset that cause the model to underperform
+on certain inputs.
+142
+
+• The model’s biases are the same as those found in the original dataset used for
+training.
+An absence of training datasets is one source of AI bias, as shown by the
+commercial face recognition system. The lack of dark-skinned women [144] in the
+training dataset is the root cause of the face recognition system’s discrepancy
+between its 99.9 percent accuracy with white males and its 35.0 percent accuracy
+with women of colour, as determined by the researchers. The issue, however, is that
+“Big Data” does not necessarily provide valid and trustworthy models. For instance,
+social media is a well-established mine for massive datasets. Conclusions obtained
+from Twitter data should be treated with caution since just 24% of internet
+teenagers utilize the platform, as reported by [145].
+An unfair model does not necessarily perform poorly on a demographic
+subset. Even if the model is correct, it is still unjust. The dataset is skewed in this
+situation, and the model repeats or amplifies the inherited bias. Natural Language
+Processing (NLP) models, for example, are often trained on a vast corpus of
+human-written text (e.g., article news). However, word embeddings trained on
+Google News articles have been observed to reflect female/male gender stereotypes.
+The models, for example, replied that a father is a doctor while the mother is a
+nurse, or that a “man” is a “computer programmer” while a “woman” is a
+“homemaker.” This kind of bias occurs when a model is trained on skewed data
+owing to unjust procedures or structures [146].
+Another example of AI bias is Yelp’s restaurant review system. Restaurants
+143
+
+may pay Yelp to promote their locations on the Yelp platform, but this inevitably
+influences how many people see adverts for a particular restaurant and, as a result,
+who decides to dine there. As a result, Yelp evaluations may be unjustly slanted in
+favor of more prominent eateries.
+7.8.2 Automation Bias in Smart Solutions of Industry 4.0
+One of the most respected psychologists in the field, Linda J. Skitka of the
+University of Illinois at Chicago, defined automation bias as “a specific class of
+errors individuals tend to make in highly automated decision-making scenarios when
+many decisions are handled by automated aids (such as computers, IoT devices, and
+smartphones) and the human actor is primarily present to monitor ongoing tasks.”
+A bias toward using automated assistance and decision support systems is
+often known as “automation bias.” When the Enbridge pipeline ruptured [147] on
+July 26, 2010, sending enormous amounts of crude oil into the Kalamazoo River and
+Talmadge Creek, automation bias was a major factor. Both complacency and a
+leaning toward automation were shown to have played significant factors in the
+Enbridge oil pipeline disaster. Therefore, businesses, governments, and regulators
+must account for automation bias while designing systems to reduce the potential
+for careless errors. “Automation bias” is humans’ propensity to favour actions
+requiring the least amount of mental effort. Similar thinking applies to the
+underlying principle of AI and automation: learning from massive amounts of data.
+Such calculations imply that future conditions will mostly stay the same. Another
+factor to consider is the possibility that faulty training data may lead to faulty
+144
+
+learning [148] that is implicitly related to AI bias.
+7.8.3 Gender Bias in O&G Industry
+According to a study report [149], the oil and gas sector is confronting a
+skilled personnel scarcity, while gender prejudice is exacerbating the problem by
+excluding female workers from recruiting. The research contains interviews with
+various male and female workers from around the globe and an analysis of their
+comments. Indeed, the oil and gas industry has a reputation for being controlled by
+males. However, while some oil and gas businesses work hard to achieve gender
+parity and worker diversity, others are allowing the gender gap to widen. Although
+many businesses strive to include gender equality in their policies, actions, and
+procedures, they still face challenges such as gender imbalance and various types of
+implicit prejudice.
+7.8.4 Cognitive Bias in Smart O&G Solutions
+Cognitive biases, a newly discovered notion, are mental faults in human
+thinking and information processing that may result in inaccurate or irrational
+assessments or decisions. Amos Tversky and Daniel Kahneman first proposed it in a
+1974 article for Science Magazine (Tversky & Kahneman, 1974 [150]). Since then, a
+great deal of literature has been produced on cognitive biases and how they impact
+human thoughts and actions.
+According to a common understanding of cognitive bias, it is a mental flaw
+that results in incorrect interpretation of external data and impairs the logic and
+precision of choices and verdicts. Biases are unconsciously occurring, automatic
+145
+
+processes that speed up and improve decision-making effectiveness. There are
+several factors that might contribute to cognitive biases, including public influence
+and emotions. There has been a growing awareness of the threats cognitive bias
+may bring to operational safety during the last several years. Biases like deviance,
+normalization, and group thinking, for instance, are now widely accepted.
+Additionally, the Deepwater Horizon [151] investigation in 2010 brought cognitive
+bias to the public’s attention, at least among those working in the offshore drilling
+industry. Consequently, the International Association of Oil and Gas Producers
+(IOGP) has brought attention to how crucial these cognitive impairments are to
+safety. Therefore, it is high time that cognitive bias should be addressed while
+building smart, automated solutions that require human decisions for complex
+industrial operations.
+7.9 Summary
+Oil and gas operations have seen dramatic changes as a result of the digital
+Industry 4.0 revolution, which has made extensive use of cutting-edge computer
+hardware and software. However, with these developments come opportunities for
+cyber criminals to improve their efficiency in locating vulnerabilities in either IT or
+OT systems, or in the hybrids that exist between the two. Another possible entry
+point for cyber criminals is provided by the heterogeneity and incompatibility of
+smart technologies, as well as the connection difficulty between them. Problems with
+human-machine and machine-to-machine interactions, as well as incompatibilities
+between technologies acquired through time, are among the most significant
+146
+
+obstacles to the widespread adoption of smart technologies in the legacy and smart
+oil and gas sectors. Though Industry 4.0 has been a boon to the oil and gas sector,
+business executives and professionals working in the sector should be wary of its
+smarts being misapplied. In the last several years, we’ve learned the hard way that
+blindly installing or embracing smart technology may open the door to a wide
+variety of risks. Researchers and practitioners must bear in mind these drawbacks
+while deciding whether or not to use smart technology. In this regard, as a part of
+this dissertation, we published a book [152] on the scope of IoT technologies with
+the rise of the Industry 4.0 revolution that addresses a detailed analysis of smart
+solutions and their drawbacks in the context of smart O&G Industry.
+147
+
+Chapter 8:
+Conclusion and Future Research Directions
+The ever-growing IoT and smart devices (e.g., smart gateway, sensors,
+controllers, actuators) produce a substantial amount of raw data that need to be
+stored, pre-processed, and analyzed to bring out potential insights that can make
+the industrial systems more efficient. Accordingly, various Industry 4.0
+latency-sensitive applications operate based on machine learning (ML) and utilize
+the generated sensor data to achieve automation and other industrial activities.
+Hence, the cloud computing platform has been offering services [153] to perform
+various operations on the ever-growing data generated in the industrial sectors.
+Privacy, centrality, and expenses have been significant constraints to utilizing cloud
+data centers effectively. As such, edge and fog computing bring the computational
+services [154, 155, 156] near the end-users closer to the data sources. However, edge
+devices may support limited computing demands due to resource limitations. In
+contrast, the fog system can be a preferable option to meet computing needs due to
+its availability of computational resources and more robust middleware compared to
+edge systems. Because, fog systems are heterogeneous and the heterogeneity is one
+factor that introduces stochasticity in the execution time of Industry 4.0
+applications that affects the completion times of these applications. To develop a
+robust solution for Industry 4.0, it is necessary to study the execution time
+behaviour of various ML-based applications in heterogeneous fog systems. As such,
+we perform statistical analysis of ML-based Industry 4.0 applications to understand
+the execution time pattern of these applications. In addition, we introduce
+148
+
+real-world Industry 4.0 smart application execution traces in fog computing systems
+that can be beneficial for the future research works.
+Even though fog systems have more computing resources than edge systems,
+the surge in computing demands at disasters can reduce performance. Therefore, in
+this dissertation, we propose federating fog computing systems (owned by private
+companies) from nearby sites to support such scenarios. Furthermore, the fog
+federation concept can be practical with system administrators’ efficient resource
+allocation mechanisms adopted by research works related to load-balancing
+methods. A real-world Industry 4.0 application execution traces on fog computing
+platforms can be crucial for devising effective resource allocation methods. As a
+result, we utilize our prior workload trace to devise a statistical resource allocation
+method across federated fog systems for Industry 4.0 latency-sensitive applications.
+In addition, the heterogeneous software methodologies (e.g., monolithic,
+micro-service) of Industry 4.0 applications can affect the execution plan of a fog
+federation due to their diverse latency constraints, resulting in decreasing system
+performance. Hence, the decomposition of micro-service applications with effective
+resource allocation methods can maintain the systems’ performance in
+oversubscribed situations (e.g., accidents, and disasters). Accordingly, the industrial
+computing platform (i.e., federated fog system) should be cognizant of stochastic
+execution behaviour, software structure, and latency requirements of micro-service
+workflow applications. We propose a resource allocation method based on
+probability estimation that partition micro-service workflows across the federated
+149
+
+fog computing systems to support their latency requirements. Furthermore, the
+concept of federating fog resources raises data security and privacy concerns for
+private fog systems participating in the federation due to having sensitive company
+data stored or processed in these fog systems. Thus, we propose a data privacy
+preserving solution that works based on federated learning method for training
+ML-based Industry 4.0 application across federated fog systems.
+8.1 Discussion
+In this dissertation, our main objective was to investigate and develop
+effective resource allocation solutions using modern distributed computing systems
+for remote Industry 4.0. As such, we first explore and identify various smart
+computing aspects in remote offshore industries (e.g., oil and gas, minerals,
+sustainable energy) where computing demand is significantly high and conventional
+computing systems are inefficient due to latency constraints. Hence, privately
+owned fog systems in remote areas can support industrial computing demands.
+Hence, we identify stochastic execution time behaviours of latency-sensitive tasks
+executing in heterogeneous fog systems. As such, we explore the execution time
+behaviour of various ML-based applications in heterogeneous execution platforms
+(e.g., amazon web service, chameleon). Consequently, we introduce a real-world
+workload of execution time in heterogeneous computing resources. Furthermore, in
+remote industries, the surge in computing demand can decrease the fog systems’
+performance at disaster times by not completing latency-sensitive task requests on
+time. Accordingly, we propose federating nearby fog systems in remote industries
+150
+
+and forming a fog federation to support surge computing demands. Thus, we enable
+the federation concept and develop a statistical resource allocation method using
+prior synthesized real-world application workload considering an oversubscribed
+situation. Hence we evaluate our proposed solution for monolithic applications
+widely used in Industry 4.0. After that, we investigate smart micro-service
+applications’ internal structure to understand the impact of the decomposition on
+application workflow completion. We suggest a probabilistic workflow partitioning
+method along with the previously proposed resource allocation method that
+improves the fog federation’s performance and ensures safety in remote Industry 4.0.
+Finally, we address the data privacy issue for sharing privately owned fog systems in
+developing accurate ML models for Industry 4.0. Hence, we explore the federated
+learning techniques across the fog federation that ensure data privacy for privately
+owned fog systems. In this context, we address the class imbalance issue in a
+federated learning setup that can reduce the robustness of the global model.
+Therefore, we propose a federated learning method that is robust against the class
+imbalance issue.
+In chapter 3, we analyze and estimate the performance of DNN-based
+applications in heterogeneous cloud and fog resources (e.g., amazon, chameleon).
+Here, we identify stochastic execution behaviours of various Industry 4.0
+applications. Thus we explore, and model the inference execution behaviours of
+various Industry 4.0 smart applications utilizing different statistical tools from two
+distinct perspectives, namely application-centric and resource-centric, respectively.
+151
+
+Furthermore, we introduce an execution time workload of four different DNN-based
+applications for Industry 4.0 with the intent of developing robust resource allocation
+methods across federated fog systems.
+In chapter 4, we explore the usability and benefits of fog federation that can
+be formed to support emergencies such as disasters (e.g., fire explosions, oil spills).
+As an example of a smart industry, we consider remote smart oil fields with multiple
+oil extraction sites in close vicinity, each with fog computing systems to support its
+local computing demands. Although in case of an emergency like an oil spill, the
+computing demands can rise due to the coordination of multiple activities (e.g.,
+drone inspection of oil spill, video camera images, sensors data processing) to
+support the situation. Hence we propose a probabilistic resource allocation method
+for monolithic latency-sensitive applications that effectively selects a relevant fog
+system from the federation by utilizing our prior workload. As the resource
+allocation method is aware of the receiving applications stochastic execution
+behaviours from our prior work, it ensures the robustness of the fog federation by
+completing majority of the receiving workload on time.
+In chapter 5, we explore modern software architecture (i.e., micro-service) of
+Industry 4.0 applications to create an efficient execution strategy over fog
+federation. In contrast, we identified legacy applications with monolithic software
+architecture are still exists in various industrial sector. Therefore, to support the
+computational demands in remote industry the execution platform (federated fog
+system) should be aware of software architectures of the Industry 4.0 applications.
+152
+
+Hence, for micro-services we consider the idea of using an application breakdown
+strategy to increase the chance of finishing the execution on time. Furthermore, for
+monolithic applications and individual micro-services we utilizes our prior
+knowledge of stochastic execution behaviour to efficiently allocate fog resources
+across the federated fog systems. As a result, we propose a statistical micro-service
+partitioning and resource allocation method that considers the underlying software
+architecture and the stochastic execution latencies of Industry 4.0 applications.
+In chapter 6, we explore the data security and privacy aspects of fog
+federation while training ML-based applications in remote Industry 4.0. In this
+case, we investigate the federated learning techniques utilizing fog federation to
+train a ML-based oil spill detection application that provide data security to
+privately owned fog systems of the federation. Accordingly, we identified low
+occurrence events in training data (i.e., class imbalance) can reduce the accuracy of
+the ML-model that can be detrimental in emergency situations. Here, we propose a
+customized federated learning technique, considering the class imbalance issue
+across fog federation to increase the safety measures of remote Industry 4.0.
+In chapter 7, we investigate the downsides and side-effects of smart solutions
+developed with the integration of various applications in the industrial sectors.
+Hence, we introduce a taxonomy of cyber threats and side-effects of smart solutions
+in the context of the O&G industry that structurally address the unsafe areas of
+these smart solutions. Accordingly, various vulnerable areas, including both
+software and hardware components, machine-human interaction issues, and different
+153
+
+forms of biases in smart solutions, are addressed with efficient resilience methods
+that would help system architects or industrial researchers to develop robust smart
+solutions for Industry 4.0.
+In conclusion, we explore and investigate the stochastic execution behaviour
+of various Industry 4.0 applications and introduce a real-world execution workload
+that has been utilized in our resource allocation research works. Then, we explore
+the federation concept using privately owned fog systems for various computing
+demands of Industry 4.0. Especially in oversubscribed situations like disasters, the
+federation could be more efficient if the load is adequately balanced. Hence we
+develop a load-balancing method to make the federation robust in emergencies that
+we consider the system administration level of our research track. Then we dive into
+the application level by investigating various software architectures (e.g.,
+monolithic, micro-service) of Industry 4.0 applications. Hence, we identify
+micro-service workflow applications can be decomposed to improve the application
+workflow completion on time. Accordingly, we propose a probabilistic micro-service
+partitioning and resource allocation method that can enhance the performance of
+the fog federation. Then, we explore the data security and privacy aspects of
+federated fog systems while training ML-based Industry 4.0 applications. Finally, in
+the end, we identify various pitfalls of smart solutions that need to be appropriately
+addressed to develop efficient and robust smart solutions for Industry 4.0
+applications.
+154
+
+8.2 Future Research Directions
+Based on our findings during the development of the resource allocation,
+micro-service workflow partitioning, and secure resource-sharing solutions, we
+identify some of the expansion areas that can improve the robustness and safety of
+Industry 4.0. There are several points where the work could be expanded.
+8.2.1 Resource Allocation Using Reinforcement Learning for Industry
+4.0 Applications across Federated Fog System
+In this dissertation, we suggest a statistical application completion time
+estimation method across the fog federation system to allocate Industry 4.0
+applications into a relevant fog system. Our estimation of task completion success
+could be coarse that sometimes leads to the deadline miss of a receiving task. We
+think that a resource allocation method operating based on the reinforcement
+learning technique [157, 158] can improve the quality of service for the federated fog
+system. The field of reinforcement learning (RL) has emerged as an important
+subset of machine learning because it enables autonomous agents to make sound
+decisions in response to changing conditions in their environment. Hence, uncertain
+execution behaviour can be addressed effectively using RL technique. In this
+scenario, RL might be used to allocate resources [159] for offloading and executing
+tasks in a federated fog computing system, leading to better overall performance.
+8.2.2 Data Locality-Aware Resource Allocation Across Federated Fog
+System
+The proliferation of IoT devices has coincided with a surge in network traffic
+155
+
+that may overwhelm the potential of the current network. Furthermore, data
+privacy and latency are important concerns when these devices analyze sensor or
+user information. Therefore, conventional methods such as cloud computing don’t
+work. Although, advanced computing platform like fog computing can fill this need.
+Understanding how the initial input data’s localization impacts on fog platforms’
+performance is critical to developing reasonable load balancing and resource
+allocation solutions [160, 161]. As a result, if several data-intensive applications
+with deadline restrictions arrive dynamically, performance evaluation of a
+heterogeneous federated fog environment is required. For example, the applications
+may need data from the IoT layer or from local fog resources (e.g., sensor data that
+have already been transferred to the fog layer or data processed by prior
+applications). In this scenario, examining the influence of input data localization on
+system performance across federated fog systems with varying data placement
+probability might influence the federation’s resource allocation efficiency. Therefore,
+we consider exploring the impact of data localization on resource allocation methods
+across federated fog systems in the oversubscribed situation for remote Industry 4.0.
+8.2.3 Dynamic Fault-Tolerant Federated Fog Systems for Industry 4.0
+Operation
+The fog computing systems provide low latency to the end users being close
+to the data sources. In contrast, the distributed characteristics of fog aid in
+processing vast amounts of sensor-generated data of Industry 4.0. Hence, federating
+fog computing resources can support latency-sensitive tasks and data processing
+156
+
+services. However, fog systems have various uncertainties (e.g., transient failures
+[162], network and power outage) that need to be considered when supporting surge
+computing demands. Especially in an emergency, those uncertainties can lead to
+unsuccessful task completion causing significant damage to the environment and
+even human lives. Hence, the resource management system for fog federation should
+be aware of the uncertainties and provide efficient fault-tolerant [163, 164, 165]
+solution to ensure successful completion of the receiving latency-sensitive tasks on
+time. Accordingly, the federation management system should consider providing a
+service that continuously monitors the fog resources and then sends the signal to all
+the participant fogs about the neighbouring fog’s state. In addition, already ongoing
+service execution can get disrupted or fail due to the fog system’s internal issues
+(e.g., software, hardware). In this case, every fog system should have a method
+(e.g., re-execution, offloading the failed task to another fog) to ensure successful
+completion of receiving task’s execution. Therefore, a fault-tolerant federated fog
+system is crucial for supporting surge computing demands in emergency or disaster
+situations, enabling the system’s robustness and leading to a safe Industry 4.0.
+Similar to super cloud [166], fog systems provide various virtual services [167]
+like application deployment, multi-tenancy, interoperability, and service migration
+across fog federation. Hence, to enable a fog federation that can avail various fog
+virtual services need to support fault-tolerant characteristics for efficient utilization
+of fog services. In addition, in oversubscribed situation, the fault-tolerant service
+needs to address the scalability of fog federation, ensuring the system’s robustness
+157
+
+in a dynamic condition. Therefore, to achieve all these requirements, we are
+considering performing research works in future on developing a fog system (super
+fog) that provides fault-tolerant fog services across the federated fog systems to
+improve the efficiency of the federation.
+The popularity of fog systems with heterogeneous resources and dynamic fog
+federation [168] concept has created the demand for developing the fog-friendly
+application that requires proper investigation of the application stack and fog
+resources. However, building this type of application is time-consuming and requires
+overcoming some major obstacles. The first is to support the dynamic nature of the
+fog network; the second is to manage the context-dependent qualities of application
+logic; the third is to cope with the system’s massive size. As a result, we must
+consider how to decompose and deploy applications to a geographically dispersed
+fog federation utilizing current software components that may be altered and reused
+to participate in fog applications [169]. Hence abstracting the application layer from
+the execution layer can be the primary objective to solve the heterogeneity challenge
+of the fog systems. Thus, we would like to perform research on developing
+fog-friendly applications for dynamic federated fog systems that are cognizant of
+super fog systems’ characteristics.
+8.2.4 The Cognitive Aspects of Human-Machine Interaction for Smart
+Industry 4.0 Solutions
+Industry 4.0, an industrial technology paradigm shift, mandates new ways in
+which human and machine (e.g., robots, drones) will work together. The
+158
+
+introduction of ever-more-advanced sensors and collaborating machines raises
+important questions about the influence on safety in the highly technical and
+inventive scenario of Industry 4.0, defined by a succession of enabling technologies
+and a strong interconnectedness of resources between human and machine. On the
+one hand, advanced software tools and machines facilitate human work
+(human-machine cooperation). On the contrary, it must communicate and share
+data with other intelligent devices (human-machine interaction) [170]. Since the
+advent of “smart” technologies, both the environment in which these innovations
+are deployed and the responsibilities of front-line human workers have changed. In
+complex industrial operations or at disaster times, the human-machine interaction
+can be challenging that is significantly related to cognitive aspects [171] of the
+human workers. When some tasks need specialized human abilities, there is genuine
+“collaboration” between humans and machines. In today’s modern industries,
+workers’ interactions with “smart machines” can make their jobs easier by making
+their tasks more automated and less prone to human error. In contrast, it makes the
+workplace more complicated by increasing information and communication flow
+between different systems. For example, using sensors and cutting-edge technology,
+we can collect the information we need to make accurate forecasts about the health
+of industrial machinery and carry out precise treatments. As humans must handle
+the massive amounts of data (big data) that need to be gathered, analyzed, and
+understood, the cognitive interaction effort of the machine operator rises from the
+skill level to the knowledge level [172]. Therefore, we would like to address various
+159
+
+cognitive aspects of human-machine interaction issues in Industry 4.0 and develop
+smart solutions for human workers to aid in a complex industrial scenario.
+8.2.5 Fog Computing and Advanced Analytics for Human-Machine
+Interaction in Industrial Sector
+The advancement in computing technology with industrial revolution has
+transformed the industrial operations using automation, robotics, artificial
+intelligence and other modern smart solutions. Although, various complex industrial
+operations (e.g., machine maintenance, oil well drilling operation, manufacturing
+machines) need human intervention and interaction [173, 174] to ensure precision
+and accuracy of the operation. Hence, human operator that communicate with
+machine sometimes need to process machine generated data to efficiently
+communicate with machines [175, 176]. In this case human operators can use a
+mobile device with them to process the data or visualize the data that is processed
+a nearby computing systems. Hence, fog computing can be a potential candidate to
+support the computing demands of human-machine interactions [177, 178, 179].
+Furthermore, fog computing utilizing various advanced analytics (e.g., machine
+learning, deep neural network, reinforcement learning) on machine generated data
+can provide useful insights to the human operators that can improve
+human-machine interactions.
+160
+
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+
+Biographical Sketch
+Razin Farhan Hussain received his Bachelor of Science in computer science
+and engineering in the fall of 2011 from Military Institute of Science and
+Technology, Bangladesh. He immediately began his career in February of 2012, as a
+programmer in one of the software company named “ERA Infotech Ltd”. After nine
+months of his first job, he joined one of the top multinational software company
+“Samsung R&D Institute” as a software engineer. Razin Farhan Hussain worked in
+Samsung for around 4 year and 5 months, and decided to enrich his academic
+knowledge by pursuing higher education. As such Razin Farhan Hussain started his
+Ph.D. journey in computer science in the fall of 2017 at the University of Louisiana
+at Lafayette. While pursuing his Ph.D. degree Razin Farhan Hussain received his
+M.Sc. degree in computer science in the spring of 2019. His research interests are:
+cloud computing, resource allocation in a fog federation, task scheduling, machine
+learning, DNN-based applications for Industry 4.0, and federated learning.
+Currently, Razin Farhan Hussain is working as a Software Engineer in one of the
+software company named “TryCycle Data Systems”, providing software solutions in
+healthcare sector of Canada & United States.
+178
+
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+page_content='Federated Fog Computing for Remote Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 Applications Razin Farhan Hussain A Dissertation presented to the Graduate Faculty in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy University of Louisiana at Lafayette Fall 2022 APPROVED: Mohsen Amini Salehi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Chair The Center for Advanced Computer Studies Sheng Chen The Center for Advanced Computer Studies Xu Yuan The Center for Advanced Computer Studies Li Chen The Center for Advanced Computer Studies Mary Farmer-Kaiser Dean of the Graduate School arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='00484v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='DC] 1 Jan 2023 © Razin Farhan Hussain 2022 All Rights Reserved Abstract Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 operates based on IoT devices, sensors, and actuators, transforming the use of computing resources and software solutions in diverse sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Various Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 latency-sensitive applications function based on machine learning and utilize the generated sensor data for automation and other industrial activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Sending sensor data to cloud systems is time consuming and detrimental to the latency constraints of the applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In this circumstance, fog computing can be used to support latency-sensitive applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Executing these applications across heterogeneous fog systems demonstrates stochastic execution time behaviour that affects the task completion time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, we investigate and model various Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 ML-based applications’ stochastic executions and introduce real-world execution time traces of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Remote offshore industries like oil and gas are prone to disasters requiring the coordination of various latency-sensitive activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accordingly, their procured fog computing resources can get oversubscribed due to the surge in the computing demands during a disaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, in this dissertation, we propose federating nearby fog computing systems and forming a fog federation to make remote Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 sites resilient against the surge in computing demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We propose a statistical resource allocation method across fog federation for latency-sensitive tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Many of the modern Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications operate based on a workflow of micro-services that are used alone within an industrial site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As such, industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 solutions need to be aware of applications’ architecture, particularly monolithic vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' iii micro-service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, we propose a probability-based resource allocation method that can partition micro-service workflows across fog federation to meet their latency constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Another concern in Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 is the data privacy of the federated fog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As such, we propose a solution based on federated learning to train industrial ML applications across federated fog systems without compromising the data confidentiality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' iv To my wife, Rezwana Mahjabeen, my daughter, Ruzainah Shehzeen Hussain, my parents, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Altaf Hussain and Fariha Akhter, my Sister, Sabrina Shahreen Sarna, my brother, Shoumin Rafsun Hussain and to all my friends and loved ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Acknowledgments I sincerely thank my supervisor, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Mohsen Amini Salehi, for his constant encouragement and passion for computer science and, especially, for his guidance, support, and cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Thanks to my dissertation committee, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Xu Yuan, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Li Chen, and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Sheng Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Thanks to Sm Zobaed, Ali Mokhtari, Davood Ghatreh Samani, and Chavit Denninart for their assistance in the work of this dissertation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Finally, thanks go to the Center for Advanced Computer Studies and the Graduate School at the University of Louisiana at Lafayette for their support and guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' vi Table of Contents Abstract .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' iii Dedication .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 178 xi List of Tables Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' DNN-based applications used in O&G Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 along with their network model, input data type, usage scope, and code base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 42 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Heterogeneous machine types and VM configurations in Amazon EC2 that are considered for performance modeling of DNN-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In this table, ML Optimized represents Inferentia machine type offered by AWS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 45 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Various VM flavors in Chameleon cloud are configured to represent a consistently heterogeneous system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 46 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The execution time distributions of DNN-based applications in AWS clouds machines using Shapiro-Wilk test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 48 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The execution time distributions of DNN applications in Chameleon cloud using Shapiro-Wilk test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 49 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Inference time distributions of DNN-based applications in AWS cloud machines using Kolmogorov-Smirnov test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 50 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Inference time distributions of DNN-based applications in Chameleon’s machines using the K-S test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 50 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The measurement of central tendency metric (µ), and data dispersion metric (σ) on the observed inference times in AWS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 52 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Central tendency metric (µ), and data dispersion metric (σ) of the inference times in the Chameleon cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 52 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' MIPS values of heterogeneous machines in AWS for each DNN-based application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 5 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 6 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', simulation, analytics, visualization, compute, machine learning, reporting) can be employed to store, process, and analyze sensor-generated data and to control industrial equipment in a smart oil and gas industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 15 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 18 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 20 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content='0 smart applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 35 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 43 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Schematic view of Temporal Convolutional Network (TCN) model that consists of six temporal blocks, the input data, and the output in form of the predicted AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 44 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The stochastic nature of inference execution time of oil spill application while running on heterogeneous VMs in the AWS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For every VM instance, the oil spill detection application is executed 30 times and those executions are plotted as number of attempts along x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The y-axis represents the inference time for every attempts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 47 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Comparative analysis of the MIPS values of AWS and Chameleon machines for various DNN-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For the sake of presentation, the MIPS values are normalized between [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 57 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' A Fog system with load balancer module that facilitates fog federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Task requests generated by sensors are received by the load balancer module and are assigned to the fog system that maximizes the likelihood of success for the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 66 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The impact of increasing oversubscription level (number of arriving tasks) on deadline miss rate using different task assignment heuristics in the load balancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 71 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' Mean communication latency overhead introduced to each task in fog federation by different heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 73 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' Average makespan time(seconds) of tasks using various task assignment heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 74 xv Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 79 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The flowchart of the workflow partitioning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 82 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Comparison of the partitioning techniques in terms of workflow deadline meet rate while utilizing proposed probabilistic partitioning technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The x-axis represents the increasing number of arriving workflow execution requests, whereas the y-axis represents the workflow deadline meet rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 90 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Comparison of resource allocation techniques while utilizing proposed workflow partitioning technique for microservice-based workflow applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 92 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Comparison of resource allocation techniques for monolithic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The proposed resource allocation technique MR outperforms other baselines in every application arrival trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 93 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Impact of scaling the fog federation for proposed partitioning and resource allocation techniques in increasing oversubscribed situations considering microservice applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The degree represents the number of neighbors each fog system has for executing the Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 94 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Impact of scaling the fog federation for proposed resource allocation techniques on monolithic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The degree represents the number of neighbors each fog system has for executing the Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 95 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' A federated learning setup in fog federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Multiple company share their fog systems to train oil spill detection DNN model where data security is preserved by federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 99 xvi Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Federated learning training considering class imbalance and global convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Tversky loss is used in the training considering class imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 103 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Comparison of FedAvg and FedBal training loss utilizing tversky loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The alpha parameter of tversky index is changed from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 107 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Comparison of FedBal with FedAvg, FedSGD, and FedProx method’s global model performance in IID setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 107 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The performance comparison of global models in terms of mIoU using FedAvg and FedBal methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The data distribution is non-IID, the number of workers are 6, and in each fed round 50 epochs of training has been performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 109 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Comparison of FedBal with FedAvg, FedProx, and FedSGD method’s global model performance in non-IID and unbalanced data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 110 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Comparison of FedAvg, and FedBal method’s global model performance in non-IID data distribution from high intensity(only 1 class per worker) to low intensity(3 classes per worker).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The difference of mIoU of FedBal, and FedAvg is plotted as barchart for 3 case scenarios (1 class, 2 class, and 3 class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 112 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The influence of federated worker on global models performance (mIoU) for FedBal, and FedAvg is measured by increasing the number of federated worker from 6 worker to 25 worker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For each case of worker pool 20 federated rounds of training are performed for both FedBal, and FedAvg method, and for each case maximum mIoU of both methods are considered for plotting as a barchart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 122 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 129 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 136 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 141 xviii Chapter 1: Introduction Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 is revolutionizing the utilization of computing resources across various industries [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' With the emergence of the Internet of Things (IoT) and modern computing systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', edge computing, fog computing, and serverless computing), industries are becoming more intelligent with smart sensors and actuators that create a large quantity of data [2] every day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, the computational resources required to store and analyze sensor-generated data are expensive and particularly scarce in remote areas [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In the industrial sector, various sorts of applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', machine learning (ML), reporting, alarm generators, and surveillance) employ sensor-generated data to automate or conduct complex operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Sometimes these data require real-time feedback to conduct fault-intolerant latency-sensitive activities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', drilling operation in an oil rig, workers’ safety operation, manufacturing products).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Alternatively, certain tasks need large computing capacity and are delay tolerance, necessitating cloud data center assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For instance, the “Fire safety” application [4] utilizes a deep neural network (DNN) model that needs extensive training in highly configured cloud data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Similarly, “reservoir simulation,” widely used in the petroleum industry to anticipate the field performance under varies producing strategies, requires a large quantity of seismic data and high-performance computing systems [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In remote or distant locations of the industrial sectors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', offshore oil extraction sites, solar fields), transferring the sensor-generated data to a cloud data 1 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Advanced computing systems in various smart industries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', oil and gas, healthcare, transportation) for real-time latency-sensitive tasks Fog systems Sensors Fog systems Cloud Computing Smart healthcare Smart oil and gas industry Smart transport system using vehicle to infrastructure (V2I) Satellite Edge system Edge system Edge system center is expensive and latency intensive, influencing the use of computing near the data sources [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Additionally, real-time applications require a faster response time, which is usually not feasible with cloud computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, bringing computational resources to the data sources near the end clients is an essential requirement for remote industries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' One of the solutions for computing near data sources is edge computing [7] as depicted in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1, which brings computational resources closer to the end devices, and data generation sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As such, edge computing can be defined as a distributed computing platform that puts industrial applications closer to data sources like IoT devices or local computer servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' This closeness to data at its 2 Base Station Task Processing Task Request nse Road vehicle moving left to right 米 Road vehicle moving right to leftsource can result in significant business benefits such as faster insights, faster reaction times, and increased bandwidth availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Although edge computing supports real-time latency-sensitive applications, edge devices are resource constraints that need efficient resource allocation [8] mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, another solution for computing platforms near end users is fog computing systems [9] that complement edge computing by having more computing resources, having more comprehensive middleware for managing workload efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The main driving force of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 revolution is machine learning (ML) or deep neural network (DNN) applications [10, 11, 12] that ensure efficient industrial operations and workplace safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence the ML or DNN-based applications encompass both the training and inference stages [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The training stage is generally carried out offline due to time and computing resource constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Whereas the inference execution can be completed utilizing general-purpose computing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The DNN-based applications are mainly trained on cloud data centers or computing servers with high configuration hardware (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', GPU, TPU, FPGA) support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In contrast, the inference operations are performed on the fog computing systems near the end users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As such, it is essential for a system engineer or system administrator to understand the performance of these DNN-based applications in fog computing systems [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Especially for fault-intolerance latency-sensitive critical DNN-based applications, the forecast of inference execution time within a computing resource can be significantly vital that sometimes save lives in a disaster situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As such, we perform a statistical analysis of the inference 3 execution time of various Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications on the cloud and fog systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Consequently, we introduce an execution time workload trace that can help system architects to develop a load-balancing solution robust against stochastic execution times of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 smart applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, efficiency, productivity, and industrial safety can be ensured by utilizing these robust solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In remote offshore industry, at times of emergencies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', disasters, accidents), the demand for task processing in the edge or even fog computing systems can be significantly high, leading to a drop in some tasks due to not meeting their latency constraints (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='a deadlines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As such, we propose to federate nearby privately owned computing resources by forming a federation of fog systems to support the surge of task processing requests in times of emergencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For instance, in a remote offshore smart oil field, as depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2, multiple oil extraction sites can be built by the respective companies that typically contain privately owned fog computing systems to support their regular computing demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' At a disaster time or other emergencies such as an explosion, the computing demands surge to support multiple recovery procedures to be coordinated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For instance, in a fire breakout event, various activities such as drone-based inspection, fire detection, and alert generation with precise fire locations need real-time coordination to neutralize the emergency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In this scenario, some latency-sensitive tasks can be offloaded [15] to other fog systems that may have more computational resources or less busy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, the federated fog systems’ resilience depends on supporting the surge in computing demands by efficient resource allocation across the federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, we 4 propose a probabilistic resource allocation method across fog federation for latency sensitive monolithic tasks to support computing demands in emergency situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' A remote offshore smart oil field consists of multiple oil rigs (oil ex- traction sites).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In this scenario, the oil rigs, drill ships, or even rescue ships have fog computing systems in the form of mobile data centers to support the oil extrac- tion computing demands along with any unpredictable emergencies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', oil spill detection, toxic gas detection) Smart applications in Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 can have various software architectures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', monolithic, micro-service [16]) that serve different purposes of industrial operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, micro-service architecture is one of the widely used software architectures that comprise various micro-level services having immense benefits on development and deployment [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For instance, as depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3, the “fire safety” micro-service-based application comprises video pre-processing, noise removal, feature extraction, fire detection, location mapping, and alert generation micro-services performing different activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=" In a typical industrial scenario, these micro-services are supported by various execution platforms that can have 5 'mmWave Wireless Link between Rigs 区 Edge Computing () Edge Computing Cooperative Monitoring Edge Computing Oil rig Sensors Smart Oil Field Sensors Smart Smart Well WellFigure 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' A typical micro-service application, “fire safety” execution scenario in edge-fog-cloud paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' video preprocessing noise removal input video feature extraction fire detection location mapping alert generation cloud data center fog system edge system stochastic execution latencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In contrast, a monolithic architecture is the conventional unified paradigm for constructing a software application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Monolithic software is intended to be self-contained, with firmly connected rather than loosely coupled components or services, as in a micro-service architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Although the industrial revolution influenced the utilization of micro-service applications, various industries have monolithic legacy applications that are still in operation and need to be supported by existing execution platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In an emergency or disaster, various application requests with different latency requirements are generated in the proximity of the disastrous area that needs distinct computational support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, 6 the nearby execution platform gets oversubscribed with the surge in demand for executing numerous applications on time, that can degrade the execution platform’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In this case, to support the high computation demands utilizing the proposed federated fog system need an efficient resource allocation method that is aware of receiving applications’ internal structure, computation, and communication latencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, the reliability of an execution platform in an oversubscribed situation depends on accommodating various computational demands that ensure industrial safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Federating computing resources in remote industrial areas imposes security concerns for each participant fog system of the federation, that is owned by private companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In addition, individual fog systems can have sensitive data that imposes privacy issues for the company owning the computing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, considering ML application training across the fog federation suffers from data scarcity, that is an obstacle to building accurate ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As such, a secure and privacy-preserving distributed ML training method should be in place to build an accurate ML model that can be crucial in emergency situations in Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Research Problem and Objectives The fundamental purpose of this research is to identify, evaluate, and manage robust execution of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications in remote areas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', offshore oil fields) across modern edge and fog computing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, we develop solutions that use fog systems in emergency and oversubscribed circumstances to satisfy the computational demand in remote industrial sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' This dissertation 7 address the following research challenges to enable a robust and QoS-efficient federated fog system for industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' How to utilize modern distributed computing systems in the context of remote smart industries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', oil and gas, energy) considering the industrial revolution, Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' What are the statistical execution behaviors of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications in fog systems?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' How to enable a robust federated fog computing system that can efficiently procure computing demands during a workload surge?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' How to support Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications with modern micro-service architecture along with monolithic legacy applications and maintain the Quality of Service (QoS) of a fog federation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' How to utilize federated fog securely to improve the performance of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Contributions We identified various obstacles as we investigated many facets of federated fog computing systems in the industrial sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, in addition to addressing significant challenges, we present state-of-the-art solutions and give exhaustive performance assessments for recommended methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In light of the research topics outlined in the preceding section, the considerable contributions of this dissertation are as follows: 8 Identifying the connection of industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 and modern distributed computing systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', edge, fog) and addressing the scope of utilizing advanced analytics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', artificial intelligence, machine learning, deep learning) in the context of the remote smart oil field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Performance analysis of ML-based Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications across fog and cloud computing systems?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Proposing a real-world workload benchmark of inference execution times for four different ML-based Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Enabling the notion of federated fog via resource allocation methods operating based on Bayesian probability utilizing fog systems for latency-sensitive tasks in an oversubscribed system that tries to recover from a disaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Proposing a statistical resource allocation solution across federated fog systems that is aware of internal software architecture and stochastic latency requirements of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 micro-service workflow applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Proposing a data privacy preserving ML-based Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 application training solution across federated fog system in remote industrial sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3 Dissertation Organisation Chapter 2 explores the related research works and provides background for edge & fog computing, fog federation systems, load balancing & task allocation techniques, and data privacy aspects of a federated fog system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 9 Hence, the scope of utilizing the modern distributed systems in the remote smart oil fields is addressed with various use case scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' – Razin Farhan Hussain, Ali Mokhtari, Mohsen Amini Salehi, and Ali Ghalambor IoT for Smart Operations in the Oil and Gas Industry published as a book by Elsevier (ISBN:9780323998444).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Chapter 3 studies the performance of ML-based Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications in heterogeneous cloud computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The statistical analysis of ML-based applications helped to generate a real-world Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 application inference execution time workload that can be beneficial for the system architect to develop robust load-balancing solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' – Razin Farhan Hussain, Alireza Pakravan, and Mohsen Amini Salehi Analyzing the Performance of Smart Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 Applications on Cloud Computing Systems published in Proceedings of the 22nd IEEE International Conference on High-Performance Computing and Communications (HPCC-2020) Chapter 4 explores the possible advantages and practicality of building a fog federation in a distant offshore smart oil field in the event of a disaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Using probabilistic load balancing heuristics across the fog federation for resource allocation can efficiently assure the system’s resiliency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Moreover, compared to baseline approaches, the advantage of employing probabilistic methods is backed by a synthetic workload created in EdgeCloudSim simulation[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 10 – Razin Farhan Hussain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Mohsen Amini Salehi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Anna Kovalenko,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' and Omid Semiari Federated Edge Computing for Disaster Management in Remote Smart Oil Fields published in Proceedings of the 21st IEEE International Conference on High Performance Computing and Communications (HPCC-2019) – Razin Farhan Hussain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Omid Semiari,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' and Mohsen Amini Salehi Robust Resource Allocation Using Edge Computing for Vehicle to Infrastructure (V2I) Networks published in Proceedings of the 3rd IEEE International Conference on Fog and Edge Computing (ICFEC’19) – Razin Farhan Hussain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Mohsen Amini Salehi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' and Omid Semiari Serverless Edge Computing for Green Oil and Gas Industry published in Proceedings of IEEE Green Technologies Conference(GreenTech) - 2019 Chapter 5 explores the advanced micro-service software architecture for Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications to enhance the robustness of remote federated fog systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, the load balancer should be aware of the software architecture of the receiving applications as well as the uncertainties of the execution platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a result, the distribution of receiving applications across the fog federation enhances the possibility of applications being completed on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' – Razin Farhan Hussain, Mohsen Amini Salehi Adapting Remote Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 Applications to Federated Fog Computing Systems prepared for submission to Future Generation Computing System journal in 2022 11 Chapter 6 explores the data privacy aspects of ML-based application training across the federated fog computing systems in Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Chapter 7 explores the downsides and side effects of smart solutions for Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' This chapter identifies and proposes various cutting-edge solutions for security issues of different industrial sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' – Razin Farhan Hussain, Ali Mokhtari, Mohsen Amini Salehi, and Ali Ghalambor IoT for Smart Operations in the Oil and Gas Industry published as a book by Elsevier (ISBN:9780323998444).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Chapter 8 concludes the dissertation with a discussion of our main findings and future research directions in the area of efficient utilization of fog computing platforms for Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 12 Chapter 2: Background and Literature Study 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Computing as a Prominent Aspect of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 Industrial systems are quickly transitioning from human-controlled processes to closed-loop control services supporting their operations autonomously using extensive sensor and computing infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' This revolutionary change is critical for supporting growing data-intensive and time-sensitive Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications, particularly at remote locations such as offshore Oil and Gas (O&G) fields where computer infrastructure is restricted and human resources are limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Realizing these systems necessitates interdisciplinary research and study at the interplay of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 in remote industry, modern computing infrastructure (such as an Edge and Cloud), and advanced analytics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', ML, DNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Consequently, this chapter aims to illustrate the challenges, prospects, and solutions for establishing a smart and robust remote industry based on the fundamentals of the Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a result of this study, researchers and practitioners can be more effective in making the remote industry safer, more sustainable, greener, automated, and, subsequently, more cost-efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' This chapter investigates several computer technologies that support the computing needs of distant industries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Furthermore, it explains how the synergy of cutting-edge computing solutions, such as the Internet of Things (IoT), Machine Learning methodologies, and distributed computing platforms, can be employed to improve industrial processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As an ideal example of a remote offshore industry, we consider Oil and Gas that has been transforming significantly with the industrial 13 revolution Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' On the other hand, the remote offshore O&G industry has been facing various disasters and catastrophes that raise concerns about production efficiency and safety measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For instance, the deepwater horizon (2010)[19], usumacinta jack-up disaster (2007) [20], mumbai high north disaster (2005) [21], and the ocean ranger disaster (1982) incidents are significantly connected with safety failures in the industrial sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence these incidents motivated us to improve the computing support in remote industries to ensure safety and productivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, this chapter explores various distributed computing technologies, federation-friendly execution platforms, software architecture, and security aspects of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0, focusing on the O&G industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Distributed Computing Systems in Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Cloud Computing Cloud computing is a concept that enables resources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', computing, storage, services) to be available as a service, on-demand, configurable, and also shareable [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Modern cloud systems provide diverse services in different levels, such as infrastructure as a service (IaaS), platform as a service (PaaS), software as a service (SaaS), and function as a service (FaaS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As presented in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1, smart O&G industry increasingly relies on cloud-based services that are hosted on remote Internet servers (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' cloud data centers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' These data centers are utilized to store and process their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' According to Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1, various sensor-generated data are sent to cloud providers to avail of different kinds of cloud services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Among these services, some of them send insightful 14 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Various cloud services (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', simulation, analytics, visualization, com- pute, machine learning, reporting) can be employed to store, process, and analyze sensor-generated data and to control industrial equipment in a smart oil and gas industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Sensors Sensor Generated Data Actuators response Smart Oil Field Cloud services Storage Analytics Machine Learning Compute Simulation Service Visualization Reports decisions to actuators to close the automation loop in the smart oil field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Cloud technology enables O&G companies to utilize various data-related and computational services (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', machine learning and visualization) without the need to maintain any computing infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, data privacy and security have remained a concern for such companies to fully embrace the cloud services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' These security concerns have caused a small pause and hesitation in adoption cloud services, particularly by major players in this industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' An alternative and more secure approach is to store the data on an on-premise computing facility that is known as a private cloud (more recently called fog computing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' On the positive side, cloud systems’ performance and ease-of-use are tempting for the O&G industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For instance, one of the main users of data-driven cloud services is the North American shale industry that drills thousands of wells 15 every year [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The scalability feature of cloud services helped the growing amount of data from these wells to be utilized efficiently, allowing the industry to expand remarkably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As such, various modern cloud-based data analytics services have emerged to help O&G companies to improve their operational workflows and make profitable decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Edge and Fog Computing for Remote Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 Due to the increasing importance of latency-sensitive applications, real-time operations close to the end user in remote offshore industries, the interest in the notion of edge computing has begun to increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Additionally, the proliferation of the Internet of Things (IoT) devices and smart sensors in the industrial sector results in a massive amount of data that need to be processed locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In a typical scenario, the data is transported to cloud data centers [24], and responses or results are transmitted back to clients through the internet, both of which take time and money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a result, a distributed computing paradigm has been introduced, which is located close to the end client and processes client data at the network’s edge [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Researchers call this type of computing “Edge Computing” since it operates at the network’s periphery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The conventional definition of edge computing is difficult to come by.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Different organizations or sources have different definitions, heavily impacted by context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The general perception of edge computing is to provide various computing services (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', application execution, data pre-processing) through distributed computer systems instead of centralized cloud data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, edge computing 16 enables analysis and knowledge collection at the point of information source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In network design, an “edge computer” is located directly next to or even on top of network endpoints (such as controllers and sensors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The data is then partially or fully processed before being transmitted to the cloud for storage or further processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Edge computing, on the other hand, may result in the direct transfer of huge volumes of data to the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' This might have an impact on system capacity, efficiency, and security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Fog computing [26] solves this problem by inserting a processing layer between the edge and the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a result, ‘fog computing’ collects and analyses data at the edge before it reaches the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, the place from the data source where computing service is offered can be a defining element in distinguishing Fog/Edge computing from cloud computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For instance, a renewable energy company geared with numerous sensors utilizes fog computing for sensor-data analysis in their operational fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accordingly, company‘s production efficiency improved by 15% by reducing data analysis latency from 10 minutes to few seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, fog computing placed near data source in remote industries can enhance efficiency in production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The emergence of edge and fog computing does not substitute the cloud computing services;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' instead, it brings some portion of those services (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', computing, storage, analytical services) near the end clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Especially with the ever-growing Internet of Things (IoT) devices, a considerable amount of data is generated [2] that is significantly valuable for Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 ML applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The generated data sometimes need immediate processing (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', edge computing 17 support), and alternatively, sometimes need complex processing (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', cloud computing support).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, a continuous computing platform (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', Edge-to-Cloud Continuum [27]) is required to support both real-time nature and complex analytical tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Edge-to-Cloud continuum for oil and gas industry as an example of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The continuum is mainly divided into four tiers, namely end devices, edge, fog, and cloud data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The bottom of the triangle has end devices that are energy limited, whereas traversing to the top, we find more energy-consuming systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Latency, Elasticity, Computing Power, Centrality, Un- trustworthy Edge Fog (Cloudlet) Cloud Data Centers Edge-to-Cloud Continuum Devices Sensors Worker Equipment Actuators PDA Smart Gateway Smart Phone Laptop ASCI Device Drone Pressure Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Smart Helmet Safety Vest Smart Watch Camera Robot Energy Limited Medium Energy Energy Hungry 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3 Edge-to-Cloud Continuum Although edge and cloud computing has a difference in terms of distance and resources, they can be utilized as a complement to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For a massive industry such as O&G, diverse operations and services are needed that require various underlying computing platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, the integration of edge or fog computing with cloud computing is a need of time that reflects the usability of the 18 edge-to-cloud continuum[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accordingly, the Edge-to-Cloud continuum is a service platform that provides various computational resources and infrastructures for supporting different types of services essential for O&G industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 demonstrates the Edge-to-Cloud continuum as a triangle where edge devices reside close to end devices (bottom of the triangle) and cloud data centers are the furthest computing entity from end devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accordingly, this is a hierarchical arrangement that is distributed vertically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence cloud computing has high latency than edge and fog computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Alternatively, cloud computing has high availability in terms of elasticity and computing power, whereas edge and fog devices are highly secure and privacy-preserving than cloud computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, various computing platforms within the continuum serve different purposes for industrial operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4 Use Case of Edge-to-Cloud Continuum in Smart O&G We investigate drone-based pipeline inspection scenarios in the oil and gas industry to understand how the edge-to-cloud continuum supports computing demands in the industrial sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Let’s consider a scenario where 4K drone-mounted cameras can collect hundreds of gigabytes of data per hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The current method of analyzing data is to transfer the massive data to the cloud data center, which is cost-prohibitive and impractical, especially if the analysis is real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence one critical question Is it feasible or scalable for the future to have any cloud vendor send their container truck with petabytes of storage?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' An example of the same scenario (presented in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3) from the oil and 19 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Drone-based inspection scenario where drone captures images and real- time analysis can be performed in edge computing resource whereas long term analysis is performed in distant cloud computing facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Drone capture images and pre- process them Edge computing: light weight processing Cloud computing: complex analysis 1 2 3 Pipeline Fracture gas industry perspective is that the drone-based inspection system could use multi-stage value extraction using an edge-to-cloud continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The O&G pipelines can be thousands of miles long and pass through an immense landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Pipe sections are generally fitted with analog gauges and smart sensors to measure pressure, flow, and other metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' By employing an edge AI-enabled surveillance drone to capture these analog gauge images presented in step 1 of figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3, it is possible to separate (step 2) the gauge images and transfer only that critical information to the next compute layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Here, data pre-processing (image separation) is real-time nature that is performed in the edge computing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore only the localized necessary data is processed for an accurate reading in the cloud data center (step 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then the output of actionable intelligence is sent to 20 the on-site maintenance team to resolve pipeline fracture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, data pre-processing, lightweight processing, and complex analysis are performed across the edge-to-cloud continuum to conduct efficient drone-based pipeline surveillance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5 Landscape of Computing in O&G Modern computing systems, such as edge, fog, or cloud enable the smooth operation of different fault-intolerant processes across different sectors of the O&G industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a cyber-physical system, the computing technology stack of the O&G industry is composed of the following components: Sensors: Numerous sensors of different types (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', to gauge pressure, emission of toxic gases, security cameras, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=') continuously procure multi-modal data in the form of signal, text, images, video, and audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The data is stored or communicated for offline or online processing to monitor the operation of the oil field or to make management decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Computer networks: In a smart oil field, short- and long-range wireless and wired computer networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', Bluetooth, CBRS, satellite, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=') have to be configured for low-latency and high data-rate communication of devices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', sensors, servers, and actuators) both for onsite and offsite communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Computing systems and middleware: All the collected data have to be eventually processed to be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' That is why, in the back-end, smart oil fields are reliant on different forms of computing systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', HPC, cloud, fog, edge, and real-time systems) to perform batch or online data processing 21 for purposes like monitoring, visualization, and human-based or automatic decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Data processing and software technologies: The rule of thumb in a smart oil field is that “the more data can be processed, the more informed decisions can be made”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The large amount of multi-modal data (text, images, video, and signals) continuously generated in a smart oil filed form what is known as big data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Such diverse formats of big data have to be processed using various algorithmic techniques, particularly Machine Learning, to provide an insight from the data or to make informed decisions upon them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Actuators: Once a decision is made, it is communicated to an actuator (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', drilling head and pressure valve) to enact the decision (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', increase or decrease the pressure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3 Smart O&G: Data and Software Aspects 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Big Data in the O&G industry The oil and gas industry generates a large volume of data on a daily basis, necessitating the need for large-scale computing resources and the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The three key sources of such considerable data in the O&G industry are as follows:: Hydrocarbon reservoirs are commonly found between 5,000 and 35,000 feet below the Earth’s surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' High-resolution images and expensive well logs are the main options for finding and characterizing reservoirs (after the wells are dug).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 22 Fluids must pass through complex rock to reach the wellbore, and the fluids themselves are complex, having many different physical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, learning about the unique characteristics of each oil well and evaluating the extracted fluid to treat it properly necessitates collecting vast amounts of data via sensors installed in the oil well and on the drill-head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Oil production entails environmental and human safety hazards, and preventing it requires significant sensor deployment across a large geographical region to gather data regularly and therefore be able to respond rapidly to any ecologically polluting discharge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Big data analytics aids in the automation of critical oil and gas operations, such as exploration, drilling, production, and delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The upstream sector, which consists of exploration and drilling, is the most dominant data source among all other sectors, owing to the increasing use of big data analytics for detecting non-conventional shale gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Furthermore, the oil and gas industry is becoming more volatile due to fluctuating oil prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a result, in addition to the engineering team, business teams are increasingly adopting a data-driven strategy to forecast the market and mitigate risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Machine Learning as a Data-driven applications in O&G The smart O&G industry is a subset of the Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 revolution, supported primarily by artificial intelligence (AI), IoT, and cutting-edge computing systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', edge, fog, and cloud computing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' An extensive range and volume of 23 relevant data are acquired from many sectors of the O&G industry due to the widespread adoption of smart sensors and actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' These data may be evaluated using machine learning models to derive valuable insights and knowledge for the industry and the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a result, in a broad sense, AI is a vital tool for transforming sensor-generated data into new and valuable information and knowledge via Edge-to-Cloud computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The term “data-driven approaches” refers to an arsenal of techniques that can be used to combine different kinds of data, evaluate uncertainties, spot trends, and recover useful facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Data-dominated software applications running on ML and Deep Neural Network (DNN) models, such as oil production control and emergency surveillance systems [29, 30, 31, 32, 33], have emerged as the fundamental pillars of the Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 revolution [34, 35, 36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Especially in remote areas where there is a need for real-time closed-loop automated processes of these applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The ML-based solutions often take the shape of micro-service processes, each of which may have one or more critical paths that together determine the latency of the whole application [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' These applications require: A large amount of data to be collected in real-time Seamless communications of sensing data despite wireless link uncertainties Dependable execution of ML applications with latency constraints in the face of unexpected load surges (for example, during emergencies) Transparent deployment and provisioning of applications and resources (also 24 known as “serverless”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Tackling these needs may be difficult, particularly in out-of-the-way places (such as offshore oil fields) with inconsistent connectivity and unstable access to cloud services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' These communication and computation constraints become crucial when a remote system must handle massive volumes of data in real-time to manage several facets of an emergency circumstance (for example, an oil spill).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' While micro datacenters (also known as fog systems) are employed to meet the computing demands of such distant systems, their capabilities are sometimes inadequate to deal with the real-time data transport, and processing demands of the load spike [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In the following subsection (ref:edgeAi), we revisit the difficulty posed by limited resources for processing surge in computing demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3 Digital Twin: Another Data-driven Applications in O&G The term “digital twin” (DT) refers to a computer simulation of an existing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Input to the twin may be set from the sensors collecting data from real-world imitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The twin may then offer real-time feedback to the management about the predicted performance or other repercussions by stimulating the physical object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' DT is a data-dominant application that operates based on Machine Learning and the scalability of cloud computing to bring the goal of data integration closer to actuality [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The importance of data in a DT system cannot be overstated since it is required for many different types of analysis, prediction, and automation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' High-quality, verified, and referenced data is required to produce a practical duplicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Since the DT operates in real-time, all previously collected data 25 and models must remain accurate, and up-to-date [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' By enabling operators and management in the O&G sector to transform enormous amounts of data into insights that might make asset failure predictable and hidden revenue opportunities revealed, DT systems can contribute to operational excellence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4 Edge-to-Cloud for AI and other Data-driven Applications in Smart O&G The wide variety of sensors that communicate through heterogeneous protocols like Modbus, CAN bus, PROFINET, and MQTT [41] makes it challenging to operationalize an Edge-to-Cloud continuity with local appliances linked to sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' It is already difficult to implement, with hundreds of agencies and oil rigs involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In addition, the next generation of cloud-native apps needs different machine learning (ML) frameworks, configurations, and requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Furthermore, applications need to be interoperable to function on a variety of devices with diverse processing capabilities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', CPU, several kinds of GPU, ASICs, and FPGAs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In addition, the human aspect of IT operational technologies, developers, and data scientists all need to join together to manage the IoT application deployed in the edge-to-cloud continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Thus, the primary difficulties throughout the Edge-to-Cloud spectrum might be summed up as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Connecting a huge number of IoT devices, as well as the edge and the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Costs associated with wireless communication technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Having access to high-quality computer resources on demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 26 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Wireless connections that are stagnant, inconsistent, or not operating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The need for real-time operation of ML-based and other data-driven applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', digital twin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Data integrity and privacy across Edge-to-Cloud systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The Edge-to-Cloud continuum problems for the O&G industry are broad, complicated, and distinct from traditional solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a result, petroleum professionals and technological specialists are the primary driving forces in developing lucrative eco-friendly solutions for the smart O&G industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Meanwhile, academic publications, research papers, and books addressing the junction of petroleum and computer science domains are uncommon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore narrowing the gap between knowing the issue space and providing efficient solutions can help the industry to be more productive and safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5 Federated Fog and It’s Challenges in Remote Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 The earlier sections of this chapter demonstrate the edge-fog-cloud continuum in a hierarchical arrangement where computing resources are distributed vertically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, multiple tiers of execution platforms can be conceptualize where higher tier (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', cloud) imposes significant latency that may not suitable for latency-sensitive tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accordingly, we investigate the horizontal scalable execution platform, fog systems, in a peer-to-peer arrangement to reduce latency issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In the industrial sector, fog computing systems are typically located in close proximity that sometimes potential candidates for forming a federation in a peer-to-peer 27 setting to support computing demands in emergencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For instance, multiple oil rigs with drillships[42] can be deployed near an offshore hydrocarbon reservoir to extract oil having their private fog systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Moreover, rescue ships with mobile data centers at disaster time comprise fog systems deployed near disastrous areas whose computing ability can be augmented by forming the fog federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, it is feasible to assume that some fog systems are underutilized and can support more task processing than their day-to-day requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Thus, efficient resource allocation across federated computing systems can increase the federated system’s quality of Service (QoS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In a related study, [43], Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='offer a resource allocation instance for edge computing platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' It uses a decoupled architecture that separates infrastructure management at Edge Computing Infrastructures (ECIs) from service delivery and administration by service providers (SPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In addition, the authors offer an auction-based resource contract mechanism and a latency-aware scheduling approach that optimizes edge computing systems and service providers’ utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, federating edge computing systems with efficient resource allocation may be used in an emergency to accommodate a spike in task requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, several other problems should be addressed to establish a robust and efficient edge federation in an emergency or disaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The main challenges can be addressed in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Real-time Services of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 To improve the response time of latency-intolerant services (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', sensor data analysis, production monitoring), fog computing systems have been exploited in the 28 literature from the network latency perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Lorenzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' [44] proposed a resource allocation model for wireless edge systems that harvest unused resources of mobile devices to mitigate network congestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The proposed model utilizes solutions at the physical, access, networking, application, and business layers to reinforce network robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' This work solely considers networking latency and not end-to-end latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In [45], Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='proposed an optimized resource migration scheme from mobile IoT devices to a heterogeneous Cloud-Fog-Edge computing environment that is aware of the resource-constrained nature of edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' It focuses on the performance gain of process migration and assigns tasks based on their run time expectations on the participating systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Heterogeneous Fog Systems in Remote Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 Fog systems in remote industries can be heterogeneous, and the research community addresses two forms of heterogeneity: consistent and inconsistent heterogeneity [46], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Consistent heterogeneity occurs when the same kind of machine has different computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Inconsistent heterogeneity occurs when various types of machines have disparate computational capacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The requested job may have different execution times depending on the heterogeneity, which substantially impacts the task completion time in an edge system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The problem of heterogeneous data acquisition from sensors in various sectors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', upstream, midstream, downstream) of smart oil fields is addressed in [47] where khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='proposed an IoT-based architecture to enable the data acquisition process more simple, secure, robust, reliable and quick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' There are several other works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', 29 [48, 49, 50]) that either do not consider the emergency (oversubscription) or ignore the uncertainties that exist in federated fog environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In another related work [51] by the same author, the main focus was on optimizing the wireless network while no resource allocation was performed at the fog system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, considering both computing and communication latencies is critical to maintaining the QoS of a fog federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3 Uncertainty of Task Completion in Fog Systems The primary uncertainty of task completion in a fog federation is influenced by execution and communication latencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, execution uncertainty mainly refers to the computational resources that execute the assigned task, whereas communication uncertainty is primarily rooted in network systems, especially the upload and download time uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Both of these uncertainly significantly influence task completion within a fog federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='addressed uncertainty in a similar study [52], mentioning that the execution uncertainty caused by performance degradation, service failure, and new service additions remains a significant barrier to the user’s service experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To overcome the uncertainty, this study proposes a software-defined network (SDN)-based fog computing architecture and a dynamic resource provisioning mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Furthermore, the nondominated sorting genetic algorithm-III is used to maximize two objectives, namely energy consumption and completion time, to produce balanced scheduling strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In another related paper [53] on resource allocation and uncertainty, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='suggested a multi-objective optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Three parallel methods have been 30 developed to increase latency, performance, and resource management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' First, a queuing model was investigated in conjunction with task buffering, offloading, and resource allocation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The authors designed the resource allocation strategy using Lyapunov drift [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' An exchange between latency and throughput is found in outcomes for improved system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Various software architecture for Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 smart applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Seis- mic analysis is represented as a monolithic application, whereas fire safety application exhibit micro-service architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' user user data pre- processing fire detection alert generation seismic analysis monolithic micro-service fire safety MS-1 MS-2 MS-3 seismic modeling Vs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4 Software Architecture of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 Applications The industrial revolution has created the demand for emerging smart applications with different software architectures, as depicted in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, smart applications consist of various micro-services that can be separately deployed with the least amount of administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For instance, a “fire safety” application based on micro-service architecture comprised of data pre-processing, fire detection, and alert generation can be deployed in remote industries to ensure the safety of 31 Thnsmission of Data centre earthquake signal (speed of light) Sesmigeations Transmission of earthguake early warning message (speed of light) S-waves (approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5km/s) Blind Zone Hypocenter/ P-waves(approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6km/s) seismicfocusonsite workers from fire hazards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, the rise of micro-service architecture was mainly introduced to reduce the complexity of large monolithic applications with huge code-base [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, the research community suggests maintaining the size of micro-service applications optimal [55, 56, 57], not too large, that can impose complexity in administrating application workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In contrast, considering old operational systems, the centralized cloud can only support legacy applications [58, 59] with huge latency-tolerant nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In contrast, modern Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications are latency sensitive that need a dynamic execution platform to enable smartness and support swift response time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As such, Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='in [60] proposed a dynamic runtime for smart industrial applications that utilize 5G technology with edge-cloud architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' This work uses application-specific knowledge to map the micro-services into the execution platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, authors consider only the critical path’s latency ignoring various generic micro-services that could play an important role in completing the smart solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Additionally, this work considers utilizing cloud data centers to ignore emergency and oversubscribed situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Similarly, Faticanti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='in a related study [61], analyzes the throughput needs of micro-service applications while offloading to various fog systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The authors in this work addressed resource allocation challenges for the fog-native application architectures built on containerized micro-service modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Two cascading algorithms make up the entirety of the answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The first one separates fog application components according to throughput, whereas the second governs application orchestration across geographically distributed data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6 The Scope of Fog Federation in Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 The smart industry’s numerous sensors create massive volumes of data that are often not analyzed due to a lack of storage and processing capabilities [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Alternatively, only some of the data is relevant to any analytical findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a result, data pre-processing and filtering of noises and anomalies may be performed in the fog federation [63], leading to effective training of ML models in cloud data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Augmented reality (AR) and real-time video analytics need a quick response and efficient, secure storage systems that fog federation can support [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For instance, a significant processing delay may confuse a process engineer to perform fault-intolerant work, leading to an accident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence AR systems supported by fog computing can maximize throughput and reduce latency in both processing and transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accordingly, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Ha, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='in [65] design and implement a wearable cognitive assistance spanning backed up by Google Glass and Cloudlet that assists the user by providing hints for social interaction via real-time scene analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To ensure security and safety, an immense amount of camera sensors are deployed in smart industries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', oil and gas, transportation, manufacturing) that perform surveillance 24/7 to detect any anomaly and monitor the hazardous area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, the captured video needs storage and computational services that can be supported by fog federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In addition, videos’ live streams, transcoding, and ML processing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', object detection, classification, object tracking) are more frequent in Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' After completing the required services with captured 33 videos, the response can be sent to users in the form of notification, events, description, or video summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence fog federation can be useful for achieving real-time processing (inference) and feedback on a huge amount of video streaming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In addition, scalability can be ensured on low-bandwidth output data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Furthermore, privacy-preserving techniques can also be applied at the fog side to ease the concern of personal privacy leakage in public surveillance systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7 Data Privacy Aspects of a Federated Fog Computing System The technological advancement in smart IoT devices and smartphones has increased the possibility of using end devices for various complex ML applications, especially training ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The ever-growing power of end devices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', mobile phones, PDAs, laptops, wearable) in computing and communication makes the complex Ml model training possible in fog devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, considering the fog federation, training with various fog systems’ local data in a distributed manner can enrich the ML model’s accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Although, data security and privacy are the major challenges in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, federated learning [66] is one solution that shares the ML model rather than data that does not leave the owner’s fog system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In federated learning, a global model is sent (global model broadcast) to the participating workers’ system to train with their local data as presented in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' After a certain training period, the updates are sent back to the central server to incorporate the updates into the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then the updated global model is again sent back to the participating FL workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The process continues until the global model achieves a 34 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' A typical federated learning scenario that consists of FL workers and a central server having the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' At the beginning of the training, the global model is broadcast to the participating workers to train with their corresponding local training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' After a period of training in FL workers, the updated model is sent back to the server for integration with the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Central server Model Updating FL workers Global model broadcasting certain accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Different techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', fedAvg, fedSGD, fedProx) can be utilized considering the global model’s accuracy to incorporate the updates from FL workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Furthermore, considering the heterogeneity of FL workers’ computation and communication capability, the updates can be generated at different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accordingly, two different types of FL techniques are considered in the literature: asynchronous and synchronous FL, respectively [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Considering the various time to generate updates by the federated worker, some stragglers need to catch up to the certain period of sending the updates to the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, asynchronous FL tries to incorporate as many updates as possible, whereas synchronous updates discard the updates that lag behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Major Challenges of FL in Fog Federation The federated learning technique in fog federation ensures the ML services 35 while preserving the privacy of the data owners to the end clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, due to heterogeneous fog devices and data anomaly, some major challenges need to be addressed that are as following: Class Imbalance Issue in Training Data: In FL technique, various FL workers’ local data are utilized for ML network model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, it is possible to have class imbalance issues within some participating workers’ local data that can impact the global models’ robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Communication Cost for Aggregating Updates into Global Model: To perform FL training, the global model needs to be transferred to participating workers via the internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' After training, the updates are sent back to the server for synchronization with the global model, and finally updated global model is sent to the FL workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' All the transfer operations utilize internet protocol which can incur a huge amount of communication cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Efficient Management of FL Workers: The number of participating FL workers can be huge where unexpected network connectivity and heterogeneous communication protocol make the management scenario nearly impossible[68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8 Downside of Smart Solutions in Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 Advances in hardware and software technology have evolved the oil and gas sector into a completely automated and machine-dependent industry [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Although 36 this digital revolution enhances production efficiency, it may produce numerous types of vulnerability and side effects that can lead to catastrophic incidents such as hazardous gas emissions, fire dangers, and oil spills [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Furthermore, the constant advancement of technology opens the potential to hack into information technology (IT) platforms [71] that deals with diverse industrial data and communication with the outside network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Another critical technology stack is the operational technology (OT) platform [72], primarily concerned with direct oil and gas production and processing operations with limited external access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, the bridge between the IT and operational technology (OT) platforms , in particular, raises cyber-threats to oil and gas operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a result, while creating smart technology for oil and gas, researchers must study or be cognizant of the drawbacks of smart solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a result, new and current smart solutions should contain better security approaches to ensure the system’s reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Furthermore, the possible side effects of smart solutions might impede operational efficiencies and become counter-productive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accordingly, it is necessary to explore and identify various vulnerable areas of IT and OT platform as well as their interplay aspects in structural categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, cyber-threats and device incompatibility should be addressed properly to identify various open doors for cyber criminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' One of the issues issue with the oil and gas industry is that it relies on systems that were not designed with network connectivity in consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Industrial plants, for example, were never designed to be connected to networks, but with the continuous digital revolution, they are today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' This can lead to a risky scenario since a cyber-attack on such a system can 37 impair operations and cause the death of life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The industrial revolution has increased the utilization of various types of machines that robots or human workers operate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Moreover, these machines sometimes communicate with other machines to complete an industrial operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, machine-machine and human-machine interactions can go wrong and create opportunities for cyber criminals to sabotage industrial processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As such, identification of industrial interaction challenges can help to build smart solutions that are safe and secure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Finally, developing any physical or software solution requires human and machine involvement that leads to the engagement of various biases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', artificial intelligence, automation, and human-related biases) in smart solutions of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' These biases can lead to unwanted accidents or loopholes for cyber criminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, addressing different forms of bias in industrial sectors can help build smart solutions that are resilient to cyber-threats and attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='9 Summary and Positioning of this Dissertation This section introduced the Edge-to-Cloud continuum and federation of fog computing paradigms and their goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' First, we discuss various scopes to utilize the Edge-Fog-Cloud continuum for different Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications, especially real-time nature and machine learning (ML) based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then in chapter 3, we analyze the performance of various Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications in widely used AWS cloud and Chameleon fog servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' After that, we investigate the challenges of federated fog systems and suggest a statistical resource allocation method across federated fog systems for monolithic workloads in remote industrial sites in chapter 38 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then in chapter 5, we explore the micro-service software architecture of the Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications and propose a probabilistic partitioning and resource allocation method to improve the robustness of the fog federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' After that, we study the data security and privacy aspects of fog federation by addressing state-of-the-art challenges in privacy-preserving ML-application training for the oil and gas industry in chapter 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Finally, in chapter 7, we identify the downsides of smart solutions and suggest state-of-the-art solutions for the remote oil and gas industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In the end, we conclude the dissertation by disclosing a summary of our findings and future avenues to explore in chapter 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 39 Chapter 3: Performance Analysis of DNN-based Application in Cloud and Fog Systems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Overview This chapter analyzes the performances of Deep Neural Network (DNN)-based Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications to study the inference execution times on cloud and fog computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Being an indispensable part of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0, DNN-based smart applications make the latency-sensitive inference that needs to be accurate and execute certain application constraints with a specific deadline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The quality of service(QoS) could be compromised due to missing each application’s deadline even if the inference accuracy is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Due to the multi-tenancy and resource heterogeneity inherent to the cloud and fog computing environments, the inference time of DNN-based applications is stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Such stochasticity, if not captured, can potentially lead to a disaster in critical sectors, such as Oil and Gas industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To make Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 robust, solution architects and researchers need to understand the behavior of DNN-based applications and capture the stochasticity that exists in their inference times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accordingly, in this study, we provide a descriptive analysis of the inference time in the popular cloud platform, Amazon, and in Chameleon as Fog system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We employ two statistical methodologies to evaluate DNN-based applications: application-centric and resource-centric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' First, we begin with an application-centric analysis in which we statistically model the inference execution time of four categorically unique DNN applications executing on both Amazon and 40 Chameleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Second, we examine a rate-based indicator known as Million Instruction Per Second (MIPS) for heterogeneous cloud and fog systems using a resource-centric approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The confidence interval of MIPS for heterogeneous cloud and fog systems is then estimated using non-parametric modeling approaches such as Jackknife and Bootstrap re-sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The findings of this work might help academics and cloud solution architects build robust solutions against the stochastic nature of inference time in the cloud, allowing them to deliver higher QoS to their users while avoiding unanticipated repercussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Furthermore, we provide a DNN-based applications benchmark a for system architects to employ in building effective resource allocation solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 DNN-Based Applications in Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 Among various DNN-based applications utilized in Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0, we consider four different applications used in the smart O&G industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The summary of the chosen applications is demonstrated in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1, which presents the abbreviated name for each application, its DNN (network) model, the type of its input data, the scope of deployment in O&G Industry [73], and the code base to build the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The applications’ code base, input data, and analysis results are publicly available for reproducibility purposes in the GitHub repository mentioned earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In the rest of this section, we explore the characteristics of each application type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Fire Detection The fire detection application is an essential component of monitoring ahttps://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='com/hpcclab/Benchmarking-DNN-applications-industry4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 41 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' DNN-based applications used in O&G Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 along with their network model, input data type, usage scope, and code base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Application Type DNN Model Input Type Scope Code Base Fire Detection (Fire) Customized Alexnet Video Segment Control & Monitoring Tensorflow (tflearn) Human Activity Recognition (HAR) Customized Sequential Neural Network Motion sensors Workers Safety keras Oil Spill Detec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' (Oil) FCN-8 SAR Images Disaster Management keras Acoustic Impedance Estimation (AIE) Temporal Convolutional Network Seismic Data Seismic Exploration PyTorch systems designed to provide safety and resilience in Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We used a convolutional neural network (CNN) to investigate a fire detection DNN-based application proposed by Dunnings and Breckon [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' It identifies fire areas (pixels) in real-time in the frames of a monitored video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We use the FireNet model, which correctly identifies and locates fire in each frame of a given video segment, among the several fire detection models offered by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' FireNet is a simplified version of the AlexNet model [75], with three convolutional layers of sizes 64, 128, and 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To obtain high-frequency features with a significant response from the preceding layer, each convolutional layer in this model is enhanced with a max-pooling layer and a local response normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We created a benchmarking dataset of 240 videos with varied backgrounds to examine the inference time of the fire detection application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' All videos are regarded as two seconds long for a fair and accurate appraisal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Human Activity Recognition Human Activity Recognition (HAR) systems are widely used in Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 to ensure workers’ safety in hazardous zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In the HAR system, various sensor 42 data are analyzed that are generated from different sensors used by human workers while performing any physical movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In this case, motion sensors, such as accelerometers and gyroscopes, that are widely available on handheld PDA devices are utilized to capture human activity-related sensor data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The HAR system we use operates based on the sequential neural network model with four layers to identify the worker’s activities (walking, walking upstairs, walking downstairs, sitting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For analysis, we use a dataset of the UCI machine learning repository, known as Human Activity Recognition Using Smartphones [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The FCN-8 model is presented in block diagram that consist of 5 fully convolutional network blocks, and 2 up-sampling blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The model receives input as a SAR image and perform pixel-wise classification to output a labeled image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' FCN-Block 1 FCN-Block 2 FCN-Block 3 FCN-Block 4 FCN-Block 5 MP-1 MP-2 MP-3 MP-4 MP-5 Convolution Convolution Upsample Upsample U*U*U Input SAR Image captured from Satellite Output Labeled Image Look-alike Oil Spill Sea surface 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3 Oil Spill Detection Detecting the oil spill is of paramount importance to have a safe and clean O&G Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The accuracy of DNN-based oil spill detection systems has been promising [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We utilize a detection system that operates based on the FCN-8 43 model [78], which is depicted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As we can see, the model contains five Fully Convolutional Network (FCN) blocks and two up-sampling blocks that collectively perform semantic segmentation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', classifying every pixel) of an input image and output a labeled image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The FCN-8 model functions based on the satellite (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' SAR) [79] images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We configure the analysis to obtain the inference time of 110 SAR images collected by MKLab [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Schematic view of Temporal Convolutional Network (TCN) model that consists of six temporal blocks, the input data, and the output in form of the predicted AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Input Seismic Traces from Marmousi Model Temporal Block (1,3) Temporal Block (3,5) Temporal Block (5,5) Temporal Block (5,5) Temporal Block (5,5) Temporal Block (5,6) Concatenation Linear Layer Temporal Convolutional Network Output Predicted Acoustic Impedance (AI) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4 Acoustic Impedance Estimation Estimating acoustic impedance (AI) from seismic data is an important step in O&G exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To estimate AI from seismic data, we utilize a solution functions based on the temporal convolutional network [80], shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The solution outperforms others in terms of gradient vanishing and overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Marmousi 2 dataset [81] is used to estimate AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3 Computing Platforms for Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Amazon Cloud AWS is a pioneer in the Cloud computing industry and provides more than 175 services, including Amazon EC2 [82], across a large set of distributed data 44 centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Amazon EC2 provides inconsistently heterogeneous machines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', CPU, GPU, and Inferentia) in form of various VM instance types (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', general purpose, compute-optimized, and machine learning (ML)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Within each VM type, a range of VM configurations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', large, xlarge, 2xlarge) are offered that reflect the consistent heterogeneity within that VM type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To realize the impact of machine heterogeneity on the inference time of various applications, we choose one representative VM type of each offered machine type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 represents the type of machines and their specification in terms of number of cores and memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We note that all the machine types use SSD storage units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Although General Purpose machines are not considered suitable for latency-sensitive DNN-based applications, the reason we study them is their similarity to the specifications of machine types often used in the fog computing platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As such, considering these types of machines (and similarly m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='small in the Chameleon cloud) makes the results of this study applicable to cases where fog computing is employed for latency-sensitive applications [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Heterogeneous machine types and VM configurations in Amazon EC2 that are considered for performance modeling of DNN-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In this table, ML Optimized represents Inferentia machine type offered by AWS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Machine Type VM Config.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' vCPU GPU Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' (GB) Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Optimized r5d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='xlarge 4 0 32 ML Optimized inf1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='xlarge 4 0 8 GPU g4dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='xlarge 4 1 16 General Purpose m5ad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='xlarge 4 0 16 Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Optimized c5d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='xlarge 4 0 8 45 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Various VM flavors in Chameleon cloud are configured to represent a consistently heterogeneous system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' VM Config.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' vCPU Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' (GB) m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='xlarge 8 16 m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='large 4 8 m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='medium 2 4 m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='small 1 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Chameleon as Fog Computing System Chameleon [84] is a large-scale public computing platform maintained by National Science Foundation (NSF) that usually utilized for academic research purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Due to Chameleon’s maintenance issues (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', transient failures, unexpected downtime, resource scarcity), less large scale VM flavors, and distributed nature, we consider Chameleon as Fog computing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Chameleon supports VM-based heterogeneous computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' It offers the flexibility to manage the compute, memory, and storage capacity of the VM instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In this study, we use the Chameleon to configure a set of consistently heterogeneous machines (Fog Systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We configure four VM flavors, namely m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='xlarge, m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='large, m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='medium, and m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='small, as detailed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We note that VMs offered by Chameleon uses HDD unit as storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4 Environmental Setup for Performance Modeling The focus of this study is on latency-sensitive DNN-based applications that are widely used in Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accordingly, we consider a dynamic (online) system that is already loaded with pre-trained DNN-based applications, explained in the previous section, and executes arriving requests on the pertinent application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' This 46 means that we measure the hot start inference time [85] in the considered applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The DNN-based applications mostly use TensorFlow, and the VMs both in AWS and Chameleon are configured to use the framework on top of Ubuntu 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The stochastic nature of inference execution time of oil spill application while running on heterogeneous VMs in the AWS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For every VM instance, the oil spill detection application is executed 30 times and those executions are plotted as number of attempts along x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The y-axis represents the inference time for every attempts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 1 10 20 30 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6 Inference Time(s) Compute Optimized 1 10 20 30 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5 General Purpose 1 10 20 30 Number of Attempts 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 GPU Instance 1 10 20 30 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 ML Optimized 1 10 20 30 16 17 Memory Optimized Our initial evaluations in AWS (shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3) demonstrate that, in different attempts, the inference execution time of an application (Oil Spill) on the same machine type can be highly stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Similar stochasticity is found for chameleon cloud while we run the oil spill detection application 30 times within same VM instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence to capture this randomness (aka consistent heterogeneity) that is caused by several factors, such as transient failures or multi-tenancy [86, 87], we base our analysis on 30 times execution of the same request within same VM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5 Application-Centric Analysis of Inference Time 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Overview In this part, we capture the inference time of the four DNN applications and try to identify their underlying statistical distributions using various statistical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then, to describe the behavior of inference execution time using a single metric, we explore the central tendency of the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Statistical Distribution of Inference Execution Time Among various statistical methods, normality tests are widely employed to understand the distribution of the collected samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, we first use the Shapiro-Wilk test [88] to verify the normality of the inference time distribution on each machine type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Next, we employ the Kolmogorov-Smirnov test [89] to find the best fit distribution based on the sampled inference execution times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Shapiro-Wilk test to verify normality of the sampled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The null hypothesis is that the inference execution times are normally distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To understand whether a random sample comes from a normal distribution, we perform the Shapiro-Wilk test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The result of this test is considered as W, whose low value (lower than wα threshold) indicates that the sampled data are not normally distributed and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The value of W is used to perform the significant testing (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', calculating P-value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The higher P-value, especially greater than a threshold value (typically 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='05), supports the null hypothesis that the sampled data are normally distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The execution time distributions of DNN-based applications in AWS clouds machines using Shapiro-Wilk test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Execution Time Distribution with Shapiro-Wilk Test in AWS Cloud App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Type Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' ML Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' GPU Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Pur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Compt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Fire Not Gaussian (P=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='73e−16) Not Gaussian (P=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='42e−16) Not Gaussian (P=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='59e−16) Not Gaussian (P=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='06e−13) Not Gaussian (P=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='9e−16) HAR Not Gaussian (P=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='12e−8) Not Gaussian (P=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='04e−8) Gaussian (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='19) Not Gaussian (P=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='76e−8) Not Gaussian (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='62e−5) Oil Not Gaussian (P=8e−4) Not Gaussian (P=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='9e−16) Not Gaussian (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='012) Not Gaussian (P=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='27e−16) Not Gaussian (P=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='86e−14) AIE Gaussian (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='46) Gaussian (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='23) Gaussian (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='08) Not Gaussian (P=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='99e−10) Gaussian (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='96) The results of Shapiro-Wilk test on the collected inference times for AWS are 48 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The execution time distributions of DNN applications in Chameleon cloud using Shapiro-Wilk test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Execution Time Distribution with Shapiro-Wilk Test in Chemeleon App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Type m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='xlarge m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='large m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='medium m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='small Fire Not Gaussian (P=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='05e−5) Not Gaussian (P=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e−4) Not Gaussian (P=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='58e−6) Not Gaussian (P=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='32e−7) HAR Gaussian (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='74) Not Gaussian (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='02) Gaussian (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='18) Gaussian (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='84) Oil Not Gaussian (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='01) Not Gaussian (P=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5e−7) Not Gaussian (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='01) N/A AIE Not Gaussian (P=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='77e−10) Not Gaussian (P= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='46e−6) Not Gaussian (P= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4e−4) Not Gaussian (P=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='46e−6) presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4, where columns present the various machine types and rows define the application types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The table reflects that our initial assumption is not totally valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The Shapiro-Wilk test result for the Chameleon cloud, depicted in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5, shows that for only three of the total cases, the normality assumption holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Considering the lack of normality in several cases, in the next section, we utilize Kolmogorov-Smirnov test to more granularly explore the best fitting distribution for the inference time of each application and also cross validate the prior results we obtained using another statistical method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Kolmogorov-Smirnov test to identify the execution time distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The null hypothesis for the Kolmogorov-Smirnov test is that the inference times of a certain application type on a given machine type follows a statistical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The Kolmogorov-Smirnov Goodness of Fit test (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' K-S test) identifies whether a set of samples derived from a population fits to a specific distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Precisely, the test measures the largest vertical distance (called test statistic D) between a known hypothetical probability distribution and the distribution generated by the observed inference times (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' empirical distribution 49 function (EDF)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then, if D is greater than the critical value obtained from the K-S test P-Value table, then the null hypothesis is rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Inference time distributions of DNN-based applications in AWS cloud machines using Kolmogorov-Smirnov test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Execution Time Distribution with Kolmogorov-Smirnov Test in AWS Cloud App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Type Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' ML Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' GPU Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Pur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Compt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Fire No Distr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' No Distr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' No Distr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' No Distr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' No Distr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' HAR Student’s t (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='08) Student’s t (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='77) Student’s t (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='99) Student’s t (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='57) Student’s t (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='95) Oil Student’s t (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='44) Student’s t (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='96) Student’s t (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5) Student’s t (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='20) Exponential (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='21) AIE Normal (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='99) Normal (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='54) Normal (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='47) Exponential (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='16) Normal (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='99) Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Inference time distributions of DNN-based applications in Chameleon’s ma- chines using the K-S test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Execution Time Distribution with Kolmogorov-Smirnov test in Chameleon App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Type m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='xlarge m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='large m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='medium m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='small Fire No Distr No Distr No Distr Log-normal HAR Normal (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='98) Student’s t (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='88) Normal (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='66) Normal (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='96) Oil Log-normal (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='36) Log-normal (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='99) Log-normal (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='81) N/A AIE Student’s t (P= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='47) Student’s t (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='12) Student’s t (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='25) Student’s t (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='83) The results of the K-S test on the observed inference times in AWS and Chameleon clouds are depicted in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' From Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6, we find that, in AWS, majority of the entries either represent Normal distribution or Student’s t-distribution that exposes similar properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, we observe that the inference time of Fire Detection application does not follow any particular distribution with an acceptable P-Value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We also observe that the inference times of both Oil Spill application on Compute Optimized machine and AIE application on General Purpose machine follow Exponential distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, low P-Value in both of these cases indicate a weak acceptance of the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 50 On the contrary, Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7 reflects the dominance of Log-normal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', the logarithm of the random variable is normally distributed) and Student’s t-distribution over other distributions in the Chameleon cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Analyzing the execution traces shows us that the inference times in Chameleon are predominantly larger than the ones in AWS that causes right-skewed property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, the distribution tends to be Log-normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Consistent with AWS observations, we see that the Fire Detection application does not follow any distribution in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Our further analysis showed that the lack of distribution is due to the input videos’ variety (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', frame rate and resolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' When we reduced the degree of freedom in the input videos and limited them to those with the same configuration (frame rate), we noticed the inference time followed a Log-normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The observation shows that the characteristics and variation of input data can be decisive in the statistical behavior of inference times (mentioned in highlighted insight).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Finally, we note that the Oil Spill application cannot be run on m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='small machine owing to its limited memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Insights: The key insights of the analysis we conducted on identifying the distribution of inference time are as follows: Shapiro-Wilk test for AWS and Chameleon rejects the null hypothesis that the inference times of DNN-based applications follow the Normal distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The K-S test reflects the dominance of Student’s t-distribution of inference time in both AWS (Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6), and Chameleon (Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Various configurations of input data is decisive on the statistical distribution of the inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 51 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The measurement of central tendency metric (µ), and data dispersion metric (σ) on the observed inference times in AWS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Mean and Standarad Deviation of Inference Execution Times (ms) in AWS App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Type Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' ML Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' GPU Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Pur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Compt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Fire µ=1461.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8 σ=457.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3 µ=1281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7 σ=387.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='93 µ=1349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5 σ=418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='9 µ =1534.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8 σ=494.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7 µ=1421.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4 σ=441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8 HAR µ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='27 σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='082 µ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='66 σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='006 µ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='51 σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='006 µ =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='17 σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='042 µ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='66 σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='003 Oil µ=269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='9 σ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='01 µ=218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8 σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='66 µ=65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='98 σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='47 µ=667.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 σ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='26 µ=242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='9 σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='68 AIE µ=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='02 σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='02 µ=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='41 σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='03 µ=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='55 σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='04 µ=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='35 σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='06 µ=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='95 σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='02 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Central tendency metric (µ), and data dispersion metric (σ) of the inference times in the Chameleon cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Mean and Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' of Inference Execution Times (ms) in Chameleon App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Type m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='xlarge m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='large m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='medium m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='small Fire µ=2155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='20 σ=725.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='48 µ=2213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='30 σ=731.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='50 µ=2330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='80 σ=742.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='20 µ=3184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='80 σ=1033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='30 HAR µ=22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='14 σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='76 µ=47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='69 σ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='26 µ=49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='24 σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='57 µ=52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='69 σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='78 Oil µ=147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='16 σ=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='23 µ=222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='22 σ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='89 µ=412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='78 σ=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='99 N/A AIE µ=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='23 σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='25 µ=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='23 σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='15 µ=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='40 σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='13 µ=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='72 σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3 Analysis of Central Tendency and Dispersion Measures Leveraging the statistical distributions of inference times, in this part, we explore their central tendency metric that summarizes the behavior of collected observations in a single value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In addition, to analyze the statistical dispersion of the observations, we measure the standard deviation of the inference times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In particular, we estimate the arithmetic mean and standard deviation of the inference times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The central tendency metric of inference times for AWS and Chameleon systems are shown in Tables 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='9, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The key insights are as follows: 52 Machine Learning Optimized and GPU instances often outperform other AWS machine types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In both clouds, the inference times of Fire and Oil experience a higher stan- dard deviation in compare with other applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The high uncertainty is attributed to the characteristics of their input data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Oil Spill input im- ages suffer from class imbalance [77], whereas, Fire input videos are highly variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In Chameleon VMs with a consistent heterogeneity, DNN applications with dense network models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', Oil and Fire) can take advantage of powerful machines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='xlarge) to significantly reduce their inference times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Overall, AWS offers a lower inference time than Chameleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The reason is utilizing SSD units in AWS rather than HDD in Chameleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In addition, we noticed that Chameleon experiences more transient failures that can slow down the applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6 Resource-Centric Analysis of Inference Time In performance analysis of computing systems, a rate-based metric [90] is defined as the normalization of number of computer instructions executed to a standard time unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' MIPS is a popular rate-based metric that allows comparison of computing speed across two or more computing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Given that computing systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', AWS ML Optimized and GPU) increasingly use instruction-level facilities for ML applications, our objective in this part is to analyze the performance of different machine types in processing DNN-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The results of this analysis can be of particular interest to researchers and cloud solution architects whose endeavor is to develop tailored resource allocation solutions for Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As for rate-based metrics we do not assume any distribution [91], we conduct a non-parametric approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In addition to MIPS, we provide the range of MIPS in form of Confidence Intervals (CI) for each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 53 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' MIPS values of heterogeneous machines in AWS for each DNN-based appli- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The MIPS for DNN Applications in AWS Cloud App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Type Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' ML Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' GPU Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Pur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Compt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Fire 1938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='63 2196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='35 2092.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='72 1862.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='04 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='56 HAR 838640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='65 1595874.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='34 2040057.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='33 891754.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='48 1581709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='12 Oil 164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='54 168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='58 331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='98 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='46 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='01 AIE 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='58 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='28 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='25 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='25 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='32 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' MIPS vales for heterogeneous machines on Chameleon cloud for each DNN- based application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The MIPS for DNN Applications in Chameleon App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content='94 N/A AIE 246366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content='49 Let application i with ni instructions have tim inference time on machine m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then, MIPS of machine m to execute the application is defined as MIPSmi = ni/(tim × 106).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, before calculating MIPS for any machine, we need to estimate the number of instructions (n) of each DNN-based application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For that purpose, we execute each task ti on a machine whose MIPS is known and estimated ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then, for each machine m, we measure tim and subsequently calculate MIPSmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Tables 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='10 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='11 show the MIPS values for AWS and Chameleon, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To measure the confidence intervals (CI) of MIPS for each application type in each machine type, we use the non-parametric statistical methods [91] that perform prediction based on the sample data without making any assumption about their underlying distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As we deal with a rate-based metric, we use harmonic mean that offers a precise analysis for this type of metric rather than the arithmetic mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We utilize Jackknife [91] re-sampling method and validate it using 54 Bootstrap [91], which is another well-known re-sampling method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Both of these methods employ harmonic mean to measure the confidence intervals of MIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The confidence intervals of MIPS values for DNN-based applications in AWS machines, resulted from Jackknife re-sampling method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' CI of MIPS using Jackknife Method in AWS cloud App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Type Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content='04] [155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='56, 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content='06, 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content='00] Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Confidence intervals of MIPS values for different DNN-based applications in Chameleon machines, resulted from Jackknife re-sampling method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' CI of MIPS using Jackknife Method in Chameleon Cloud App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content='15] N/A AIE [237710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='12, 252686.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content='73, 251673.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='68] [168804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='58, 268273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='11] [199676.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='71, 203681.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='17] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Estimating Confidence Interval using Jackknife Method Let p be the number of observed inference times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The Jackknife method calculates the harmonic mean in p iterations, each time by eliminating one sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' That is, each time it creates a new sample (re-sample) with size p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Let xj be the jth observed inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then, the harmonic mean of re-sample i is called the pseudo-harmonic value (denoted as yi) and is calculated based on Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' yi = p − 1 p� j=1,j̸=i 1 xj (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1) 55 Next, the arithmetic mean (denoted ¯y) of the p pseudo-harmonic values is computed, and is used to estimate the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Finally, the t-distribution table is used to calculate the CI boundaries with a 95% confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The result of the Jackknife method for AWS machines is shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='12 that conforms with the MIPS calculation in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Similarly, the results of analysis for Chameleon cloud using Jackknife method, shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='13, validate the prior MIPS calculations in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, in the next part, we cross-validate these results using Bootstrap method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Estimating Confidence Interval using Bootstrap Method Bootstrap repeatedly performs random sampling with a replacement technique [91] on the observed inference times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The random sampling refers to the selection of a sample with the chance of non-zero probability and the number (represented as k) of re-sample data depends on the user’s consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' After re-sampling, the harmonic means of k number of samples are calculated and sorted in ascending order to estimate the confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Finally, for a specific confidence level, the (α/2 × k)th and ((1 − α/2) × k)th values are selected from the sorted samples as the lower and upper bounds of the CI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We set the k value to 100 and α to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='05 for 95% confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For both AWS and Chameleon, the results of CI analysis using the Bootstrap method are similar to, thus validate, the ranges estimated by the Jackknife method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, due to the shortage of space, we do not report the table of MIPS values for the Bootstrap method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, we note that the CI ranges provided by the Bootstrap method are shorter (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', have less uncertainty), regardless of the 56 application type and the cloud platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The reason for the shorter range is that Bootstrap performs re-sampling with a user-defined number of samples that can be larger than the original sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Comparative analysis of the MIPS values of AWS and Chameleon machines for various DNN-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For the sake of presentation, the MIPS values are normalized between [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Fire HAR Oil AIE DNN-based Applications 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Normalized MIPS AWS Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Optimized Chameleon m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='large To perform a cross-platform analysis of the MIPS values, in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4, we compare the range of MIPS values for AWS Compute Optimized against m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='large that is a compatible machine type in Chameleon (see Tables 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The horizontal axis of this figure shows different application types and the vertical axis shows the MIPS values, normalized based on MinMax Scaling in the range of [0,1], for the sake of better presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Due to high variation in the input videos, we observe a broad CI range for Fire detection across both cloud platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, for HAR, Oil Spill, and AIE applications, we observe that the first and third 57 quartiles of the CI range in Chameleon (whose machines are prone to more transient failures [92]) is larger than those in AWS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' This wide range indicates that, apart from variations in the input data, the reliability of underlying resources is also decisive on the stochasticity of the inference times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7 Summary and Discussion Accurately estimating the inference time of latency-sensitive DNN-based applications plays a critical role in robustness and safety of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Such accurate estimations enable cloud providers and solution architects to devise resource allocation and load balancing solutions that are robust against uncertainty exists in the execution time of DNN-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In this work, we provide application- and resource-centric analyses on the uncertainty exists in the inference times of several DNN-based applications deployed on heterogeneous machines of two computing platforms, namely AWS and Chameleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In the first part, we utilized the Shapiro-Wilk test to verify if the assumption of Normal distribution for the inference time holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We observed that the inference times often do not follow a Normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, in the second part, we broaden our distribution testing investigation and utilized the Kolmogorov-Smirnov test to verify the underlying distributions in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The analysis showed that inference times across the two computing platforms often follow Student’s t-distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, in several cases in Chameleon system we observed the Log-normal distribution that we attribute it to the uncertain performance of VMs in this platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Next, to conduct a resource-centric analysis, we modeled MIPS (as a rate-based performance 58 metric) of the heterogeneous machines for each application type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In the analysis, we took a non-parametric approach, which is suitable for rate-based metrics, and utilized the Jackknife and Bootstrap re-sampling methods with harmonic mean to determine the range of confidence intervals of the MIPS values in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The calculated MIPS values and their CI ranges reflect the behavior of different DNN-based applications under various machine types of cloud and fog systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' A comparative analysis of the CI ranges across AWS and Chameleon demonstrate that the uncertainty in the inference time is because of variations in the input data and unreliability of the underlying platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In the future, we plan to incorporate the findings of this research to devise accurate resource allocation methods in IoT and edge computing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In addition, we plan to develop a predictive analysis to determine the execution of each inference task upon arrival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 59 Chapter 4: The Benefits of Federated Fog to Manage Monolithic Workload in Remote Industrial Sites 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Overview In the previous chapters, our preliminary research found that fog federation can be a potential computational platform for remote smart industries with stochastic execution behaviors for Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, the stochastic execution of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications has an influence on task completion times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In this case, an efficient resource allocation and load balancing technique that is aware of stochastic execution behaviors of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications can ensure the system’s robustness by enabling the on-time completion of receiving tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accordingly, in this chapter, we first strategically develop a load-balancing method for allocating arriving tasks to a fog federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, our primary goal is to ensure the system’s robustness (fog federation) in terms of meeting the deadlines of arriving tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To achieve the goal, we estimate the end-to-end latency of a receiving task in a fog system and utilize the latency to predict the task completion time across the fog federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, we propose a probabilistic task allocation method in the load balancer of each fog system that is aware of the latency constraints of the receiving tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then, in the second part, we evaluate our proposed load balancing method using the synthetic workload (customized to industrial tasks workload) of EdgeCloudSim [18] simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 End-to-End Latency in Federated Fog Systems When a task request arrives at a fog system’s load balancer, communication and computational latencies combine to generate the end-to-end latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Furthermore, several factors impact each of these latencies, causing them to behave stochastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For these reasons, calculating end-to-end latency and capturing its stochastic character in fog computing systems is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In the following sections, we go over the elements that influence communication and processing latencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In addition, we present a model for estimating end-to-end latency while accounting for its stochastic character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Estimating Communication Latency The time it takes to process and return a response to a task request is the communication latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' More specifically, communication latency is caused by transmission latency and propagation latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The transmission latency between any two points m and n (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', two fog systems in the fog federation) for task t of type i, denoted Θi(m, n), is defined as the sum of uplink transmission latency, denoted τu(m, n, i), and downlink transmission latency, denoted τd(m, n, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' That is, we have Θi(m, n) = τu(m, n, i) + τd(m, n, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Let Iu(i) be the size of data payload (in bits), originally captured by a sensor, serving as input for task type i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Note that, for some sensors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', cameras), there can be randomness in the size of captured data, in every sensor reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Also, let Ru(m, n) represent the uplink bandwidth, through which the data is transmitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' T is the time required to transmit each data packet to the uplink channel (known as Transmission Time Intervals (TTI)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then, the 61 uplink latency is calculated based on Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' τu(m, n, i) = ⌈ Iu(i) Ru(m, n)· T ⌉ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1) Similarly, the downlink latency is defined as Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' τd(m, n, i) = ⌈ Id(i) Rd(m, n)· T ⌉ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2) An orthogonal frequency-division multiplexing (OFDM) with total bandwidth W is divided equally into a set of k sub-channels (where k ∈ K) each with bandwidth w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accordingly, the downlink bandwidth is defined based on Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Rd(m, n) = w· � k∈K ymnk log2(1 + γd(m, n, k)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3) where ymnk = 1, if sub-channel k is allocated, otherwise ymnk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As the wireless communication is prone to noise and interference from other fog systems in the federation, the value of Rd(m, n) also depends on downlink signal to noise plus interference ratio (also known as SINR [93]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' SINR is defined as the power of a particular signal divided by the sum of the interference power (from all the other interfering signals) along with the power of background noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We note that, details of calculating uplink transmission latency (τu(m, n, i)) is similar to those for downlink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In fog federation, due to the vicinity, the propagation latency between fog systems is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In contrast, the communication between fog systems and cloud datacenters is commonly achieved via satellite that introduces a substantial propagation latency [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The propagation latency, denoted τp, is calculated based 62 on Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' τp = 2· d(n, st) Sl (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4) In the Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4, d(n, st) is the distance between fog n to satellite st and Sl is the propagation speed in medium or link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To calculate propagation latency in the round trip time, the fraction value should be doubled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Once we know propagation latency, the overall communication latency, denoted dcomm, to access cloud datacenter is calculated based on Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' dcomm = Θi(m, n) + τp (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5) As we noticed, there are several factors that collectively form the communication latency with stochastic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To capture this stochastic behavior, we treat communication latency as a random variable and model it using statistical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' That is, we represent the communication latency between any two points (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', two fog systems in the federation) using a probability density function (PDF), built upon historical communication information [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Based on the central limit theorem, communication latency can be modeled using Normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Estimating Computational Latency Once the load balancer assigns arriving task request t to a fog system, the task has to wait in the scheduling queues of the fog system before its execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For a given task t of type i, denoted ti, its completion time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', computational latency) is influenced by the waiting time in the queue (queuing latency), plus the task’s execution time (execution latency) on the machines of the assigned fog system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 63 Importantly, both of these factors are stochastic, as a result, the task completion time exhibits a stochastic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The queuing latency of task ti is dependent on the number and execution times of tasks ahead of it in the fog system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The stochasticity in execution time can be due to different task types and characteristics of machines in different fog systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Even the execution time of tasks from the same type on homogeneous machines of the same fog system is stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' This can be because of variations in the size of data to be processed and multi-tenancy of tasks in the fog system [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Other factors, such as machine failure, can also be reasons for stochastic task execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To capture the stochasticity in computational latency, we consider the task completion time of each task type on each fog system as a random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then, we model the computational latency using statistical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' That is, the computational latency is modeled using PDF, built upon historical completion time information of each task type on each fog system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Based on the central limit theorem, the computational latency of each task type on each fog system can be modeled using Normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3 Estimating End-to-End Latency Once we estimate the communication and computational latencies, their compound latency forms the end-to-end latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' More specifically, the compound latency can be obtained by convolving the PDF of communication latency with the PDF of the computational latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For an arriving task ti to a load balancer, let Ni 64 be PDF of its communication latency to another fog system in the federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Also, let Mi be PDF of the computational latency of ti on the other fog system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then, the end-to-end latency for ti, denoted Ei, is calculated as Ei = Ni ⊛ Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3 Robust Resource Allocation in the Federated Fog Computing System The synopsis of the proposed resource allocation model in the federated fog computing system is demonstrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The resource allocation model utilizes a load balancer module that is the main enabler of fog federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Every fog system is equipped with a load balancer that, for each arriving task, it determines the appropriate fog system (either the receiving fog or to a neighboring one) where the task has the highest likelihood of completion before its deadline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The functionality of load balancer is particularly prominent to cope with the uncertainty exists in task arrivals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', during disaster time) and make the fog system robust against it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The load balancer operates in immediate mode [96] and assigns arriving tasks to the appropriate fog system, immediately upon task arrival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The appropriateness is characterized based on the fog system that maximizes the probability of the task meeting its deadline (known as the probability of success).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The probability of success for task ti with deadline δi can be calculated for each neighboring fog system, by leveraging the end-to-end latency distribution of executing task ti on that system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To avoid repetitive task reassignment and compound latency, we determine that once a task assignment decision is made, the task cannot be re-allocated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The resource allocation of each fog system leverages the historical 65 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' A Fog system with load balancer module that facilitates fog federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Task requests generated by sensors are received by the load balancer module and are assigned to the fog system that maximizes the likelihood of success for the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Batch Queue Scheduler Fog system 1 Load Balancer Pressure Sensor Flow rate monitor Sensor H2S Gas Sensor Load Balancer Actuators Actuators ETT Matrix ETC Matrix ETC Matrix ETT Matrix Batch Queue Scheduler Fog system2 information of computational and communication latencies to build PDF of their distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For that purpose, each load balancer maintains two matrices, namely Estimated Task Completion (ETC) [97] and Estimated Task Transfer (ETT), to keep track of computational and communication latencies for each task type on each neighboring fog system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Entry ETC(i, j) keeps the PDF of computational latency for task type i on fog system j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Similarly, entry ETT(i, j) keeps the PDF of communication latency for task type i to reach fog system j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The entries of ETC and ETT matrices are periodically updated in an offline manner and they do not interfere with the real-time operation of the load balancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 66 Upon arriving task ti, load balancer of the receiving fog can calculate the end-to-end latency distribution of ti on any neighboring fog j, using ETC(i, j) and ETT(i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The end-to-end distribution can be used to obtain the probability of completing ti before its deadline, denoted pj(ti), on any of those fog systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We have: pj(ti) = P(Ei ≤ δi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We note that the probability calculation for task ti on the receiving fog does not imply further communication latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As such, for the receiving fog r we have: pr(ti) = P(Mi ≤ δi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In the next step, the fog system that provides the highest probability of success is chosen as a suitable destination to assign task ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' This implies that task ti is assigned to a neighboring fog system, only if even after considering the communication latency, the neighboring fog provides a higher probability of success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' It is noteworthy that the probability of success on a neighboring fog can be higher than the receiving fog by a non-significant amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In practice, a task should be assigned to a neighboring fog, only if the neighboring fog system offers a substantially higher probability of success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To understand if the difference between the probabilities is substantial, we leverage confidence intervals (CI) of the underlying end-to-end distributions, from which the probability of success for receiving and remote fogs are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' More specifically, we determine a neighboring fog offers a significantly higher probability of success for a given task, only if CI of end-to-end distribution of the neighboring fog does not overlap with the CI of end-to-end distribution of the receiving fog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The pseudo-code provided in Algorithm 1 expresses the robust task 67 Algorithm 1: Task assignment algorithm for load balancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Input : Task ti;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' ETC and ETT matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' G (set of neighboring fog systems) Output: Chosen fog j ∈ G to assign ti 1 pr(ti) ← Probability of success on receiving fog r 2 foreach fog system j ∈ G do 3 pj(ti) ← Probability of success on neighbor fog j 4 if pj(ti) > pr(ti) then 5 Add pj(ti) to P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' as a potential fog for assignment 6 end 7 end 8 Sort elements of set P in descending order 9 Consider receiving fog r as default assignment for ti 10 foreach pj ∈ P do 11 if CI of Ej does not overlap with CI of Nr then 12 Choose fog j as destination and assign ti to it 13 Exit the loop 14 end 15 end assignment heuristic that load balancer utilizes to take advantage of federated fog system and increase the robustness of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The heuristic is called Maximum Robustness (MR) and invoked upon arrival of a new task ti to the load balancer of a fog system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Based on the deadline of the arriving task (δi), the algorithm first calculates the probability of success for ti on the receiving fog and on its neighboring fog systems (Step 1-7 in Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then, in Step 8, the calculated probabilities are sorted in the descending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' If the probability of success on the receiving fog is higher, then the task is allocated to the receiving fog system (Step 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Otherwise, CI of the end-to-end latency distribution for the neighbor with the highest probability of success is compared against receiving fog CI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' If the CIs do not overlap, then task ti is assigned to the neighboring fog (Step 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Otherwise, the 68 same procedure is performed for the rest of the neighbors of the receiving fog system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' If there is no no-overlap neighbor found then, task ti is assigned to the receiving fog system (default assignment in Step 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4 Performance Evaluation of Federated Fog We have used EdgeCloudSim [18], which is a discrete event simulator for performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We simulate five fog systems (micro-datacenters) each one with eight cores and [1500, 2500] Million Instructions Per seconds (MIPs) computational capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Cores of each fog system are homogeneous: however, different fog systems have different MIPs that represents the heterogeneity across the fog systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We also consider a cloud datacenter with 40,000 MIPs to process non-urgent tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Task within each fog is mapped in the first come first serve manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The bandwidth to access cloud is based on satellite communication and set to 200 Mbps, and the propagation delay is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='57 seconds [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In each workload trial, generated to simulate load of a smart oil field, we consider half of the tasks represent urgent and the other half represent non-urgent tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Each task is of a certain type that represents its service type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In each workload trial, urgent tasks are instantiated from two different task types and non-urgent tasks are instantiated from two other task types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The execution time of each task instantiated from a certain type is sampled from a normal distribution, representing that particular task type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Each task is considered to be sequential (requires one core) and its execution time is simulated in the form of MIPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Poisson distribution (with different means for different task types) is used to generate the 69 inter-arrival rate of the tasks and simulate task arrival during oversubscription periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The number of tasks in each workload trial is varied to represent different oversubscription levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Deadline for task i in a workload trial is generated as: δi = arri + β· avgi comp + α· avgi comm + ϵ, where arri is the task arrival time, avgi comp is average computational latency of the task type across fog systems, and avgi comm is average communication latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' β and α are coefficients, respectively, represent computation and communication uncertainties, and ϵ is the slack of other uncertainties exist in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We consider maintaining ETC and ETT matrices in every fog system and update them in every 10% of the workload execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The entries of these matrices are considered as normal distribution as mentioned in the system model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For accuracy, each experiment was conducted 30 times and the mean and 95% confidence interval of the results are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Baseline Task Assignment Heuristics for Load Balancer Minimum Expected Completion Time (MECT): This heuristic [46] uses the ETC matrix to calculate the average expected completion time for the arriving task on each fog system and selects the fog system with the minimum expected completion time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Maximum Computation Certainty (MCC): This heuristic (used in [99]) utilizes ETC matrix to calculate the difference between the task’s deadline and average completion time (called certainty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then, the task is assigned to the fog that offers the highest certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 70 Edge Cloud (EC): This heuristic operates based on conventional fog computing model where no federation is recognized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Specifically, urgent tasks are assigned to the receiving fog and non-urgent tasks are assigned to the cloud datacenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Experimental Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Analyzing the impact of oversubscription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The main metric to measure the robustness of an oversubscribed fog system in a smart oil field is the deadline miss rate of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In this experiment, we study the performance of our system by increasing the number of tasks sensors generate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', oversubscription level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 shows the results of varying the number of arriving tasks (from 1,500 to 7,500 in the horizontal axis) on deadline miss rate (vertical axis) when different task assignment heuristics is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The impact of increasing oversubscription level (number of arriving tasks) on deadline miss rate using different task assignment heuristics in the load balancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 1500 3000 4500 6000 7500 Number of arriving tasks 0 20 40 60 80 100 Tasks deadline miss rate (%) MR MECT MCC EC 71 In Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2, it is visible that as the number of tasks increases, the deadline miss rate grows for all of the heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Under low oversubscription level (1,500 tasks), MR, MECT, and MCC perform similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, as the system gets more oversubscribed (4,500 tasks) the difference becomes substantial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' With 7,500 tasks, MR offers around 16% lower deadline miss rate than MECT and MCC and approximately 21% better than EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The reason is that MR captures end-to-end latency and proactively utilizes federation, only if it has a remarkable impact on the probability of success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Nonetheless, EC does not consider federation, and other baseline heuristics only consider the computational latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We can conclude that considering end-to-end latency and capturing its underlying uncertainties can remarkably improve the robustness, particularly, when the system is oversubscribed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', at a disaster time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Analyzing communication overhead of fog federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Although we showed in the previous experiment that using federation improves system robustness, we are unaware of the communication overhead of task assignment in the federated environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, in this experiment, we evaluate the communication latency imposed as a result of applying different task assignment heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Specifically, we measure the mean communication latency overhead (vertical axis in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3) induced to each task, for the various number of arriving tasks (horizontal axis in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3 shows that MECT and MCC cause higher average communication latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The reason is that these heuristics do not consider the communication 72 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Mean communication latency overhead introduced to each task in fog federation by different heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 2000 3000 4000 5000 6000 7000 Number of arriving tasks 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='35 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='40 Mean network latency overhead (s) 1e 2 MECT MR MCC latency and aggressively redirect tasks to the same fog system, making that particular network link (between receiving fog and redirected fog system) congested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In contrast, MR that considers communication latency and redirect tasks more conservatively, only if the improvement in the probability of success is substantial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3 Analyzing average makespan of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Different task assignment heuristics cause various computational latencies for the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To understand the computational latency, we measure the average makespan of tasks, resulted by applying various task assignment heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4 demonstrates that EC leads to the maximum average makespan time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The reason is that EC does not utilize federation, making the receiving fog system highly oversubscribed while other neighboring fog systems are underutilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 73 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Average makespan time(seconds) of tasks using various task assignment heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 2000 3000 4000 5000 6000 7000 Number of arriving tasks 0 10 20 30 40 50 60 70 Average makespan time of tasks (s) MECT MR EC MCC Hence, average makespan time rapidly rises after the receiving fog is saturated with 3,000 tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' MECT and MCC do not consider the stochastic nature of task completion time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' hence, they can potentially assign arriving tasks to one fog and oversubscribe that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a result, the average makespan of tasks rises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In contrast, MR considers stochastic nature of end-to-end latency and calculates the probability of success on neighboring fog systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Besides, it assigns tasks to a neighboring fog system, only if it offers a sufficiently higher probability of success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, MR offers the lowest average makespan time than other heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5 Summary In this chapter, we explored the usability of a fog federation for a smart Industry (Oil and Gas) in a disastrous situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To support the computational demands in an emergency situation allocating various tasks in suitable fog system is 74 challenging due to heterogeneity across fog systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, maintaining the robustness of the system in terms of every real-time urgent tasks deadline can be difficult unless any efficient load balancing technique adopted by the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To achieve that, we presented dynamic federation of fog computing systems, exist in nearby industries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Within the federated environment, we captured two sources of uncertainty, namely communication and computation, that are otherwise detrimental to the real-time services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The federation is achieved by a load-balancer module in each fog system that is aware of the end-to-end latency between fog systems and can capture the stochasticity in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The load balancer leverages this awareness to find the fog system that can substantially improve the probability of success for each arriving task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Experimental results demonstrate that our proposed federated system can enhance the robustness of fog computing systems against uncertainties in arrival time, communication, and computational latencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We concluded that the load balancer could be particularly useful (by up to 27%) for higher levels of oversubscription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Even for na¨ıve load balancing methods (MCC and MECT) in the federation, the performance improvement is approximately 13%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 75 Chapter 5: Adapting Remote Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 Smart Micro-Service Applications to Federated Fog Computing Systems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Overview The advancement of IoT technologies with smart applications drives the wheel of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 [71] revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Various smart sensors, actuators, and smart devices are deployed in different industries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', manufacturing, food processing, oil & gas) to control the operational technology platform [100, 101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accordingly, sensors utilized in industrial operations frequently produce tons of data every day [102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The oil and gas industry is an example of generating enormous amounts of sensor data and the necessity for processing close to the data source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For instance, a typical offshore oil rig produces 1 to 2 TB of data daily [103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The majority of this data is fed to advanced computing applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', machine learning, report generation, automation) that can make smart latency-sensitive decisions to improve energy efficiency, production, and safety measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For example, applications like workplace air quality estimation [104] for workers’ safety utilize environmental sensors that measure the quality (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', the existence of harmful particles in the air) of breathable air in the surrounding of the workplace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, the air quality estimation must be fast to avoid potential occurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In the remote offshore industry, several services (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', data acquisition, alert generation, object tracking) are critical for complex or safety-related operations that need to be performed synchronously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The situation can worsen when any unwanted emergency brings many more computational activities completed within limited 76 time frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In this case, our motivation is the smart Oil and Gas industry that has been facing various disasters and catastrophes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', the deepwater horizon (2010)[19], usumacinta jack-up disaster (2007) [20], mumbai high north disaster (2005) [21], the ocean ranger disaster (1982) [105] ) due to complex fault intolerant industrial processes in exploration, drilling, and production operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, remote offshore industries need latency-aware support [106] that can not be feasible with typical cloud data centers due to the remote locations of the industrial operation sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The current solution utilizes fluctuating satellite communication [107] for sending data to mainland cloud data centers reducing the quality of service (QoS) and increasing the industrial safety risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, the high-level challenge is the lack of computational resources to support over-subscribed situations in remote industries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The structure of a microservice-based workflow is presented in a block diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Every microservice need to be processed to complete the fire safety appli- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' fire detection input video noise removal feature extraction alert generation location mapping video preprocessing expansion prediction 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Smart Micro-Service Applications for Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 smart applications typically follow modern software architecture [16, 108] where various micro-services [17] need to be executed in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, 77 micro-services can be separately deployed using an automated deployment process, require the least amount of administration, can be developed using a variety of programming languages and data storage techniques, and can each be independently updated, changed, and scaled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Thus, we concentrate on micro-services applications frequently used in remote industries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For instance, as depicted in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1, a “fire safety” application can include micro-services for capturing video surveillance data, pre-processing captured video, noise removal, feature extraction, fire detection, location mapping, alert generation, and expansion prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In contrast, many industries have previously deployed legacy applications [109] with inflexible software architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, the execution platform should support both monolithic legacy applications and modern micro-services to ensure industrial safety and fault-tolerant operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, modern industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications are comprised of micro-services that pose new challenges for the execution platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Under this arrangement, an application‘s latency constraint is subject to the completion time of the micro-services defined by the underlying software architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, to develop a robust execution platform for industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0, system architects need to understand the software architecture of the receiving applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Federated Fog Systems for Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 Micro-service Applications The emerging industrial IoT and advancements in communication technology have brought computational resources near the data sources and end devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For instance, nowadays, fog computing systems [110] in remote industries typically 78 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Offshore oil and gas industry has the fog federation infrastructure that can support smart microservice-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' G1 G2 G3 Monolithic Seismic Analysis Fire Detection Fed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Fog Platform Wireless Fed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Fog Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 Applications AP Fog Oil Spill Oil Spill Detection execute the industrial computational process to enable smooth production and workplace safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, as depicted in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2, a federated fog platform can be conceptualized from chapter 4 that can form by connecting through wireless gateways denoted as Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, various applications with heterogeneous latency constraints require computational support from federated fog computing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accordingly, the federation should be cognizant of communications and computing uncertainties, as well as the applications’ software structure and latency requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Thus, an application execution plan needs to perform for monolithic and micro-service software structures, considering the stochastic execution times and uncertainties that derive from the execution platform and communication technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Consequently, in our previous work [38] presented in chapter 4, the resource 79 allocation methods are explored intensively for monolithic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Furthermore, considering a complex operational process performed by various micro-services, one of the main problems is ensuring the completion of the whole application workflow within the time limit known as the deadline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, it is crucial to know the optimal point to partition the application workflow so that it can be completed on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accordingly, the question that needs to be addressed is “How to distribute Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', monolithic, micro-service) across fog federation so that the application workflow can be completed within the given time frame?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, from a system administration perspective executing the smart micro-service applications raises two more questions, and they are 1) How to partition the micro-service workflows so that its deadline constraint can be realized?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 2) How do we allocate partitioned micro-services across fog federation so that it has the highest likelihood of completing on time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Our prior work [38] suggests that federating nearby resources is one solution to the resource restrictions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', oversubscribe) encountered by edge computing systems in distant sectors like Oil and Gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Furthermore, we explore new challenges imposed by smart software architecture, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='a micro-service workflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, to address the difficulties faced by the offshore O&G sector at large, we propose a resource provisioning method for Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications across the federated fog system that is aware of both the software architecture and the underlying execution platform’s structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' More so, the solution maintains the deadline limitations of the micro-service workflow, which in turn makes the execution platform more reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 80 As a result, our approach consists of two stages: understanding the software architecture of the receiving applications and allocating computational resources for the successful completion of these applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, the following are the contribution of this research: Proposing a probabilistic partitioning method that is aware of the underlying software architecture of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Proposing a statistical resource allocation heuristic considering the time constraints of the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Providing extensive evaluation of partitioning technique along with resource allocation across fog federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The suggested solution can serve as a foundation upon which system architects or industry-focused research associates might construct more elaborate solutions referring to distant offshore sectors at peak demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In addition, the solution is compatible with monolithic legacy applications, which may aid conventional industries in transitioning to and adapting to the changes brought about by Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Partitioning Method for Micro-service Application Workflow Maintaining latency constraints of a smart application comprising multiple micro-services depends on underlying software architecture and mapping of computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For example, executing a micro-service application into a single fog system may not be possible or may not maintain its deadline constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 81 On the other hand, a monolithic application can not be partitioned and can be considered an application with a single micro-service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For micro-service software architecture, partitioning the application into multiple partitions and allocating them across fog federation can increase the likelihood of its completion within the latency constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Furthermore, allocating appropriate computational resources to the partitioned micro-services also ensures the completion of the whole application workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The flowchart of the workflow partitioning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The partitioned workflow is sent to the resource allocation module, which is denoted as the end box for this flow chart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' estimate the chance of on time completion for workflow on the local fog no yes submit to the resource allocation module partition workflow into two sub-graphs and estimate the chance of sucess for each partition , across fog federation rollback to yes no yes no OR OR The main goal of the partitioning method is to partition the micro-service application in a way such that the application can meet its deadline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, we 82 considered an application having micro-service architecture as a set of micro-services that are connected together in some manner to form a graph G = (V, E), where the set of vertices V = (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='.mn) denotes the micro-services and edge e(mi, mj) ∈ E represents the communication between micro-service mi and mj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As the first step of the partitioning method, we consider executing the whole micro-service workflow into the local fog system without partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As such, the partitioning method estimates the chance of on-time completion for workflow w on the local fog, which is the first processing box of flowchart 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To estimate the deadline for the whole application workflow w, we perform a summation of the deadlines for the micro-services that can be defined as δw = mδ 1 + mδ 2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' + mδ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, each micro-service has a deadline (mδ i) known in advance to the load balancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Furthermore, for each micro-service type, we have computational latency distribution (md i ) that represents the execution times across fog federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, to estimate the probability of success for the entire w, we convolve the computational latency distributions of the application’s micro-services that can be defined as Dw = md 1 ⊛ md 2 ⊛ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='.md n (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1) Finally, using the convolved distribution DA, we measure the probability of success as follows, P(w) = P(DA ≤ δA) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2) The output of equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 is compared with a conditional variable α as depicted in first condition of flowchart 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We choose an average success rate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', 50%) for α 83 as our experimental evaluation scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' When the likelihood of completing the workflow is less than α, the partitioning service takes place using the min-cut [111] graph partitioning algorithm, which is the partition workflow w into two sub-graphs i and j process box in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, considering the flow of actions within the application, we employ one of the widely utilized graph theorems, max-flow min-cute [112] in our proposed solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Due to finding the minimum number of partitions which is an np-hard problem, we developed our customized solution for Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 micro-service applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Thus, the partitions resulting from the min-cut are estimated for the chance of success across fog federation using equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 in the third process box of flowchart 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As we utilize probability to determine the partitions, we named the proposed partitioning method as Probabilistic Paritioning (ProPart).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' If the new sub-graphs completion success is less than the prior success rate (2nd condition of the flowchart), we consider earlier partitions as optimal (rollback to w process box).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accordingly, the resource allocation methods for those partitions are started, which is the end process box of the flowchart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' On the other hand, if the latest partition’s chance of on-time completion is greater than the prior success rate, then we evaluate each partition’s micro-service architecture, which is the third condition of the flowchart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' If the condition fails (“no” line from the third condition), the partitioning process is halted for partitions with only one micro-service, and the resource allocation service takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In contrast, for partitions with more than one micro-service (“yes” line from the third 84 condition), the partitioning process is repeated until each has a single micro-service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, the partitioning method is a repeated process where the output is the optimal number of partitions that are submitted to the resource allocation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3 Resource Allocation Method for Partitioned Micro-service Applica- tions Across Fog Federation Resource allocation occurs when the partitioning is completed with an optimum number of partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The partitioning method returns the whole application as one part to the resource allocation module for a monolithic application that is considered a single micro-service workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The efficacy of the resource allocation approach is significant in dealing with the unpredictability that occurs in applications’ arrival (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', during disaster time) and making the fog system resilient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The resource allocation module runs in immediate mode [96] and quickly allocates incoming applications or micro-service partitions to the relevant fog system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The relevance is defined by the fog system, which increases the likelihood of the micro-services achieving their deadlines (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='a the probability of success).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, the likelihood of on-time completion for a micro-service mi on a particular fog system can be estimated using the historical end-to-end latency distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Furthermore, to minimize frequent application reassignment and compound delay, we have decided that the micro-service cannot be relocated once an assignment choice is made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Each fog system’s resource allocation module uses prior data on computational and communication latencies of various micro-service types across 85 fog federation to generate PDFs of their distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To that end, each load balancer keeps track of computational and communication latencies for each micro-service type on each nearby fog using two matrices: Estimated Task Completion (ETC) [97] and Estimated Task Transfer (ETT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The PDF of computational delay for micro-service type i on fog system j is stored in entry ETC(i, j) that is previously used in the partitioning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Similarly, the item ETT(i, j) maintains the PDF of communication delay for micro-service type i to reach fog system j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, the resource allocation module is aware of communication latencies as well, whereas the partitioning method is only aware of computation latencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The entries of the ETC and ETT matrices are regularly updated offline and do not interfere with the load balancer’s real-time functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The resource allocation module can compute the end-to-end latency distribution across fog federation upon the arrival of a partition of micro-services using convolution of ETC(i, j) and ETT(i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' On any fog system j, the end-to-end distribution can be used to calculate the probability of completing each micro-service partition mpi before its deadline, denoted pj(mpi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, we estimate the deadline δi for the given partition mpi by adding each micro-service’s deadline within that partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then we convolve each micro-service’s computational latency distribution dcomp with communication distribution dcomm to measure the completion time ei in a particular fog system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To estimate the completion time of the partition mpi denoted as Ei, we convolve the completion time distribution for each micro-service within a given partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We have: pj(mpi) = P(Ei ≤ δi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We see 86 that the probability of mpi on the receiving fog does not entail any additional communication delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Consequently, for receiving fog system, we don’t convolve communication latency distribution to completion time estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In the subsequent stage, the fog system with the greatest likelihood of completion is selected as a viable destination to allocate mpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' This assignment entails that the micro-service partition mpi is only given to an adjacent fog system if the surrounding fog offers a greater chance of on-time completion after accounting for communication delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' It‘s important to note that the success rate on a neighboring fog could be greater than on the receiving fog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' This is because assigning a micro-service partition to a fog system in close proximity should only be done if doing so significantly increases the likelihood of the partition being completed successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, we use confidence intervals (CI) of the underlying end-to-end distributions, from which we derive the likelihood of success for receiving and distant fogs, to assess the significance of the discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In particular, we find that the CI of the end-to-end distribution of the nearby fog does not overlap with the CI of the receiving fog only if the neighboring fog gives a much better likelihood of success for a given micro-service partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The pseudo-code provided in Algorithm 2 expresses the resource allocation method that the load balancer utilizes to take advantage of the federated fog system and increase the system’s robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The method is called Maximum Robustness (MR) and is invoked when the partitioning method sends micro-service partitions 87 Algorithm 2: Resource allocation algorithm Input : Micro-service partition set M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' ETC and ETT matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' G (set of neighboring fog systems) Output: Chosen fog f ∈ G to assign micro-service partitions mpn ∈ M 1 foreach micro-service partition mpi ∈ M do 2 pr(mpi) ← Probability of success on receiving fog r 3 foreach fog system f ∈ G do 4 pf(mpi) ← Probability of success on neighbor fog f 5 if pf(mpi) > pr(mpi) then 6 Add pf(mpi) to P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' as a potential fog for assignment 7 end 8 end 9 Sort elements of set P in descending order 10 Consider receiving fog r as default assignment for mpi 11 foreach pf ∈ P do 12 if CI of Ef does not overlap with CI of Nr then 13 Choose fog f as destination and assign mpi to it 14 Exit the loop 15 end 16 end 17 end M for resource allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' At first, the micro-service partitions are separated for further processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then based on the deadline (δi) of each micro-service partition mpi, the algorithm calculates the probability of success on the receiving fog and on its neighboring fog systems (Step 2-8 in Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Next, step 9 sorts the calculated probabilities in descending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' If the probability of success on the receiving fog is higher, then the micro-service partition mpi is considered for allocation to the receiving fog system (Step 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Otherwise, the CI of the end-to-end latency distribution for the neighbor with the highest probability of success is compared against receiving fog’s CI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' If the CIs do not overlap, then partition mpi is assigned to the neighboring fog (Step 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Otherwise, the same procedure is 88 performed for the rest of the neighbors of the receiving fog system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' If no non-overlap neighbor is found, then partition mpi is assigned to the receiving fog system (default assignment in Step 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4 Performance Evaluation of Software Architecture-Aware Federated Fog Systems The partitioning and resource allocation components of the proposed technique occur one after the other within the load balancer module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a consequence, we evaluate each component separately in various experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The recommended partitioning approach is compared to different baselines in the first experiment to examine how the deadline constraints for workflow applications based on microservices have improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Following partitioning, the allocation of resources to those partitions must be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We execute the second category of trials to assess the system’s efficacy, which compares our proposed resource allocation techniques to alternative baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The third experiment is then performed to determine how scaling the fog federation impacts the suggested solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Finally, for microservice and monolithic applications, we examine the computational latencies resulting from partitioning and resource allocation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The experiments are thoroughly described in the subsections that follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Comparison of Micro-service Workflow Partitioning Methods In this experiment, we use the suggested partitioning technique (Probabilistic Partitioning, defined as ProPart) for accepting microservice-based workflow applications and compare it to the other two baselines (Min-cut, Least 89 data transfer, for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In this experiment, we increase the number of microservice applications submitted to the system to generate oversubscribed conditions and record the applications’ deadline meet rate in each round of request submission, shown as a bar chart in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The figure’s x-axis indicates the number of microservice-based applications received by the system, while the y-axis reflects the rate at which application deadlines have been met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Comparison of the partitioning techniques in terms of workflow deadline meet rate while utilizing proposed probabilistic partitioning technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The x-axis represents the increasing number of arriving workflow execution requests, whereas the y-axis represents the workflow deadline meet rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 100 200 300 400 Number of Receiving Microservice-based Workflows 0 20 40 60 80 100 Workflow deadline meet rate (%) ProPart Min-Cut Least data transfer The results of this experiment, shown in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4, indicate that the deadline meet rate decreases as the number of workflow requests to the system increases for all partitioning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, in every round of submissions, ProPart surpasses other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For less overloaded scenarios (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', 100 & 200 requests), the performance gap between the least efficient strategy (least data transfer) and the suggested technique ProPart is greater than for completely oversubscribed 90 conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', 300 & 400 requests).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The primary reason for ProPart’s superior performance is its statistical assessment of each partition’s success likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For up to 200 application requests, the min-cut strategy performed better than the least data transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In contrast, in totally overloaded scenarios, the least data transfer performed marginally better than the min-cut because it considers the connection that generates the least output data for splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Min-cut, in contrast, examines the smallest communication channel when partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Finally, due to the repeated probabilistic calculation of deadline fulfillment for all microservices, ProPart performed better in totally oversubscribed conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Comparison of Resource Allocation Methods The load balancer in every fog system utilizes a resource allocation technique after the partitioning steps for microservice-based workflow applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In contrast, for monolithic applications, as soon as load balancer receives a request, it performs resource allocation using probabilistic estimation across fog federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As such, to compare the proposed resource allocation technique, we performed the following experiments with three different resource allocation methods for microservice and monolithic applications respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Microservice-based Workflow Applications: Similar to the previous experiment, the number of receiving microservice-based application is incremented to create more oversubscribed situations(i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', the x-axis of the graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To visualize the performance of the resource allocation techniques, the deadline meet rates of receiving applications are captured and plotted in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 91 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Comparison of resource allocation techniques while utilizing proposed workflow partitioning technique for microservice-based workflow applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 100 200 300 400 Number of Receiving Microservice-based Workflows 0 20 40 60 80 100 Workflow deadline meet rate (%) MR MECT MCC The result represents a downward trend for all the resource allocation techniques with increasing oversubscribed situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, it is visible that the proposed resource allocation technique, MR outperforms other baselines in every oversubscribed situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' This is because MR is aware of uncertainty in computation and communication of receiving microservices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In contrast, MECT is only aware of computation, and Certainty utilizes deadlines in its resource allocation technique which lacks communication information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Monolithic Independent Applications: In this experiment, we investigate the performance of our system by increasing the number of monolithic applications generated by sensors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', the oversubscription level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6 shows the effects of altering the number of incoming applications (from 400 to 1000 on the horizontal axis) on the deadline meet rate (vertical axis) when various resource allocation heuristics are employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 92 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Comparison of resource allocation techniques for monolithic applica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The proposed resource allocation technique MR outperforms other baselines in every application arrival trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 400 600 800 1000 Number of Receiving Monolithic Tasks 0 20 40 60 80 100 120 Tasks deadline meet rate (%) MR MECT MCC In figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6, it is visible that as the number of applications increases, the deadline meets rate decreases for all of the heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Under low oversubscription levels (400 tasks), MR, MECT, and MCC perform similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, the difference becomes substantial as the system gets more oversubscribed (800 applications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' With 1000 applications, MR offers around 18-20% higher deadline-meeting rates than MECT and MCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The reason is that MR captures end-to-end latency and proactively utilizes federation only if it remarkably impacts the probability of success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Nonetheless, other baseline heuristics only consider computational latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, we can conclude that for monolithic applications considering end-to-end latency and capturing its underlying uncertainties can remarkably improve the robustness, particularly when the system is oversubscribed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', at a disaster time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3 Fog Federation Scaling Impact Fog federation in remote offshore areas can be scaled up in times of 93 emergencies by utilizing mobile fog systems mounted on a boat or other vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In contrast, a scaled-down fog federation can decrease the system’s robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, to understand the impact of federation scaling over the proposed solution, we increase the fog federation degree that represents the number of neighbors and captures the deadline meet rates of the received applications within the increasing oversubscribed situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The result of this experiment is presented in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In addition, we performed a similar experiment for monolithic applications, where we fixed the number of receiving tasks and incremented the fog federation degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The result for monolithic applications is presented in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Impact of scaling the fog federation for proposed partitioning and resource allocation techniques in increasing oversubscribed situations considering mi- croservice applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The degree represents the number of neighbors each fog system has for executing the Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 100 200 300 400 Number of Receiving Microservice-based Workflows 0 20 40 60 80 100 Workflow deadline meet rate (%) Fog Fed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Degree: 2 Fog Fed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Degree: 3 Fog Fed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Degree: 4 Fog Fed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Degree: 1 Microservice-based Application: The result shown in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7 demonstrates the advantages of scaling up the fog federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a result, in any overcrowded circumstance, the federation with the greatest number of neighbors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', fog fed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' degree 4) excels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Despite this, considerable performance improvements are seen in 94 most oversubscribed circumstances (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', a system processing 400 microservice-based workflows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For less overloaded scenarios (for example, a system with 100 - 200 receiving microservice-based workflows), the performance difference for minor scale-up fog federation is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' This is due to the suggested method, particularly the partitioning technique, attempting to put the whole application into a fog system rather than partitioning and distributing them around the federation in less oversubscribed conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a result, the performance increase is substantial in the fully oversubscribed scenario with the most neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Impact of scaling the fog federation for proposed resource allocation techniques on monolithic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The degree represents the number of neighbors each fog system has for executing the Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Degree 1 Degree 2 Degree 3 Degree 4 Fog Federation Scaling 0 20 40 60 80 100 120 Tasks deadline meet rate (%) MR MECT MCC Monolithic Applications: In this experiment, we compare the resource allocation techniques for monolithic applications while scaling up the fog federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Similar to microservice-based workflows, the monolithic applications positively impact federation scaling, which is visible from figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The result reflected a significant performance improvement when the federation scaled up from degree 1 to 95 degree 2 for all heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, degree 1 defines only one neighbor, and the federation is formed with two fog systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, none of the heuristics performed well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Even though the proposed method MR, performed better than baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Whereas for the highest degree of the federation, the proposed MR heuristic performed approximately 18-20% better than MECT and MCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, for all of the federation scales up, the proposed MR heuristic outperforms others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The main reason is that MR, efficiently utilizes fog federation resources, considering the communication and computation latencies to complete every monolithic application on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5 Summary The advancement in software and hardware stack has brought the industrial revolution, Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0, that changed many legacy system architectures and imposed latency constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As such, complex industrial processes are adopting smart solutions every day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, computation near the data source supports smart microservice-based solutions that significantly face resource scarcity and latency constraints challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Especially in remote offshore Industries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', Oil and Gas, mining ), the latency issue can be critical for complex fault-intolerant industrial processes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', hydrocarbon exploration, drilling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Moreover, in emergency situations, the computational execution platform gets oversubscribed with various types of microservices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To overcome challenges enforced by the smart microservice-based solutions, a robust task allocation scheme proposed in this research work that is aware of the software architecture of the solution as well as 96 uncertainties imposed by fog federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, the proposed solution works on two levels within a load balancer module that exists in every fog system of the federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The first level considers the software architecture of the receiving application and performs partitioning if necessary, utilizing the probabilistic success rate to complete the applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then in the second level, the received applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', monolithic applications or partitioned microservices) are mapped across fog federation, considering the computation and communication constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The evaluation results reflect the benefits of using the proposed solution in oversubscribed situations that are approximately 15∼20% better than the baseline partitioning and resource allocation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In the future, we plan to incorporate an ML-based resource provisioning method to improve the robustness of the federated fog system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 97 Chapter 6: Data Security & Privacy Aspects in Federated Fog Computing System 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Overview The rise of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 [113] elevates the utilization of IoT devices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', sensors, actuators) and fog computing for developing deep neural network (DNN) applications in various industrial sectors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', smart oil field, smart farms, smart factory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The DNN-based applications mainly backed up by ML network models that are supposed to train with huge amount of data for achieving relatively high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Although the training data could be privacy preserving (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', sensitive to any company), and sometimes data acquisition (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', Satellite image data, high resolution camera data) is expensive, and time consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The expense of developing these DNN-based applications could also increase with data transfer to cloud datacenter using internet for training operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence fog federation (formed by multiple private companies fog systems) can be a potential candidate for supporting computational demand of ML-model training where data security and privacy of the participant private fog systems in the federation need to be addressed to efficient ML training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In this case, federated Learning (FL) techniques [66] that brings ML-model to participant end user without leaving the data their source device, can be applied to overcome the privacy constrains of the fog systems owned by private companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Although the privacy preservation constraints are mitigated by FL as depicted in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1, it can impose some new challenges for the ML models training operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Here, the problem is that data are coming from various 98 sources, and it is feasible to assume data distribution tends to be non-identical and independent distribution (non-IID).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As such, lack of any priority class (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', consider oil spill class in oil spill detection problem) that is termed as class imbalance [114] can reduce the performance of the global DNN model in a FL setup as presented in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, ignoring the class imbalance issue, current federated learning methods [66] are providing less robust DNN model for oil spill detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In this case, an object detection model can show misleading high accuracy for all other classes while providing low performance for the desired class (oil spill).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' A federated learning setup in fog federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Multiple company share their fog systems to train oil spill detection DNN model where data security is pre- served by federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Global DNN model Local DNN model 1 Local DNN model 2 Local data Fog computing (micro-data center) Aggregator Fog System 5G communication Oil spill Oil spill Drone capturing images Federated learning is a special branch of distributed machine learning where the global model needs to be converged at a constraint rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, the convergence of FL mainly depends on the local workers’ aggregation that affects the global model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Among two types of federated learning (synchronize and 99 0asynchronous), we propose to utilize the synchronize FL method as it is a proven model, especially for class imbalance issue [115, 116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As such to overcome the challenges of FL for oil spill detection, we have adopted an objective function (loss function) to train the local model considering the class imbalance problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Considering the priority class (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', oil spill), we introduce a weight for each participating worker that intensifies or attenuates its influence over the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The relevant worker selection based on the worker weights is verified by empirical evaluation in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Finally, a dynamic threshold mechanism has been proposed to select relevant workers efficiently considering the global model’s performance and fast convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Problem Formulation for Federated Learning The oil spill detection problem can be well defined in semantic segmentation domain of deep learning where various classes are identified in pixel level from original source image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In oil spill detection training various classes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', oil-spill, look-alike, land, ship, sea-surface) are found in real world satellite image data set [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Here, each class is labeled as an individual color in ground truth image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For training a deep neural network (DNN) model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', Unet) with federated learning settings a set of workers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', fog systems of a fog federation) S = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', S are considered with its own local data set DL where L ∈ S with nL samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Here, D = � L∈S DL is the full training data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The total size of these workers’ data set for a random set of workers S′ is N(S′) = � L∈S′ nL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The objective loss function over a model m and a sample z can be denoted as L(m, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 100 Then in most prior FL work, the goal is to solve the following min w f(w) = M � m=1 pmFm(w) where pm = nm D is the fraction of the total data worker, and thus, � m pm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The local objective Fm is typically defined by the empirical loss over local data, Fm(w) = 1 nm nm � j=1 Lj(m, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Here, w is the model parameter that used for predicting loss over a sample data, and the goal is to find the optimal w for which the loss should be minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accordingly, we focus on utilizing a loss function that consider class imbalance problem in local data samples, and select a set of client worker’s (fog system) models to aggregate that have certain level of accuracy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', mean intersection over union (mIoU) for semantic segmentation) to ensure the robustness of the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, our new objective for this work would be as following: min w f(w) = M � m=1 pmFm(w) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' mIoU(m) >= γ, θ > 1 Where, γ is a dynamic threshold (initial value set to 50% or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5) for checking the local trained model’s mIoU with auxiliary test data, and θ is the user defined worker’s weight with respect to oil spill class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Both of this parameters are used to select the relevant worker’s model for aggregation into the global model that ensure the robustness, and consistency of the convergence for the aggregated model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 101 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Pixel distribution for each of the class in oil spill detection data set Class Pixels Sea Surface 797.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7 M Oil Spill 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 M Look-alike 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4 M Ship 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3 M Land 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7 M 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3 Federated Learning to Mitigate the Class Imbalance In a typical federated learning setup, the server (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', fog device, cloud) stores a global ML model for training with local data of the participating workers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', fog systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We use one of the popular semantic segmentation DNN models named as Unet [117] model for oil spill detection in the FL setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 represents a pictorial view of our solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' At first, some fog nodes agree to participate in the FL training, and they download the global model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', Unet) from the fog server presented in step 1 of figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then, downloaded ML models are trained with their local data in step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, we utilize tversky loss function for local training that work efficiently for class imbalance issue proven by the research community [118, 119, 120, 121, 122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In step 3, ML models are checked for relevant worker model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Finally, in step 4, selected workers updated models are aggregated (new model), and the previous global model is updated accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' This whole process is considered a federated round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The updated model is again downloaded by participating worker for the next federated round, and the training continues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The proposed solution is presented in algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Usually, the aggregator fog server provides the global model and aggregates 102 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Federated learning training considering class imbalance and global convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Tversky loss is used in the training considering class imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' After training of each epoch, mean intersection over union (mIoU) is checked with a dynamic threshold for global convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' GPU Global DNN model Aggregator fog server Model training Local data Download global model Check mIoU & worker weight Check mIoU & worker weight Check mIoU & worker weight Fog worker 1 Fog worker 2 Fog worker 3 Download global model Updated model send Model aggregartion Update global model Tversky loss Tversky loss Tversky loss Relevant worker selection Local Level Global Level Step 1 Step 2 Step 3 Step 4 the updates sent by the worker fogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The FedBal algorithm starts with initializing global model mg, relevant worker list, rf, and setting the threshold, th value to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' After that, the federated round continues as a for loop that is presented with variable f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then m number of workers are selected from K participating worker, and assigned to selected worker list, St for training (“ClientUpdate” function) with their local data in the second for loop of the algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Finally, relevant workers are selected using function “selectionCriteria”, and aggregate into the new global model, mg using “averageModel” function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The “ClientUpdate” function performs the training with the tversky loss function and defined number of epochs to reduce the class imbalance at the local level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The proposed solution’s global level is 103 Algorithm 3: The K workers are indexed by k, C is the initial worker selection percentage 1 Initialize global model, mg, test data, Dtest, relevant Worker List, rf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 2 Set threshold, th = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 3 for each federated round f = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' do 4 m ← max(C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='K, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 5 St ← (random set of m workers);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 6 for each worker k ∈ St in parallel do 7 ClientUpdate(k, mg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 8 rf = selectionCriteria(St, Dtest, th);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 9 mg = averageModels(mg, rf);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' triggered in the “selectionCriteria” function, where trained worker models are evaluated according to their weight, θ, and mIoU value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The dynamic threshold mechanism also takes place in the “selectionCriteria” function to ensure the robustness of the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In this way, in every federated round,f, the global model updates and converges to a model robust against class imbalance with guaranteeing performance for our priority class, oil spill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4 Experimental Setup The Federated learning setup can be synthesized by PyTorch’s one of the popular library pysyft [123], and TensorFlow’s federated learning library named as tff [124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Due to pysyft’s customization capability, we have selected pysyft as our development library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The oil spill detection is considered a semantic segmentation problem that typically uses real-world SAR image data sets (in this work, the data set is collected from MKLab [77], a research institute in Greece) for training a DNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To execute the DNN training operation, we used Google’s Colab [125] run-time environment that provides a GPU platform with a high-speed ram of size 104 24 GB with storage of 128 GB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The Colab provides Tesla P100, T4, or similar GPUs for the paid “pro” version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' It also has the high-RAM option for faster execution while using GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We utilize pysyft’s virtual worker’s concept to synthesize fog devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Our primary focus in this work is to reduce class imbalance issues and ensure a robust global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence we concentrate on the computation part of FL and ignore the communication (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', network) of conventional FL setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Our federated learning setup can be utilized for any aggregation algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', FedAvg, FedSGD, FedProx), and as such, we develop our codebase on top of these baseline algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As our FL setup works on reducing the class imbalance, we named this setup “FedBal”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In most of our experiments, we use 20 federated rounds where each round consists of 50 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The reason behind choosing these values for the training parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', number of epoch, number of federated rounds) is to observe a significant difference among the aggregation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Finally, due to time constraints, we bound our experiments within 20 federated rounds of aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5 Performance Evaluation The federated learning setup is always beneficial for fog devices where data tends to be generated frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, to understand the advantage of utilizing federated learning, we perform an experiment capturing the loss found in each epoch of training using federated learning and single machine training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The federated learning setup (a) could use more data as there are four workers perform the DNN model(Unet) training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' On the other hand, a non-federated learning setup 105 uses fewer data to train the model with a single fog device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Moreover, the uncertainty in federated learning setup is less severe than non-federated learning that we found in our initial experiment, (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Considering the convergence of the training model, FL is also faster than non-fl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Although FL has better performance than typical machine learning, the class imbalance issue in the local data can make the global model’s performance degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, our local worker level solution utilizes the tversky loss function where α for penalizing false negative and β for penalizing false positive parameters need to be tuned for better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The experiments with these parameters are provided in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Tuning Loss Function To find the optimal Tversky loss function, we change the alpha parameter value from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8 and capture each training epoch’s loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The main goal is to find the optimum alpha value for which the loss will be minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The results of these experiments are demonstrated in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3 represents the training loss (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', y-axis of the figure) for each epoch (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', the x-axis of the figure) while using alpha values 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8 respectively within FedAvg, and FedBal FL setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' From figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3, we find that FedBal performed similar in comparison with FedAvg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' It is also visible that for alpha value 0f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7, FedBal has minimum training loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' When we increase the alpha value to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8, the training loss does not decrease, which means we can penalize false negatives up to a certain point (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The reason is that while we are penalizing false negatives, the false positive predictions are ignored (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', α + β = 1) 106 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Comparison of FedAvg and FedBal training loss utilizing tversky loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The alpha parameter of tversky index is changed from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8 (left to right) and the loss per epoch is captured for both FedAvg and FedBal algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' FedBal FedAvg Alpha = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6 training loss training loss Alpha = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7 Alpha = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8 epochs as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence for a higher value of alpha, we get less benefit by penalizing false-negative predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, we use α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7 for the rest of our experiments throughout this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Comparison of FedBal with FedAvg, FedSGD, and FedProx method’s global model performance in IID setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Global model mIoU comparison in IID setup (a) Comparison with FedAvg (b) Comparison with FedSGD (c) Comparison with FedProx 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 The Impact of Using IID Data Distribution The benefit of a federated learning setup is reflected in the performance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', accuracy (mIoU)) of the global model after the aggregation step of FL is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content='0000 0 20 40 60 80 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content='0000 0 20 40 60 80 100FedProx ★- FedBalFedSGD ★ FedBalFedAvg ★ FedBal NNHence, we measure the mIoU of the global model after every communication or federated round of FedBal with FedAvg, FedSGD, and FedProx, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then, we plot the result as a line graph in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For this experiment, we consider the data distribution among the FL workers is identical and independent distribution (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='a IID) which means every worker gets all the classes of images in their local data for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The x-axis of the figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4 represents the federated rounds, whereas the y-axis presents the mIoU of the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' From the figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4 (a), it is visible that FedBal has outperformed FedAvg in most of the fed rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Considering FedSGD, in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4 (b), FedBal performed significantly well in the last few rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Although, in the initial rounds, FedSGD performed better than FedBal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Finally, from figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4 (c), we find that comparing FedProx, our FedBal method performed significantly well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' This improvement mainly comes from the utilization of left-out workers in the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In addition, the worker selection in FedBal considers the class imbalance issue and the priority class (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', oil spill class) for aggregation into the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In contrast, other methods randomly select active workers for aggregation, leading to a less robust global model than FedBal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3 The Impact of Using non-IID Data Distribution In a real-world scenario, data distribution among FL workers is typically non-IID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' That means every worker will get some fixed number of classes (not all the classes) for local training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, we consider providing two classes for each worker, and these classes are different for every worker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Similar to our previous experiment, 108 we measure the mIoU of the global models for FedAvg and FedBal algorithms in each federated round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The result is provided in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5 for 20 federated rounds with six federated fog workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The performance comparison of global models in terms of mIoU using FedAvg and FedBal methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The data distribution is non-IID, the number of workers are 6, and in each fed round 50 epochs of training has been performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 Fed Rounds 38 40 42 44 46 mean intersection over union (mIoU) % Global Model Performance nonIID FedAvg FedBal The figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5 reflects that FedBal has a consistent performance (mIoU) for 20 federated rounds then FedAvg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The FedBal method has less uncertainty (fewer spikes in orange line of figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5) across the federated rounds for selecting relevant workers in every federated round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Although FedBal has less significant performance improvement than FedAvg, the average mIoU of the global model of FedBal is higher than FedAvg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' This consistent performance of FedBal represents the robustness of our method across the federated rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4 The Impact of Using non-IID and Unbalanced Data Distribution The non-IID and unbalance data distribution means each FL worker has a different number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For instance, worker one can have two classes, whereas worker two can have only one class in its training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, we measure the 109 mIoU of the global model and compare our method (FedBal) with the other three baseline methods named FedAvg, FedProx, and FedSGD, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As our method is considered an improvement of any federated learning aggregation method (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', FedAvg, FedProx, FedSGD), we compare the baselines separately in three different sub-figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The result demonstrates as a line plot in the figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6 where the x-axis represents the federated rounds, and the y-axis represents the mean intersection over union (mIoU) of the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Comparison of FedBal with FedAvg, FedProx, and FedSGD method’s global model performance in non-IID and unbalanced data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Global model mIoU comparison in non-IID and unbalanced setup (a) Comparison with FedAvg (b) Comparison with FedProx (c) Comparison with FedSGD Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6 represents that FedBal outperforms FedAvg, FedProx, and FedSGD respectively in the final round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Although, in the 19th federated round, FedSGD and FedBal perform similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The performance improvement for FedBal is significant for FedProx, due to the utilization of the left out workers in the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' It is also visible that baseline methods performance has severe uncertainty (more spikes than FedBal), whereas FedBal has comparatively consistent performance throughout the federated rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The main reason behind this consistency is the relevant worker selection in FedBal with a dynamic threshold mechanism that maintains a certain performance and provides a robust global 110 FedSGD ★-FedBalFedProx ★- FedBalFedAvg ★ FedBalmodel against class imbalance issues in the local data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5 The Impact of Class Imbalance Intensity The class imbalance is a common phenomenon in the oil spill data set where the intensity of the imbalance can be severe within the FL setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As such, we consider our solution, FedBal, to be performed consistently well than the baseline FedAvg algorithm in all levels of class imbalance intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To explore the class imbalance intensity, we distribute the classes from high imbalance to low imbalance using a non-IID setup and measure the mIoU of the global model for FedBal and FedAvg across the federated rounds of FL training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We estimate the difference of mIoU values for each federated round for three cases (one class, two classes, and three classes distribution) of imbalanced data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Furthermore, then we plot a bar chart presented in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7 where positive values indicate FedBal’s improvement over FedAvg, and negative values represent the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7 represents the advantage or disadvantage of FedBal over FedAvg algorithm across 20 federated rounds of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For the first nine federated rounds, the improvement of FedBal over FedAvg is not significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In the 10th federated round, the difference values are all positive, and in the final federated round (20th), we find the highest performance of FedBal over FedAvg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We also notice that for 2 class per worker distribution, FedBal constantly outperforms FedAvg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For high intense class distribution (only 1 class per worker), FedBal starts to perform well after the 10th round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In the final round, we find that for 3 class distribution FedBal has the most remarkable improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The main reason behind 111 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Comparison of FedAvg, and FedBal method’s global model performance in non-IID data distribution from high intensity(only 1 class per worker) to low inten- sity(3 classes per worker).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The difference of mIoU of FedBal, and FedAvg is plotted as barchart for 3 case scenarios (1 class, 2 class, and 3 class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 1 5 10 15 20 Fed Rounds 3 2 1 0 1 2 3 4 5 mIoU Class Imbalance Intensity - Advantage over FedAvg 1 Class Per Worker 2 class Per Worker 3 class Per Worker the less significant performance could be the dynamic threshold mechanism that starts with a good mIoU value (50%) and dynamically change over federated rounds to increase the performance of the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' After the 10th round, the threshold becomes stable with a sufficient number of relevant workers, and we see performance improvement for the last ten rounds of federated training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6 The Impact of Number of Workers on the Global Model To understand the influence of FL workers in our federated learning method, we measure the maximum mIoU of the global model for FedBal and FedAvg methods by gradually increasing workers from 6 to 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The main focus of this experiment is to compare the performance of our method, FedBal over FedAvg, 112 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The influence of federated worker on global models performance (mIoU) for FedBal, and FedAvg is measured by increasing the number of federated worker from 6 worker to 25 worker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For each case of worker pool 20 federated rounds of training are performed for both FedBal, and FedAvg method, and for each case maximum mIoU of both methods are considered for plotting as a barchart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 6 worker 10 worker 15 worker 20 worker 25 worker Fed Rounds 0 10 20 30 40 50 60 mIoU Influence of workers on performance FedBal FedAvg while increasing the FL workers gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The result of this experiment is provided in the figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8 presents that the performance improvement of FedBal is significant when we have an increased number of FL workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' It is visible that up to 15 workers FedBal does not show performance improvement then FedAvg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The reason is that FedBal selects relevant workers from the active workers’ pool, and sometimes the relevant workers are very few, leading to a less improved global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' On the contrary, FedAvg always selects the same number of FL workers throughout the federated round, thus having better performance than the low number of FL workers pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, an increased number of workers can significantly improve the global model’s performance using the FedBal method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7 Verifying the Impact of Aggregation Scheme on Global Model The selection technique of the FedBal algorithm considers the relevant 113 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The impact of workers weight (averaged on each of the federated round) on global model’s mIoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 2 4 6 8 10 Fed Rounds 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='9 Workers Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' weight weight 37 38 39 40 41 42 43 44 45 mIoU The influence of weights on Global Model mIoU mIoU worker by checking their weights defined by the priority class (oil spill class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, workers with significant performance have a positive impact on the global model’s mIoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To explore the impact of the workers’ weight in the global model, we measure the average workers’ weight in each federated round and estimate the global model’s mIoU after the aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The result of this experiment is presented in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='9, where the x-axis presents the federated round, the left y-axis presents the average workers’ weight, and the right y-axis presents the global model’s mIoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='9 presents that selected worker’s weight has a positive impact on the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' It is visible that with the increase of the average weight, the mIoU of the global model goes up, and with the decrease, the mIoU goes down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, the FedBal method’s relevant workers are supported by the user-defined weight based on priority class, oil spill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 114 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6 Summary The federated learning technique has revolutionized distributed machine learning, especially considering data privacy and computational flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' With the emergence of IoT, Edge, and Fog computing, the data generation is getting faster and expensive when needed to be transferred utilizing network bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence federated learning can bring the ML model to the data generation sources that is less expensive and secure than cloud data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' This case is more applicable than conventional centralized DNN model training considering ML support in remote areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In addition, the data captured or collected in remote oil fields are sensitive, and privacy preservation is of significant importance for oil and gas companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Although federated learning can overcome these challenges, the class imbalance issue can degrade the DNN model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As such, we focus on reducing the effect of class imbalance at the local level while training the model, and the global level while aggregating the federated worker into the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' At the local level, we use the tversky loss function with appropriate parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', α, β) tuning to train each federated workers model considering the class imbalance issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then we assign each worker a weight, considering our priority class, oil spill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Finally, we check each federated worker’s model mIoU with a predefined widely accepted mIoU value (50% or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='50) and dynamically change the threshold to ensure the robustness of the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In the empirical evaluation considering the global model’s mIoU we find that for IID setup, FedBal has around 3% performance improvement than FedAvg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For 115 non-IID setup, we find similar performance in the final federated round (20th) compared to FedAvg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Although, FedBal’s average performance for the non-IID setup is better than FedAvg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For non-IID and unbalanced setup, FedBal outperforms FedAvg, FedProx, and FedSGD respectively in the 20th federated round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Although, FedSGD has similar performance compared to FedBal, it has more uncertainty than FedBal (figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6 (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In the class imbalance intensity, we find FedBal performs better than FedAvg in the final federated round (20th round) in three of the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Although, for high-class imbalance (only one class per worker) intensity FedBal has less significant improvement (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='25%) whereas for low-class imbalance (three class per worker) intensity FedBal shows significant performance improvement (more than 2%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The experiment with the increasing number of federated workers reflects that FedBal’s performance can be improved (up to 2%) with an increased number of federated workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Due to time constraints and network vulnerability, we could not scale up the experiments, especially with the increased number of workers and class imbalance intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Finally, the impact of FL workers weight in FedBal method’s global model reflects a positive relation that verifies the selection methods acceptability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The ML training parameters (number of epochs per federated rounds, optimizer, batch size) can be tuned in a more granular way to explore the areas of improvement using the FedBal method that is considered as the future work of this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accordingly in future, we also plan to develop a custom loss function for the semantic segmentation field of deep learning and enhance our method (FedBal) as a service plugin that can be used on 116 top of any federated learning algorithm to improve the robustness of the ML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 117 Chapter 7: Threats and Side-Effects of Smart Solutions in Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Overview The convergence of new IoT technologies, cloud computing systems, improved wireless networks, and machine learning solutions have enabled smooth operations of large-scale cyber-physical industrial systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Wireless connection, in particular, has altered operating paradigms to the point that most, if not all, production activity may now be managed remotely using a variety of sensors and actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Furthermore, these technologies have significantly increased the production and efficiency of various complex industrial operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, not everything about the digitization and smartness paradigm shift is positive!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' There are some disadvantages to consider as well—digital transformation and pervasive connection present weaknesses that criminals might use to launch cyber-attacks, thus jeopardizing industrial production, distribution, and even safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As we have noticed in several recent instances, such as colonial pipeline [126], Amsterdam-Rotterdam-Antwerp (ARA) cyber-attack [127], and Norwegian energy company [128], malicious software systems (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' malware) have been able to take over the control of a system and block its regular operation until the intruders’ demand has been fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Indeed, these recent cyber-attacks have proven that cyber-attacks can be as harmful as physical attacks in terms of both implications and severity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Smart sectors, such as O&G, have unique security challenges that can only be addressed through in-depth research and diagnostics of the entire system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, 118 specific security solutions for smart industries have yet to be available due to their high implementation complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' This security vacuum has allowed countless cyberattacks to flourish in recent years, endangering people and communities worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, it is imperative that, as part of the Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 revolution, all-encompassing security solutions be investigated for smart industries due to the crucial nature of these sectors and the managerial and technical gaps between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As the oil and gas industry becomes increasingly complicated and digitized, we are considering researching key areas of smart O& G that pose a security risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The upstream, midstream and downstream deployment of a vast network of linked “things” (IoT devices) presents a significant security risk for the oil and gas industry as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Predictive maintenance and on-site worker safety are just two examples of the kinds of efficiency gains that may be made possible by processing the massive amounts of real-time, real-world data generated by smart sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' There is a risk that the use of internet-connected devices might compromise the physical security and safety of O&G infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For instance, interconnected cameras equipped with object-tracking capabilities, geofencing perimeter protection solutions, third-party infiltration, and other access control systems can cause security breaches in operational sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, this thesis section focuses on the adverse outcomes of smart solutions for Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 and the strategies for minimizing those outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 119 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' A taxonomy reflecting the downsides of smart solutions implemented with advanced technology is organized using box flow-chart form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The main three levels are colored in orange, blue, and yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The white boxes represent different types (examples) of its parent box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Cyber-Threats Device Incompatibilities Interaction Challenges Bias in Smart Solutions Unauthorized Data Exposure Information Technology (IT) Platform Operational Technology (OT) Platform Hardware Software Data Pipeline Human-to-Machine Interaction Machine-to-Machine Interaction Biased AI Automation Bias Gender,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Age,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Language,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' & Culture Bias Vulnerability of Smart Solutions Side-Effects Bridging IT & OT Platforms Malware Direct attack on actuators Ransomware SCADA attacks Man-in-the-middle (MIMT) attack Denial-of-service (DoS) attack Biased Data Biased Model Machine Bias Format Workflow Data Accuracy Downsides of Smartness in O&G Version Visual Interface Connection Lack of Conyrol 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Taxonomy of Cyber-Threats and Side-Effects in the Smart O&G In- dustry To categorically explore various drawbacks of smart solutions in the O&G industry, we develop a taxonomy that is presented in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We separate the possible drawbacks of smart solutions into two groups in this taxonomy: vulnerabilities and side-effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The vulnerability section investigates cyber-threats 120 and challenges caused by device incompatibilities in a smart O&G system, focusing on software, hardware, infrastructure, and data-related vulnerabilities in the oil and gas industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' On the other hand, the side-effect category focuses on difficulties that develop as a result of interactions with smart solutions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', human-machine and machine-machine interactions) and biases in a smart system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 categorizes various drawbacks of smart solutions implemented or will be implemented in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' This taxonomy serves as the blueprint for this chapter, enabling readers to keep track of sophisticated smart solutions and their accompanying outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, we will traverse major taxonomy sections in the following parts to comprehend the magnitude of smart solutions’ drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3 Vulnerabilities caused by the Interplay of Informational and Opera- tional technologies A smart oil and gas industry’s technological operations are organized into two key technological platforms: information technology (IT) and operational technology (OT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 depicts an overview of an oil and gas company’s IT and OT components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As seen in the diagram, the IT component is primarily concerned with the movement of data and information throughout the company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' IT components frequently access outside networks due to their operational context, which is mainly business logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In contrast, the OT component is involved with the operation of physical processes of oil and gas production and the machinery needed to carry them out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a result, cyber thieves primarily target IT and OT platforms to meet their needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Traditionally, the IT component has been more susceptible than 121 the OT platforms because IT has numerous open windows (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', operating systems, email servers, direct communication applications) that attackers may exploit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' On the other hand, OT platforms mainly deal with direct oil and gas production and processing activities with limited external access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Notably, the junction of IT and OT platforms is frequently a target for cyber attacks that system architects must effectively handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Furthermore, smart IoT solutions based on sophisticated computing technologies are opening up access to OT platforms with the rise of IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a result, we explore the extent of the vulnerabilities in these two platforms as well as their overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Information technology (IT) and operational technology (OT) platforms of a smart oil and gas company that operates using different networks to run the entire operation of smart O&G industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The IT platform is significantly related to business applications and the financial side of O&G, whereas the OT platform directly involves with oil or gas extraction and production operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Both IT and OT platform is connected at some point which creates the sweet spot for cyber-attackers to penetrate into the whole system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Information Technology (IT) Operational Technology (OT) ERP Solution Refinery operation Pipeline management Hydrocarbon extraction Vulnerability Smart O&G Supervison system Control system Database management wireless network router Sensor users applications network users operations applications network The OT platform is comprised of technologies that are actively engaged in the production of petroleum end products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The activities include extraction, refining, pipelines, production, control, and monitoring systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' On the other hand, 122 兰 $the oil and gas IT commodity primarily deals with finance, database administration, digital asset management, and other business operations using different computer platforms and communication protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In this case, the OT entity provides the petroleum end products, while the IT entity develops commercial prospects and financial policies by exploiting the OT entity’s output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, compared to the OT entity, the contact with the outside network from the O&G company’s internal network is substantially greater for the IT entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Because of this relationship, a petroleum company might become a victim of ransomware and other cyber-attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' OT was traditionally an “air-gapped” environment, which was not linked to public networks or other digital technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For decades, traditional OT has depended on computers to monitor or modify a system’s physical state, such as employing SCADA systems to monitor and control equipment to increase operational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Traditional OT security largely comprises simple physical tasks, such as ensuring that a machine performs the same operation correctly and that an assembly line continues to run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Nonetheless, the emergence of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 in recent years has altered the conventional OT environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Companies have started to deploy new digital solutions in their networks to boost automation via the addition of “smart devices” that can gather data more effectively and have network access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The IT and OT systems were integrated as a consequence of this connection and to process/analyze the OT data as it was generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Although this technological paradigm change (referred to as IT-OT Convergence [129, 130]) has generated new possibilities and unlocked new use cases, it has also offered scope for 123 cybersecurity vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For example, Colonial Pipeline’s assault [131] demonstrates how poor password management may harm the country’s largest gasoline pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The hackers found the password for an old but still working VPN account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In light of this threat, oil and gas companies should establish strict cybersecurity safeguards, including employee training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' STUXNET [132] was the first specialized hack into industrial control system (ICS) to attract considerable attention, although not being the first cyberattack against an industrial environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' STUXNET is a computer worm that is accused of creating havoc on Iran’s nuclear programme, damaging more than 20% of the country’s nuclear centrifuges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Since then, cyber-attacks on industrial organizations have progressively risen, affecting a wide range of industries, including power grids (Industroyer), energy (Black Energy), petrochemicals (Havex), and oil and gas (Havex) (TRISIS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hackers are hacking into industrial networks, among other things, to shut down machines, demand ransom, and steal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4 Cyber Threats in Smart Oil and Gas Industry The challenge with the oil and gas industry is that its systems need to be designed with network connections in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For instance, plants were never designed to be network-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, they are today as a result of the developing digital revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' This may create a dangerous scenario since a cyberattack on such a system can damage operations and cause loss of life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In terms of cybersecurity, the O&G industry lags behind other industries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Even though cybersecurity is vital to the company’s sustainability, many companies still need to 124 spend more on robust systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The remainder of this section discusses some security problems the O&G industry confronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Vulnerabilities of Sensitive Data When stored on industrial IoT devices (sensors and actuators), sensitive information must be protected by rigorous security protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a result, oil and gas companies now routinely examine private information gathered from a wide range of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Here are some examples of such data sources: Historical oil & gas exploration, delivery, and pricing data Demographic data Response data from job postings Web browsing patterns (on informational websites) Social Media Traditional enterprise data from operational systems Data from sensors during oil and gas drilling exploration, production, transportation, and refining The aforementioned are examples of highly confidential information for any private corporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Various confidential information belonging to one company might be precious to a company’s competitors in the oil and gas industry due to the intense rivalry in this sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a result, hackers with questionable ethics increasingly focus on gaining access to these sensitive records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 125 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Vulnerabilities of Smart Systems In earlier chapters, we covered smart solutions that empower Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Although these are intriguing and future technologies that might assist the oil and gas sector as a whole, their weaknesses should also be acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The following are some of the ways a smart solution might fail or be compromised: Inherent bias in a machine learning method: The quality of the training dataset is crucial to the success of any machine learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' A biased dataset is one that has been selected in such a way that some types of examples are given more weight than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For example, suppose the photo dataset used to train the model for pipeline leakage detection by drones mostly covers bright weather settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In that case, the trained model will perform badly in rainy or snowy weather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Predictive maintenance may also be used when a model is taught to work with a certain brand of equipment under certain conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Consequently, the model failed to generalize previously observed data correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, it would need to improve in accuracy before it could be used as a predictive maintenance model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Uncertainty exists in the machine learning model: Machine learning models are susceptible due to their inherent ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, it is feasible that the model may provide false-positive or false-negative findings, which might have disastrous repercussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For example, if a refinery’s smart fire detection system overlooks a fire, it might cause severe damage quite rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Failure in the workflow of a smart application: Smart solutions are usually composed of many parts that work together to build a directed acyclic graph or 126 DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Face detection on an oil rig, for example, entails capturing videos, removing frames, and then analyzing each frame individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Interrupting such a smart application cycle at any point might cause the whole application to fail, making it vulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Similarly, if the command is not sent to the actuator, the whole workflow may be deactivated, resulting in a loss of control over the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3 Malware and Vulnerability of Information Technology (IT) Malware, an abbreviation for “malicious software,” refers to any invasive program created by cyber criminals (also referred to as “hackers”) to steal data and damage or destroy computers and computer systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Examples of malware include viruses, worms, trojans, spyware, adware, and ransomware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Recent malware attacks have resulted in massive data leaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, the malicious actor(s) must be identified swiftly to remove malware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Among many forms of malwares, we discuss four major types in the following paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Virus: In order to infect other computers, viruses often attach themselves to files that can run macros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The virus will remain latent inside the downloaded file until it is opened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Viruses are malicious programmes that interfere with normal system functioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' This means that infections may interrupt operations and lead to lost data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Worm: A worm is a piece of malicious software that can quickly copy itself and infect any system on a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In contrast to viruses, worms may spread without the help of any host software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For example, a worm may infect a device by a file download or a network connection, then rapidly replicate and spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Worms, 127 like viruses, may drastically impair a device’s functionality and lead to data loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Trojan: Often, Trojan malware may mask as seemingly valuable pieces of software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, once downloaded, the Trojan virus may access the user’s private information and make changes, prevent access, or even erase it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The device’s functionality may suffer severely as a result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In contrast to common viruses and worms, Trojan viruses are not programmed to multiply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Spyware: Spyware is malicious software that works surreptitiously on a computer and feeds data back to an outside source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Spyware is especially hazardous since it affects device performance, targets sensitive data, and allows would-be attackers remote access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Spyware often targets financial or personal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' A key-logger, for example, is a kind of spyware that records users’ keystrokes in order to steal passwords and other confidential information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Ransomware: Ransomware is a kind of malicious software that infiltrates a system, encrypts its data so that the user cannot access it, and then demands payment in exchange for decrypting the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The use of ransomware is often associated with a phishing scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3 depicts the stages of an actual ransomware attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As can be seen, in these assaults, the victim downloads the ransomware by accidentally clicking on a spoofed link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The attacker then encrypts the targeted data using a cryptographic key that is known only to the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Finally, in exchange for money, the hacker will release the information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We then go on to analyze this threat in further depth because of its rising prevalence over the last several years, especially in the oil and gas sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 128 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Ransomware attack incidents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' During a targeted cyberattack, a single virus may be used for a variety of reasons, including data theft, spread, and penetration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The threat actor’s goal is to maintain persistence inside the victim’s network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, they have to constantly communicate with and update their virus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Using the DNS protocol, a process known as DNS tunneling [133] transmits information between malware and the controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Additionally, email and cloud services have greatly expanded the scope of modern-day communication, which creates a wide door for ransomware criminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The anatomy of ransomware from start to end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The ransomware client enters the IT platform through malicious email or other external mediums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The client then communicate with hacker’s command and control server to download the encryption key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The user’s data encrypted by the ransomware client, and finally extortion notice is sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=" ransomware encrypts data and post extortion notice ransomware self install, contact attacker C2 server, and download public key user must pay the ransom in exchange of dycription key malicous code from email, flash drive or other external medium The command and control (C2) server 1 public key hacker's control server that store encryption key 2 3 4 5 129 Historically, hackers used spam botnets to infiltrate as many systems as possible and propagate ransomware." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' reference mandal2020digital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Although ransomware has always been a huge problem for everyone with digital files, it has become an even bigger problem as criminals have begun to specifically target businesses in assaults that may have devastating effects on operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The following are examples of some of the most notable ransomware attacks: BitPaymer19: BitPaymer19 [134] is a particularly deadly type of ransomware that recently attacked a U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' firm providing oil well drilling services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' BitPaymer actors often employ phishing emails to infect their victim with first malware before moving laterally throughout the network to compromise other sensitive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' When IT personnel are unavailable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', on weekends and holidays), the ransomware attacks are APT33: One well-known actor group’s primary concentration is on the oil sector and its supply networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Organizations in the energy sector with linkages to petrochemical manufacturing and the aviation industry, where APT33 is involved in both military and commercial capacities, have been targeted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' APT33 has also hit energy companies in Europe and Asia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' From October 2018 through December 2018 and into 2019, a Powerton C&C server was hosted on the C&C timesync.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='com website and communicated with a small number of IP addresses belonging to oil corporations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Over the course of three weeks in late November and early December of 2019, a database server run by a European oil company in India spoke with a Powerton C&C server used by APT33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In the fall of 2018, it was discovered that a 130 significant UK-based corporation offering specialized services to oil refineries and petrochemical plants might have been penetrated by APT33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Email phishing was APT33’s most common method of infiltration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For many years, this scam has relied on the same bait: an email that seems legitimate but is a spear phishing attempt to offer a job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Other campaigns were directed against the recruiting procedure in the aviation and oil industries [135].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Additionally, a link to the malicious “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='hta” file is provided in the email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To further entrench themselves in the target’s network, APT33 may use the PowerShell script downloaded with the “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='hta” file to download further malware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5 Incompatible IoT Devices Among the smart O&G sector’s most common vulnerabilities is the use of incompatible Internet of Things (IoT) devices, as seen in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In fact, automated systems that take data from a wide variety of Internet of Things (IoT) devices and sensors, a process that data using machine learning or statistical models, and then implement their decisions via a variety of actuation operations are the real engines behind a smart industry like oil and gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In reality, the sensors and other linked IoT devices are created and purchased by a wide variety of companies, making them inherently heterogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' This diversity may cause incompatibilities and can be exploited by cyber-attackers or lead to gaps in service during times of crisis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For effective data transmission and offering real-time communication during emergency scenarios (for example, poisonous gas detection), it is crucial to configure linked and suitable IoT devices that can interact seamlessly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 131 Since acquiring uniform and completely compatible IoT equipment is difficult, if not impossible, researchers are looking at other solutions, such as the development of standard protocols that would enable effective communication across all industrial IoT devices—because of this, leading IT firms are collaborating to create a single protocol (called matter protocol [136]) that will be compatible with any and all Internet of Things (IoT) gadgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The issue is a new protocol for inter-network communication between smart homes that are being backed by the Connectivity Standards Alliance, which includes tech giants like Apple, Google, Amazon, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The problem is the lack of a standard, IP-based communication protocol that is based on tried and true technologies to construct safe and secure IoT ecosystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In smart O&G and other smart industrial contexts, we might examine three different kinds and degrees of incompatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In the following sections, we will discuss the incompatibilities that exist: those at the hardware, the software, and the data pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Hardware-level Incompatibility Commonly available products are increasingly being utilized to replace specialized equipment in the oil and gas sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' They are more susceptible to security problems than traditional process control systems because of their adaptability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Because they are so widely used and deployed, the attack surface is widened significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' There are several methods to assault an oil field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For instance, a smart real-time video camera may be employed to keep an eye on a 132 potentially dangerous region for anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Still, an unauthorized user might be able to use the control system to open a valve that lets poisonous or explosive gas escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Sensors, actuators, cameras, and their supporting hardware might be protected against this kind of assault if they all used the same protocol to check for vulnerabilities and flag any unusual activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Software-level Incompatibility Compatibility issues at the software level might arise from the usage of outdated or unsupported software, which can lead to system failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Furthermore, there is a risk that malicious viruses will be introduced into internal systems through third-party software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' But antiquated software must be updated to work with modern hardware and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Systems are more likely to be attacked if they haven’t been kept up-to-date or are using enhancements that weren’t made for their operating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Companies in the oil and gas industry often purchase digital items on the assumption that they are secure and can be integrated into the more extensive system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, it is common practice for them to verify that everything else in the system is compatible with the new component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Also, oil and gas companies may not always have the resources available to verify incompatibility at the software level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' That’s how cybercriminals get into oil and gas firms’ private networks via vulnerabilities in “smart” technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a result of the Internet of Things (IoT) smartness, business leaders in the petroleum supply chain must come up with novel approaches to preventing 133 cybersecurity concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In recent hacks, vulnerabilities in the software were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In 2017, a cyberattack known as NotPetya hit a variety of institutions, including a single electricity provider, banks, public transportation networks, and a large international container shipping firm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Interestingly, the virus propagated via Ukrainian companies’ updated accounting software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' When the infection spread to other computers, it caused crashes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' in this case, the cybercriminals had infected customer-ready, certified software with spyware known as “SolarWinds” (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In both cases, hackers used vulnerabilities in software to get into connected vendors’ systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In addition, they put in place loopholes that might be used to steal IP financial data or propagate malware among user machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6 Blockchain to Overcome Cyber-Threats in Smart O&G Blockchain technology, which has recently risen to prominence as the foundation of cryptocurrencies such as BitCoin and Etherium, is an effective security method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As the data is stored, it is linked in a series of blocks, and the hash value of the preceding block is kept in each block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Since the hash value of a tampered data block would no longer be consistent with that of the succeeding block, the attack could be traced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Several subsystems of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 and smart O&G are now using blockchain technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Blockchain-based Control Systems (SCADA) The Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 movement has transformed the role of IT and OT in the modern industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The SCADA systems that gather information from the smart IoT devices and send it to the servers where it is analyzed constitute the backbone of 134 most OT platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, this kind of data collection is inherently unsafe and unreliable, providing an opening for hackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For this reason, edge and fog computing-based blockchain security procedures have been suggested to safeguard SCADA systems’ data collection transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The gathered sensor data are encrypted in data blocks before being processed on a cloud-based SCADA system, and a high-level overview of this method is shown graphically in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Data hashes from the previous and current blocks are stored in each block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then, the Data Aggregator (DA) and all relay servers participate in the block verification procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The servers will answer many times for the purpose of verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Upon consensus that the block is legitimate, the DA will forward the request to all participating servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The DA adds the new block to the blockchain and then successfully sends the updated blockchain to the command center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Both the mining node selection technique and a more secure consensus process that is compatible with Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 have been suggested in a recent paper [137], and these are discussed in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Consensus mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Simply put, a consensus mechanism is a process by which validators/miners verify the authenticity of freshly released blocks before adding them to the blockchain, hence preserving the integrity of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Various blockchain networks have spent considerable time and energy developing various consensus techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Both public and private blockchains use some consensus mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Proof-of-Work (PoW), Proof-of-Stake (PoS), Delegated Proof-of-Stake (DPoS), Practical Byzantine Fault Tolerance (PBFT), 135 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Blockchain based data transmission within end-to-end SCADA system of an oil and gas company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Blockchain enable encryption while transmitting the data for processing that increase the data security even data is hijacked while transmitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Pressure sensor Flow Rate Sensor Satellite Sensor layer Edge systems Cellular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Radio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Wireless long haul ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='communication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Tank ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Sensor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Temparature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='sensor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Human Machine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Interface ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Database ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Service ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Machine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Reporting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Service ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Cloud-based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='SCADA system ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='RTU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Data Flow ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Elastic Load ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Balancing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Analytic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Service ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Corporate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Lan ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Hash of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='current ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='block ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Hash of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='previous ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='block ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Blockchain based data acquisition with edge systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Data transmission ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Data processing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Hash of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='current ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='block ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Hash of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content='block ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Blockchain block ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Fog-based data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='processin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Fog systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Proof-of-Authority (PoA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' and RAFT [138] are all examples of popular consensus techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Both the benefits and drawbacks of each consensus technique are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' PoW, for instance, is unjust to new entrants since it has a large processing expense and favors the richest validators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' On the other hand, DPoS is less robust and decentralized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Due to its lack of anonymity, PBFT has restricted to permission (non-public) blockchains [139].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Mining node selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' A machine that participates in a blockchain network by hosting blockchain software and facilitating data transfer is called a “node.” Nodes in a network might be anything from a laptop to a phone to a router.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' “mining nodes” are the nodes that participate in the processing and verification of blockchain transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Any participant in the blockchain network may choose to take part in the mining process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a term, “mining” refers to the activity of adding new transactions to a blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4 shows the Data 136 TEMPERATURE SENS Serial No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='13 signal TypC AADIAAND二二Aggregator (DA) edge server collecting data, processing it, and coordinating mining node selection and verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' If you want to save as much time and processing power as possible, place the DA on the same private network as the relay servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Because of this, the fog servers used in the pre-processing stage of the data shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4 are hosted inside the DAs’ own private network In [137], the authors provide a specialized method for selecting mining nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To begin, the DA server initiates a data request to the relay servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Once the DA collects all of the readings from the various relays, it will produce a random number and send it out across the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To count how many times a random number appears, relay servers hash their data and compare the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' At this point, the DA’s server statistics are identical to every other server’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Ultimately, every server casts a vote for the one that has made the most random appearances during the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The DA server will choose the relay with the highest count as the mining node for the current cycle if all other relay servers agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' On the other hand, let’s pretend that a large number of relay hosts have the same highest count or that they all have 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The DA here selects the mining node at random using a cryptographically sound process [140].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Blockchain to Enable Trust Across Industrial IoT The problem of trust is one of the barriers to the security of the industrial Internet of Things (IIoT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The conventional Public Key Infrastructure (PKI) design, which is built on a single root of trust, does not operate well in this heterogeneous dispersed IoT environment, which may be subject to several administrative 137 domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, a distributed trust model that can be constructed on top of current trust domains and produce end-to-end trust across IoT devices without depending on a single root of trust is necessary for this sort of scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a result, establishing a credit-based Blockchain with an integrated reputation system might be beneficial [141].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Another potential use of blockchain in the oil and gas sector is the storage of credentials required to operate safety-critical industrial machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For example, employee and contractor qualifications, such as H2S training, first aid, and welding, may be securely recorded and preserved on a company’s blockchain network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' By storing such information in a blockchain network [142], all members may perform verification of credentials and standard operating procedures at any time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7 Risks of Smart Solutions in industrial IoT As technology improves and more industries and products are connected to the internet, it is important to understand the risks of industrial IoT installations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Any business that wants to use IoT in manufacturing or industry or connect existing technologies for automated and remote monitoring should consider the advantages and disadvantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In the next section, we’ll discuss about the possible inadequate performances of smart solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Human-Machine Interaction Issues The industrial IoT has come a long way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' machines can now process data from connected devices automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In addition, various automated sensors and actuators (like video cameras, smart glasses, and automatic valves with audio 138 input/output) are in place to help or replace the human worker in order to make sure that production runs smoothly and/or that workers are safe when using different machines to do their jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Human-machine interaction workflow from sensing to control operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Sensing Information Processing Controlling Controls Operation Display Human Machine Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5 shows how the human and machine sides of human-machine interaction work together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As this picture shows, people use different senses (like sight, smell, and hearing) to look at the machine’s results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' So, a human worker uses information processing to run or control the machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then, machines do their 139 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='jobs, and the results are shown to people so they can figure out what they mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The whole process of how people and machines work together is called the human-machine interaction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Production and safety on the job site may be jeopardized if the interdependent machines fail to operate as intended or are not user-friendly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To perform vital industrial tasks or, more crucially, to ”emphasize overrule” the choice of a smart system, human interactions are often required beyond those with a computational interface through input/output devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Take the case of a drone or ROV sent to a politically sensitive location (like a border region) to conduct autonomous oil and gas surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, inefficiently or a glitch in its algorithm may cause it to survey regions beyond the designated zone and prevent the operator from navigating the survey route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Unforeseen repercussions on the political or military front may result from such a glitch in human-machine interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Machine-to-Machine Interaction Issues Machines communicate with one another in networked autonomous systems to complete various activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In these systems, an automated sequence of actions is carried out using multiple devices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' if anything goes wrong, it could be due to (A) the devices producing misleading output (for instance, automated valve shutdown with wrong anomaly detection or automatic door closing that traps onsite workers with false alarm), or (B) incompatibility across devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accidents or catastrophes may arise due to machine-to-machine interface issues in a production setting with fault-intolerant operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 140 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' A small fire breakout accident occurs in a closed oil production area in a compressor unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The fire alarm generates, and water sprinkler starts to sprinkle water that causes power failure in power generator that made the electric door locked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Unfortunately, workers were working on pipeline maintenance, and were trapped in- side the facility due to door closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Here, machine to machine interaction cause the safety issue of the onsite worker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' electric door locked trapped worker fire alarm water sprinkler power generator compressor unit fire 1 2 3 4 5 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6 depicts one scenario illustrating the repercussions of the machine-to-machine interaction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Consider pipeline maintenance and communication with numerous pieces of equipment in a production scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Consider a pipeline linked to a machine (for example, a distillation unit) in an enclosed space that requires repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a result, maintenance staff works within the enclosed space when a fire danger occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a precaution, the gas sensor detects smoke (Step 2 in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='6), activates the water sprinkler (Step 3), and sends an alert to the controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The electricity generator shuts down due to sprinkler water 141 fimtgroup(step 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To safeguard the safety of the outside employees, the controller instantly sends the automatic door to shut (Step 5) while disregarding the workers within the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In this scenario, the controller cannot recognize personnel within the facility and executes a safety action for outside workers, putting the workers inside at risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We can see that machine-to-machine interaction concerns may sometimes lead to scenarios that must be solved by evaluating a smart solution for the O&G industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8 Bias in Smart Industry Human intervention at different stages of software development may introduce a wide range of biases that might undermine the quality of otherwise intelligent software solutions [143].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accidents and potentially dangerous situations have occurred as a result of prejudices in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Many types of bias exist, including those based on age, race, sexual orientation, disability, and other demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In addition, many onsite team members (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', workers, engineers, and coordinators) rely on software tools and simulations that are prone to the above mentioned flaws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Our discussion here will center on the different kinds of bias and the damage they might do to the smart O&G business and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Biases Caused by the Artificial Intelligence (AI) Solutions Even though AI systems have shown to be revolutionary in several contexts, they are prone to the following two types of bias: There are gaps in the training dataset that cause the model to underperform on certain inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 142 The model’s biases are the same as those found in the original dataset used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' An absence of training datasets is one source of AI bias, as shown by the commercial face recognition system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The lack of dark-skinned women [144] in the training dataset is the root cause of the face recognition system’s discrepancy between its 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='9 percent accuracy with white males and its 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 percent accuracy with women of colour, as determined by the researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The issue, however, is that “Big Data” does not necessarily provide valid and trustworthy models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For instance, social media is a well-established mine for massive datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Conclusions obtained from Twitter data should be treated with caution since just 24% of internet teenagers utilize the platform, as reported by [145].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' An unfair model does not necessarily perform poorly on a demographic subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Even if the model is correct, it is still unjust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The dataset is skewed in this situation, and the model repeats or amplifies the inherited bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Natural Language Processing (NLP) models, for example, are often trained on a vast corpus of human-written text (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', article news).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, word embeddings trained on Google News articles have been observed to reflect female/male gender stereotypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The models, for example, replied that a father is a doctor while the mother is a nurse, or that a “man” is a “computer programmer” while a “woman” is a “homemaker.” This kind of bias occurs when a model is trained on skewed data owing to unjust procedures or structures [146].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Another example of AI bias is Yelp’s restaurant review system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Restaurants 143 may pay Yelp to promote their locations on the Yelp platform, but this inevitably influences how many people see adverts for a particular restaurant and, as a result, who decides to dine there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a result, Yelp evaluations may be unjustly slanted in favor of more prominent eateries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Automation Bias in Smart Solutions of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 One of the most respected psychologists in the field, Linda J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Skitka of the University of Illinois at Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' defined automation bias as “a specific class of errors individuals tend to make in highly automated decision-making scenarios when many decisions are handled by automated aids (such as computers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' IoT devices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' and smartphones) and the human actor is primarily present to monitor ongoing tasks.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' A bias toward using automated assistance and decision support systems is often known as “automation bias.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' When the Enbridge pipeline ruptured [147] on July 26,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 2010,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' sending enormous amounts of crude oil into the Kalamazoo River and Talmadge Creek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' automation bias was a major factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Both complacency and a leaning toward automation were shown to have played significant factors in the Enbridge oil pipeline disaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, businesses, governments, and regulators must account for automation bias while designing systems to reduce the potential for careless errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' “Automation bias” is humans’ propensity to favour actions requiring the least amount of mental effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Similar thinking applies to the underlying principle of AI and automation: learning from massive amounts of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Such calculations imply that future conditions will mostly stay the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Another factor to consider is the possibility that faulty training data may lead to faulty 144 learning [148] that is implicitly related to AI bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3 Gender Bias in O&G Industry According to a study report [149], the oil and gas sector is confronting a skilled personnel scarcity, while gender prejudice is exacerbating the problem by excluding female workers from recruiting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The research contains interviews with various male and female workers from around the globe and an analysis of their comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Indeed, the oil and gas industry has a reputation for being controlled by males.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, while some oil and gas businesses work hard to achieve gender parity and worker diversity, others are allowing the gender gap to widen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Although many businesses strive to include gender equality in their policies, actions, and procedures, they still face challenges such as gender imbalance and various types of implicit prejudice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4 Cognitive Bias in Smart O&G Solutions Cognitive biases, a newly discovered notion, are mental faults in human thinking and information processing that may result in inaccurate or irrational assessments or decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Amos Tversky and Daniel Kahneman first proposed it in a 1974 article for Science Magazine (Tversky & Kahneman, 1974 [150]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Since then, a great deal of literature has been produced on cognitive biases and how they impact human thoughts and actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' According to a common understanding of cognitive bias, it is a mental flaw that results in incorrect interpretation of external data and impairs the logic and precision of choices and verdicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Biases are unconsciously occurring, automatic 145 processes that speed up and improve decision-making effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' There are several factors that might contribute to cognitive biases, including public influence and emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' There has been a growing awareness of the threats cognitive bias may bring to operational safety during the last several years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Biases like deviance, normalization, and group thinking, for instance, are now widely accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Additionally, the Deepwater Horizon [151] investigation in 2010 brought cognitive bias to the public’s attention, at least among those working in the offshore drilling industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Consequently, the International Association of Oil and Gas Producers (IOGP) has brought attention to how crucial these cognitive impairments are to safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, it is high time that cognitive bias should be addressed while building smart, automated solutions that require human decisions for complex industrial operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='9 Summary Oil and gas operations have seen dramatic changes as a result of the digital Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 revolution, which has made extensive use of cutting-edge computer hardware and software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, with these developments come opportunities for cyber criminals to improve their efficiency in locating vulnerabilities in either IT or OT systems, or in the hybrids that exist between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Another possible entry point for cyber criminals is provided by the heterogeneity and incompatibility of smart technologies, as well as the connection difficulty between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Problems with human-machine and machine-to-machine interactions, as well as incompatibilities between technologies acquired through time, are among the most significant 146 obstacles to the widespread adoption of smart technologies in the legacy and smart oil and gas sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Though Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 has been a boon to the oil and gas sector, business executives and professionals working in the sector should be wary of its smarts being misapplied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In the last several years, we’ve learned the hard way that blindly installing or embracing smart technology may open the door to a wide variety of risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Researchers and practitioners must bear in mind these drawbacks while deciding whether or not to use smart technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In this regard, as a part of this dissertation, we published a book [152] on the scope of IoT technologies with the rise of the Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 revolution that addresses a detailed analysis of smart solutions and their drawbacks in the context of smart O&G Industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 147 Chapter 8: Conclusion and Future Research Directions The ever-growing IoT and smart devices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', smart gateway, sensors, controllers, actuators) produce a substantial amount of raw data that need to be stored, pre-processed, and analyzed to bring out potential insights that can make the industrial systems more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accordingly, various Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 latency-sensitive applications operate based on machine learning (ML) and utilize the generated sensor data to achieve automation and other industrial activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, the cloud computing platform has been offering services [153] to perform various operations on the ever-growing data generated in the industrial sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Privacy, centrality, and expenses have been significant constraints to utilizing cloud data centers effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As such, edge and fog computing bring the computational services [154, 155, 156] near the end-users closer to the data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, edge devices may support limited computing demands due to resource limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In contrast, the fog system can be a preferable option to meet computing needs due to its availability of computational resources and more robust middleware compared to edge systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Because, fog systems are heterogeneous and the heterogeneity is one factor that introduces stochasticity in the execution time of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications that affects the completion times of these applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' To develop a robust solution for Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0, it is necessary to study the execution time behaviour of various ML-based applications in heterogeneous fog systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As such, we perform statistical analysis of ML-based Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications to understand the execution time pattern of these applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In addition, we introduce 148 real-world Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 smart application execution traces in fog computing systems that can be beneficial for the future research works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Even though fog systems have more computing resources than edge systems, the surge in computing demands at disasters can reduce performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, in this dissertation, we propose federating fog computing systems (owned by private companies) from nearby sites to support such scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Furthermore, the fog federation concept can be practical with system administrators’ efficient resource allocation mechanisms adopted by research works related to load-balancing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' A real-world Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 application execution traces on fog computing platforms can be crucial for devising effective resource allocation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a result, we utilize our prior workload trace to devise a statistical resource allocation method across federated fog systems for Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 latency-sensitive applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In addition, the heterogeneous software methodologies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', monolithic, micro-service) of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications can affect the execution plan of a fog federation due to their diverse latency constraints, resulting in decreasing system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, the decomposition of micro-service applications with effective resource allocation methods can maintain the systems’ performance in oversubscribed situations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', accidents, and disasters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accordingly, the industrial computing platform (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', federated fog system) should be cognizant of stochastic execution behaviour, software structure, and latency requirements of micro-service workflow applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We propose a resource allocation method based on probability estimation that partition micro-service workflows across the federated 149 fog computing systems to support their latency requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Furthermore, the concept of federating fog resources raises data security and privacy concerns for private fog systems participating in the federation due to having sensitive company data stored or processed in these fog systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Thus, we propose a data privacy preserving solution that works based on federated learning method for training ML-based Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 application across federated fog systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Discussion In this dissertation, our main objective was to investigate and develop effective resource allocation solutions using modern distributed computing systems for remote Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As such, we first explore and identify various smart computing aspects in remote offshore industries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', oil and gas, minerals, sustainable energy) where computing demand is significantly high and conventional computing systems are inefficient due to latency constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, privately owned fog systems in remote areas can support industrial computing demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, we identify stochastic execution time behaviours of latency-sensitive tasks executing in heterogeneous fog systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As such, we explore the execution time behaviour of various ML-based applications in heterogeneous execution platforms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', amazon web service, chameleon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Consequently, we introduce a real-world workload of execution time in heterogeneous computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Furthermore, in remote industries, the surge in computing demand can decrease the fog systems’ performance at disaster times by not completing latency-sensitive task requests on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accordingly, we propose federating nearby fog systems in remote industries 150 and forming a fog federation to support surge computing demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Thus, we enable the federation concept and develop a statistical resource allocation method using prior synthesized real-world application workload considering an oversubscribed situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence we evaluate our proposed solution for monolithic applications widely used in Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' After that, we investigate smart micro-service applications’ internal structure to understand the impact of the decomposition on application workflow completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We suggest a probabilistic workflow partitioning method along with the previously proposed resource allocation method that improves the fog federation’s performance and ensures safety in remote Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Finally, we address the data privacy issue for sharing privately owned fog systems in developing accurate ML models for Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, we explore the federated learning techniques across the fog federation that ensure data privacy for privately owned fog systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In this context, we address the class imbalance issue in a federated learning setup that can reduce the robustness of the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, we propose a federated learning method that is robust against the class imbalance issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In chapter 3, we analyze and estimate the performance of DNN-based applications in heterogeneous cloud and fog resources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', amazon, chameleon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Here, we identify stochastic execution behaviours of various Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Thus we explore, and model the inference execution behaviours of various Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 smart applications utilizing different statistical tools from two distinct perspectives, namely application-centric and resource-centric, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 151 Furthermore, we introduce an execution time workload of four different DNN-based applications for Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 with the intent of developing robust resource allocation methods across federated fog systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In chapter 4, we explore the usability and benefits of fog federation that can be formed to support emergencies such as disasters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', fire explosions, oil spills).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As an example of a smart industry, we consider remote smart oil fields with multiple oil extraction sites in close vicinity, each with fog computing systems to support its local computing demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Although in case of an emergency like an oil spill, the computing demands can rise due to the coordination of multiple activities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', drone inspection of oil spill, video camera images, sensors data processing) to support the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence we propose a probabilistic resource allocation method for monolithic latency-sensitive applications that effectively selects a relevant fog system from the federation by utilizing our prior workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As the resource allocation method is aware of the receiving applications stochastic execution behaviours from our prior work, it ensures the robustness of the fog federation by completing majority of the receiving workload on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In chapter 5, we explore modern software architecture (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', micro-service) of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications to create an efficient execution strategy over fog federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In contrast, we identified legacy applications with monolithic software architecture are still exists in various industrial sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, to support the computational demands in remote industry the execution platform (federated fog system) should be aware of software architectures of the Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 152 Hence, for micro-services we consider the idea of using an application breakdown strategy to increase the chance of finishing the execution on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Furthermore, for monolithic applications and individual micro-services we utilizes our prior knowledge of stochastic execution behaviour to efficiently allocate fog resources across the federated fog systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a result, we propose a statistical micro-service partitioning and resource allocation method that considers the underlying software architecture and the stochastic execution latencies of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In chapter 6, we explore the data security and privacy aspects of fog federation while training ML-based applications in remote Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In this case, we investigate the federated learning techniques utilizing fog federation to train a ML-based oil spill detection application that provide data security to privately owned fog systems of the federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accordingly, we identified low occurrence events in training data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', class imbalance) can reduce the accuracy of the ML-model that can be detrimental in emergency situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Here, we propose a customized federated learning technique, considering the class imbalance issue across fog federation to increase the safety measures of remote Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In chapter 7, we investigate the downsides and side-effects of smart solutions developed with the integration of various applications in the industrial sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, we introduce a taxonomy of cyber threats and side-effects of smart solutions in the context of the O&G industry that structurally address the unsafe areas of these smart solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accordingly, various vulnerable areas, including both software and hardware components, machine-human interaction issues, and different 153 forms of biases in smart solutions, are addressed with efficient resilience methods that would help system architects or industrial researchers to develop robust smart solutions for Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In conclusion, we explore and investigate the stochastic execution behaviour of various Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications and introduce a real-world execution workload that has been utilized in our resource allocation research works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then, we explore the federation concept using privately owned fog systems for various computing demands of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Especially in oversubscribed situations like disasters, the federation could be more efficient if the load is adequately balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence we develop a load-balancing method to make the federation robust in emergencies that we consider the system administration level of our research track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then we dive into the application level by investigating various software architectures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', monolithic, micro-service) of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, we identify micro-service workflow applications can be decomposed to improve the application workflow completion on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accordingly, we propose a probabilistic micro-service partitioning and resource allocation method that can enhance the performance of the fog federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Then, we explore the data security and privacy aspects of federated fog systems while training ML-based Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Finally, in the end, we identify various pitfalls of smart solutions that need to be appropriately addressed to develop efficient and robust smart solutions for Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 154 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Future Research Directions Based on our findings during the development of the resource allocation, micro-service workflow partitioning, and secure resource-sharing solutions, we identify some of the expansion areas that can improve the robustness and safety of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' There are several points where the work could be expanded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='1 Resource Allocation Using Reinforcement Learning for Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 Applications across Federated Fog System In this dissertation, we suggest a statistical application completion time estimation method across the fog federation system to allocate Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 applications into a relevant fog system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Our estimation of task completion success could be coarse that sometimes leads to the deadline miss of a receiving task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' We think that a resource allocation method operating based on the reinforcement learning technique [157, 158] can improve the quality of service for the federated fog system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The field of reinforcement learning (RL) has emerged as an important subset of machine learning because it enables autonomous agents to make sound decisions in response to changing conditions in their environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, uncertain execution behaviour can be addressed effectively using RL technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In this scenario, RL might be used to allocate resources [159] for offloading and executing tasks in a federated fog computing system, leading to better overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2 Data Locality-Aware Resource Allocation Across Federated Fog System The proliferation of IoT devices has coincided with a surge in network traffic 155 that may overwhelm the potential of the current network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Furthermore, data privacy and latency are important concerns when these devices analyze sensor or user information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, conventional methods such as cloud computing don’t work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Although, advanced computing platform like fog computing can fill this need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Understanding how the initial input data’s localization impacts on fog platforms’ performance is critical to developing reasonable load balancing and resource allocation solutions [160, 161].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a result, if several data-intensive applications with deadline restrictions arrive dynamically, performance evaluation of a heterogeneous federated fog environment is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For example, the applications may need data from the IoT layer or from local fog resources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', sensor data that have already been transferred to the fog layer or data processed by prior applications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In this scenario, examining the influence of input data localization on system performance across federated fog systems with varying data placement probability might influence the federation’s resource allocation efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, we consider exploring the impact of data localization on resource allocation methods across federated fog systems in the oversubscribed situation for remote Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='3 Dynamic Fault-Tolerant Federated Fog Systems for Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 Operation The fog computing systems provide low latency to the end users being close to the data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In contrast, the distributed characteristics of fog aid in processing vast amounts of sensor-generated data of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, federating fog computing resources can support latency-sensitive tasks and data processing 156 services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, fog systems have various uncertainties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', transient failures [162], network and power outage) that need to be considered when supporting surge computing demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Especially in an emergency, those uncertainties can lead to unsuccessful task completion causing significant damage to the environment and even human lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, the resource management system for fog federation should be aware of the uncertainties and provide efficient fault-tolerant [163, 164, 165] solution to ensure successful completion of the receiving latency-sensitive tasks on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Accordingly, the federation management system should consider providing a service that continuously monitors the fog resources and then sends the signal to all the participant fogs about the neighbouring fog’s state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In addition, already ongoing service execution can get disrupted or fail due to the fog system’s internal issues (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', software, hardware).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In this case, every fog system should have a method (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', re-execution, offloading the failed task to another fog) to ensure successful completion of receiving task’s execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, a fault-tolerant federated fog system is crucial for supporting surge computing demands in emergency or disaster situations, enabling the system’s robustness and leading to a safe Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Similar to super cloud [166], fog systems provide various virtual services [167] like application deployment, multi-tenancy, interoperability, and service migration across fog federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, to enable a fog federation that can avail various fog virtual services need to support fault-tolerant characteristics for efficient utilization of fog services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In addition, in oversubscribed situation, the fault-tolerant service needs to address the scalability of fog federation, ensuring the system’s robustness 157 in a dynamic condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, to achieve all these requirements, we are considering performing research works in future on developing a fog system (super fog) that provides fault-tolerant fog services across the federated fog systems to improve the efficiency of the federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The popularity of fog systems with heterogeneous resources and dynamic fog federation [168] concept has created the demand for developing the fog-friendly application that requires proper investigation of the application stack and fog resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' However, building this type of application is time-consuming and requires overcoming some major obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The first is to support the dynamic nature of the fog network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' the second is to manage the context-dependent qualities of application logic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' the third is to cope with the system’s massive size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As a result, we must consider how to decompose and deploy applications to a geographically dispersed fog federation utilizing current software components that may be altered and reused to participate in fog applications [169].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence abstracting the application layer from the execution layer can be the primary objective to solve the heterogeneity challenge of the fog systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Thus, we would like to perform research on developing fog-friendly applications for dynamic federated fog systems that are cognizant of super fog systems’ characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='4 The Cognitive Aspects of Human-Machine Interaction for Smart Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 Solutions Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0, an industrial technology paradigm shift, mandates new ways in which human and machine (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', robots, drones) will work together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' The 158 introduction of ever-more-advanced sensors and collaborating machines raises important questions about the influence on safety in the highly technical and inventive scenario of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0, defined by a succession of enabling technologies and a strong interconnectedness of resources between human and machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' On the one hand, advanced software tools and machines facilitate human work (human-machine cooperation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' On the contrary, it must communicate and share data with other intelligent devices (human-machine interaction) [170].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Since the advent of “smart” technologies, both the environment in which these innovations are deployed and the responsibilities of front-line human workers have changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In complex industrial operations or at disaster times, the human-machine interaction can be challenging that is significantly related to cognitive aspects [171] of the human workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' When some tasks need specialized human abilities, there is genuine “collaboration” between humans and machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In today’s modern industries, workers’ interactions with “smart machines” can make their jobs easier by making their tasks more automated and less prone to human error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In contrast, it makes the workplace more complicated by increasing information and communication flow between different systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' For example, using sensors and cutting-edge technology, we can collect the information we need to make accurate forecasts about the health of industrial machinery and carry out precise treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As humans must handle the massive amounts of data (big data) that need to be gathered, analyzed, and understood, the cognitive interaction effort of the machine operator rises from the skill level to the knowledge level [172].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Therefore, we would like to address various 159 cognitive aspects of human-machine interaction issues in Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0 and develop smart solutions for human workers to aid in a complex industrial scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='5 Fog Computing and Advanced Analytics for Human-Machine Interaction in Industrial Sector The advancement in computing technology with industrial revolution has transformed the industrial operations using automation, robotics, artificial intelligence and other modern smart solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Although, various complex industrial operations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', machine maintenance, oil well drilling operation, manufacturing machines) need human intervention and interaction [173, 174] to ensure precision and accuracy of the operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, human operator that communicate with machine sometimes need to process machine generated data to efficiently communicate with machines [175, 176].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' In this case human operators can use a mobile device with them to process the data or visualize the data that is processed a nearby computing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Hence, fog computing can be a potential candidate to support the computing demands of human-machine interactions [177, 178, 179].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Furthermore, fog computing utilizing various advanced analytics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=', machine learning, deep neural network, reinforcement learning) on machine generated data can provide useful insights to the human operators that can improve human-machine interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
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+page_content=' 177 Biographical Sketch Razin Farhan Hussain received his Bachelor of Science in computer science and engineering in the fall of 2011 from Military Institute of Science and Technology, Bangladesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' He immediately began his career in February of 2012, as a programmer in one of the software company named “ERA Infotech Ltd”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' After nine months of his first job, he joined one of the top multinational software company “Samsung R&D Institute” as a software engineer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Razin Farhan Hussain worked in Samsung for around 4 year and 5 months, and decided to enrich his academic knowledge by pursuing higher education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' As such Razin Farhan Hussain started his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' journey in computer science in the fall of 2017 at the University of Louisiana at Lafayette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' While pursuing his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' degree Razin Farhan Hussain received his M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' degree in computer science in the spring of 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' His research interests are: cloud computing, resource allocation in a fog federation, task scheduling, machine learning, DNN-based applications for Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content='0, and federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' Currently, Razin Farhan Hussain is working as a Software Engineer in one of the software company named “TryCycle Data Systems”, providing software solutions in healthcare sector of Canada & United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
+page_content=' 178' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfnPg_/content/2301.00484v1.pdf'}
diff --git a/wNFJT4oBgHgl3EQffyxP/content/tmp_files/2301.11558v1.pdf.txt b/wNFJT4oBgHgl3EQffyxP/content/tmp_files/2301.11558v1.pdf.txt
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@@ -0,0 +1,2725 @@
+Published as a conference paper at ICLR 2023
+ACCELERATING GUIDED DIFFUSION SAMPLING WITH
+SPLITTING NUMERICAL METHODS
+Suttisak Wizadwongsa, Supasorn Suwajanakorn
+VISTEC, Thailand
+{suttisak.w s19, supasorn.s}@vistec.ac.th
+ABSTRACT
+Guided diffusion is a technique for conditioning the output of a diffusion model at
+sampling time without retraining the network for each specific task. One drawback
+of diffusion models, however, is their slow sampling process. Recent techniques
+can accelerate unguided sampling by applying high-order numerical methods to
+the sampling process when viewed as differential equations. On the contrary, we
+discover that the same techniques do not work for guided sampling, and little
+has been explored about its acceleration. This paper explores the culprit of this
+problem and provides a solution based on operator splitting methods, motivated by
+our key finding that classical high-order numerical methods are unsuitable for the
+conditional function. Our proposed method can re-utilize the high-order methods
+for guided sampling and can generate images with the same quality as a 250-
+step DDIM baseline using 32-58% less sampling time on ImageNet256. We also
+demonstrate usage on a wide variety of conditional generation tasks, such as text-
+to-image generation, colorization, inpainting, and super-resolution.
+1
+INTRODUCTION
+A family of generative models known as diffusion models has recently gained a lot of attention
+with state-of-the-art image generation quality (Dhariwal & Nichol, 2021). Guided diffusion is an
+approach for controlling the output of a trained diffusion model for conditional generation tasks
+without retraining its network. By engineering a task-specific conditional function and modifying
+only the sampling procedure, guided diffusion models can be used in a variety of applications, such
+as class-conditional image generation (Dhariwal & Nichol, 2021; Kawar et al., 2022), text-to-image
+generation (Nichol et al., 2022), image-to-image translation (Zhao et al., 2022), inpainting (Chung
+et al., 2022a), colorization (Song et al., 2020b), image composition (Sasaki et al., 2021), adversarial
+purification (Wang et al., 2022; Wu et al., 2022) and super-resolution (Choi et al., 2021).
+One common drawback of both guided and regular “unguided” diffusion models is their slow sam-
+pling processes, usually requiring hundreds of iterations to produce a single image. Recent speed-
+up attempts include improving the noise schedule (Nichol & Dhariwal, 2021; Watson et al., 2021),
+redefining the diffusion process to be non-Markovian, thereby allowing a deterministic sampling
+process Song et al. (2020a), network distillation that teaches a student model to simulate multiple
+sampling steps of a teacher model Salimans & Ho (2022); Luhman & Luhman (2021), among oth-
+ers. Song et al. (2020a) show how each sampling step can be expressed as a first-order numerical
+step of an ordinary differential equation (ODE). Similarly, Song et al. (2020b) express the sam-
+pling of a score-based model as solving a stochastic differential equation (SDE). By regarding the
+sampling process as an ODE/SDE, many high-order numerical methods have been suggested, such
+as Liu et al. (2022), Zhang & Chen (2022), and Zhang et al. (2022) with impressive results on un-
+guided diffusion models. However, when applied to guided diffusion models, these methods produce
+surprisingly poor results (see Figure 1)—given a few number of steps, those high-order numerical
+methods actually perform worse than low-order methods.
+Guided sampling differs from the unguided one by the addition of the gradients of the conditional
+function to its sampling equation. The observed performance decline thus suggests that classical
+high-order methods may not be suitable for the conditional function and, consequently, the guided
+sampling equation as a whole. Our paper tests this hypothesis and presents an approach to accel-
+1
+arXiv:2301.11558v1 [cs.CV] 27 Jan 2023
+
+Published as a conference paper at ICLR 2023
+Number
+of steps
+8
+16
+32
+64
+128
+256
+DDIM
+PLMS4
+STSP4
+(Ours)
+Figure 1: Generated samples of a classifier-guided diffusion model trained on ImageNet256 using 8-
+256 sampling steps from different sampling methods. Our technique, STSP4, produces high-quality
+results in a fewer number of steps.
+erating guided diffusion sampling. The key idea is to use an operator splitting method to split the
+less well-behaved conditional function term from the standard diffusion term and solve them sepa-
+rately. This approach not only allows re-utilizing the successful high-order methods on the diffusion
+term but also provides us with options to combine different specialized methods for each term to
+maximize performance. Splitting method can be used to solve diffusion SDE in Dockhorn et al.
+(2021).
+Our design process includes comparing different splitting methods and numerical methods for each
+split term. When tested on ImageNet, our approach achieves the same level of image quality as a
+DDIM baseline while reducing the sampling time by approximately 32-58%. Compared with other
+sampling methods using the same sampling time, our approach provides better image quality as
+measured by LPIPS, FID, and Perception/Recall. With only minimal modifications to the sampling
+equation, we also show successful acceleration on various conditional generation tasks.
+2
+BACKGROUND
+This section provides a high-level summary of the theoretical foundation of diffusion models as well
+as numerical methods that have been used for diffusion models. Here we briefly explain a few that
+contribute to our method.
+2.1
+DIFFUSION MODELS
+Assuming that x0 is a random variable from the data distribution we wish to reproduce, diffusion
+models define a sequence of Gaussian noise degradation of x0 as random variables x1, x2, ..., xT ,
+where xt ∼ N(√1 − βtxt−1, βtI) and βt ∈ [0, 1] are parameters that control the noise levels.
+With a property of Gaussian distribution, we can express xt directly as a function of x0 and noise
+ϵ ∼ N(0, I) by xt = √¯αtx0 +√1 − ¯αtϵ, where ¯αt = �t
+i=1(1−βi). By picking a sufficiently large
+T (e.g., 1,000) and an appropriate set of βt, we can assume xT is a standard Gaussian distribution.
+The main idea of diffusion model generation is to sample a Gaussian noise xT and use it to reversely
+sample xT −1, xT −2, ... until we obtain x0, which belongs to our data distribution.
+Ho et al. (2020) propose Denoising Diffusion Probabilistic Model (DDPM) and explain how to
+employ a neural network ϵθ(xt, t) to predict the noise ϵ that is used to compute xt. To train the
+network, we sample a training image x0, t, and ϵ to compute xt using the above relationship. Then,
+we optimize our network ϵθ to minimize the difference between the predicted and real noise, i.e.,
+∥ϵ − ϵθ(xt, t)∥2.
+2
+
+Published as a conference paper at ICLR 2023
+Song et al. (2020a) introduce Denoising Diffusion Implicit Model (DDIM), which uses the network
+ϵθ to deterministically obtain xt−1 given xt. The DDIM generative process can be written as
+xt−1 =
+� ¯αt−1
+¯αt
+�
+xt −
+√
+1 − ¯αtϵθ(xt, t)
+�
++
+�
+1 − ¯αt−1ϵθ(xt, t).
+(1)
+This formulation could be used to skip many sampling steps and boost sampling speed. To turn this
+into an ODE, we rewrite Equation 1 as:
+xt−∆t
+√¯αt−∆t
+=
+xt
+√¯αt
++
+��
+1 − ¯αt−∆t
+¯αt−∆t
+−
+�
+1 − ¯αt
+¯αt
+�
+ϵθ(xt, t),
+(2)
+which is now equivalent to a numerical step in solving an ODE. To derive the corresponding ODE,
+we can re-parameterize σt = √1 − ¯αt/√¯αt, ¯x(t) = xt/√¯αt and ¯ϵσ(¯x) = ϵθ(xt, t), yielding
+¯x(t − ∆t) − ¯x(t) = (σt−∆t − σt)¯ϵσ(¯x). By letting (σt−∆t − σt) → 0, the ODE becomes:
+d¯x
+dσ = ¯ϵσ(¯x).
+(3)
+Note that this change of variables is equivalent to an exponential integrator technique described in
+both Zhang & Chen (2022) and Lu et al. (2022). Since xt and ¯x(t) have the same value at t = 0,
+our work can focus on solving ¯x(t) rather than xt. Many numerical methods can be applied to the
+ODE Equation 3 to accelerate diffusion sampling. We next discuss some of them that are relevant.
+2.2
+NUMERICAL METHODS
+Euler’s Method is the most basic numerical method. A forward Euler step is given by ¯xn+1 =
+¯xn + ∆σ¯ϵσ(¯xn). When we apply the forward Euler step to the ODE Equation 3, we get the DDIM
+formulation (Song et al., 2020a).
+Heun’s Method, also known as the trapezoid rule or improved Euler, is given by: ¯xn+1 = ¯xn +
+∆σ
+2 (e1 + e2), where e1 = ¯ϵσ(¯xn) and e2 = ¯ϵσ(¯xn + ∆σe1). This method modifies Euler’s method
+into a two-step method to improve accuracy. Many papers have used this method on diffusion
+models, including Algorithm 1 in Karras et al. (2022) and DPM-Solver-2 in Lu et al. (2022). This
+method is also the simplest case of Predictor-Corrector methods used in Song et al. (2020b).
+Runge-Kutta Methods represent a class of numerical methods that integrate information from mul-
+tiple hidden steps and provide high accuracy results. Heun’s method also belongs to a family of
+2nd-order Runge-Kutta methods (RK2). The most well-known variant is the 4th-order Runge-Kutta
+method (RK4), which is written as follows:
+e1 = ¯ϵσ(¯xn),
+e2 = ¯ϵσ
+�
+¯xn + ∆σ
+2 e1
+�
+,
+e3 = ¯ϵσ
+�
+¯xn + ∆σ
+2 e2
+�
+,
+e4 = ¯ϵσ (¯xn + ∆σe3) ,
+¯xn+1 = ¯xn + ∆σ
+6 (e1 + 2e2 + 2e3 + e4).
+(4)
+This method has been tested on diffusion models in Liu et al. (2022) and Salimans & Ho (2022), but
+it has not been used as the main proposed method in any paper.
+Linear Multi-Step Method, similar to the Runge-Kutta methods, aims to combine information from
+several steps; however, rather than evaluating new hidden steps, this method uses the previous steps
+to estimate the new step. The 1st-order formulation is the same as Euler’s method. The 2nd-order
+formulation is given by
+¯xn+1 = ¯xn + ∆σ
+2 (3e0 − e1) ,
+(5)
+while the 4th-order formulation is given by
+¯xn+1 = ¯xn + ∆σ
+24 (55e0 − 59e1 + 37e2 − 9e3),
+(6)
+where ek = ¯ϵσ(¯xn−k). These formulations are designed for a constant ∆σ in each step. However,
+our experiments and previous work that uses this method (e.g., Liu et al. (2022); Zhang & Chen
+3
+
+Published as a conference paper at ICLR 2023
+(2022)) still show good results when this assumption is not strictly satisfied, i.e., when ∆σ is not
+constant. We will refer to these formulations as PLMS (Pseudo Linear Multi-Step) for the rest of
+the paper, like in Liu et al. (2022). A similar linear multi-step method for non-constant ∆σ can also
+be derived using a technique used in Zhang & Chen (2022), which we detail in Appendix B. The
+method can improve upon PLMS slightly, but it is not as flexible because we have to re-derive the
+update rule every time the σ schedule changes.
+3
+SPLITTING METHODS FOR GUIDED DIFFUSION MODELS
+This section introduces our technique that uses splitting numerical methods to accelerate guided
+diffusion sampling. We first focus our investigation on classifier-guided diffusion models for class-
+conditional generation and later demonstrate how this technique can be used for other conditional
+generation tasks in Section 4.3. Like any guided diffusion models, classifier-guided models (Dhari-
+wal & Nichol, 2021) share the same training objective with regular unguided models with no mod-
+ifications to the training procedure; but the sampling process is guided by an additional gradient
+signal from an external classifier to generate class-specific output images. Specifically, the sampling
+process is given by
+ˆϵ = ϵθ(xt) −
+√
+1 − ¯αt∇x log pφ(c|xt),
+xt−1 = √¯αt−1
+�xt − √1 − ¯αtˆϵ
+√¯αt
+�
++
+�
+1 − ¯αt−1ˆϵ, (7)
+where pφ(c|xt) is a classifier model trained to output the probability of xt belonging to class c. As
+discussed in the previous section, we can rewrite this formulation as a “guided ODE”:
+d¯x
+dσ = ¯ϵσ(¯x) − ∇fσ(¯x),
+(8)
+where fσ(¯x) =
+σ
+√
+σ2+1 log pφ(c|xt). We refer to fσ as the conditional function, which can be
+substituted with other functions for different tasks. After obtaining the ODE form, any numerical
+solver mentioned earlier can be readily applied to accelerate the sampling process. However, we
+observe that classical high-order numerical methods (e.g., PLMS4, RK4) fail to accelerate this task
+(see Figure 1) and even perform worse than the baseline DDIM.
+We hypothesize that the two terms in the guided ODE may have different numerical behaviors
+with the conditional term being less suitable to classical high-order methods. We speculate that the
+difference could be partly attributed to how they are computed: ∇fσ(¯x) is computed through back-
+propagation, whereas ¯ϵσ(¯x) is computed directly by evaluating a network. One possible solution to
+handle terms with different behaviors is the so-called operator splitting method, which divides the
+problem into two subproblems:
+dy
+dσ = ¯ϵσ(y),
+dz
+dσ = −∇fσ(z).
+(9)
+We call these the diffusion and condition subproblems, respectively. This method allows separating
+the hard-to-approximate ∇fσ(z) from ¯ϵσ(y) and solving them separately in each time step. Impor-
+tantly, this helps reintroduce the effective use of high-order methods on the diffusion subproblem as
+well as provides us with options to combine different specialized methods to maximize performance.
+We explore two most famous first- and second-order splitting techniques for our task:
+3.1
+LIE-TROTTER SPLITTING (LTSP)
+Our first example is the simple first-order Lie-Trotter splitting method (Trotter, 1959), which ex-
+presses the splitting as
+dy
+dσ = ¯ϵσ(y),
+y(σn) = ¯xn,
+σ ∈ [σn+1, σn]
+(10)
+dz
+dσ = −∇fσ(z),
+z(σn) = y(σn+1),
+σ ∈ [σn+1, σn]
+(11)
+with the solution of this step being ¯xn+1 = z(σn+1). Note that σn is a decreasing sequence in
+sampling schedule. Here Equation 10 is the same as Equation 3, which can be solved using any
+4
+
+Published as a conference paper at ICLR 2023
+Algorithm 1: Lie-Trotter Splitting (LTSP)
+sample ¯x0 ∼ N(0, σ2
+maxI) ;
+for n ∈ {0, ..., N − 1} do
+yn+1 = PLMS(¯xn, σn, σn+1, ¯ϵσ);
+¯xn+1 = yn+1 −(σn+1 −σn)∇f(yn+1) ;
+end
+Result: ¯xN
+Algorithm 2: Strang Splitting (STSP)
+sample ¯x0 ∼ N(0, σ2
+maxI) ;
+for n ∈ {0, ..., N − 1} do
+zn+1 = ¯xn − (σn+1−σn)
+2
+∇f(¯xn) ;
+yn+1 = PLMS(zn+1, σn, σn+1, ¯ϵσ);
+¯xn+1 = yn+1 − (σn+1−σn)
+2
+∇f(yn+1) ;
+end
+Result: ¯xN
+high-order numerical method, e.g., PLMS. For Equation 11, we can use a forward Euler step:
+zn+1 = zn − ∆σ∇fσ(zn).
+(12)
+This is equivalent to a single iteration of standard gradient descent with a learning rate ∆σ. This
+splitting scheme is summarized by Algorithm 1. We investigate different numerical methods for
+each subproblem in Section 4.1.
+3.2
+STRANG SPLITTING (STSP)
+Strang splitting (or Strang-Marchuk) (Strang, 1968) is one of the most famous and widely used
+operator splitting methods. This second-order splitting works as follows:
+dz
+dσ = −∇fσ(z),
+z(σn) = ¯xn,
+σ ∈
+�1
+2(σn + σn+1), σn
+�
+(13)
+dy
+dσ = ¯ϵσ(y),
+y(σn) = z
+�1
+2(σn + σn+1)
+�
+,
+σ ∈ [σn+1, σn]
+(14)
+d˜z
+dσ = −∇fσ(˜z),
+˜z
+�1
+2(σn + σn+1)
+�
+= y(σn+1),
+σ ∈
+�
+σn+1, 1
+2(σn + σn+1)
+�
+(15)
+Instead of solving each subproblem for a full step length, we solve the condition subproblem for
+half a step before and after solving the diffusion subproblem for a full step. In theory, we can swap
+the order of operations without affecting convergence, but it is practically cheaper to compute the
+condition term twice rather than the diffusion term twice because fσ is typically a smaller network
+compared to ¯ϵσ. The Strange splitting algorithm is shown in Algorithm 2. This method can be
+proved to have better accuracy than the Lie-Trotter method using the Banker-Campbell-Hausdorff
+formula (Tuckerman, 2010), but it requires evaluating the condition term twice per step in exchange
+for improved image quality. We assess this trade-off in the experiment section.
+4
+EXPERIMENTS
+Extending on our observation that classical high-order methods failed on guided sampling, we con-
+ducted a series of experiments to investigate this problem and evaluate our solution. Section 4.1
+uses a simple splitting method (first-order LTSP) to study the effects that high-order methods have
+on each subproblem, leading to our key finding that only the conditional subproblem is less suited to
+classical high-order methods. This section also determines the best combination of numerical meth-
+ods for the two subproblems under LTSP splitting. Section 4.2 explores improvements from using a
+higher-order splitting method and compares our best scheme to previous work. Finally, Section 4.3
+applies our approach to a variety of conditional generation tasks with minimal changes.
+For our comparison, we use pre-trained state-of-the-art diffusion models and classifiers from Dhari-
+wal & Nichol (2021), which were trained on the ImageNet dataset (Russakovsky et al., 2015) with
+1000 total sampling step. We treat full-path samples from a classifier-guided DDIM at 1,000 steps as
+reference solutions. Then the performance of each configuration is measured by the image similar-
+ity between its generated samples using fewer steps and the reference DDIM samples, both starting
+from the same initial noise maps. Given the same sampling time, we expect configurations with
+better performance to better match the full DDIM. We measure image similarity using Learned Per-
+ceptual Image Patch Similarity (LPIPS) (Zhang et al., 2018) (lower is better) and measure sampling
+time using a single NVIDIA RTX 3090 and a 24-core AMD Threadripper 3960x.
+5
+
+Published as a conference paper at ICLR 2023
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+22.5
+Sampling time (sec.)
+10
+2
+10
+1
+LPIPS
+DDIM 250 steps
+3 × 10
+2
+3 × 10
+1
+Euler (DDIM)
+PLMS4
+[PLMS1,PLMS1]
+[PLMS2,PLMS1]
+[PLMS4,PLMS1]
+[RK2, PLMS1]
+[RK4, PLMS1]
+(a) Varying the method for the diffusion subproblem
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+22.5
+Sampling time (sec.)
+10
+2
+10
+1
+LPIPS
+DDIM 250 steps
+3 × 10
+2
+3 × 10
+1
+Euler (DDIM)
+[PLMS1, PLMS1]
+[PLMS1, PLMS2]
+[PLMS1, PLMS4]
+[PLMS1, RK2]
+[PLMS1, RK4]
+(b) Varying the method for the condition subproblem
+Figure 2: Comparison of different combinations of numerical methods under LTSP splitting for
+guided diffusion sampling. We plot LPIPS against the sampling time. [A, B] denotes the use of
+method A in the diffusion subproblem and method B in the condition subproblem. The red dotted
+lines indicate a reference DDIM score obtained from 250 sampling steps, which produce images
+visually close to those from 1,000 steps.
+4.1
+FINDING A SUITABLE NUMERICAL METHOD FOR EACH SUBPROBLEM
+To study the effects of different numerical methods on each subproblem of the guided ODE (Equa-
+tion 8), we use the simplest Lie-Trotter splitting, which itself requires no additional network evalu-
+ations. This controlled experiment has two setups: a) we fix the numerical method for the condition
+subproblem (Equation 11) to first-order PLMS1 (Euler’s method) and vary the numerical method
+for the diffusion subproblem (Equation 10), and conversely b) we fix the method for the diffusion
+subproblem and vary the method for the condition subproblem. The numerical method options
+are Euler’s method (PLMS1), Heun’s method (RK2), 4th order Runge-Kutta’s method (RK4), and
+2nd/4th order pseudo linear multi-step (PLMS2/PLMS4). We report LPIPS vs. sampling time of
+various numerical combinations on a diffusion model trained on ImageNet 256×256 in Figure 2.
+The red dotted lines indicate a reference DDIM score obtained from 250 sampling steps, a common
+choice that produces good samples that are perceptually close to those from a full 1,000-step DDIM
+(Dhariwal & Nichol, 2021; Nichol & Dhariwal, 2021).
+Given a long sampling time, non-split PLMS4 performs better than the DDIM baseline. However,
+when the sampling time is reduced, the image quality of PLMS4 rapidly decreases and becomes
+much worse than that of DDIM, especially under 15 seconds in Figure 2. When we split the ODE
+and solve both subproblems using first-order PLMS1 (Euler), the performance is close to that of
+DDIM, which is also considered first-order but without any splitting. This helps verify that merely
+splitting the ODE does not significantly alter the sampling speed.
+In the setup a), when RK2 and RK4 are used for the diffusion subproblem, they also perform worse
+than the DDIM baseline. This slowdown is caused by the additional evaluations of the network by
+these methods, which outweigh the improvement gained in each longer diffusion step. Note that if
+we instead measure the image quality with respect to the number of diffusion steps, RK2 and RK4
+can outperform other methods (Appendix E); however, this is not our metric of interest. On the
+other hand, PLMS2 and PLMS4, which require no additional network evaluations, are about 8-10%
+faster than DDIM and can achieve the same LPIPS score as the DDIM that uses 250 sampling steps
+in 20-26 fewer steps. Importantly, when the sampling time is reduced, their performance does not
+degrade rapidly like the non-split PLMS4 and remains at the same level as DDIM.
+In the setup b) where we vary the numerical method for the condition subproblem, the result re-
+veals an interesting contrast—none of the methods beats DDIM and some even make the sampling
+diverged [PLMS1, RK4]. These findings suggest that the gradients of conditional functions are less
+“compatible” with classical high-order methods, especially when used with a small number of steps.
+This phenomenon may be related to the “stiffness” condition of ODEs, which we discuss further in
+Section 5. For the remainder of our experiments, we will use the combination [PLMS4, PLMS1] for
+the diffusion and condition subproblems, respectively.
+6
+
+Published as a conference paper at ICLR 2023
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+22.5
+Sampling time (sec.)
+10
+2
+10
+1
+LPIPS
+DDIM 250 steps
+3 × 10
+2
+3 × 10
+1
+Euler (DDIM)
+PLMS4
+RK2
+RK4
+LTSP4 [PLMS4,PLMS1]
+STSP4 [PLMS4,PLMS1]
+Figure 3: Comparison of different numerical
+methods for guided diffusion sampling.
+Sampling time within
+5 sec.
+10 sec.
+15 sec.
+20 sec.
+DDIM
+0.116
+0.062
+0.043
+0.033
+PLMS4
+0.278
+0.141
+0.057
+0.026
+RK2
+0.193
+0.059
+0.036
+0.028
+RK4
+0.216
+0.054
+0.039
+0.028
+LTSP4
+0.121
+0.058
+0.037
+0.028
+STSP4
+0.079
+0.035
+0.022
+0.013
+Table 1: Average LPIPS when the sampling time is
+limited to be under 5 - 20 seconds.
+4.2
+IMPROVED SPLITTING METHOD
+This experiment investigates improvements from using a high-order splitting method, specifically
+the Strang splitting method, with the numerical combination [PLMS4, PLMS1] and compares our
+methods to previous work. Note that besides DDIM Dhariwal & Nichol (2021), no previous work
+is specifically designed for accelerating guided sampling, thus the baselines in this comparison are
+only adaptations of the core numerical methods used in those papers. And to our knowledge, no prior
+guided-diffusion work uses splitting numerical methods. Non-split numerical method baselines are
+PLMS4, which is used in Liu et al. (2022), RK2, which is used in Karras et al. (2022); Lu et al.
+(2022), and higher-order RK4. We report the LPIPS scores of these methods with respect to the
+sampling time in Figure 3 and Table 1.
+Without any splitting, PLMS4, RK2 and RK4 show significantly poorer image quality when used
+with short sampling times < 10 seconds. The best performer is our Strang splitting (STSP4), which
+can reach the same quality as 250-step DDIM while using 32-58% less sampling time. STSP4 also
+obtains the highest LPIPS scores for sample times of 5, 10, 15, and 20 seconds. More statistical
+details and comparison with other split combinations are in Appendix F, G.
+In addition, we perform a quantitative evaluation for class-conditional generation by sampling
+50,000 images based on uniformly chosen class conditions with a small number of sampling steps
+and evaluating the Fenchel Inception Distance (FID) Heusel et al. (2017) (lower is better) and the
+improved precision/recall Kynk¨a¨anniemi et al. (2019) (higher is better) against an ImageNet test
+set. Following (Dhariwal & Nichol, 2021), we use a 25-step DDIM as a baseline, which already
+produces visually reasonable results. As PLMS and LTSP require the same number of network eval-
+uations as the DDIM, they are used also with 25 steps. For STSP with a longer network evaluation
+time, it is only allowed 20 steps, which is the highest number of steps such that its sampling time
+is within that of the baseline 25-step DDIM. Here LTSP2 and STSP2 are Lie-Trotter and Strang
+splitting methods with the combination [PLMS2, PLMS1]. In Table 2, we report the results of three
+different ImageNet resolutions and the average sampling time per image in seconds.
+Our STSP4 performs best on all measurements except Recall on ImageNet512. On ImageNet512,
+PLMS4 has the highest Recall score but a poor FID of 16, indicating that the generated images have
+good distribution coverage but may poorly represent the real distribution. On ImageNet256, STSP4
+can yield 4.49 FID in 20 steps, compared to 4.59 FID in 250 steps originally reported in the paper
+(Dhariwal & Nichol, 2021); our STSP4 is about 9.4× faster when tested on the same machine.
+4.3
+SPLITTING METHODS IN OTHER TASKS
+Besides class-conditional generation, our approach can also accelerate any conditional image gen-
+eration as long as the gradient of the conditional function can be defined. We test our approach on
+four tasks: text-to-image generation, image inpainting, colorization, and super-resolution.
+Text-to-image generation: We use a pre-trained text-to-image Disco-Diffusion (Letts et al., 2021)
+based on Crowson (2021), which substitutes the classifier output with the dot product of the image
+and caption encodings from CLIP (Radford et al., 2021). For more related experiments on Stable-
+Diffusion (Rombach et al., 2022), please refer to Appendix L, M.
+7
+
+Published as a conference paper at ICLR 2023
+Method
+Steps
+Time
+FID
+Prec
+Rec
+ImageNet128
+DDIM
+25
+0.55
+6.69
+0.78
+0.49
+PLMS2
+25
+0.57
+5.71
+0.80
+0.51
+PLMS4
+25
+0.57
+4.97
+0.80
+0.53
+LTSP2
+25
+0.55
+5.14
+0.81
+0.51
+LTSP4
+25
+0.55
+3.85
+0.81
+0.54
+STSP2
+20
+0.54
+5.33
+0.80
+0.52
+STSP4
+20
+0.54
+3.78
+0.81
+0.54
+ADM-G
+250
+5.59*
+2.97
+0.78
+0.59
+Method
+Steps
+Time
+FID
+Prec
+Rec
+ImageNet256
+DDIM
+25
+1.99
+5.47
+0.80
+0.47
+PLMS4
+25
+2.05
+4.71
+0.82
+0.49
+STSP4
+20
+1.95
+4.49
+0.83
+0.50
+ADM-G
+250
+20.9*
+4.59
+0.82
+0.50
+ImageNet512
+DDIM
+25
+5.56
+9.07
+0.81
+0.42
+PLMS4
+25
+5.78
+16.00
+0.75
+0.51
+STSP4
+20
+5.13
+8.24
+0.83
+0.45
+ADM-G
+250
+56.2*
+7.72
+0.87
+0.42
+Table 2: Comparison of different numerical methods using a few steps on guided diffusion sampling.
+Our methods and the best scores are highlighted in bold. We provide the reported scores from
+Dhariwal & Nichol (2021) using 250 sampling steps, referred to as ADM-G in their paper. *ADM-
+G’s sampling times are measured using our machine.
+“A beautiful painting of a singular
+lighthouse, shining its light across
+a tumultuous sea of blood,
+trending on artstation.”
+“A beautiful painting of a starry
+night, over a sunflower sea,
+trending on artstation.”
+Full DDIM
+DDIM
+PLMS4
+LTSP4
+STSP4
+(1,000 steps)
+(45 steps)
+(45 steps)
+(45 steps)
+(30 steps)
+(approximately using the same sampling time)
+Figure 4: Text-to-image generation using different sampling methods.
+Image inpainting & colorization: For these two tasks, we follow the techniques proposed in Song
+et al. (2020b) and Chung et al. (2022a), which improves the conditional functions of both tasks
+with “manifold constraints.” We use the same diffusion model Dhariwal & Nichol (2021) trained on
+ImageNet as our earlier Experiments 4.1, 4.2.
+Super-resolution: We follow the formulation from ILVR (Choi et al., 2021) combined with the
+manifold contraints Chung et al. (2022a), and also use our earlier ImageNet diffusion model.
+Figure 4 compares our techniques, LTSP4 and STSP4, with the DDIM baseline and PLMS4 on
+text-to-image generation. Each result is produced using a fixed sampling time of about 26 seconds.
+STSP4, which uses 30 diffusion steps compared to 45 in the other methods, produces more realistic
+results with color contrast that is more similar to the full DDIM references’. Figure 5 shows that
+our STSP4 produces more convincing results than the DDIM baseline with fewer artifacts on the
+other three tasks while using the same 5 second sampling time. Implementation details, quantitative
+evaluations, and more results are in Appendix J, K.
+5
+DISCUSSION
+Our findings show that when the sampling ODE consists of multiple terms from different networks,
+their numerical behaviors can be different and treating them separately can be more optimal. Another
+promising direction is to improve the behavior of the gradient of the conditional function / classifier
+itself and study whether related properties such as adversarial robustness or gradient smoothness can
+induce the desirable temporal smoothness in the sampling ODE. However, it is not yet clear what
+specific characteristics of the behavior play an important role. This challenge may be related to a
+8
+
+Published as a conference paper at ICLR 2023
+Original
+Input
+STSP4 (Ours)
+DDIM
+Inpainting
+Colorization
+8xSuper-
+Resolution
+Figure 5: Guided-diffusion results of our STSP4 and DDIM on inpainting, colorization, and super-
+resolution. Both methods were limited to use approximately the same sampling time.
+condition called “stiffness” in solving ODEs Ernst & Gerhard (2010), which lacks a clear definition
+but describes the situation where explicit numerical methods, such as RK or PLMS, require a very
+small step size even in a region with smooth curvature.
+As an alternative to the classifier-guided model, Ho & Salimans (2021) propose a classifier-free
+model that can perform conditional generation without a classifier while remaining a generative
+model. This model can utilize high-order methods as no classifier is involved, but it requires evalu-
+ating the classifier-free network twice per step, which is typically more expensive than evaluating a
+normal diffusion model and a classifier. It is important to note that our accelerating technique and
+classifier-free models are not mutually exclusive, and one can still apply a conditional function and
+our splitting technique to guide a classifier-free model in a direction it has not been trained for.
+While our paper only focuses on ODEs derived from the deterministic sampling of DDIM, one can
+convert SDE-based diffusion models to ODEs (Karras et al., 2022) and still use our technique. More
+broadly, we can accelerate any diffusion model that can be expressed as a differential equation with
+a summation of two terms. When these terms behave differently, the benefit from splitting can be
+substantial. Nevertheless, our findings are based on common, existing models and σ schedule from
+Dhariwal & Nichol (2021). Further investigation into the impact of the σ schedule or different types
+and architectures of diffusion models is still required.
+6
+CONCLUSION
+In this paper, we investigate the failure to accelerate guided diffusion sampling of classical high-
+order numerical methods and propose a solution based on splitting numerical methods. We found
+that the gradients of conditional functions are less suitable to classical high-order numerical meth-
+ods and design a technique based on Strang splitting and a combination of forth- and first-order
+numerical methods. Our method achieves better LPIPS and FID scores than previous work given
+the same sampling time and is 32-58% faster than a 250-step DDIM baseline. Our technique can
+successfully accelerate a variety of tasks, such as text-to-image generation, inpainting, colorization,
+and super-resolution.
+9
+
+Published as a conference paper at ICLR 2023
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+Published as a conference paper at ICLR 2023
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+models. arXiv preprint arXiv:2206.05564, 2022.
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+Published as a conference paper at ICLR 2023
+Appendix
+Table of Contents
+A Implementation
+13
+B
+Improving PLMS
+13
+C Numerical Methods for Unguided Diffusion
+15
+D Comparing PLMS and DEIS
+15
+E
+LPIPS vs. the number of sampling step
+15
+F
+More Statistics for Experiement 4.2
+17
+G STSP4 vs. More Numerical Methods Combination
+17
+H Comparison with DEIS and DPM-solver
+18
+I
+Experiment on FID vs. Sampling Time
+19
+J
+Text-guided image generation
+19
+K Controllable generation
+19
+L
+Dreambooth Stable Diffusion
+20
+M CLIP-Guided Stable Diffusion
+22
+N Convergence Orders of Methods
+23
+O Toy Example
+25
+P
+Stability Analysis
+25
+A
+IMPLEMENTATION
+Our implementation is available here1.
+The implementation is based on Katherine Crowson’s
+guided-diffusion 2, which is inspired by OpenAI’s guided-diffusion3. All of the pre-trained dif-
+fusion and classifier models are available here4. For evaluation, we use OpenAI’s measurement
+implementation with their reference image batch, which can be found here5.
+B
+IMPROVING PLMS
+Initial points are required for using high-order PLMS. The fourth-order formulation, for example,
+requires three initial points. The original paper (Liu et al., 2022) employs the Runge-Kutta method
+to compute the initial points. However, Runge-Kutta’s method has high computational costs and
+1https://github.com/sWizad/split-diffusion
+2https://github.com/crowsonkb/guided-diffusion
+3https://github.com/openai/guided-diffusion
+4https://github.com/openai/guided-diffusion/blob/main/model-card.md
+5https://github.com/openai/guided-diffusion/tree/main/evaluations
+13
+
+Published as a conference paper at ICLR 2023
+is inconvenient to use when the number of steps is small. To reduce the computation costs, we
+compute the starting points of the higher-order PLMS using lower-order PLMS. Our PLMS can be
+summarized using Algorithm 3.
+Algorithm 3: PLMS
+input: ¯xn (previous result), σn+1, σn,
+{ei}i ∆t(s + 1) > 0.
+Let us substitute ∆t = 1/N, where N is the number of steps. Now, we can conclude that if N is
+lower than s+1
+2 , the solution of Euler’s method in Equation 36 diverges from the exact solution.
+PLMS2: Consider a second-order linear multistep method on the same test Equation 36:
+yn+1 = yn + ∆t
+�
+−3
+2(s + 1)yn + 1
+2yn−1(s + 1)
+�
+(37)
+=
+�
+1 − ∆t3
+2(s + 1)
+�
+yn + ∆t1
+2(s + 1).
+(38)
+After solving the linear recurrence relation, we obtain
+yn = a1rn
+1 + a2rn
+2 ,
+(39)
+where r1 = 1
+2
+�
+1 − 3
+2∆t(s + 1) +
+�
+1 − ∆t(s + 1) + 9
+4(∆t)2(s + 1)2
+�
+,
+(40)
+and r2 = 1
+2
+�
+1 − 3
+2∆t(s + 1) −
+�
+1 − ∆t(s + 1) + 9
+4(∆t)2(s + 1)2
+�
+.
+(41)
+The numerical solution yn → 0 as n → ∞ when both |r1| < 1 and |r2| < 1, which is equivalent to
+������
+1
+2
+�
+�1 − 3
+2
+(s + 1)
+N
+±
+�
+1 − (s + 1)
+N
++ 9
+4
+�(s + 1)
+N
+�2
+�
+�
+������
+< 1.
+(42)
+In Table 10, we report the lowest number N for each s before Inequality 42 is not satisfied. In other
+words, if the number of steps is below the lowest number N in the table, the solution of the method
+26
+
+Published as a conference paper at ICLR 2023
+in Equation 36 is guaranteed to diverge from the exact solution. The analysis of the higher-order
+methods can be done in a similar fashion.
+LTSP2: We analyze the Lie-Trotter splitting method similarly. In this case, the test Equation 36
+needs to also be split into
+ˆy′ = − ˆy,
+(43)
+˜y′ = − s˜y.
+(44)
+Let us apply the second order linear multistep method (PLMS2) in Equaiton 43 and Euler’s method
+(PLMS1) on Equation 44. We have
+ˆyn+1 = ˆyn − ∆t
+�3
+2 ˆyn − 1
+2 ˆyn−1
+�
+,
+˜yn+1 = ˜yn − ∆ts˜yn.
+(45)
+Thus, a single combining step of LTSP2 can be formulated by
+yn+1 = (1 − s∆t)
+��
+1 − 3
+2∆t
+�
+yn + ∆t
+2 yn−1
+�
+.
+(46)
+Similar to the above, we solve the linear recurrence relation and obtain the following condition
+������
+1
+2
+�
+�
+�
+1 − s
+N
+� �
+1 − 3
+2
+s
+N
+�
+±
+��
+1 − s
+N
+�2 �
+1 − 3
+2
+s
+N
+�2
++ 2
+N
+�
+1 − s
+N
+�
+�
+�
+������
+< 1.
+(47)
+We report the lowest integer number N for each s before Inequality 47 is not satisfied in Table 10.
+STSP2: We analyze the Strang splitting method by splitting the test Equation 36 into
+¯y′ = − s¯y
+(48)
+ˆy′ = − ˆy
+(49)
+˜y′ = − s˜y
+(50)
+We apply the second-order linear multistep method (PLMS2) in Equaiton 49 and Euler’s method on
+Equation 48 and 50.
+¯yn+1 =
+�
+1 − ∆t
+2 s
+�
+¯yn
+(51)
+ˆyn+1 =
+�
+1 − 3
+2∆t
+�
+ˆyn + ∆t
+2 ˆyn−1
+(52)
+˜yn+1 =
+�
+1 − ∆t
+2 s
+�
+˜yn
+(53)
+We combine Equation 51-53 into
+yn+1 =
+�
+1 −
+s
+2N
+�2 �
+1 − 3
+2N
+�
+yn + 1
+2N
+�
+1 −
+s
+2N
+�2
+yn−1.
+(54)
+After solving the linear recurrence relation, we obtain the following condition
+�����
+1
+2
+�
+b ±
+�
+b2 + 2
+N c
+������ < 1,
+(55)
+where b =
+�
+1 −
+s
+2N
+�2 �
+1 −
+3
+2N
+�
+and c =
+�
+1 −
+s
+2N
+�2. In Table 10, we report the lowest number of
+steps N for each s before Inequality 55 is not satisfied.
+In Table 10, we compare the lowest number of steps N before each method is guaranteed to diverge
+from our analysis. We also show numerical solutions of our toy example in Figure 16 to compare to
+our analysis. It is important to note that if the number of steps exceeds Table 10, we cannot presume
+that the numerical solution will function properly.
+27
+
+Published as a conference paper at ICLR 2023
+s = 5
+s = 10
+s = 15
+s = 20
+s = 30
+s = 40
+s = 60
+s = 80
+Euler
+4
+6
+9
+11
+16
+21
+31
+41
+PLMS2
+6
+11
+16
+22
+32
+42
+63
+83
+LTSP2
+2
+3
+7
+9
+14
+19
+29
+39
+STSP2
+2
+3
+4
+5
+8
+10
+15
+20
+Table 10: The lowest number of steps before we can guarantee by theory that each numerical method
+will fail to solve Equation 36. Notice that LTSP2 and STSP2 are having lower number which mean
+they are also harder to fail when the number of step are reduced than Euler and PLMS2.
+s = 10
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+x1
+0.8
+0.6
+0.4
+0.2
+0.0
+x2
+exact
+Euler (PLMS1)
+PLMS2
+LTSP2
+STSP2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+x1
+0.8
+0.6
+0.4
+0.2
+0.0
+x2
+exact
+Euler (PLMS1)
+PLMS2
+LTSP2
+STSP2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+x1
+0.8
+0.6
+0.4
+0.2
+0.0
+x2
+exact
+Euler (PLMS1)
+PLMS2
+LTSP2
+STSP2
+s = 15
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+x1
+0.8
+0.6
+0.4
+0.2
+0.0
+x2
+exact
+Euler (PLMS1)
+PLMS2
+LTSP2
+STSP2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+x1
+0.8
+0.6
+0.4
+0.2
+0.0
+x2
+exact
+Euler (PLMS1)
+PLMS2
+LTSP2
+STSP2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+x1
+0.8
+0.6
+0.4
+0.2
+0.0
+x2
+exact
+Euler (PLMS1)
+PLMS2
+LTSP2
+STSP2
+s = 20
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+x1
+0.8
+0.6
+0.4
+0.2
+0.0
+x2
+exact
+Euler (PLMS1)
+PLMS2
+LTSP2
+STSP2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+x1
+0.8
+0.6
+0.4
+0.2
+0.0
+x2
+exact
+Euler (PLMS1)
+PLMS2
+LTSP2
+STSP2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+x1
+0.8
+0.6
+0.4
+0.2
+0.0
+x2
+exact
+Euler (PLMS1)
+PLMS2
+LTSP2
+STSP2
+10 steps
+15 steps
+20 steps
+.
+Figure 16: This figure show how numerical solutions look like when the number of steps are close
+to the number in Table 10
+28
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf,len=1403
+page_content='Published as a conference paper at ICLR 2023 ACCELERATING GUIDED DIFFUSION SAMPLING WITH SPLITTING NUMERICAL METHODS Suttisak Wizadwongsa, Supasorn Suwajanakorn VISTEC, Thailand {suttisak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='w s19, supasorn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='s}@vistec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='th ABSTRACT Guided diffusion is a technique for conditioning the output of a diffusion model at sampling time without retraining the network for each specific task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' One drawback of diffusion models, however, is their slow sampling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Recent techniques can accelerate unguided sampling by applying high-order numerical methods to the sampling process when viewed as differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' On the contrary, we discover that the same techniques do not work for guided sampling, and little has been explored about its acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' This paper explores the culprit of this problem and provides a solution based on operator splitting methods, motivated by our key finding that classical high-order numerical methods are unsuitable for the conditional function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Our proposed method can re-utilize the high-order methods for guided sampling and can generate images with the same quality as a 250- step DDIM baseline using 32-58% less sampling time on ImageNet256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' We also demonstrate usage on a wide variety of conditional generation tasks, such as text- to-image generation, colorization, inpainting, and super-resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' 1 INTRODUCTION A family of generative models known as diffusion models has recently gained a lot of attention with state-of-the-art image generation quality (Dhariwal & Nichol, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Guided diffusion is an approach for controlling the output of a trained diffusion model for conditional generation tasks without retraining its network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' By engineering a task-specific conditional function and modifying only the sampling procedure, guided diffusion models can be used in a variety of applications, such as class-conditional image generation (Dhariwal & Nichol, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Kawar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', 2022), text-to-image generation (Nichol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', 2022), image-to-image translation (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', 2022), inpainting (Chung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', 2022a), colorization (Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', 2020b), image composition (Sasaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', 2021), adversarial purification (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', 2022) and super-resolution (Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' One common drawback of both guided and regular “unguided” diffusion models is their slow sam- pling processes, usually requiring hundreds of iterations to produce a single image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Recent speed- up attempts include improving the noise schedule (Nichol & Dhariwal, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Watson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', 2021), redefining the diffusion process to be non-Markovian, thereby allowing a deterministic sampling process Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (2020a), network distillation that teaches a student model to simulate multiple sampling steps of a teacher model Salimans & Ho (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Luhman & Luhman (2021), among oth- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (2020a) show how each sampling step can be expressed as a first-order numerical step of an ordinary differential equation (ODE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Similarly, Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (2020b) express the sam- pling of a score-based model as solving a stochastic differential equation (SDE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' By regarding the sampling process as an ODE/SDE, many high-order numerical methods have been suggested, such as Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (2022), Zhang & Chen (2022), and Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (2022) with impressive results on un- guided diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' However, when applied to guided diffusion models, these methods produce surprisingly poor results (see Figure 1)—given a few number of steps, those high-order numerical methods actually perform worse than low-order methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Guided sampling differs from the unguided one by the addition of the gradients of the conditional function to its sampling equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' The observed performance decline thus suggests that classical high-order methods may not be suitable for the conditional function and, consequently, the guided sampling equation as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Our paper tests this hypothesis and presents an approach to accel- 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='11558v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='CV] 27 Jan 2023 Published as a conference paper at ICLR 2023 Number of steps 8 16 32 64 128 256 DDIM PLMS4 STSP4 (Ours) Figure 1: Generated samples of a classifier-guided diffusion model trained on ImageNet256 using 8- 256 sampling steps from different sampling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Our technique, STSP4, produces high-quality results in a fewer number of steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' erating guided diffusion sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' The key idea is to use an operator splitting method to split the less well-behaved conditional function term from the standard diffusion term and solve them sepa- rately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' This approach not only allows re-utilizing the successful high-order methods on the diffusion term but also provides us with options to combine different specialized methods for each term to maximize performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Splitting method can be used to solve diffusion SDE in Dockhorn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Our design process includes comparing different splitting methods and numerical methods for each split term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' When tested on ImageNet, our approach achieves the same level of image quality as a DDIM baseline while reducing the sampling time by approximately 32-58%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Compared with other sampling methods using the same sampling time, our approach provides better image quality as measured by LPIPS, FID, and Perception/Recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' With only minimal modifications to the sampling equation, we also show successful acceleration on various conditional generation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' 2 BACKGROUND This section provides a high-level summary of the theoretical foundation of diffusion models as well as numerical methods that have been used for diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Here we briefly explain a few that contribute to our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='1 DIFFUSION MODELS Assuming that x0 is a random variable from the data distribution we wish to reproduce, diffusion models define a sequence of Gaussian noise degradation of x0 as random variables x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', xT , where xt ∼ N(√1 − βtxt−1, βtI) and βt ∈ [0, 1] are parameters that control the noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' With a property of Gaussian distribution, we can express xt directly as a function of x0 and noise ϵ ∼ N(0, I) by xt = √¯αtx0 +√1 − ¯αtϵ, where ¯αt = �t i=1(1−βi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' By picking a sufficiently large T (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', 1,000) and an appropriate set of βt, we can assume xT is a standard Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' The main idea of diffusion model generation is to sample a Gaussian noise xT and use it to reversely sample xT −1, xT −2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' until we obtain x0, which belongs to our data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (2020) propose Denoising Diffusion Probabilistic Model (DDPM) and explain how to employ a neural network ϵθ(xt, t) to predict the noise ϵ that is used to compute xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' To train the network, we sample a training image x0, t, and ϵ to compute xt using the above relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Then, we optimize our network ϵθ to minimize the difference between the predicted and real noise, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', ∥ϵ − ϵθ(xt, t)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' 2 Published as a conference paper at ICLR 2023 Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (2020a) introduce Denoising Diffusion Implicit Model (DDIM), which uses the network ϵθ to deterministically obtain xt−1 given xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' The DDIM generative process can be written as xt−1 = � ¯αt−1 ¯αt � xt − √ 1 − ¯αtϵθ(xt, t) � + � 1 − ¯αt−1ϵθ(xt, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (1) This formulation could be used to skip many sampling steps and boost sampling speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' To turn this into an ODE, we rewrite Equation 1 as: xt−∆t √¯αt−∆t = xt √¯αt + �� 1 − ¯αt−∆t ¯αt−∆t − � 1 − ¯αt ¯αt � ϵθ(xt, t), (2) which is now equivalent to a numerical step in solving an ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' To derive the corresponding ODE, we can re-parameterize σt = √1 − ¯αt/√¯αt, ¯x(t) = xt/√¯αt and ¯ϵσ(¯x) = ϵθ(xt, t), yielding ¯x(t − ∆t) − ¯x(t) = (σt−∆t − σt)¯ϵσ(¯x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' By letting (σt−∆t − σt) → 0, the ODE becomes: d¯x dσ = ¯ϵσ(¯x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (3) Note that this change of variables is equivalent to an exponential integrator technique described in both Zhang & Chen (2022) and Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Since xt and ¯x(t) have the same value at t = 0, our work can focus on solving ¯x(t) rather than xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Many numerical methods can be applied to the ODE Equation 3 to accelerate diffusion sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' We next discuss some of them that are relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='2 NUMERICAL METHODS Euler’s Method is the most basic numerical method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' A forward Euler step is given by ¯xn+1 = ¯xn + ∆σ¯ϵσ(¯xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' When we apply the forward Euler step to the ODE Equation 3, we get the DDIM formulation (Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Heun’s Method, also known as the trapezoid rule or improved Euler, is given by: ¯xn+1 = ¯xn + ∆σ 2 (e1 + e2), where e1 = ¯ϵσ(¯xn) and e2 = ¯ϵσ(¯xn + ∆σe1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' This method modifies Euler’s method into a two-step method to improve accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Many papers have used this method on diffusion models, including Algorithm 1 in Karras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (2022) and DPM-Solver-2 in Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' This method is also the simplest case of Predictor-Corrector methods used in Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Runge-Kutta Methods represent a class of numerical methods that integrate information from mul- tiple hidden steps and provide high accuracy results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Heun’s method also belongs to a family of 2nd-order Runge-Kutta methods (RK2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' The most well-known variant is the 4th-order Runge-Kutta method (RK4), which is written as follows: e1 = ¯ϵσ(¯xn), e2 = ¯ϵσ � ¯xn + ∆σ 2 e1 � , e3 = ¯ϵσ � ¯xn + ∆σ 2 e2 � , e4 = ¯ϵσ (¯xn + ∆σe3) , ¯xn+1 = ¯xn + ∆σ 6 (e1 + 2e2 + 2e3 + e4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (4) This method has been tested on diffusion models in Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (2022) and Salimans & Ho (2022), but it has not been used as the main proposed method in any paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Linear Multi-Step Method, similar to the Runge-Kutta methods, aims to combine information from several steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' however, rather than evaluating new hidden steps, this method uses the previous steps to estimate the new step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' The 1st-order formulation is the same as Euler’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' The 2nd-order formulation is given by ¯xn+1 = ¯xn + ∆σ 2 (3e0 − e1) , (5) while the 4th-order formulation is given by ¯xn+1 = ¯xn + ∆σ 24 (55e0 − 59e1 + 37e2 − 9e3), (6) where ek = ¯ϵσ(¯xn−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' These formulations are designed for a constant ∆σ in each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' However, our experiments and previous work that uses this method (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Zhang & Chen 3 Published as a conference paper at ICLR 2023 (2022)) still show good results when this assumption is not strictly satisfied, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', when ∆σ is not constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' We will refer to these formulations as PLMS (Pseudo Linear Multi-Step) for the rest of the paper, like in Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' A similar linear multi-step method for non-constant ∆σ can also be derived using a technique used in Zhang & Chen (2022), which we detail in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' The method can improve upon PLMS slightly, but it is not as flexible because we have to re-derive the update rule every time the σ schedule changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' 3 SPLITTING METHODS FOR GUIDED DIFFUSION MODELS This section introduces our technique that uses splitting numerical methods to accelerate guided diffusion sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' We first focus our investigation on classifier-guided diffusion models for class- conditional generation and later demonstrate how this technique can be used for other conditional generation tasks in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Like any guided diffusion models, classifier-guided models (Dhari- wal & Nichol, 2021) share the same training objective with regular unguided models with no mod- ifications to the training procedure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' but the sampling process is guided by an additional gradient signal from an external classifier to generate class-specific output images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Specifically, the sampling process is given by ˆϵ = ϵθ(xt) − √ 1 − ¯αt∇x log pφ(c|xt), xt−1 = √¯αt−1 �xt − √1 − ¯αtˆϵ √¯αt � + � 1 − ¯αt−1ˆϵ, (7) where pφ(c|xt) is a classifier model trained to output the probability of xt belonging to class c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' As discussed in the previous section, we can rewrite this formulation as a “guided ODE”: d¯x dσ = ¯ϵσ(¯x) − ∇fσ(¯x), (8) where fσ(¯x) = σ √ σ2+1 log pφ(c|xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' We refer to fσ as the conditional function, which can be substituted with other functions for different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' After obtaining the ODE form, any numerical solver mentioned earlier can be readily applied to accelerate the sampling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' However, we observe that classical high-order numerical methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', PLMS4, RK4) fail to accelerate this task (see Figure 1) and even perform worse than the baseline DDIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' We hypothesize that the two terms in the guided ODE may have different numerical behaviors with the conditional term being less suitable to classical high-order methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' We speculate that the difference could be partly attributed to how they are computed: ∇fσ(¯x) is computed through back- propagation, whereas ¯ϵσ(¯x) is computed directly by evaluating a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' One possible solution to handle terms with different behaviors is the so-called operator splitting method, which divides the problem into two subproblems: dy dσ = ¯ϵσ(y), dz dσ = −∇fσ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (9) We call these the diffusion and condition subproblems, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' This method allows separating the hard-to-approximate ∇fσ(z) from ¯ϵσ(y) and solving them separately in each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Impor- tantly, this helps reintroduce the effective use of high-order methods on the diffusion subproblem as well as provides us with options to combine different specialized methods to maximize performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' We explore two most famous first- and second-order splitting techniques for our task: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='1 LIE-TROTTER SPLITTING (LTSP) Our first example is the simple first-order Lie-Trotter splitting method (Trotter, 1959), which ex- presses the splitting as dy dσ = ¯ϵσ(y), y(σn) = ¯xn, σ ∈ [σn+1, σn] (10) dz dσ = −∇fσ(z), z(σn) = y(σn+1), σ ∈ [σn+1, σn] (11) with the solution of this step being ¯xn+1 = z(σn+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Note that σn is a decreasing sequence in sampling schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Here Equation 10 is the same as Equation 3, which can be solved using any 4 Published as a conference paper at ICLR 2023 Algorithm 1: Lie-Trotter Splitting (LTSP) sample ¯x0 ∼ N(0, σ2 maxI) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' for n ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', N − 1} do yn+1 = PLMS(¯xn, σn, σn+1, ¯ϵσ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' ¯xn+1 = yn+1 −(σn+1 −σn)∇f(yn+1) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' end Result: ¯xN Algorithm 2: Strang Splitting (STSP) sample ¯x0 ∼ N(0, σ2 maxI) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' for n ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', N − 1} do zn+1 = ¯xn − (σn+1−σn) 2 ∇f(¯xn) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' yn+1 = PLMS(zn+1, σn, σn+1, ¯ϵσ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' ¯xn+1 = yn+1 − (σn+1−σn) 2 ∇f(yn+1) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' end Result: ¯xN high-order numerical method, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', PLMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' For Equation 11, we can use a forward Euler step: zn+1 = zn − ∆σ∇fσ(zn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (12) This is equivalent to a single iteration of standard gradient descent with a learning rate ∆σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' This splitting scheme is summarized by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' We investigate different numerical methods for each subproblem in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='2 STRANG SPLITTING (STSP) Strang splitting (or Strang-Marchuk) (Strang, 1968) is one of the most famous and widely used operator splitting methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' This second-order splitting works as follows: dz dσ = −∇fσ(z), z(σn) = ¯xn, σ ∈ �1 2(σn + σn+1), σn � (13) dy dσ = ¯ϵσ(y), y(σn) = z �1 2(σn + σn+1) � , σ ∈ [σn+1, σn] (14) d˜z dσ = −∇fσ(˜z), ˜z �1 2(σn + σn+1) � = y(σn+1), σ ∈ � σn+1, 1 2(σn + σn+1) � (15) Instead of solving each subproblem for a full step length, we solve the condition subproblem for half a step before and after solving the diffusion subproblem for a full step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' In theory, we can swap the order of operations without affecting convergence, but it is practically cheaper to compute the condition term twice rather than the diffusion term twice because fσ is typically a smaller network compared to ¯ϵσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' The Strange splitting algorithm is shown in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' This method can be proved to have better accuracy than the Lie-Trotter method using the Banker-Campbell-Hausdorff formula (Tuckerman, 2010), but it requires evaluating the condition term twice per step in exchange for improved image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' We assess this trade-off in the experiment section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' 4 EXPERIMENTS Extending on our observation that classical high-order methods failed on guided sampling, we con- ducted a series of experiments to investigate this problem and evaluate our solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='1 uses a simple splitting method (first-order LTSP) to study the effects that high-order methods have on each subproblem, leading to our key finding that only the conditional subproblem is less suited to classical high-order methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' This section also determines the best combination of numerical meth- ods for the two subproblems under LTSP splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='2 explores improvements from using a higher-order splitting method and compares our best scheme to previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Finally, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='3 applies our approach to a variety of conditional generation tasks with minimal changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' For our comparison, we use pre-trained state-of-the-art diffusion models and classifiers from Dhari- wal & Nichol (2021), which were trained on the ImageNet dataset (Russakovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', 2015) with 1000 total sampling step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' We treat full-path samples from a classifier-guided DDIM at 1,000 steps as reference solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Then the performance of each configuration is measured by the image similar- ity between its generated samples using fewer steps and the reference DDIM samples, both starting from the same initial noise maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Given the same sampling time, we expect configurations with better performance to better match the full DDIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' We measure image similarity using Learned Per- ceptual Image Patch Similarity (LPIPS) (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', 2018) (lower is better) and measure sampling time using a single NVIDIA RTX 3090 and a 24-core AMD Threadripper 3960x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' 5 Published as a conference paper at ICLR 2023 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='5 Sampling time (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=') 10 2 10 1 LPIPS DDIM 250 steps 3 × 10 2 3 × 10 1 Euler (DDIM) PLMS4 [PLMS1,PLMS1] [PLMS2,PLMS1] [PLMS4,PLMS1] [RK2, PLMS1] [RK4, PLMS1] (a) Varying the method for the diffusion subproblem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='5 Sampling time (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=') 10 2 10 1 LPIPS DDIM 250 steps 3 × 10 2 3 × 10 1 Euler (DDIM) [PLMS1, PLMS1] [PLMS1, PLMS2] [PLMS1, PLMS4] [PLMS1, RK2] [PLMS1, RK4] (b) Varying the method for the condition subproblem Figure 2: Comparison of different combinations of numerical methods under LTSP splitting for guided diffusion sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' We plot LPIPS against the sampling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' [A, B] denotes the use of method A in the diffusion subproblem and method B in the condition subproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' The red dotted lines indicate a reference DDIM score obtained from 250 sampling steps, which produce images visually close to those from 1,000 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='1 FINDING A SUITABLE NUMERICAL METHOD FOR EACH SUBPROBLEM To study the effects of different numerical methods on each subproblem of the guided ODE (Equa- tion 8), we use the simplest Lie-Trotter splitting, which itself requires no additional network evalu- ations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' This controlled experiment has two setups: a) we fix the numerical method for the condition subproblem (Equation 11) to first-order PLMS1 (Euler’s method) and vary the numerical method for the diffusion subproblem (Equation 10), and conversely b) we fix the method for the diffusion subproblem and vary the method for the condition subproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' The numerical method options are Euler’s method (PLMS1), Heun’s method (RK2), 4th order Runge-Kutta’s method (RK4), and 2nd/4th order pseudo linear multi-step (PLMS2/PLMS4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' We report LPIPS vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' sampling time of various numerical combinations on a diffusion model trained on ImageNet 256×256 in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' The red dotted lines indicate a reference DDIM score obtained from 250 sampling steps, a common choice that produces good samples that are perceptually close to those from a full 1,000-step DDIM (Dhariwal & Nichol, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Nichol & Dhariwal, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Given a long sampling time, non-split PLMS4 performs better than the DDIM baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' However, when the sampling time is reduced, the image quality of PLMS4 rapidly decreases and becomes much worse than that of DDIM, especially under 15 seconds in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' When we split the ODE and solve both subproblems using first-order PLMS1 (Euler), the performance is close to that of DDIM, which is also considered first-order but without any splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' This helps verify that merely splitting the ODE does not significantly alter the sampling speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' In the setup a), when RK2 and RK4 are used for the diffusion subproblem, they also perform worse than the DDIM baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' This slowdown is caused by the additional evaluations of the network by these methods, which outweigh the improvement gained in each longer diffusion step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Note that if we instead measure the image quality with respect to the number of diffusion steps, RK2 and RK4 can outperform other methods (Appendix E);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' however, this is not our metric of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' On the other hand, PLMS2 and PLMS4, which require no additional network evaluations, are about 8-10% faster than DDIM and can achieve the same LPIPS score as the DDIM that uses 250 sampling steps in 20-26 fewer steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Importantly, when the sampling time is reduced, their performance does not degrade rapidly like the non-split PLMS4 and remains at the same level as DDIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' In the setup b) where we vary the numerical method for the condition subproblem, the result re- veals an interesting contrast—none of the methods beats DDIM and some even make the sampling diverged [PLMS1, RK4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' These findings suggest that the gradients of conditional functions are less “compatible” with classical high-order methods, especially when used with a small number of steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' This phenomenon may be related to the “stiffness” condition of ODEs, which we discuss further in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' For the remainder of our experiments, we will use the combination [PLMS4, PLMS1] for the diffusion and condition subproblems, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' 6 Published as a conference paper at ICLR 2023 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='5 Sampling time (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=') 10 2 10 1 LPIPS DDIM 250 steps 3 × 10 2 3 × 10 1 Euler (DDIM) PLMS4 RK2 RK4 LTSP4 [PLMS4,PLMS1] STSP4 [PLMS4,PLMS1] Figure 3: Comparison of different numerical methods for guided diffusion sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Sampling time within 5 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' 10 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' 15 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' 20 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' DDIM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='033 PLMS4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='278 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='141 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='057 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='026 RK2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='193 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='059 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='028 RK4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='216 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='028 LTSP4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='121 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='028 STSP4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='079 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='013 Table 1: Average LPIPS when the sampling time is limited to be under 5 - 20 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='2 IMPROVED SPLITTING METHOD This experiment investigates improvements from using a high-order splitting method, specifically the Strang splitting method, with the numerical combination [PLMS4, PLMS1] and compares our methods to previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Note that besides DDIM Dhariwal & Nichol (2021), no previous work is specifically designed for accelerating guided sampling, thus the baselines in this comparison are only adaptations of the core numerical methods used in those papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' And to our knowledge, no prior guided-diffusion work uses splitting numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Non-split numerical method baselines are PLMS4, which is used in Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (2022), RK2, which is used in Karras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (2022), and higher-order RK4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' We report the LPIPS scores of these methods with respect to the sampling time in Figure 3 and Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Without any splitting, PLMS4, RK2 and RK4 show significantly poorer image quality when used with short sampling times < 10 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' The best performer is our Strang splitting (STSP4), which can reach the same quality as 250-step DDIM while using 32-58% less sampling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' STSP4 also obtains the highest LPIPS scores for sample times of 5, 10, 15, and 20 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' More statistical details and comparison with other split combinations are in Appendix F, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' In addition, we perform a quantitative evaluation for class-conditional generation by sampling 50,000 images based on uniformly chosen class conditions with a small number of sampling steps and evaluating the Fenchel Inception Distance (FID) Heusel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (2017) (lower is better) and the improved precision/recall Kynk¨a¨anniemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (2019) (higher is better) against an ImageNet test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Following (Dhariwal & Nichol, 2021), we use a 25-step DDIM as a baseline, which already produces visually reasonable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' As PLMS and LTSP require the same number of network eval- uations as the DDIM, they are used also with 25 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' For STSP with a longer network evaluation time, it is only allowed 20 steps, which is the highest number of steps such that its sampling time is within that of the baseline 25-step DDIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Here LTSP2 and STSP2 are Lie-Trotter and Strang splitting methods with the combination [PLMS2, PLMS1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' In Table 2, we report the results of three different ImageNet resolutions and the average sampling time per image in seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Our STSP4 performs best on all measurements except Recall on ImageNet512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' On ImageNet512, PLMS4 has the highest Recall score but a poor FID of 16, indicating that the generated images have good distribution coverage but may poorly represent the real distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' On ImageNet256, STSP4 can yield 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='49 FID in 20 steps, compared to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='59 FID in 250 steps originally reported in the paper (Dhariwal & Nichol, 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' our STSP4 is about 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='4× faster when tested on the same machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='3 SPLITTING METHODS IN OTHER TASKS Besides class-conditional generation, our approach can also accelerate any conditional image gen- eration as long as the gradient of the conditional function can be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' We test our approach on four tasks: text-to-image generation, image inpainting, colorization, and super-resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Text-to-image generation: We use a pre-trained text-to-image Disco-Diffusion (Letts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', 2021) based on Crowson (2021), which substitutes the classifier output with the dot product of the image and caption encodings from CLIP (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' For more related experiments on Stable- Diffusion (Rombach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', 2022), please refer to Appendix L, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' 7 Published as a conference paper at ICLR 2023 Method Steps Time FID Prec Rec ImageNet128 DDIM 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='55 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='49 PLMS2 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='57 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
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+page_content='53 LTSP2 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
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+page_content='51 LTSP4 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
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+page_content='54 STSP2 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
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+page_content='52 STSP4 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
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+page_content='54 ADM-G 250 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='59* 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
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+page_content='59 Method Steps Time FID Prec Rec ImageNet256 DDIM 25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
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+page_content='47 PLMS4 25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
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+page_content='49 STSP4 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='95 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='50 ADM-G 250 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='9* 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='50 ImageNet512 DDIM 25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='56 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='42 PLMS4 25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='78 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
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+page_content='51 STSP4 20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='13 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
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+page_content='45 ADM-G 250 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='2* 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='42 Table 2: Comparison of different numerical methods using a few steps on guided diffusion sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Our methods and the best scores are highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' We provide the reported scores from Dhariwal & Nichol (2021) using 250 sampling steps, referred to as ADM-G in their paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' *ADM- G’s sampling times are measured using our machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' “A beautiful painting of a singular lighthouse, shining its light across a tumultuous sea of blood, trending on artstation.” “A beautiful painting of a starry night, over a sunflower sea, trending on artstation.” Full DDIM DDIM PLMS4 LTSP4 STSP4 (1,000 steps) (45 steps) (45 steps) (45 steps) (30 steps) (approximately using the same sampling time) Figure 4: Text-to-image generation using different sampling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Image inpainting & colorization: For these two tasks, we follow the techniques proposed in Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (2020b) and Chung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (2022a), which improves the conditional functions of both tasks with “manifold constraints.” We use the same diffusion model Dhariwal & Nichol (2021) trained on ImageNet as our earlier Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Super-resolution: We follow the formulation from ILVR (Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', 2021) combined with the manifold contraints Chung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' (2022a), and also use our earlier ImageNet diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Figure 4 compares our techniques, LTSP4 and STSP4, with the DDIM baseline and PLMS4 on text-to-image generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Each result is produced using a fixed sampling time of about 26 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' STSP4, which uses 30 diffusion steps compared to 45 in the other methods, produces more realistic results with color contrast that is more similar to the full DDIM references’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Figure 5 shows that our STSP4 produces more convincing results than the DDIM baseline with fewer artifacts on the other three tasks while using the same 5 second sampling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Implementation details, quantitative evaluations, and more results are in Appendix J, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' 5 DISCUSSION Our findings show that when the sampling ODE consists of multiple terms from different networks, their numerical behaviors can be different and treating them separately can be more optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Another promising direction is to improve the behavior of the gradient of the conditional function / classifier itself and study whether related properties such as adversarial robustness or gradient smoothness can induce the desirable temporal smoothness in the sampling ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' However, it is not yet clear what specific characteristics of the behavior play an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' This challenge may be related to a 8 Published as a conference paper at ICLR 2023 Original Input STSP4 (Ours) DDIM Inpainting Colorization 8xSuper- Resolution Figure 5: Guided-diffusion results of our STSP4 and DDIM on inpainting, colorization, and super- resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Both methods were limited to use approximately the same sampling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' condition called “stiffness” in solving ODEs Ernst & Gerhard (2010), which lacks a clear definition but describes the situation where explicit numerical methods, such as RK or PLMS, require a very small step size even in a region with smooth curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' As an alternative to the classifier-guided model, Ho & Salimans (2021) propose a classifier-free model that can perform conditional generation without a classifier while remaining a generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' This model can utilize high-order methods as no classifier is involved, but it requires evalu- ating the classifier-free network twice per step, which is typically more expensive than evaluating a normal diffusion model and a classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' It is important to note that our accelerating technique and classifier-free models are not mutually exclusive, and one can still apply a conditional function and our splitting technique to guide a classifier-free model in a direction it has not been trained for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' While our paper only focuses on ODEs derived from the deterministic sampling of DDIM, one can convert SDE-based diffusion models to ODEs (Karras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', 2022) and still use our technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' More broadly, we can accelerate any diffusion model that can be expressed as a differential equation with a summation of two terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' When these terms behave differently, the benefit from splitting can be substantial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Nevertheless, our findings are based on common, existing models and σ schedule from Dhariwal & Nichol (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Further investigation into the impact of the σ schedule or different types and architectures of diffusion models is still required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' 6 CONCLUSION In this paper, we investigate the failure to accelerate guided diffusion sampling of classical high- order numerical methods and propose a solution based on splitting numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' We found that the gradients of conditional functions are less suitable to classical high-order numerical meth- ods and design a technique based on Strang splitting and a combination of forth- and first-order numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Our method achieves better LPIPS and FID scores than previous work given the same sampling time and is 32-58% faster than a 250-step DDIM baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Our technique can successfully accelerate a variety of tasks, such as text-to-image generation, inpainting, colorization, and super-resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' 9 Published as a conference paper at ICLR 2023 REFERENCES Jooyoung Choi, Sungwon Kim, Yonghyun Jeong, Youngjune Gwon, and Sungroh Yoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' ILVR: Conditioning method for denoising diffusion probabilistic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
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+page_content=' Hyungjin Chung, Byeongsu Sim, and Jong Chul Ye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
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+page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' 12413–12422, 2022b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Katherine Crowson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
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+page_content=' arXiv preprint arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
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+page_content=' Quanlin Wu, Hang Ye, and Yuntian Gu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Guided diffusion model for adversarial purification from random noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' arXiv preprint arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='10875, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Qinsheng Zhang and Yongxin Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Fast sampling of diffusion models with exponential integrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' In NeurIPS 2022 Workshop on Score-Based Methods, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' 11 Published as a conference paper at ICLR 2023 Qinsheng Zhang, Molei Tao, and Yongxin Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' gDDIM: Generalized denoising diffusion implicit models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' arXiv preprint arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='05564, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Richard Zhang, Phillip Isola, Alexei A Efros, Eli Shechtman, and Oliver Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' The unreasonable effectiveness of deep features as a perceptual metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' 586–595, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Min Zhao, Fan Bao, Chongxuan Li, and Jun Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' EGSDE: Unpaired image-to-image translation via energy-guided stochastic differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' In Advances in Neural Information Processing Systems, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' 12 Published as a conference paper at ICLR 2023 Appendix Table of Contents A Implementation 13 B Improving PLMS 13 C Numerical Methods for Unguided Diffusion 15 D Comparing PLMS and DEIS 15 E LPIPS vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' the number of sampling step 15 F More Statistics for Experiement 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='2 17 G STSP4 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' More Numerical Methods Combination 17 H Comparison with DEIS and DPM-solver 18 I Experiment on FID vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Sampling Time 19 J Text-guided image generation 19 K Controllable generation 19 L Dreambooth Stable Diffusion 20 M CLIP-Guided Stable Diffusion 22 N Convergence Orders of Methods 23 O Toy Example 25 P Stability Analysis 25 A IMPLEMENTATION Our implementation is available here1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' The implementation is based on Katherine Crowson’s guided-diffusion 2, which is inspired by OpenAI’s guided-diffusion3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' All of the pre-trained dif- fusion and classifier models are available here4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' For evaluation, we use OpenAI’s measurement implementation with their reference image batch, which can be found here5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' B IMPROVING PLMS Initial points are required for using high-order PLMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' The fourth-order formulation, for example, requires three initial points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' The original paper (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=', 2022) employs the Runge-Kutta method to compute the initial points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' However, Runge-Kutta’s method has high computational costs and 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='com/sWizad/split-diffusion 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='com/crowsonkb/guided-diffusion 3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='com/openai/guided-diffusion 4https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='com/openai/guided-diffusion/blob/main/model-card.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='md 5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content='com/openai/guided-diffusion/tree/main/evaluations 13 Published as a conference paper at ICLR 2023 is inconvenient to use when the number of steps is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' To reduce the computation costs, we compute the starting points of the higher-order PLMS using lower-order PLMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Our PLMS can be summarized using Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFJT4oBgHgl3EQffyxP/content/2301.11558v1.pdf'}
+page_content=' Algorithm 3: PLMS input: ¯xn (previous result), σn+1, σn, {ei}idp1. As seen, the Mohr-Coulomb circle shifted more to
+the left with increasing pressure. It indicates that the normal
+and shear stress tended to reach closer to the failure line and
+shear stress axis. Therefore, the failure may occur easily.
+Li B, et al. [19] provided a model to assess the cement
+sheath integrity in the carbon sequestration fields. Then the
+model was verified with Atkinson’s model and Carter’s finite
+element model. A case study showed that that maximum
+injection pressure would be under-evaluated if cement
+sheath-induced stress is not examined. One of the conclusions
+is that the critical failure point lied at the cement-sandstone/
+shale formation interface.
+Omosebi O, et al. [20] performed an experiment to
+investigate the cement integrity during CO2 injection. An
+important conclusion was that class H cement turned into
+more elastic than class G cement with CO2. The differences in
+composition and mineralogy can explain it.
+Thermal Property
+The changes in temperature amid CO2 injection may
+induce stress variation, which can devastate the well. The
+cold injected CO2 in contact with warm wellbore, reservoir
+and the thermal properties difference between wellbore
+casing, lithology, and the cement will stress the vicinity of a
+wellbore. As a result, the chances of creating more leakage
+paths are very high.
+Thermal stress, which results in deterioration in cement
+sheath, relies on the thermo-mechanical properties of the
+material, the azimuthal extent, and radial position of fracture
+in addition to injection temperature and effective in-situ
+horizontal stress. The thermal-mechanical properties consist
+of elastic modulus, thermal conductivity, and the thermal
+expansion coefficient of cement. Thermal conductivity is
+used to measure how much heat is transmitted through the
+
+Failure Zone
+(f)= To + otan()
+Q3
+3
+SafeZone
+D2B
+4b
+3
+Q1Radial Crack
+Inner Debonding
+Outer debonding
+Shear cracksshear stress
+Failure envelope for
+intact rock
+Failure envelope for
+pre-existing fracture
+tdp
+initial stress
+dp2
+Normal effective stress
+after injection
+stressPetroleum & Petrochemical Engineering Journal
+4
+Nguyen V, et al. Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A
+Literature Review. Pet Petro Chem Eng J 2021, 5(3): 000269.
+Copyright© Nguyen V, et al.
+wellbore materials. The thermal expansion measure how
+much cement would experience shrinkage or swell due to
+the change in temperature.
+Experimental and numerical models have been
+used to study the thermal effect on wellbore integrity.
+The experiments were performed by Shadravan A, et al.
+[21] by applying different pressure on the casing at high
+temperatures. Teodoriu C, et al. [22] designed a ring similar
+to a cement sample which was exposed to various internal
+pressure at high temperatures. Boukhelifa L, et al. [23]
+performed studies about sealants under different wellbore
+conditions, including the pressure, temperature, and
+geometry changes. The cracks tending to be perpendicular
+were observed. It is in accordance with the theory. The
+casing radial displacement is more prominent than its
+axial displacement, so the possibility for cement sheath
+cracking to be orthogonal with its radius is high. One of
+the most significant constraints is the mechanical loading
+corresponding to the temperature changes, which has not
+been fully investigated. Furthermore, the thermal property of
+the material was not taken into serious account. Todorovic J, et
+al. [24] revealed that water saturation is a critical parameter
+to damage wellbore integrity in harsh cooling conditions.
+This concludes that during the CO2 injection, the possibility
+for cement and formation failure would surge. Aursand P, et
+al. [25] used a mathematical model to couple two-phase flow
+of CO2 and radial heat transfer between the CO2 flow and the
+well geometry. The study shows that the most considerable
+downhole temperature variations take place in the bottom
+part of the well. It also states that the parameters such as
+injection temperature, injection flow rate, injection duration,
+and downtime would affect the thermal stress leading to the
+damage of wellbore integrity. Lund H, et al. [26] introduced
+a heat-conduction model to compute the radial heat transfer
+from the well to the casing, annular seal, and rock formation.
+The model reveals that displacing cement with an annular
+sealant material with higher thermal conductivity would
+reduce the temperature variation between the casing/seal
+interface and the seal/rock interface. Ruan B, et al. [27]
+set up a two-dimensional radial wellbore flow model and
+solve the mass equation, momentum equations, and energy
+equation to investigate the thermal behavior of CO2. The
+study showed the temperature profile along the radial and
+axial direction, which is crucial information to predict the
+thermal stresses along the casing pipe and the outer cement.
+Lavrov A, et al. [28] initiated the study, which investigated
+that the most sensitive part of tensile cracking during CO2
+injection is cement adjacent to the casing pipe. The study
+suggested that reducing stiffness and increasing the thermal
+conductivity of damaged materials would inhibit the number
+of tensile cracks.
+The thermal stress is proportional to Young’s modulus
+E(MPa), and the linear thermal expansion coefficient α;
+inversely proportional to Poisson’s ratio ν. The relationship
+between them is shown in Equation 10:
+1
+T
+E T
+α
+σ
+ν
+∆
+∆
+=
+−
+ (10)
+
+
+Figure 4: Growth of the minimum in situ stress and reservoir pressure compared to the bottom hole pressure (BHP) during
+CO2 injection [16].
+
+Minimum in situ stress, reservoir pressure, and BHP in the injection well during COz injection
+220
+Min hor. stress.Expected case
+Averagereservoirpressure
+200
+Min hor.stress.Low case
+Shut in
+180
+BHP high injection rates
+Min hor. stress. Low case + thermal effects
+160
+BHP for low injection rates
+Stress (bar)
+140
+120
+100
+80
+60
+40
+20
+0
+2
+3
+4
+5
+6
+8
+Time (years)Petroleum & Petrochemical Engineering Journal
+5
+Nguyen V, et al. Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A
+Literature Review. Pet Petro Chem Eng J 2021, 5(3): 000269.
+Copyright© Nguyen V, et al.
+To illustrate the thermal effects on the stability of
+wellbore, the simulation was performed at sandstone depth
+of 1300m and the temperature difference of 20oC between
+CO2 and reservoir rock. As shown in Figure 4 for the high
+injection rate, the minimum horizontal stress with low case
+and thermal effects is lower than the bottom hole pressure.
+It signifies that the fracture would take place.
+Coefficient of thermal expansion
+10-5 (1/oC)
+Poisson’s ratio
+0.2-0.21
+Modulus of elasticity
+14-41 Gpa
+Table 2: Typical properties of Portland cement concrete
+[29].
+The typical properties of Portland cement concrete
+were presented in Table 2. The values of thermal expansion
+coefficient and Poisson’s ratio are small while big for
+modulus of elasticity. Therefore, based on Equation 10,
+reducing the cement elastic modulus is an effective way to
+decrease thermal stress. The relationship between thermal
+stress and elastic modulus was also studied thoroughly in
+Thiercelin, et al. [30]. This study concluded that increasing
+the temperature change would advance the thermal stress
+and a low cement elastic modulus would adapt better than
+a high one.
+Chemical Effect
+The injection of CO2 also affects rock stability in terms
+of chemical effects. Previous studies were carried out to
+demonstrate the chemical effects. One of them can be found
+in Orlic B, et al. [16]. The Permian Zechstein formation was
+selected for investigating the effect of CO2 storage. The design
+composed of circular elements of rosettes (60% volume) in
+the size of millimeters implanted in a matrix of 3 types of
+anhydrite circular element with the size of 50-83 micrometer.
+The conditions for simulation are vertical stress= 50 MPa,
+horizontal stress=40 MPa, T=80oC for 50,000 years. The
+reaction of anhydrite with CO2 and water is in Equation 11:
+
+4
+2
+2
+3
+2
+4
+CaSO
+CO
+H O
+CaCO
+H SO
++
++
+→
++
+ (11)
+According to the authors, the composition of anhydrite
+caprock was selected from the Permian Zechstein, which was
+already studied by Hangx SJT, et al. [31], in order to compare
+the results. The condition for vertical stress, horizontal
+stress, and temperature for this model has emulated the
+condition for caprock buried at 2.5 km depth.
+The anhydrite failure strength was significantly reduced
+by 25% over the course of 50,000 years in a CO2-rich
+environment. For 1000 years, the anhydrite failure strength
+reduction is inconsiderable. These results match with Hangx
+SJT, et al. [31].
+For injecting CO2 into a deep aquifer and depleted
+oil reservoir, CO2 dissolves into brine to create carbonic
+acid, H2CO3, then reacts with rock formation (carbonate).
+The chemical reactions are presented in Equation 12 and
+Equation 13. The dissolution of rock by carbonic acid causes
+the rock properties changes such as geomechanical and
+petrophysical, which were investigated in Kim K, et al. [32];
+Charalampidou EM, et al. [33]; Luquot L, et al. [34]; Rohmer
+J, et al. [35]; Bemer E, et al. [36]; Vanorio TV, et al. [37], Iyer
+J, et al. [38]; Dávila G, et al. [39]. These studies revealed that
+injecting CO2 would increase the rock porosity and decrease
+the elastic moduli.
+2
+2
+3
+CO
+H O
+H
+HCO
++
+−
++
+↔
++
+ (12)
+2
+3
+3
+MCO
+H
+M
+HCO
++
++
+−
++
+↔
++
+ (13)
+Where M is metal such as Ca, Mg.
+Tang Y, et al. [40] carried the dynamic and static
+experiments to interpret the impact of CO2-brine- rock
+interaction in gas reservoir with an aquifer. The results
+discovered that CO2 brine-rock interaction takes place in
+both gas zone and water zone because water vaporizes
+into gas zone to contact with CO2 to form carbonic acid. Six
+cores representing three reservoir types that are different in
+length, diameter, porosity, and permeability, were selected
+to perform the investigation. In general, the core porosity
+would be increased, and the core permeability could be
+decreased, as presented in Figure 5 and Figure 6. It can be
+demonstrated by mineral dissolution and particle migration
+in the pore space. Mineral dissolution induces the increasing
+of rock porosity, and particle migration in the pore space
+retards the flow resulting in the decreasing of rock
+permeability. However, two irregular cases were observed.
+The porosity of core #1 decreases, and the permeability of
+core #6 increases. It can be explained by the characteristics
+of core # 1 and # 6. The pore diameter in core #1 is small, so
+minerals dissolution cannot be driven out easily, causing the
+decreasing of porosity. In contrast, the pore size of core #6 is
+large, the free grains move out the pore smoothly, resulting in
+the decreasing of rock permeability.
+
+
+Petroleum & Petrochemical Engineering Journal
+6
+Nguyen V, et al. Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A
+Literature Review. Pet Petro Chem Eng J 2021, 5(3): 000269.
+Copyright© Nguyen V, et al.
+Figure 5: The difference of porosity before and after experiment [40].
+ Figure 6: The difference of permeability before and after experiment [40].
+There are still several studies Santra A, et al. [41]; Ojala
+IO [42]; Morris JP, et al. [43]; Karimnezhad M, et al. [44])
+which were discussed in detail the CO2-injection induced
+the changes of mechanical, chemical, and thermal property.
+These studies revealed that the tensile strength is a rock
+porosity dependence. Particularly, increasing porosity would
+decrease the tensile strength by an exponential function.
+Corrosion
+The corrosion costs billions of dollars annually and
+damages our environment due to the leakage of unwanted
+fluid. An understanding of the process is very crucial to
+reduce costs and handle the process. There are two different
+corrosion mechanism that occur in the wellbore cement: the
+first due to electrochemical reaction and the second caused
+by the carbonation reaction [45,46].
+Electrochemical Corrosion Mechanism
+This is metal corrosion. The oxidation-reduction reactions
+take place at anode and cathode. Particularly, the anodic
+reaction (Equation 14) is the oxidation of iron and the
+cathodic reaction (Equation 15) is hydrogen evolution.
+Anode:
+2
+2
+Fe
+Fe
+e
++
+−
+↔
++
+ (14)
+Cathode:
+2
+2
+2
+H
+e
+H
++
+−
++
+→
+ (15)
+
+0.35
+0.3
+0.25
+orosity
+0.2
+0.15
+0.1
+0.05
+0
+1
+2
+3
+4
+5
+6
+Core #
+pre_experiment
+after experiment70
+09
+50
+40
+30
+20
+10
+0
+1
+2
+3
+4
+5
+6
+Core #
+pre experiment
+after experimentPetroleum & Petrochemical Engineering Journal
+7
+Nguyen V, et al. Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A
+Literature Review. Pet Petro Chem Eng J 2021, 5(3): 000269.
+Copyright© Nguyen V, et al.
+Several researchers have carried out studies to inspect
+the metal corrosion rate. Nevertheless, the corrosion process
+under high pressure of CO2 (above the critical point at 7.38
+MPa at 31.1oC as seen in Figure 7) needs to be investigated
+further.
+
+Figure 7: Carbon dioxide phase diagram [47].
+Russick EM, et al. [48] carried the experiment on
+stainless steels (304L and 316), copper (CDA 101), aluminum
+alloys (2024, 6061, and 7075), and carbon steel (1018) in
+contact with pure supercritical CO2, water-saturated CO2, the
+mixture of supercritical CO2 with 10 wt% of methanol, and
+supercritical CO2 with 4 wt % of tetrahydrofurfuryl alcohol
+(THFA) at 3500 psi and 50 oC. No sign of corrosion on any
+metal was observed when they are in contact with pure
+supercritical CO2. For water-saturated CO2, only carbon steel
+1018 was sensitive, while the others were not influenced. The
+copper CDA 101 and aluminum 2024 got corrosive with the
+combination of supercritical CO2 and 10 wt% of methanol.
+The mixture of supercritical CO2 with 4 wt% of THFA almost
+did not cause corrosion on any metal. The THFA comprise
+an organic additive, Polygard, which acts like a corrosion
+inhibitor.
+Seiersten M, et al. [49] studied the impact of the pressure
+of CO2 up to 80 bar and temperature up to 50oC to corrosion
+rate of carbon steel X65. The study presented that dry CO2
+and non-CO2 saturated with water did not cause corrosion to
+carbon steel. At 50oC in the systems consisting of only water,
+the corrosion rate positively correlates with CO2 partial
+pressure. The corrosion rate reaches the maximum value of
+6.9 mm/year at 40 bar. Seiersten M [50] revealed that at 4
+oC, increasing the CO2 partial pressure would decrease the
+corrosion rate. The maximum value for corrosion rate is
+about 5.6 mm/year at 10 bar. The difference between the two
+observations above can be demonstrated by the formation
+of the film FeCO3. At 40oC, increasing CO2 partial pressure
+would decrease the saturation, which leads to non-creation
+of film. In contrast, at 50 oC the solution saturation has a
+positive correlation with CO2 partial pressure.
+
+Choi YS, et al. [51] investigated the behavior of carbon
+steel under the CO2- saturated water phase and the water-
+saturated CO2 phase with and without the presence of
+oxygen. The research exhibited that oxygen would make the
+corrosion rate faster. The presence of oxygen would inhibit
+the formation of the defensive film layer FeCO3, which leads
+to an increase in the corrosion rate. The increasing corrosion
+rate with the presence of oxygen is also demonstrated by
+the oxidation-reduction reaction mechanism. Oxygen acts
+as an oxidizing agent, which would make the redox reaction
+between iron, oxygen, and water happen. The SEM and EDS
+techniques also were conducted to justify the results.
+
+Lin G, et al. [52] examined the influence of CO2 at
+different temperatures and pressure in autoclaves on three
+types of carbon steel N80, P110, and J55. At 6.89 MPa and
+90oC, the corrosion rate are 1.752 mm/y, 2.403 mm/y, and
+1.854 mm/y for N80, P110, an J55, respectively. On the other
+hand, at higher pressure 10.34 MPa and 90 oC, those values
+seem likely to decrease. They are 0.922 mm/y, 1.054 mm/y,
+1.105 mm/y. All values were plotted in Figure 8. The average
+decreasing percentage of corrosion rate for N80, P110, and
+J55 is 95 %, and the most decreasing corrosion rate is for
+P110 with 127%. P110 is also the most corrosive steel in
+this study. The illustration to explain the most corrosive
+characteristic of P110 lies in its composition as presented
+in Table 3. Steel P110 contains the most manganese among
+three types of steel N80, P110, and J55, and manganese is a
+very strong oxidizing agent. Therefore, the corrosion rate on
+P110 is almost higher comparing to N80 and J55.
+Steel Grade
+C
+Si
+Mn
+P
+S
+Cr
+Mo
+Ni
+Ti
+Cu
+N80
+0.24
+0.22
+1.19
+0.013
+0.004
+0.036
+0.021
+0.028
+0.011
+<0.019
+P110
+0.26
+0.2
+1.4
+0.009
+0.003
+0.15
+0.01
+0.12
+0.03
+<0.01
+J55
+0.19
+0.31
+1.39
+0.014
+0.004
+0.19
+0.092
+0.017
+0.04
+<0.01
+Table 3: Steel composition (wt%) [33].
+
+10
+6
+Supercritical
+phase
+8
+Liquid
+phase
+7
+Solid
+P
+Critical point
+Phase
+Pressure
+6
+Gas phase
+P
+4
+3
+2
+1
+Triple point
+-100-80-60-40-20
+204060
+Temperature(C)Petroleum & Petrochemical Engineering Journal
+8
+Nguyen V, et al. Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A
+Literature Review. Pet Petro Chem Eng J 2021, 5(3): 000269.
+Copyright© Nguyen V, et al.
+Figure 8: Corrosion rate for N80, P110, J55 on different conditions [52].
+Cement Corrosion Mechanism
+Understanding the cement corrosion during the CO2
+geological storage is necessary to handle the process correctly.
+This includes the CO2 leakage paths and the carbonation time.
+Recognizing all possible leakage paths would aid in looking
+for a reasonable solution to cracking problems. Interpreting
+the carbonation time could contribute to evaluating the
+safety of the process for an extended period.
+CO2 Leakage Path: There are many leakages pathways for
+CO2. It could be an interface formation-cement or cement-
+casing, or from cement cracking as shown in Figure 9.
+Figure 9: The possible leakage paths of CO2 [53].
+Estimated Time for Carbonation Process: Many research
+groups have predicted the carbonation times of cement
+exposed to CO2 by using experimental and mathematical
+models. The carbonation is a substantial variable to evaluate
+the quality of the Carbon Capture Storage. Therefore, it
+is essential to obtain it. The experimental and numerical
+methods have their advantages and limitations. The empirical
+model in some situations will produce inaccurate results due
+to equipment error, and occasionally it is hard to manage the
+procedure properly due to external conditions beyond our
+observation, and most likely, the cost to do experiments is
+more expensive and time-consuming. In contrast, on average,
+the numerical model is much easier to do. It will eliminate
+the errors from equipment and human. It will produce the
+result faster. The most important thing for using a numerical
+model is to set up a governing equation with correct initial
+and boundary conditions. Although showing the limitation
+on both experimental and numerical models, they are used
+to verify one another. That is a reason why it is necessary to
+study both.
+Experimental Measurements: Duguid A [53] designed an
+experiment to forecast the time to deteriorate the cement
+sheath in a well exposed to carbonated brine. The samples
+were created by drilling the stone cylinder 55 mm in diameter,
+10 mm in height with a 25-mm axial hole. The minimum
+depth from the outside of the cylinder to the boundary of
+the hole was 3mm. According to Duguid A [53], the depth
+of reaction was quantified at five different locations: 0, 45,
+90,135,180 degrees as presented in Figure 10 in order to
+evaluate how different the surface cement would react with
+carbonate brine.
+The relationship between carbonation depth and time1/2/
+radius is linear in most cases. The most linear relationship is
+at the condition pH=3 and T=50 oC, and this condition also
+causes the most carbonation for cement. It is in agreement
+with Duguid A, et al. [54]. The prediction time for 25 mm
+cement sheath to be deteriorated is approximately from
+30,000 to 70,000 years if the favorable cement is selected
+and the good cementing job is done.
+
+J55
+Type
+Steel
+P110
+At 10.34 Mpa,90oC
+At 6.89 Mpa,90oC
+N80
+0
+0.5
+1.5
+2
+2.5
+3
+Corrosion rate(mm/year)Cementlayer
+Formation
+Steel
+Casing
+Cement
+Plug
+Cement
+cracking
+path
+Cement-formation interface
+Cement-casing interface
+leaking
+leakingPetroleum & Petrochemical Engineering Journal
+9
+Nguyen V, et al. Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A
+Literature Review. Pet Petro Chem Eng J 2021, 5(3): 000269.
+Copyright© Nguyen V, et al.
+Figure 10: The top view of sample of experiment [53].
+Kutchko BG [55] showed the reaction rate when the
+cement was exposed to supercritical CO2 and CO2-saturated
+brine. Supercritical CO2 is a separate free phase causing
+hydrodynamic trapping. On the other hand, some would
+dissolve in the brine existing in CO2 saturated brine form,
+which induces the solubility trapping. The cement samples
+were embedded in 1% NaCl at 30.3 MPa and 50 oC under
+static conditions. The estimation for penetration depth is
+1±0.07 mm for the CO2 -saturated brine and 2.9±0.89 mm
+for the supercritical CO2 after 30 years. It indicated that the
+supercritical CO2 would degrade Portland cement faster than
+the CO2 saturated brine. This is comprehensible because the
+condition of supercritical CO2 is at high pressure and high
+temperature. The penetration depth over time for the CO2-
+saturated brine and supercritical can be found in Figure 11.
+
+Figure 11: The penetration depth over time when cement
+exposed to CO2 Saturated brine and Supercritical CO2 [55].
+Zhang L, et al. [56] introduced another approach to
+estimate the penetration depth over time. Fick’s diffusion and
+Elovich’s equation were fit to experimental data. Elovich’s
+equation (Equation 16) can be shown by Allen JA, et al. [57]
+and Kutchko BG, et al. [58].
+(
+)
+*
+dL
+a exp
+bL
+dt =
+−
+ (16)
+Where L is the penetration depth(mm) at time t (days) of
+exposure and a, b are constants decided from experiment
+data.
+Integrating the Equation 16 above with respect to t
+yields Equation 17:
+( )
+(
+)
+1
+1
+
+L
+ln t
+ln ab
+b
+b
+=
++
+ (17)
+Where a, b can be estimated to fit the data a=2.47, b=22.08
+Estimation of penetration depth with Elovich’s equation
+is more accurate than Fick’s diffusion as shown in Figure 12.
+Elovich’s equation has been used in several kinetic studies
+[59-61]. However, the outcomes reaffirmed Elovich’s equation
+as a powerful method to measure the CO2 penetration depth.
+Figure 12: Fitting data with Fick’s law(left) and Elovich’s
+equation (right) [56].
+In Duguid A, et al. [54], the experiment for limestone
+and sandstone- like condition were executed. Class H cement
+pastes were exposed to temperatures from 20 to 50oC and pH
+2.4 to 5. Then the samples were interpreted by using multiple
+techniques such as Inductively Coupled Plasma Optical
+Emission Spectroscopy (ICP-OES), optical microscopy, X-ray
+diffraction, and Electron Probe Microanalysis (EPMA). The
+experimental model was designed as presented in Figure 13.
+The CO2 air was percolated into the carbonated brine and
+then pumped to the reactor vessel, which contains the cement
+sample. The purpose of the recirculated flow is to assure that it
+was saturated with CaCO3 before reaching the reaction vessel.
+No observable degradation in the limestone-like condition
+was observed. Under the sandstone-like condition, there are
+
+180°
+Sandstone sample
+135°
+90°
+Cement
+class H
+45°
+0°0.9
+0.8
+y=0.04824x, Fick's law
+Penetration Depth (mm)
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+0
+5
+10
+15
+20
+Time0.5 (Day ^0.5)
++cO2SaturatedBrine
+Supercriticalco20.6
+Penetration depth (mm)
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+0
+20
+40
+60
+80
+100
+Exposuretime (Days)
+Experimentdata
+Fick'slawfit
+ElovichPetroleum & Petrochemical Engineering Journal
+10
+Nguyen V, et al. Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A
+Literature Review. Pet Petro Chem Eng J 2021, 5(3): 000269.
+Copyright© Nguyen V, et al.
+5 distinct layers that appeared: orange, brown, white, gray,
+and core. Each layer would expose the different behavior
+to carbonated brine. The orange and brown part display a
+leached region. The white layer shows a carbonated region.
+The gray section depicts a calcium hydroxide dissolution
+region, and the core section is no change. The outer layer
+was degraded fully at pH=2.4, 3.7, and temperatures 20oC
+and 50oC. The sample got the most damage at pH=2.4 and
+T=20oC, and the least degradation occurred at pH= 5 and
+T=50oC. This conclusion is illustrated by the dissolution of
+carbon dioxide in water. The carbon dioxide solubility in
+water increases with decreasing temperature, and more
+carbonic acid is formed. Therefore, the most degradation
+was occurred at pH=2.4 and temperature 20 oC.
+Figure 13: Experimental system for sandstone like
+condition (top) and limestone like condition(bottom)
+reactor [54].
+Carey JW, et al. [62] and Carey JW, et al. [63] investigated
+the behavior of wellbore integrity and CO2-brine flow along
+the casing-cement micro annulus. The core-flood examination
+was performed at 40oC, 14 MPa pore pressure, and 28 MPa
+confining pressure. The experimental system included a 10
+cm length of limestone with a combination of rectangular
+steel embedded in the cement. The blended solution, 50%
+supercritical CO2 and 50% brine, was run through limestone/
+cement combination. There are two corrosion processes:
+steel corrosion and cement corrosion. The corrosion on steel
+occurs by the electrochemical mechanism. The film FeCO3
+layer was formed from the CO2-rich fluid in contact with steel.
+The film layer protected the steel from deeper penetration of
+the CO2-rich fluid. However, the solubility of the FeCO3 layer
+would increase if the pH decreases. As a result, corrosion
+rate is increased, with increasing flow rate of the CO2-rich
+fluid. For cement degradation, the rate is dependent on
+cement properties and the flow rate of CO2 -rich fluid. The
+diffusion coefficient was discovered in the interval from 10-
+12 to 10-10 cm2/sec by assuming a 1D diffusion problem
+with characteristic diffusion time 2√Dt and penetration
+depth from 50-250μm.
+Adeoye JT, et al. [64] carried out experiments of a novel
+engineered cementitious composite (ECC) exposed to CO2-
+saturated water under static and flow conditions at 10 MPa
+and 50oC. The depth of alteration estimated was 72 mm over
+50 years by Fick’s law. In a similar study was performed
+by Kutchko BG, et al. [65] at 15 MPa and 50oC, the depth of
+alteration was 224 mm over 50 years. The higher pressure of
+CO2 in the study Kutchko BG, et al. [65] was not probably the
+main reason to lead to a 3 three times increase of carbonation
+depth. Nonetheless, the exciting finding lay on the pozzolan
+to cement ratio. The material used in the study Kutchko BG,
+et al. [65] has the pozzolan to cement ratio of 65:35, while
+this ratio in the study Adeoye JT, et al. [64] is 45:55. Thus, the
+increase of the pozzolan would lead to faster carbonation.
+Mathematical Prediction: Along with experimental works,
+there are very few studies using the robust mathematical
+model to predict the penetration depth during geological
+storage CO2. The mathematical model may accompany
+experimental work, which will conserve resources compared
+to purely experimental work. Below are some remarkable
+studies which have been done so far.
+Tao Q, et al. [66,67] developed a mathematical model to
+investigate some relationships during geological CO2 storage.
+The leaking pathway of CO2 includes two parts, as shown in
+Figure 14a: the bottom due to cement degradation, the top
+is the water-saturated porous medium where an assumption
+of no resistance was made. Figure 14b showed how the
+pressure of the reservoir changes during CO2 injection. The
+pressure of the reservoir increased as CO2 was injected and
+decreased after the injection was stopped.
+
+Figure 14: a) Two different leaking pathways. b) Deviated
+reservoir pressure during CO2 injection [67].
+If there is only buoyancy force causing CO2 flow, the
+potential gradient of CO2 is calculated by Equation 18:
+
+Reacted brine out
+CO2
+Pump
+Cement sample
+Carbonate brine
+Reacted brine out
+CO2
+Calcite
+dund
+Cement sample
+Carbonate brineLeakage path due
+to water saturated
+p_max
+porousmedium
+pressure
+Earth
+Reservoir
+Leakage
+Ap
+path due
+to cement
+defect
+p_hydro
+Cement
+CO2
+t begin
+t_end
+t after
+plume
+Time
+(a)
+(b)Petroleum & Petrochemical Engineering Journal
+11
+Nguyen V, et al. Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A
+Literature Review. Pet Petro Chem Eng J 2021, 5(3): 000269.
+Copyright© Nguyen V, et al.
+(
+)
+gz
+ϕ
+ρ
+∆
+∇
+= ∇
+ (18)
+Where ∆ρ is the density difference between H2O and CO2
+z and g are the depth and gravitational, respectively.
+During the injection period, both buoyancy and pressure
+elevation contribute to drive CO2. Therefore, the potential
+gradient of CO2 is computed by Equation 19:
+(
+)
+c
+gz
+p
+ϕ
+ρ
+∇
+=
++ ∇
+∆
+∇
+ (19)
+Where pc is the capilary pressure
+Several wells were investigated during the geological
+CO2 storage. The CO2 flux was discovered to be responsive
+to pressure elevation and the leakage depth. The CO2 flux
+decreases while increasing the leakage depth due to the
+increasing of CO2 density. The relationship between the CO2
+leakage flux and injection pressure at shallow leak (4000 ft)
+and deep leak (10000 ft) was investigated. The CO2 leakage
+flux increases with increasing injection pressure, but the
+rates are different. The CO2 leakage flux rate is more at deep
+leak than it does at the shallow leak. According to the author,
+this is because the injection pressure would overcome the
+buoyancy pressure at a shallow leak. In contrast, at a deep
+leak, the buoyancy pressure decreases more gradually due to
+the increasing CO2 density.
+
+Figure 15: The Calcite solubility diagram [68].
+
+Deremble L, et al. [68] simulated the evolution of layers
+under CO2 -rich brine flow. The development of layer calcite
+or silica gel will be proportional, while the flux of calcium or
+CO2 is inversely proportional to the square root of time. An
+increase in CO2 flow rate increases calcium dissolution rate
+until calcite equilibrium is reached at point λ, as presented in
+Figure 15. Then ion calcium cannot be discharged anymore.
+As a result, CO2 would react with other species in the cement
+until it approaches point δ where all species have gone,
+and then the mathematical model takes into consideration
+the additional physical aspects including: micro-annulus
+geometry, the Peclet number, and the characteristic length
+scales of a defect. The model uses the implicit algorithm to
+solve for a solution. It also affirms that the penetration depth
+is proportional to the square root of time.
+Huet BM, et al. [69] connected the geochemical and
+transport module to simulate the degradation of cement
+during geological CO2 storage. The Dynaflow was adopted
+to solve a non-linear system of partial differential equations.
+Both Galerkin finite element and vertex centered finite
+volume space discretization of transport equations were
+executed. An implicit backward finite difference time stepping
+of the transport equation was applied to produce the results.
+The effective diffusion coefficient needs to be assumed in
+order of 10-11 m2/s by Bentz DP, et al. [70]. The thickness
+of the calcite layer is also proportional to the square root
+of time. The difference between experimental data and the
+model occurred due to the diffusion coefficient estimated.
+The growth of calcite carbonate concentration over time
+as presented in Figure 16 would provide insight into the
+transport of CO2 to cement. The carbonate concentration
+increases very rapidly while its radius decreases with
+time elapsed. There are two distinct regions with different
+transport mechanisms. In the first region with time exposure
+less than 60 days, carbonate species (CO2, HCO3-) disperse
+into the sample through the calcite and the silica gel layer.
+The second region is where the calcite layer dissolves.
+
+Figure 16: Carbonate concentration profile over time [69].
+Also, well integrity has been investigated extensively in
+Hawkes C, et al. [71]; Hawkes CD, et al. [72]; Scherer GW, et
+al. [73]; Neuville N, et al. [74] to evaluate for long-term CO2
+storage. The downhole testing programs, such as cement
+sheath pressure transient testing, mini-frac testing, cement
+
+Calcite equilibrium curve
+[Ca?+]
+[CO2]
+0
+Calcite solubility diagram6x101
+a. 1 day
+b. 7 days
+Concentration (mmol/L)
+c. 15 days
+5x101
+d. 30 days
+e. 60 days
+6
+e
+p
+c
+b
+4x101
+f. 120 days
+g. 180 days
+3x101
+2x101
+101
+0
+0
+1
+2
+3
+4
+Radius (mm)Petroleum & Petrochemical Engineering Journal
+12
+Nguyen V, et al. Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A
+Literature Review. Pet Petro Chem Eng J 2021, 5(3): 000269.
+Copyright© Nguyen V, et al.
+sampling, and fluid sampling, were performed.
+Outlook
+The traditional Portland cement carries many advantages
+such as low cost, high compressive strength, low alkali
+content, and long-term stability. Nevertheless, cement is
+sensitive in an acidic environment, and the cement industry is
+a source of CO2 emissions. Therefore, Portland cement is not
+the most excellent sealant material to serve in CCS project.
+Based on the nature of CCS project, the most optimal cement
+could prevent corrosion under the acidic environment. Also,
+the cement should possess low permeability, porosity, and
+high mechanical strength.
+Mahmoud AA, et al. [75] proposed to add Synthetic
+Polypropylene Fiber (PPF) into Class G cement to improve it.
+Four samples with 0%(PPF0), 0.125%(PPF1), 0.25%(PPF2),
+0.375%(PPF3) of PPF were arranged for the experiment.
+The results of this study showed that the carbonation depth
+and carbonation rate decreased while compressive strength
+and tensile strength increased, as presented in Figures
+17-20, respectively. The decreasing of carbonation depth
+and carbonation rate indicated the reduction of cement
+permeability.
+
+ Figure 17: Carbonation Depth after experiment 10 days [75].
+
+Figure 18: Carbonation Rate after experiment 10 days [75].
+
+
+3000
+2586
+2500
+ Depth(μm)
+2182
+2000
+1842
+1588
+Carbonation
+1500
+1000
+500
+0
+PPFO
+PPF1
+PPF2
+PPF3300
+259
+250
+Carbonation Rate(μm/Day)
+218
+200
+184
+11
+159
+150
+100
+50
+0
+PPFO
+PPF1
+PPF2
+PPF3Petroleum & Petrochemical Engineering Journal
+13
+Nguyen V, et al. Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A
+Literature Review. Pet Petro Chem Eng J 2021, 5(3): 000269.
+Copyright© Nguyen V, et al.
+Figure 19: Compressive strength after experiment 10 days [75].
+
+Figure 20: Tensile Strength after experiment 10 days [75].
+Nanomaterial such as nano-silica (nano SiO2) [8],
+nano-alumina (nano Al2O3) [76], nano-titanium dioxide
+(nanoTiO2) [9], carbon nanotubes (CNTs) [77], Polymer/
+clay nanocomposites [78], nanoglass flake (NGFS) [79] were
+considered as good additive to improve cement quality
+because of their large surface area and reactivity. The
+nanomaterials can solidify the cement microstructure and
+lessen the porosity, then advance the mechanical strength.
+Ponzi GGD, et al. [80] proposed basalt powder as an
+additive material in cement formulation because the basalt
+powder has low pozzolanic activity, large inert fraction,
+and small particle size. The basalt power plays a role as a
+filling-substance to the porous cement networks to curb
+the fluid intrusion. The experimental results explored that
+the formulation with low basalt powder content (≤ 0.5 w.%)
+exhibited more resistance to CO2 degradation, lower porosity
+and permeability, and stronger mechanical properties.
+Other sealant materials can replace traditional Portland
+cement, such as geopolymer cement, resins, biofilms
+barriers, foams [81]. Geopolymer, such as zeolites, was
+discovered to have better resistance with CO2- rich brine
+because it contains less calcium oxide than Portland cement
+does. Resins are particle-free fluids with low mobility, hard,
+rigid, and impermeable materials. They include phenolic,
+
+90
+81.9
+80
+71.7
+70
+8'S9
+61.1
+60
+50
+40
+30
+20
+10
+0
+PPFO
+PPF1
+PPF2
+PPF38
+6.86
+7
+Tensile Strength(MPa)
+5.75
+5.83
+6
+4.89
+5
+4
+3
+2
+1
+0
+PPFO
+PPF1
+PPF2
+PPF3Petroleum & Petrochemical Engineering Journal
+14
+Nguyen V, et al. Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A
+Literature Review. Pet Petro Chem Eng J 2021, 5(3): 000269.
+Copyright© Nguyen V, et al.
+epoxy, and furan resins. Biofilm sealants include urea,
+Ca2+, nutrient feed, and micro-organism. The principle is to
+accelerate calcium to form calcite and seal fractures. Foam
+is a gas-liquid blend, and it can block the flow rate of CO2 in
+porous media and increase the CO2 viscosity.
+Because carbon capture and storage projects are
+expanding, so the cement consumption is increasing. Those
+materials introduced above have many advantages, but
+the disadvantages still exist. Therefore, the new research
+direction should focus on improving the cement quality.
+For example, geopolymer is detrimental to human health.
+Thus, new components should be studied to make it become
+human friendly.
+Conclusions
+This study’s major goal was to review the previous papers
+for carbon capture and storage project. Many experiments
+have been performed to assess the well integrity and predict
+the time degradation of geological CO2 storage, but there are
+very few mathematical models. Research predicts 30,000-
+70,000 years for 25 mm cement to be carbonated. Some
+studies tried to match the experimental data to a particular
+equation and introduce that the penetration depth is
+proportional to the square root of time. Furthermore, the
+debonding interface between casing/cement or cement/
+formation is the primary cause for leakage of CO2. However,
+it has not been thoroughly investigated. Some substantial
+conclusions can be drawn from this review:
+•
+The need for a more accurate mathematical model to
+evaluate the well integrity and anticipate the corrosion
+rate during geological CO2 storage is very crucial.
+•
+A further investigation should identify the debonding
+interface between casing/cement or cement/formation
+issue and predict how the micro-annuli of case/cement
+or cement/formation behaves with the variation of
+temperature, stress, and chemical reactions during
+geological CO2 storage.
+•
+The diffusion coefficient is one of the most crucial
+parameters in the corrosion process. However, it has not
+been studied sufficiently in petroleum corrosion. Hence,
+it should be an interesting topic for future studies.
+•
+Improving the cementing property is one of the means
+to curb the corrosion rate. Future studies should
+investigate more how to reduce reactive species and add
+more inhibitors to advance Portland cement quality.
+If these tasks are done properly, it will clear up a
+considerable concern to make the operation more predictable
+and administered.
+Author Contribution
+Writing- original draft preparation, Nguyen V; writing-
+review and editing, Nguyen V, Olatunji O, Guo B and Ning Liu.
+All authors have read and agreed to the published version of
+the manuscript.
+Funding
+This research received no external funding
+Conflicts of Interest
+The authors declare no conflict of interest.
+References
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+Choolaei M, Rashidi AM, Ardjmand M, Yadegari A,
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+15
+Nguyen V, et al. Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A
+Literature Review. Pet Petro Chem Eng J 2021, 5(3): 000269.
+Copyright© Nguyen V, et al.
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\ No newline at end of file
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf,len=876
+page_content='Petroleum & Petrochemical Engineering Journal ISSN: 2578-4846 MEDWIN PUBLISHERS Committed to Create Value for Researchers Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A Literature Review Pet Petro Chem Eng J Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A Literature Review Nguyen V*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Olatunji O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Guo B and Ning Liu Department of Petroleum Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' University of Louisiana at Lafayette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' USA Corresponding author: Nguyen V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Department of Petroleum Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' University of Louisiana at Lafayette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' LA 70504,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' USA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Tel: 6787903709;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Email: vu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='nguyen1@louisiana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='edu Review Article Volume 5 Issue 3 Received Date: July 21, 2021 Published Date: July 28, 2021 DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='23880/ppej-16000269 Abstract Carbon capture and storage (CCS) has emerged as the most effective method to curb the CO2 concentration in the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' It can store up to 5 billion tons of CO2 per year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' To guarantee a safe and economical geological storage, the well cement degradation and wellbore integrity need to be studied thoroughly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' This review paper is designed to provide a fundamental background of well cement degradation and wellbore integrity in geological CO2 storages to support the researchers in further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The review mainly focuses on mechanical, thermal, chemical property changes and corrosion time for cement in experiments and simulation during geological CO2 storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' However, the debonding interface between casing/cement or cement/formation has not been addressed profoundly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' A further investigation should inspect how pressure, temperature, and chemical reaction affect the micro-annuli of casing/cement or cement/formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Also, a mathematical model should be established to predict the corrosion rate in geological CO2 storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Keywords: Geological;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Corrosion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' CO2 concentration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Storage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Well cement Introduction Portland cement is composed of four major components: tricalcium silicate (Ca3SiO5 or C3S), dicalcium silicate (Ca2SiO4 or C2S), tricalcium aluminate (Ca3Al2O6 or C3A), tetracalcium aluminoferrite (Ca4Al2Fe2O10 or C4AF) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The composition of Portland cement [2] can be found in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Corresponding to MacLaren DC [3], the Portland cement dissolves in water through the following reactions in Equation1 and Equation 2: ( ) 3 5 2 3 2 7 2 2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='3 3 Ca SiO H O Ca Si O H O Ca OH + → + (1) ( ) 2 4 2 3 2 7 2 2 2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='3 Ca SiO H O Ca Si O H O Ca OH + → + (2) An understanding of cement corrosion during the geologic storage of CO2 is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The mechanism of cement corrosion takes place in the following sequence of reaction: CO2 will first dissolve in Calcium hydroxide (CH) to form calcite [4] ( calcium carbonate) by the reaction in Equation 3: ( ) 2 3 2 2 CO Ca OH CaCO H O + → + (3) This reaction benefits the mechanical property of the wellbore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Specifically, calcite would enhance the mechanical strength of the cement layer and reduce porosity, which results in the permeability decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The decline of permeability characteristics will retard the possibility of CO2 leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Then the Calcium Silicate Hydrate (Ca3SiO7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='3H2O or C-H-S) in cement reacts with CO2 by Equation 4 MPetroleum & Petrochemical Engineering Journal 2 Nguyen V, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A Literature Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Pet Petro Chem Eng J 2021, 5(3): 000269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Copyright© Nguyen V, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' 3 2 7 2 2 3 2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='3 3 3 2 Ca Si O H O CO CaCO SiO H O + → + + (4) Cement quality is reduced when it is converted to bicarbonate in the presence of excess CO2 [5] in Equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' ( ) 2 3 2 3 2 CO CaCO H O Ca HCO + + → (5) Dissolution of the bicarbonate in the presence of water leaves the material more porous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' This reaction is a great issue to stop CO2 from leaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The challenge is how to control the bicarbonate reaction and predict the penetration depth of carbonation to protect the cement layer sheath from damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Several groups have been conducting experiments to enhance the trait of cement to inhibit the corrosion process [6-10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Others used the mathematical model to simulate how quickly CO2 would penetrate by diffusion mechanism through the cement layer [11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Some groups carried out an experiment by measuring the penetration depth versus time with support from techniques such as Secondary Energy Microscopy, X-Ray Diffraction, Scanning Electron Microscope (SEM)-Back Scattered Electrons (BSE) micrograph, and Energy Dispersive Spectroscopy (EDS) map, etc [13,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The penetration depth was found as a function of the square root of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' However, these studies are time consuming, expensive, and prone to error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Therefore, an accurate mathematical model is required to govern the diffusion process to reduce any drawback by experiment- induced and increase the answer’s reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The diffusion equation (Equation 6) by Fick’s second law will suit this requisition: 2 2 2 / ( ) / ( ) C t D C x ∂ ∂ = ∂ ∂ (6) Where C is the carbonate concentration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' t is the cement exposure time, D is the diffusion coefficient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' x is the penetration depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' ASTM type Description C3S C2S C3A C3AF CS2 I General purpose 55 17 10 7 6 II Moderate sulfate resistant 55 20 6 10 5 III High early strength 55 17 9 8 7 IV Low heat of hydration 35 40 4 12 4 V Sulfate resistant 55 20 4 12 4 (C=Cao, S=SiO2, A=Al2O3, and F= Fe2O2) Table 1: Composition of Portland cement (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Cement and Wellbore Property One of the crucial tasks of a wellbore is well cementing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The purposes of well cementing are to reinforce the casing strength, prevent undesirable formations away from the borehole, inhibit casing corrosion rate, constraint irregular pore pressure, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The most substantial function of cementing is to obtain zonal isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The other duty of cementing that is also important is to acquire a good bond between cement and pipe or cement and formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' However, the cement characteristics are changed over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' During CO2 storage, the three most common changes in the cement and in the rock that need to be concerned are mechanical, thermal, and chemical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Mechanical Property The pressure build-up due to CO2 injection may result in the poro-mechanical effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The change in pressure leads to a change in the stress field, which causes a mechanical impact on the fault and top seals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The mechanical defect on seals and faults is a reason not only for the migration of CO2 out of the well but also for leading ground movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The most prevalent criteria to evaluate the failure of rock and cement in geomechanics is Mohr-Coulomb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' This hypothesis assessed the failure of material based on the combination of normal and shear stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The failure line is computed by Equation 7: ( ) f o tan τ τ σ φ = + (7) Where: τf, τo, σ and ϕ are shear strength, cohesion, effective normal stress, and internal friction angle, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The normal and shear effective stresses are given in Equation 8 and Equation 9 and demonstrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The shear failure occurs as (σ,τ) hits or crosses the failure line fτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The tensile failure takes place as (σ, τ) reaches or crosses the shear axis, τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' ( ) 1 3 1 sin 2 2 τ σ σ β = (8) ( ) ( ) 1 3 1 3 1 1 2 2 2 cos σ σ σ σ σ β = + + − (9) where σ1,σ3 are the maximum and minimum principal stresses, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Petroleum & Petrochemical Engineering Journal 3 Nguyen V, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A Literature Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Pet Petro Chem Eng J 2021, 5(3): 000269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Copyright© Nguyen V, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Figure 1: Mohr- Coulomb failure criteria in a cylindrical sample (left) and Mohr circle (right) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Numerous failure mechanisms can take place in a cement sheath, such as inner debonding, outer debonding, radial cracks, shear cracks, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Figure 2 would interpret these mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Figure 2: The cement sheath failure mechanism [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Orlic B, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [16] examined the poro-mechanical effects for geological CO2 storage in a depleted gas field [17,18] and a deep saline aquifer in the Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Figure 3: Stress and pressure deviation because of CO2 injection in a deep saline aquifer [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Figure 3 presented the shear stress and the normal stress before and after CO2-injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The simulation was performed by increasing pressure dp1 and dp2 with dp2>dp1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' As seen, the Mohr-Coulomb circle shifted more to the left with increasing pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' It indicates that the normal and shear stress tended to reach closer to the failure line and shear stress axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Therefore, the failure may occur easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Li B, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [19] provided a model to assess the cement sheath integrity in the carbon sequestration fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Then the model was verified with Atkinson’s model and Carter’s finite element model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' A case study showed that that maximum injection pressure would be under-evaluated if cement sheath-induced stress is not examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' One of the conclusions is that the critical failure point lied at the cement-sandstone/ shale formation interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Omosebi O, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [20] performed an experiment to investigate the cement integrity during CO2 injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' An important conclusion was that class H cement turned into more elastic than class G cement with CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The differences in composition and mineralogy can explain it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Thermal Property The changes in temperature amid CO2 injection may induce stress variation, which can devastate the well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The cold injected CO2 in contact with warm wellbore, reservoir and the thermal properties difference between wellbore casing, lithology, and the cement will stress the vicinity of a wellbore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' As a result, the chances of creating more leakage paths are very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Thermal stress, which results in deterioration in cement sheath, relies on the thermo-mechanical properties of the material, the azimuthal extent, and radial position of fracture in addition to injection temperature and effective in-situ horizontal stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The thermal-mechanical properties consist of elastic modulus, thermal conductivity, and the thermal expansion coefficient of cement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Thermal conductivity is used to measure how much heat is transmitted through the Failure Zone (f)= To + otan() Q3 3 SafeZone D2B 4b 3 Q1Radial Crack Inner Debonding Outer debonding Shear cracksshear stress Failure envelope for intact rock Failure envelope for pre-existing fracture tdp initial stress dp2 Normal effective stress after injection stressPetroleum & Petrochemical Engineering Journal 4 Nguyen V, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A Literature Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Pet Petro Chem Eng J 2021, 5(3): 000269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Copyright© Nguyen V, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' wellbore materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The thermal expansion measure how much cement would experience shrinkage or swell due to the change in temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Experimental and numerical models have been used to study the thermal effect on wellbore integrity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The experiments were performed by Shadravan A, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [21] by applying different pressure on the casing at high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Teodoriu C, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [22] designed a ring similar to a cement sample which was exposed to various internal pressure at high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Boukhelifa L, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [23] performed studies about sealants under different wellbore conditions, including the pressure, temperature, and geometry changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The cracks tending to be perpendicular were observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' It is in accordance with the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The casing radial displacement is more prominent than its axial displacement, so the possibility for cement sheath cracking to be orthogonal with its radius is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' One of the most significant constraints is the mechanical loading corresponding to the temperature changes, which has not been fully investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Furthermore, the thermal property of the material was not taken into serious account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Todorovic J, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [24] revealed that water saturation is a critical parameter to damage wellbore integrity in harsh cooling conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' This concludes that during the CO2 injection, the possibility for cement and formation failure would surge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Aursand P, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [25] used a mathematical model to couple two-phase flow of CO2 and radial heat transfer between the CO2 flow and the well geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The study shows that the most considerable downhole temperature variations take place in the bottom part of the well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' It also states that the parameters such as injection temperature, injection flow rate, injection duration, and downtime would affect the thermal stress leading to the damage of wellbore integrity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Lund H, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [26] introduced a heat-conduction model to compute the radial heat transfer from the well to the casing, annular seal, and rock formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The model reveals that displacing cement with an annular sealant material with higher thermal conductivity would reduce the temperature variation between the casing/seal interface and the seal/rock interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Ruan B, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [27] set up a two-dimensional radial wellbore flow model and solve the mass equation, momentum equations, and energy equation to investigate the thermal behavior of CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The study showed the temperature profile along the radial and axial direction, which is crucial information to predict the thermal stresses along the casing pipe and the outer cement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Lavrov A, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [28] initiated the study, which investigated that the most sensitive part of tensile cracking during CO2 injection is cement adjacent to the casing pipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The study suggested that reducing stiffness and increasing the thermal conductivity of damaged materials would inhibit the number of tensile cracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The thermal stress is proportional to Young’s modulus E(MPa), and the linear thermal expansion coefficient α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' inversely proportional to Poisson’s ratio ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The relationship between them is shown in Equation 10: 1 T E T α σ ν ∆ ∆ = − (10) Figure 4: Growth of the minimum in situ stress and reservoir pressure compared to the bottom hole pressure (BHP) during CO2 injection [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Minimum in situ stress, reservoir pressure, and BHP in the injection well during COz injection 220 Min hor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='Expected case Averagereservoirpressure 200 Min hor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='Low case Shut in 180 BHP high injection rates Min hor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Low case + thermal effects 160 BHP for low injection rates Stress (bar) 140 120 100 80 60 40 20 0 2 3 4 5 6 8 Time (years)Petroleum & Petrochemical Engineering Journal 5 Nguyen V, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A Literature Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Pet Petro Chem Eng J 2021, 5(3): 000269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Copyright© Nguyen V, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' To illustrate the thermal effects on the stability of wellbore, the simulation was performed at sandstone depth of 1300m and the temperature difference of 20oC between CO2 and reservoir rock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' As shown in Figure 4 for the high injection rate, the minimum horizontal stress with low case and thermal effects is lower than the bottom hole pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' It signifies that the fracture would take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Coefficient of thermal expansion 10-5 (1/oC) Poisson’s ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='21 Modulus of elasticity 14-41 Gpa Table 2: Typical properties of Portland cement concrete [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The typical properties of Portland cement concrete were presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The values of thermal expansion coefficient and Poisson’s ratio are small while big for modulus of elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Therefore, based on Equation 10, reducing the cement elastic modulus is an effective way to decrease thermal stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The relationship between thermal stress and elastic modulus was also studied thoroughly in Thiercelin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' This study concluded that increasing the temperature change would advance the thermal stress and a low cement elastic modulus would adapt better than a high one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Chemical Effect The injection of CO2 also affects rock stability in terms of chemical effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Previous studies were carried out to demonstrate the chemical effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' One of them can be found in Orlic B, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The Permian Zechstein formation was selected for investigating the effect of CO2 storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The design composed of circular elements of rosettes (60% volume) in the size of millimeters implanted in a matrix of 3 types of anhydrite circular element with the size of 50-83 micrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The conditions for simulation are vertical stress= 50 MPa, horizontal stress=40 MPa, T=80oC for 50,000 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The reaction of anhydrite with CO2 and water is in Equation 11: 4 2 2 3 2 4 CaSO CO H O CaCO H SO + + → + (11) According to the authors, the composition of anhydrite caprock was selected from the Permian Zechstein, which was already studied by Hangx SJT, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [31], in order to compare the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The condition for vertical stress, horizontal stress, and temperature for this model has emulated the condition for caprock buried at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='5 km depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The anhydrite failure strength was significantly reduced by 25% over the course of 50,000 years in a CO2-rich environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' For 1000 years, the anhydrite failure strength reduction is inconsiderable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' These results match with Hangx SJT, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' For injecting CO2 into a deep aquifer and depleted oil reservoir, CO2 dissolves into brine to create carbonic acid, H2CO3, then reacts with rock formation (carbonate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The chemical reactions are presented in Equation 12 and Equation 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The dissolution of rock by carbonic acid causes the rock properties changes such as geomechanical and petrophysical, which were investigated in Kim K, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [32];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Charalampidou EM, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [33];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Luquot L, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [34];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Rohmer J, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [35];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Bemer E, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [36];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Vanorio TV, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [37], Iyer J, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [38];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Dávila G, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' These studies revealed that injecting CO2 would increase the rock porosity and decrease the elastic moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' 2 2 3 CO H O H HCO + − + ↔ + (12) 2 3 3 MCO H M HCO + + − + ↔ + (13) Where M is metal such as Ca, Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Tang Y, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [40] carried the dynamic and static experiments to interpret the impact of CO2-brine- rock interaction in gas reservoir with an aquifer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The results discovered that CO2 brine-rock interaction takes place in both gas zone and water zone because water vaporizes into gas zone to contact with CO2 to form carbonic acid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Six cores representing three reservoir types that are different in length, diameter, porosity, and permeability, were selected to perform the investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' In general, the core porosity would be increased, and the core permeability could be decreased, as presented in Figure 5 and Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' It can be demonstrated by mineral dissolution and particle migration in the pore space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Mineral dissolution induces the increasing of rock porosity, and particle migration in the pore space retards the flow resulting in the decreasing of rock permeability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' However, two irregular cases were observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The porosity of core #1 decreases, and the permeability of core #6 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' It can be explained by the characteristics of core # 1 and # 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The pore diameter in core #1 is small, so minerals dissolution cannot be driven out easily, causing the decreasing of porosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' In contrast, the pore size of core #6 is large, the free grains move out the pore smoothly, resulting in the decreasing of rock permeability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Petroleum & Petrochemical Engineering Journal 6 Nguyen V, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A Literature Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Pet Petro Chem Eng J 2021, 5(3): 000269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Copyright© Nguyen V, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Figure 5: The difference of porosity before and after experiment [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Figure 6: The difference of permeability before and after experiment [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' There are still several studies Santra A, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [41];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Ojala IO [42];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Morris JP, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [43];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Karimnezhad M, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [44]) which were discussed in detail the CO2-injection induced the changes of mechanical, chemical, and thermal property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' These studies revealed that the tensile strength is a rock porosity dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Particularly, increasing porosity would decrease the tensile strength by an exponential function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Corrosion The corrosion costs billions of dollars annually and damages our environment due to the leakage of unwanted fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' An understanding of the process is very crucial to reduce costs and handle the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' There are two different corrosion mechanism that occur in the wellbore cement: the first due to electrochemical reaction and the second caused by the carbonation reaction [45,46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Electrochemical Corrosion Mechanism This is metal corrosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The oxidation-reduction reactions take place at anode and cathode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Particularly, the anodic reaction (Equation 14) is the oxidation of iron and the cathodic reaction (Equation 15) is hydrogen evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Anode: 2 2 Fe Fe e + − ↔ + (14) Cathode: 2 2 2 H e H + − + → (15) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='25 orosity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='05 0 1 2 3 4 5 6 Core # pre_experiment after experiment70 09 50 40 30 20 10 0 1 2 3 4 5 6 Core # pre experiment after experimentPetroleum & Petrochemical Engineering Journal 7 Nguyen V, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A Literature Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Pet Petro Chem Eng J 2021, 5(3): 000269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Copyright© Nguyen V, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Several researchers have carried out studies to inspect the metal corrosion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Nevertheless, the corrosion process under high pressure of CO2 (above the critical point at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='38 MPa at 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='1oC as seen in Figure 7) needs to be investigated further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Figure 7: Carbon dioxide phase diagram [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Russick EM, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [48] carried the experiment on stainless steels (304L and 316), copper (CDA 101), aluminum alloys (2024, 6061, and 7075), and carbon steel (1018) in contact with pure supercritical CO2, water-saturated CO2, the mixture of supercritical CO2 with 10 wt% of methanol, and supercritical CO2 with 4 wt % of tetrahydrofurfuryl alcohol (THFA) at 3500 psi and 50 oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' No sign of corrosion on any metal was observed when they are in contact with pure supercritical CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' For water-saturated CO2, only carbon steel 1018 was sensitive, while the others were not influenced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The copper CDA 101 and aluminum 2024 got corrosive with the combination of supercritical CO2 and 10 wt% of methanol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The mixture of supercritical CO2 with 4 wt% of THFA almost did not cause corrosion on any metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The THFA comprise an organic additive, Polygard, which acts like a corrosion inhibitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Seiersten M, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [49] studied the impact of the pressure of CO2 up to 80 bar and temperature up to 50oC to corrosion rate of carbon steel X65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The study presented that dry CO2 and non-CO2 saturated with water did not cause corrosion to carbon steel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' At 50oC in the systems consisting of only water, the corrosion rate positively correlates with CO2 partial pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The corrosion rate reaches the maximum value of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='9 mm/year at 40 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Seiersten M [50] revealed that at 4 oC, increasing the CO2 partial pressure would decrease the corrosion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The maximum value for corrosion rate is about 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='6 mm/year at 10 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The difference between the two observations above can be demonstrated by the formation of the film FeCO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' At 40oC, increasing CO2 partial pressure would decrease the saturation, which leads to non-creation of film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' In contrast, at 50 oC the solution saturation has a positive correlation with CO2 partial pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Choi YS, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [51] investigated the behavior of carbon steel under the CO2- saturated water phase and the water- saturated CO2 phase with and without the presence of oxygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The research exhibited that oxygen would make the corrosion rate faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The presence of oxygen would inhibit the formation of the defensive film layer FeCO3, which leads to an increase in the corrosion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The increasing corrosion rate with the presence of oxygen is also demonstrated by the oxidation-reduction reaction mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Oxygen acts as an oxidizing agent, which would make the redox reaction between iron, oxygen, and water happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The SEM and EDS techniques also were conducted to justify the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Lin G, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [52] examined the influence of CO2 at different temperatures and pressure in autoclaves on three types of carbon steel N80, P110, and J55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' At 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='89 MPa and 90oC, the corrosion rate are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='752 mm/y, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='403 mm/y, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='854 mm/y for N80, P110, an J55, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' On the other hand, at higher pressure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='34 MPa and 90 oC, those values seem likely to decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' They are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='922 mm/y, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='054 mm/y, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='105 mm/y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' All values were plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The average decreasing percentage of corrosion rate for N80, P110, and J55 is 95 %, and the most decreasing corrosion rate is for P110 with 127%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' P110 is also the most corrosive steel in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The illustration to explain the most corrosive characteristic of P110 lies in its composition as presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Steel P110 contains the most manganese among three types of steel N80, P110, and J55, and manganese is a very strong oxidizing agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Therefore, the corrosion rate on P110 is almost higher comparing to N80 and J55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Steel Grade C Si Mn P S Cr Mo Ni Ti Cu N80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='011 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='019 P110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='03 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='01 J55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='04 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='01 Table 3: Steel composition (wt%) [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' 10 6 Supercritical phase 8 Liquid phase 7 Solid P Critical point Phase Pressure 6 Gas phase P 4 3 2 1 Triple point 100-80-60-40-20 204060 Temperature(C)Petroleum & Petrochemical Engineering Journal 8 Nguyen V, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A Literature Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Pet Petro Chem Eng J 2021, 5(3): 000269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Copyright© Nguyen V, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Figure 8: Corrosion rate for N80, P110, J55 on different conditions [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Cement Corrosion Mechanism Understanding the cement corrosion during the CO2 geological storage is necessary to handle the process correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' This includes the CO2 leakage paths and the carbonation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Recognizing all possible leakage paths would aid in looking for a reasonable solution to cracking problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Interpreting the carbonation time could contribute to evaluating the safety of the process for an extended period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' CO2 Leakage Path: There are many leakages pathways for CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' It could be an interface formation-cement or cement- casing, or from cement cracking as shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Figure 9: The possible leakage paths of CO2 [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Estimated Time for Carbonation Process: Many research groups have predicted the carbonation times of cement exposed to CO2 by using experimental and mathematical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The carbonation is a substantial variable to evaluate the quality of the Carbon Capture Storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Therefore, it is essential to obtain it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The experimental and numerical methods have their advantages and limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The empirical model in some situations will produce inaccurate results due to equipment error, and occasionally it is hard to manage the procedure properly due to external conditions beyond our observation, and most likely, the cost to do experiments is more expensive and time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' In contrast, on average, the numerical model is much easier to do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' It will eliminate the errors from equipment and human.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' It will produce the result faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The most important thing for using a numerical model is to set up a governing equation with correct initial and boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Although showing the limitation on both experimental and numerical models, they are used to verify one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' That is a reason why it is necessary to study both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Experimental Measurements: Duguid A [53] designed an experiment to forecast the time to deteriorate the cement sheath in a well exposed to carbonated brine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The samples were created by drilling the stone cylinder 55 mm in diameter, 10 mm in height with a 25-mm axial hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The minimum depth from the outside of the cylinder to the boundary of the hole was 3mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' According to Duguid A [53], the depth of reaction was quantified at five different locations: 0, 45, 90,135,180 degrees as presented in Figure 10 in order to evaluate how different the surface cement would react with carbonate brine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The relationship between carbonation depth and time1/2/ radius is linear in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The most linear relationship is at the condition pH=3 and T=50 oC, and this condition also causes the most carbonation for cement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' It is in agreement with Duguid A, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The prediction time for 25 mm cement sheath to be deteriorated is approximately from 30,000 to 70,000 years if the favorable cement is selected and the good cementing job is done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' J55 Type Steel P110 At 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='34 Mpa,90oC At 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='89 Mpa,90oC N80 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='5 3 Corrosion rate(mm/year)Cementlayer Formation Steel Casing Cement Plug Cement cracking path Cement-formation interface Cement-casing interface leaking leakingPetroleum & Petrochemical Engineering Journal 9 Nguyen V, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A Literature Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Pet Petro Chem Eng J 2021, 5(3): 000269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Copyright© Nguyen V, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Figure 10: The top view of sample of experiment [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Kutchko BG [55] showed the reaction rate when the cement was exposed to supercritical CO2 and CO2-saturated brine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Supercritical CO2 is a separate free phase causing hydrodynamic trapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' On the other hand, some would dissolve in the brine existing in CO2 saturated brine form, which induces the solubility trapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The cement samples were embedded in 1% NaCl at 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='3 MPa and 50 oC under static conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The estimation for penetration depth is 1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='07 mm for the CO2 -saturated brine and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='89 mm for the supercritical CO2 after 30 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' It indicated that the supercritical CO2 would degrade Portland cement faster than the CO2 saturated brine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' This is comprehensible because the condition of supercritical CO2 is at high pressure and high temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The penetration depth over time for the CO2- saturated brine and supercritical can be found in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Figure 11: The penetration depth over time when cement exposed to CO2 Saturated brine and Supercritical CO2 [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Zhang L, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [56] introduced another approach to estimate the penetration depth over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Fick’s diffusion and Elovich’s equation were fit to experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Elovich’s equation (Equation 16) can be shown by Allen JA, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [57] and Kutchko BG, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' ( ) dL a exp bL dt = − (16) Where L is the penetration depth(mm) at time t (days) of exposure and a, b are constants decided from experiment data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Integrating the Equation 16 above with respect to t yields Equation 17: ( ) ( ) 1 1 L ln t ln ab b b = + (17) Where a, b can be estimated to fit the data a=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='47, b=22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='08 Estimation of penetration depth with Elovich’s equation is more accurate than Fick’s diffusion as shown in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Elovich’s equation has been used in several kinetic studies [59-61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' However, the outcomes reaffirmed Elovich’s equation as a powerful method to measure the CO2 penetration depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Figure 12: Fitting data with Fick’s law(left) and Elovich’s equation (right) [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' In Duguid A, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [54], the experiment for limestone and sandstone- like condition were executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Class H cement pastes were exposed to temperatures from 20 to 50oC and pH 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='4 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Then the samples were interpreted by using multiple techniques such as Inductively Coupled Plasma Optical Emission Spectroscopy (ICP-OES), optical microscopy, X-ray diffraction, and Electron Probe Microanalysis (EPMA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The experimental model was designed as presented in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The CO2 air was percolated into the carbonated brine and then pumped to the reactor vessel, which contains the cement sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The purpose of the recirculated flow is to assure that it was saturated with CaCO3 before reaching the reaction vessel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' No observable degradation in the limestone-like condition was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Under the sandstone-like condition, there are 180° Sandstone sample 135° 90° Cement class H 45° 0°0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='8 y=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content="04824x, Fick's law Penetration Depth (mm) 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='1 0 0 5 10 15 20 Time0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='5 (Day ^0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='5) +cO2SaturatedBrine Supercriticalco20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='6 Penetration depth (mm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content="1 0 0 20 40 60 80 100 Exposuretime (Days) Experimentdata Fick'slawfit ElovichPetroleum & Petrochemical Engineering Journal 10 Nguyen V, et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A Literature Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Pet Petro Chem Eng J 2021, 5(3): 000269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Copyright© Nguyen V, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' 5 distinct layers that appeared: orange, brown, white, gray, and core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Each layer would expose the different behavior to carbonated brine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The orange and brown part display a leached region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The white layer shows a carbonated region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The gray section depicts a calcium hydroxide dissolution region, and the core section is no change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The outer layer was degraded fully at pH=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='4, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='7, and temperatures 20oC and 50oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The sample got the most damage at pH=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='4 and T=20oC, and the least degradation occurred at pH= 5 and T=50oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' This conclusion is illustrated by the dissolution of carbon dioxide in water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The carbon dioxide solubility in water increases with decreasing temperature, and more carbonic acid is formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Therefore, the most degradation was occurred at pH=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='4 and temperature 20 oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Figure 13: Experimental system for sandstone like condition (top) and limestone like condition(bottom) reactor [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Carey JW, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [62] and Carey JW, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [63] investigated the behavior of wellbore integrity and CO2-brine flow along the casing-cement micro annulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The core-flood examination was performed at 40oC, 14 MPa pore pressure, and 28 MPa confining pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The experimental system included a 10 cm length of limestone with a combination of rectangular steel embedded in the cement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The blended solution, 50% supercritical CO2 and 50% brine, was run through limestone/ cement combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' There are two corrosion processes: steel corrosion and cement corrosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The corrosion on steel occurs by the electrochemical mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The film FeCO3 layer was formed from the CO2-rich fluid in contact with steel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The film layer protected the steel from deeper penetration of the CO2-rich fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' However, the solubility of the FeCO3 layer would increase if the pH decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' As a result, corrosion rate is increased, with increasing flow rate of the CO2-rich fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' For cement degradation, the rate is dependent on cement properties and the flow rate of CO2 -rich fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The diffusion coefficient was discovered in the interval from 10- 12 to 10-10 cm2/sec by assuming a 1D diffusion problem with characteristic diffusion time 2√Dt and penetration depth from 50-250μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Adeoye JT, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [64] carried out experiments of a novel engineered cementitious composite (ECC) exposed to CO2- saturated water under static and flow conditions at 10 MPa and 50oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The depth of alteration estimated was 72 mm over 50 years by Fick’s law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' In a similar study was performed by Kutchko BG, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [65] at 15 MPa and 50oC, the depth of alteration was 224 mm over 50 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The higher pressure of CO2 in the study Kutchko BG, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [65] was not probably the main reason to lead to a 3 three times increase of carbonation depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Nonetheless, the exciting finding lay on the pozzolan to cement ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The material used in the study Kutchko BG, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [65] has the pozzolan to cement ratio of 65:35, while this ratio in the study Adeoye JT, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [64] is 45:55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Thus, the increase of the pozzolan would lead to faster carbonation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Mathematical Prediction: Along with experimental works, there are very few studies using the robust mathematical model to predict the penetration depth during geological storage CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The mathematical model may accompany experimental work, which will conserve resources compared to purely experimental work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Below are some remarkable studies which have been done so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Tao Q, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [66,67] developed a mathematical model to investigate some relationships during geological CO2 storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The leaking pathway of CO2 includes two parts, as shown in Figure 14a: the bottom due to cement degradation, the top is the water-saturated porous medium where an assumption of no resistance was made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Figure 14b showed how the pressure of the reservoir changes during CO2 injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The pressure of the reservoir increased as CO2 was injected and decreased after the injection was stopped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Figure 14: a) Two different leaking pathways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' b) Deviated reservoir pressure during CO2 injection [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' If there is only buoyancy force causing CO2 flow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' the potential gradient of CO2 is calculated by Equation 18: Reacted brine out CO2 Pump Cement sample Carbonate brine Reacted brine out CO2 Calcite dund Cement sample Carbonate brineLeakage path due to water saturated p_max porousmedium pressure Earth Reservoir Leakage Ap path due to cement defect p_hydro Cement CO2 t begin t_end t after plume Time (a) (b)Petroleum & Petrochemical Engineering Journal 11 Nguyen V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A Literature Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Pet Petro Chem Eng J 2021, 5(3): 000269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Copyright© Nguyen V, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' ( ) gz ϕ ρ ∆ ∇ = ∇ (18) Where ∆ρ is the density difference between H2O and CO2 z and g are the depth and gravitational, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' During the injection period, both buoyancy and pressure elevation contribute to drive CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Therefore, the potential gradient of CO2 is computed by Equation 19: ( ) c gz p ϕ ρ ∇ = + ∇ ∆ ∇ (19) Where pc is the capilary pressure Several wells were investigated during the geological CO2 storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The CO2 flux was discovered to be responsive to pressure elevation and the leakage depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The CO2 flux decreases while increasing the leakage depth due to the increasing of CO2 density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The relationship between the CO2 leakage flux and injection pressure at shallow leak (4000 ft) and deep leak (10000 ft) was investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The CO2 leakage flux increases with increasing injection pressure, but the rates are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The CO2 leakage flux rate is more at deep leak than it does at the shallow leak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' According to the author, this is because the injection pressure would overcome the buoyancy pressure at a shallow leak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' In contrast, at a deep leak, the buoyancy pressure decreases more gradually due to the increasing CO2 density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Figure 15: The Calcite solubility diagram [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Deremble L, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [68] simulated the evolution of layers under CO2 -rich brine flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The development of layer calcite or silica gel will be proportional, while the flux of calcium or CO2 is inversely proportional to the square root of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' An increase in CO2 flow rate increases calcium dissolution rate until calcite equilibrium is reached at point λ, as presented in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Then ion calcium cannot be discharged anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' As a result, CO2 would react with other species in the cement until it approaches point δ where all species have gone, and then the mathematical model takes into consideration the additional physical aspects including: micro-annulus geometry, the Peclet number, and the characteristic length scales of a defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The model uses the implicit algorithm to solve for a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' It also affirms that the penetration depth is proportional to the square root of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Huet BM, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [69] connected the geochemical and transport module to simulate the degradation of cement during geological CO2 storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The Dynaflow was adopted to solve a non-linear system of partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Both Galerkin finite element and vertex centered finite volume space discretization of transport equations were executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' An implicit backward finite difference time stepping of the transport equation was applied to produce the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The effective diffusion coefficient needs to be assumed in order of 10-11 m2/s by Bentz DP, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The thickness of the calcite layer is also proportional to the square root of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The difference between experimental data and the model occurred due to the diffusion coefficient estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The growth of calcite carbonate concentration over time as presented in Figure 16 would provide insight into the transport of CO2 to cement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The carbonate concentration increases very rapidly while its radius decreases with time elapsed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' There are two distinct regions with different transport mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' In the first region with time exposure less than 60 days, carbonate species (CO2, HCO3-) disperse into the sample through the calcite and the silica gel layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The second region is where the calcite layer dissolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Figure 16: Carbonate concentration profile over time [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Also, well integrity has been investigated extensively in Hawkes C, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [71];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Hawkes CD, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [72];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Scherer GW, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [73];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Neuville N, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [74] to evaluate for long-term CO2 storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The downhole testing programs, such as cement sheath pressure transient testing, mini-frac testing, cement Calcite equilibrium curve [Ca?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='+] [CO2] 0 Calcite solubility diagram6x101 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' 1 day b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' 7 days Concentration (mmol/L) c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' 15 days 5x101 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' 30 days e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' 60 days 6 e p c b 4x101 f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' 120 days g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' 180 days 3x101 2x101 101 0 0 1 2 3 4 Radius (mm)Petroleum & Petrochemical Engineering Journal 12 Nguyen V, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A Literature Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Pet Petro Chem Eng J 2021, 5(3): 000269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Copyright© Nguyen V, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' sampling, and fluid sampling, were performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Outlook The traditional Portland cement carries many advantages such as low cost, high compressive strength, low alkali content, and long-term stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Nevertheless, cement is sensitive in an acidic environment, and the cement industry is a source of CO2 emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Therefore, Portland cement is not the most excellent sealant material to serve in CCS project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Based on the nature of CCS project, the most optimal cement could prevent corrosion under the acidic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Also, the cement should possess low permeability, porosity, and high mechanical strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Mahmoud AA, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [75] proposed to add Synthetic Polypropylene Fiber (PPF) into Class G cement to improve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Four samples with 0%(PPF0), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='125%(PPF1), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='25%(PPF2), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='375%(PPF3) of PPF were arranged for the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The results of this study showed that the carbonation depth and carbonation rate decreased while compressive strength and tensile strength increased, as presented in Figures 17-20, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The decreasing of carbonation depth and carbonation rate indicated the reduction of cement permeability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Figure 17: Carbonation Depth after experiment 10 days [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Figure 18: Carbonation Rate after experiment 10 days [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' 3000 2586 2500 Depth(μm) 2182 2000 1842 1588 Carbonation 1500 1000 500 0 PPFO PPF1 PPF2 PPF3300 259 250 Carbonation Rate(μm/Day) 218 200 184 11 159 150 100 50 0 PPFO PPF1 PPF2 PPF3Petroleum & Petrochemical Engineering Journal 13 Nguyen V, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A Literature Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Pet Petro Chem Eng J 2021, 5(3): 000269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Copyright© Nguyen V, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Figure 19: Compressive strength after experiment 10 days [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Figure 20: Tensile Strength after experiment 10 days [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Nanomaterial such as nano-silica (nano SiO2) [8], nano-alumina (nano Al2O3) [76], nano-titanium dioxide (nanoTiO2) [9], carbon nanotubes (CNTs) [77], Polymer/ clay nanocomposites [78], nanoglass flake (NGFS) [79] were considered as good additive to improve cement quality because of their large surface area and reactivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The nanomaterials can solidify the cement microstructure and lessen the porosity, then advance the mechanical strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Ponzi GGD, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' [80] proposed basalt powder as an additive material in cement formulation because the basalt powder has low pozzolanic activity, large inert fraction, and small particle size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The basalt power plays a role as a filling-substance to the porous cement networks to curb the fluid intrusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The experimental results explored that the formulation with low basalt powder content (≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='5 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='%) exhibited more resistance to CO2 degradation, lower porosity and permeability, and stronger mechanical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Other sealant materials can replace traditional Portland cement, such as geopolymer cement, resins, biofilms barriers, foams [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Geopolymer, such as zeolites, was discovered to have better resistance with CO2- rich brine because it contains less calcium oxide than Portland cement does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Resins are particle-free fluids with low mobility, hard, rigid, and impermeable materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' They include phenolic, 90 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='9 80 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content="7 70 8'S9 61." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='1 60 50 40 30 20 10 0 PPFO PPF1 PPF2 PPF38 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='86 7 Tensile Strength(MPa) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='75 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='83 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content='89 5 4 3 2 1 0 PPFO PPF1 PPF2 PPF3Petroleum & Petrochemical Engineering Journal 14 Nguyen V, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Well Cement Degradation and Wellbore Integrity in Geological CO2 Storages: A Literature Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Pet Petro Chem Eng J 2021, 5(3): 000269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Copyright© Nguyen V, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' epoxy, and furan resins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Biofilm sealants include urea, Ca2+, nutrient feed, and micro-organism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The principle is to accelerate calcium to form calcite and seal fractures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Foam is a gas-liquid blend, and it can block the flow rate of CO2 in porous media and increase the CO2 viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Because carbon capture and storage projects are expanding, so the cement consumption is increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Those materials introduced above have many advantages, but the disadvantages still exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Therefore, the new research direction should focus on improving the cement quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' For example, geopolymer is detrimental to human health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Thus, new components should be studied to make it become human friendly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Conclusions This study’s major goal was to review the previous papers for carbon capture and storage project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Many experiments have been performed to assess the well integrity and predict the time degradation of geological CO2 storage, but there are very few mathematical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Research predicts 30,000- 70,000 years for 25 mm cement to be carbonated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Some studies tried to match the experimental data to a particular equation and introduce that the penetration depth is proportional to the square root of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Furthermore, the debonding interface between casing/cement or cement/ formation is the primary cause for leakage of CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' However, it has not been thoroughly investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Some substantial conclusions can be drawn from this review: The need for a more accurate mathematical model to evaluate the well integrity and anticipate the corrosion rate during geological CO2 storage is very crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' A further investigation should identify the debonding interface between casing/cement or cement/formation issue and predict how the micro-annuli of case/cement or cement/formation behaves with the variation of temperature, stress, and chemical reactions during geological CO2 storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' The diffusion coefficient is one of the most crucial parameters in the corrosion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' However, it has not been studied sufficiently in petroleum corrosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Hence, it should be an interesting topic for future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Improving the cementing property is one of the means to curb the corrosion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Future studies should investigate more how to reduce reactive species and add more inhibitors to advance Portland cement quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' If these tasks are done properly, it will clear up a considerable concern to make the operation more predictable and administered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Author Contribution Writing- original draft preparation, Nguyen V;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' writing- review and editing, Nguyen V, Olatunji O, Guo B and Ning Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' All authors have read and agreed to the published version of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
+page_content=' Funding This research received no external funding Conflicts of Interest The authors declare no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
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+page_content=' Proceeding of annual Carbon Management Technology Conference, Texas, USA, 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
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+page_content=' SPE Drill Complet 30: 68-75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfcACj/content/2301.02357v1.pdf'}
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+Argoverse 2: Next Generation Datasets for
+Self-Driving Perception and Forecasting
+Benjamin Wilson∗†,1, William Qi∗†, Tanmay Agarwal∗†, John Lambert†, Jagjeet Singh†,
+Siddhesh Khandelwal2, Bowen Pan†,3, Ratnesh Kumar†, Andrew Hartnett†,
+Jhony Kaesemodel Pontes†, Deva Ramanan†,4, Peter Carr†, James Hays†,1
+1Georgia Tech, 2UBC, 3MIT, 4CMU
+Abstract
+We introduce Argoverse 2 (AV2) — a collection of three datasets for perception and
+forecasting research in the self-driving domain. The annotated Sensor Dataset con-
+tains 1,000 sequences of multimodal data, encompassing high-resolution imagery
+from seven ring cameras, and two stereo cameras in addition to lidar point clouds,
+and 6-DOF map-aligned pose. Sequences contain 3D cuboid annotations for 26
+object categories, all of which are sufficiently-sampled to support training and
+evaluation of 3D perception models. The Lidar Dataset contains 20,000 sequences
+of unlabeled lidar point clouds and map-aligned pose. This dataset is the largest
+ever collection of lidar sensor data and supports self-supervised learning and the
+emerging task of point cloud forecasting. Finally, the Motion Forecasting Dataset
+contains 250,000 scenarios mined for interesting and challenging interactions be-
+tween the autonomous vehicle and other actors in each local scene. Models are
+tasked with the prediction of future motion for “scored actors" in each scenario
+and are provided with track histories that capture object location, heading, velocity,
+and category. In all three datasets, each scenario contains its own HD Map with 3D
+lane and crosswalk geometry — sourced from data captured in six distinct cities.
+We believe these datasets will support new and existing machine learning research
+problems in ways that existing datasets do not. All datasets are released under the
+CC BY-NC-SA 4.0 license.
+1
+Introduction
+In order to achieve the goal of safe, reliable autonomous driving, a litany of machine learning tasks
+must be addressed, from stereo depth estimation to motion forecasting to 3D object detection. In recent
+years, numerous high quality self-driving datasets have been released to support research into these and
+other important machine learning tasks. Many datasets are annotated “sensor” datasets [4, 45, 39, 40,
+24, 33, 18, 14, 41, 36] in the spirit of the influential KITTI dataset [17]. The Argoverse 3D Tracking
+dataset [6] was the first such dataset with “HD maps” — maps containing lane-level geometry. Also
+influential are self-driving “motion prediction” datasets [12, 22, 34, 4, 52] — containing abstracted
+object tracks instead of raw sensor data — of which the Argoverse Motion Forecasting dataset [6]
+was the first.
+In the last two years, the Argoverse team has hosted six competitions on 3D tracking, stereo depth
+estimation, and motion forecasting. We maintain evaluation servers and leaderboards for these tasks,
+*Equal contribution.
+†Work completed while at Argo AI.
+35th Conference on Neural Information Processing Systems (NeurIPS 2021) Track on Datasets and Benchmarks.
+arXiv:2301.00493v1 [cs.CV] 2 Jan 2023
+
+as well as 3D detection. The leaderboards collectively contain thousands of submissions from four
+hundred teams1. We also maintain the Argoverse API and have addressed more than one hundred
+issues2. From these experiences we have formed the following guiding principles to guide the creation
+of the next iteration of Argoverse datasets.
+1. Bigger isn’t always better. Self-driving vehicles capture a flood of sensor data which is logisti-
+cally difficult to work with. Sensor datasets are several terabytes in size, even when compressed.
+If standard benchmarks grow further, we risk alienating much of the academic community and
+leaving progress to well-resourced industry groups. For this reason, we match but do not exceed
+the scale of sensor data in nuScenes [4] and Waymo Open [45].
+2. Make every instance count. Much of driving is boring. Datasets should focus on the difficult,
+interesting scenarios where current forecasting and perception systems struggle. Therefore we
+mine for especially crowded, dynamic, and kinematically unusual scenarios.
+3. Diversity matters. Training on data from wintertime Detroit is not sufficient for detecting objects
+in Miami — Miami has 15 times the frequency of motorcycles and mopeds. Behaviors differ
+as well, so learned pedestrian motion behavior might not generalize. Accordingly, each of our
+datasets are drawn from six diverse cities — Austin, Detroit, Miami, Palo Alto, Pittsburgh, and
+Washington D.C. — and different seasons, as well, from snowy to sunny.
+4. Map the world. HD maps are powerful priors for perception and forecasting. Learning-based
+methods that found clever ways to encode map information [31] performed well in Argoverse
+competitions. For this reason, we augment our HD map representation with 3D lane geometry,
+paint markings, crosswalks, higher resolution ground height, and more.
+5. Self-supervise. Other machine learning domains have seen enormous success from self-supervised
+learning in recent years. Large-scale lidar data from dynamic scenes, paired with HD maps, could
+lead to better representations than current supervised approaches. For this reason, we build the
+largest dataset of lidar sensor data.
+6. Fight the heavy tail. Passenger vehicles are common, and thus we can assess our forecasting
+and detection accuracy for cars. However, with existing datasets, we cannot assess forecasting
+accuracy for buses and motorcycles with their distinct behaviors, nor can we evaluate stroller and
+wheel chair detection. Thus we introduce the largest taxonomy to date for sensor and forecasting
+datasets, and we ensure enough samples of rare objects to train and evaluate models.
+With these guidelines in mind we built the three Argoverse 2 (AV2) datasets. Below, we highlight
+some of their contributions.
+1. The 1,000 scenario Sensor dataset has the largest self-driving taxonomy to date – 30 categories.
+26 categories contain at least 6,000 cuboids to enable diverse taxonomy training and testing. The
+dataset also has stereo imagery, unlike recent self-driving datasets.
+2. The 20,000 scenario Lidar dataset is the largest dataset for self-supervised learning on lidar. The
+only similar dataset, concurrently developed ONCE [36], does not have HD maps.
+3. The 250,000 scenario Motion Forecasting Dataset has the largest taxonomy – 5 types of dynamic
+actors and 5 types of static actors – and covers the largest mapped area of any such dataset.
+We believe these datasets will support research into problems such as 3D detection, 3D tracking,
+monocular and stereo depth estimation, motion forecasting, visual odometry, pose estimation, lane
+detection, map automation, self-supervised learning, structure from motion, scene flow, optical flow,
+time to contact estimation, and point cloud forecasting.
+2
+Related Work
+The last few years have seen rapid progress in self-driving perception and forecasting research,
+catalyzed by many high quality datasets.
+Sensor datasets and 3D Object Detection and Tracking. New sensor datasets for 3D object
+detection [4, 45, 39, 40, 24, 33, 18, 14, 41, 36] have led to influential detection methods such as
+1This count includes private submissions not posted to the public leaderboards.
+2https://github.com/argoverse/argoverse-api
+2
+
+anchor-based approaches like PointPillars [27], and more recent anchor-free approaches such as
+AFDet [16] and CenterPoint [51]. These methods have led to dramatic accuracy improvements on all
+datasets. In turn, these improvements have made isolation of object-specific point clouds possible,
+which has proven invaluable for offboard detection and tracking [42], and for simulation [8], which
+previously required human-annotated 3D bounding boxes [35]. New approaches explore alternate
+point cloud representations, such as range images [5, 2, 46]. Streaming perception [29, 21] introduces
+a paradigm to explore the tradeoff between accuracy and latency. A detailed comparison between the
+AV2 Sensor Dataset and recent 3D object detection datasets is provided in Table 1.
+Motion Forecasting. For motion forecasting, the progress has been just as significant. A transition
+to attention-based methods [28, 38, 37] has led to a variety of new vector-based representations for
+map and trajectory data [15, 31]. New datasets have also paved the way for new algorithms, with
+nuScenes [4], Lyft L5 [22], and the Waymo Open Motion Dataset [12] all releasing lane graphs
+after they proved to be essential in Argoverse 1 [6]. Lyft also introduced traffic/speed control data,
+while Waymo added crosswalk polygons, lane boundaries (with marking type), speed limits, and stop
+signs to the map. More recently, Yandex has released the Shifts [34] dataset, which is the largest (by
+scenario hours) collection of forecasting data available to date. Together, these datasets have enabled
+exploration of multi-actor, long-range motion forecasting leveraging both static and dynamic maps.
+Following upon the success of Argoverse 1.1, we position AV2 as a large-scale repository of high-
+quality motion forecasting scenarios - with guarantees on data frequency (exactly 10 Hz) and diversity
+(>2000 km of unique roadways covered across 6 cities). This is in contrast to nuScenes (reports data
+at just 2 Hz) and Lyft (collected on a single 10 km segment of road), but is complementary to Waymo
+Open Motion Dataset (employs a similar approach for scenario mining and data configuration).
+Complementary datasets are essential for these safety critical problems as they provide opportunities
+to evaluate generalization and explore transfer learning. To improve ease of use, we have also
+designed AV2 to be widely accessible both in terms of data size and format — a detailed comparison
+vs. other recent forecasting datasets is provided in Table 2.
+Broader Problems of Perception for Self-Driving. Aside from the tasks of object detection and
+motion forecasting, new, large-scale sensor datasets for self-driving present opportunities to explore
+dozens of new problems for perception, especially those that can be potentially solved via self-
+supervision. A number of new problems have been recently proposed; real-time 3D semantic
+segmentation in video has received attention thanks to SemanticKITTI [1]. HD map automation
+[54, 30] and HD map change detection [26] have received additional attention, along with 3D
+scene flow and pixel-level scene simulation [50, 8]. Datasets exist with unique modalities such as
+thermal imagery [10, 9]. Our new Lidar Dataset enables large-scale self-supervised training of new
+approaches for freespace forecasting [23] or point cloud forecasting [48, 49].
+3
+The Argoverse 2 Datasets
+3.1
+Sensor Dataset
+The Argoverse 2 Sensor Dataset is the successor to the Argoverse 1 3D Tracking Dataset. AV2 is
+larger, with 1,000 scenes, up from 113 in Argoverse 1, but each AV2 scene is also richer – there
+are 23x as many non-vehicle, non-pedestrian cuboids in AV2. The constituent 30 s scenarios in
+the Argoverse 2 Sensor Dataset were manually selected by the authors to contain crowded scenes
+with under-represented objects, noteworthy weather, and interesting behaviors, e.g., cut ins and
+jaywalking. Each scenario is fifteen seconds in duration. Table 1 compares the AV2 Sensor Dataset
+with a selection of self-driving datasets. Figures 1, 2, and 3 plot how the scenarios of AV2 compare
+favorably to other datasets in terms of annotation range, object diversity, object density, and scene
+dynamism.
+The most similar sensor dataset to ours is the highly influential nuScenes [4] – both datasets have
+1,000 scenarios and HD maps, although Argoverse is unique in having ground height maps. nuScenes
+contains radar data while AV2 contains stereo imagery. nuScenes has a large taxonomy – twenty-three
+object categories of which ten have suitable data for training and evaluation. Our dataset contains
+thirty object categories of which twenty-six are well sampled enough for training and evaluation.
+nuScenes spans two cities, while our proposed dataset spans six.
+3
+
+Table 1: Comparison of the Argoverse 2 Sensor and Lidar datasets with other sensor datasets.
+Name
+# Scenes
+Cities
+Lidar?
+# Cameras
+Stereo
+HD Maps?
+# Classes
+# Evaluated Classes
+Argoverse 1 [6]
+113
+2
+✓
+7
+✓
+✓
+15
+3
+KITTI [17]
+22
+1
+✓
+2
+✓
+3
+3
+nuScenes [4]
+1,000
+2
+✓
+6
+✓
+23
+10
+ONCE [36]
+581
+–
+✓
+7
+5
+3
+Waymo Open [45]
+1,150
+3
+✓
+5
+4
+4
+Argoverse 2 Sensor
+1,000
+6
+✓
+9
+✓
+✓
+30
+26
+Argoverse 2 Lidar
+20,000
+6
+✓
+-
+✓
+-
+-
+Regular vehicle
+Pedestrian
+Bollard
+Construction cone
+Construction barrel
+Stop sign
+Bicycle
+Large vehicle
+Wheeled device
+Bus
+Box truck
+Sign
+Truck
+Motorcycle
+Bicyclist
+Vehicular trailer
+Truck cab
+Motorcyclist
+Dog
+School bus
+Wheeled rider
+Stroller
+Articulated bus
+Message board trailer
+Mobile pedestrian sign
+Wheelchair
+Railed vehicle
+Official signaler
+Traffic light trailer
+Animal
+100
+2
+5
+1000
+2
+5
+10k
+2
+5
+100k
+2
+5
+1M
+2
+5
+10M
+Argoverse 1
+Argoverse 2
+nuScenes
+ONCE
+Waymo Open
+Number of 3D cuboids
+Figure 1: Number of annotated 3D cuboids per category for Argoverse 1 3D Tracking, Argoverse
+2 Sensor Dataset, nuScenes, ONCE, and Waymo Open. The nuScenes annotation rate is 2 Hz,
+compared to 10 Hz for Argoverse, but that does not account for the relative increase in object diversity
+in Argoverse 2.
+Sensor Suite.
+Lidar sweeps are collected at 10 Hz, along with 20 fps imagery from 7 cameras
+positioned to provide a fully panoramic field of view. In addition, camera intrinsics, extrinsics and
+6-DOF ego-vehicle pose in a global coordinate system are provided. Lidar returns are captured by
+two 32-beam lidars, spinning at 10 Hz in the same direction, but separated in orientation by 180°.
+The cameras trigger in-sync with both lidars, leading to a 20 Hz frame-rate. The seven global shutter
+cameras are synchronized to the lidar to have their exposure centered on the lidar sweeping through
+their fields of view. In the Appendix, we provide a a schematic figure illustrating the car sensor suite
+and its coordinate frames.
+Lidar synchronization accuracy. In AV2, we improve the synchronization of cameras and lidars
+significantly over Argoverse 1. Our synchronization accuracy is within [−1.39, 1.39] ms, which
+compares favorably to the Waymo Open Dataset, which is reported as [−6, 7] ms [45].
+Annotations. The AV2 Sensor Dataset contains 10 Hz 3D cuboid annotations for objects within our
+30 class taxonomy (Figure 1). Cuboids have track identifiers that are consistent over time for the
+0
+50
+100
+150
+200
+250
+0
+50k
+100k
+150k
+200k
+250k
+Waymo Open
+Argoverse 2
+nuScenes
+Argoverse 1
+ONCE
+Range (m)
+Number of 3D cuboids
+0
+50
+100
+150
+200
+250
+300
+0
+500
+1000
+1500
+2000
+2500
+Waymo Open
+Argoverse 2
+nuScenes
+Argoverse 1
+ONCE
+Number of 3D cuboids
+Number of lidar frames
+Figure 2: Left: Number of annotated 3D cuboids by range in the Argoverse 2 Sensor Dataset. About
+14% of the Argoverse 2 cuboids are beyond 75 m – Waymo Open, nuScenes, and ONCE have less
+than 1%. Right: Number of 3D cuboids per lidar frame. Argoverse 2 has an average of 75 3D
+cuboids per lidar frame – Waymo Open has an average of 61, nuScenes 33, and ONCE 30.
+4
+
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9 10 11 12 13 14 15 16 17 18 19
+0
+20k
+40k
+60k
+80k
+100k
+Waymo Open
+Argoverse 2
+Argoverse 1
+nuScenes
+ONCE
+Number of different categories
+Number of lidar frames
+5
+10
+15
+20
+25
+0
+20k
+40k
+60k
+80k
+Argoverse 2
+Waymo Open
+Argoverse 1
+nuScenes
+Speed (m/s)
+Number of 3D vehicle cuboids
+ (speed > 0.5 m/s)
+Figure 3: Left: Number of annotated categories per lidar frame in the Argoverse 2 Sensor Dataset.
+Per scene, Argoverse 2 is about 2× more diverse than Argoverse 1 and 2.3× more diverse than
+Waymo Open. Right: Speed distribution for the vehicle category. We consider only moving vehicles
+with speeds greater than 0.5 m/s. Argoverse 2 has about 1.3× more moving vehicles than Waymo
+Open. About 28% of the vehicles in Argoverse 2 are moving with an average speed of 7.27 m/s. We
+did not compare against the ONCE dataset because it does not provide tracking information for the
+3D cuboids.
+same object instance. Objects are annotated if they are within the “region of interest” (ROI) – within
+five meters of the mapped “driveable” area.
+Privacy. All faces and license plates, whether inside vehicles or outside of the driveable area, are
+blurred extensively to preserve privacy.
+Sensor Dataset splits. We randomly partition the dataset with train, validation, and test splits of 700,
+150, and 150 scenarios, respectively.
+3.2
+Lidar Dataset
+The Argoverse 2 Lidar Dataset is intended to support research into self-supervised learning in the
+lidar domain as well as point cloud forecasting [48, 49]. Because lidar data is more compact than the
+full sensor suite, we can include double-length scenarios (30 s instead of 15 s), and far more – 20,000
+instead of 1,000 – equating to roughly 40x as many driving hours, for 5x the space budget. The AV2
+Lidar Dataset is mined with the same criteria as the Forecasting Dataset (Section 3.3.2) to ensure that
+each scene is interesting. While the Lidar Dataset does not have 3D object annotations, each scenario
+carries an HD map with rich, 3D information about the scene.
+Our dataset is the largest such collection to date with 20,000 thirty second sequences. The only
+similar dataset, concurrently released ONCE [36], contains 1 M lidar frames compared to 6 M lidar
+frames in ours. Our dataset is sampled at 10 Hz instead of 2 Hz, as in ONCE, making our dataset
+more suitable for point cloud forecasting or self-supervision tasks where point cloud evolution over
+time is important.
+Lidar Dataset splits. We randomly partition the dataset with train, validation, and test splits of
+16,000, 2,000, and 2,000 scenarios, respectively.
+3.3
+Motion Forecasting Dataset
+Motion forecasting addresses the problem of predicting future states (or occupancy maps) for dynamic
+actors within a local environment. Some examples of relevant actors for autonomous driving include:
+vehicles (both parked and moving), pedestrians, cyclists, scooters, and pets. Predicted futures
+generated by a forecasting system are consumed as the primary inputs in motion planning, which
+conditions trajectory selection on such forecasts. Generating these forecasts presents a complex,
+multi-modal problem involving many diverse, partially-observed, and socially interacting agents.
+However, by taking advantage of the ability to “self-label” data using observed ground truth futures,
+motion forecasting becomes an ideal domain for application of machine learning.
+Building upon the success of Argoverse 1, the Argoverse 2 Motion Forecasting dataset provides
+an updated set of prediction scenarios collected from a self-driving fleet. The design decisions
+enumerated below capture the collective lessons learned from both our internal research/development,
+5
+
+Table 2: Comparison between the Argoverse 2 Motion Forecasting dataset and other recent motion
+forecasting datasets. Hyphens "-" indicate that attributes are either not applicable, or not available.
+We define “mined for interestingness” to be true if interesting scenarios/actors are mined after data
+collection, instead of taking all/random samples. † Public leaderboard counts as retrieved on Aug. 27,
+2021.
+ARGOVERSE [6]
+INTER [52]
+LYFT [22]
+WAYMO [12]
+NUSCENES [4]
+YANDEX [34]
+OURS
+# SCENARIOS
+324k
+-
+170k
+104k
+41k
+600k
+250k
+# UNIQUE TRACKS
+11.7M
+40k
+53.4M
+7.6M
+-
+17.4M
+13.9M
+AVERAGE TRACK LENGTH
+2.48 s
+19.8 s
+1.8 s
+7.04 s
+-
+-
+5.16 s
+TOTAL TIME
+320 h
+16.5 h
+1118 h
+574 h
+5.5 h
+1667 h
+763 h
+SCENARIO DURATION
+5 s
+-
+25 s
+9.1 s
+8 s
+10 s
+11 s
+TEST FORECAST HORIZON
+3 s
+3 s
+5 s
+8 s
+6 s
+5 s
+6 s
+SAMPLING RATE
+10 Hz
+10 Hz
+10 Hz
+10 Hz
+2 Hz
+5 Hz
+10 Hz
+# CITIES
+2
+6
+1
+6
+2
+6
+6
+UNIQUE ROADWAYS
+290 km
+2 km
+10 km
+1750 km
+-
+-
+2220 km
+AVG. # TRACKS PER SCENARIO
+50
+-
+79
+-
+75
+29
+73
+# EVALUATED OBJECT CATEGORIES
+1
+1
+3
+3
+1
+2
+5
+MULTI-AGENT EVALUATION
+×
+✓
+✓
+✓
+×
+✓
+✓
+MINED FOR INTERESTINGNESS
+✓
+×
+-
+✓
+×
+×
+✓
+VECTOR MAP
+✓
+×
+×
+✓
+✓
+×
+✓
+DOWNLOAD SIZE
+4.8 GB
+-
+22 GB
+1.4 TB
+48 GB
+120 GB
+58 GB
+# PUBLIC LEADERBOARD ENTRIES†
+194
+-
+935
+23
+18
+3
+-
+as well as feedback from more than 2,700 submissions by nearly 260 unique teams3 across 3
+competitions [43]:
+1. Motion forecasting is a safety critical system in a long-tailed domain. Consequently, our
+dataset is biased towards diverse and interesting scenarios containing different types of focal
+agents (see section 3.3.2). Our goal is to encourage the development of methods that ensure safety
+during tail events, rather than to optimize the expected performance on “easy miles”.
+2. There is a “Goldilocks zone” of task difficulty. Performance on the Argoverse 1 test set has
+begun to plateau, as shown in Figure 10 of the appendix. Argoverse 2 is designed to increase
+prediction difficulty incrementally, spurring productive focused research for the next few years.
+These changes are intended to incentivize methods that perform well on extended forecast horizons
+(3 s → 6 s), handle multiple types of dynamic objects (1 → 5), and ensure safety in scenarios
+from the long tail. Future Argoverse releases could continue to increase the problem difficulty by
+reducing observation windows and increasing forecasting horizons.
+3. Usability matters. Argoverse 1 benefited from a large and active research community—in large
+part due to the simplicity of setup and usage. Consequently, we took care to ensure that existing
+Argoverse models can be easily ported to run on Argoverse 2. In particular, we have prioritized
+intuitive access to map elements, encouraging methods which use the lane graph as a strong prior.
+To improve training and generalization, all poses have also been interpolated and resampled at
+exactly 10 Hz (Argoverse 1 was approximate). The new dataset includes fewer, but longer and
+more complex scenarios; this ensures that total dataset size remains large enough to train complex
+models but small enough to be readily accessible.
+3.3.1
+Data Representation
+The dataset consists of 250,000 non-overlapping scenarios (80/10/10 train/val/test random splits)
+mined from six unique urban driving environments in the United States. It contains a total of 10
+object types, with 5 from each of the dynamic and static categories (see Figure 4). Each scenario
+includes a local vector map and 11 s (10 Hz) of trajectory data (2D position, velocity, and orientation)
+for all tracks observed by the ego-vehicle in the local environment. The first 5 s of each scenario is
+denoted as the observed window, while the subsequent 6 s is denoted as the forecasted horizon.
+Within each scenario, we mark a single track as the “focal agent”. Focal tracks are guaranteed to
+be fully observed throughout the duration of the scenario and have been specifically selected to
+maximize interesting interactions with map features and other nearby actors (see Section 3.3.2). To
+evaluate multi-agent forecasting, we also mark a subset of tracks as “scored actors” (as shown in
+Figure 5), with guarantees for scenario relevance and minimum data quality.
+3This count includes private submissions not posted to the public leaderboards.
+6
+
+Figure 4: Object type and geographic histograms for the Motion Forecasting Dataset. Left: Histogram
+of object types over the “focal” and “scored” categories. Center: Histogram of object types over all
+tracks present in the dataset. The fine grained distinctions between different static object types (e.g.
+Construction Cone vs Riderless Bicycle) are unique among forecasting datasets. Right: Histogram of
+metropolitan areas included in the dataset.
+Figure 5: Visualization of a few interesting scenarios from the Motion Forecasting Dataset. The
+scenarios demonstrate a mix of the various object types (Vehicle, Pedestrian, Bus, Cyclist, or Motor-
+cyclist). The ego-vehicle is indicated in green, the focal agent is purple, and scored actors are orange.
+Other un-scored tracks are shown in blue. Object positions are captured at the last timestep of the
+observed history. For visualization purposes the full 5 s history and 6 s future are rendered for the
+focal agent, while only 1.5 s of future are shown for the other scored actors. Left shows a pedestrian
+crossing in front of the ego-vehicle, while center and right depict a motorcyclist weaving through
+traffic.
+3.3.2
+Mining Interesting Scenarios
+The source data for Argoverse 2 was drawn from fleet logs tagged with annotations consistent
+with interesting or difficult-to-forecast events. Each log was trimmed to 30 s and run through an
+interestingness scoring module in order to bias data selection towards examples from the long-tail of
+the natural distribution. We employ heuristics to score each track in the scene across five dimensions:
+object category, kinematics, map complexity, social context, and relation to the ego-vehicle (details
+in Appendix).
+The final scenarios are generated by extracting non-overlapping 11 s windows where at least one
+candidate track is fully observed for the entire duration. The highest scoring candidate track is
+denoted as the “focal agent”; all other fully observed tracks within 30 m of the ego-vehicle are
+denoted as “scored actors”. The resulting dataset is diverse, challenging, and still right-sized for
+widespread use (see the download size in Table 2). In Figure 6, we show that the resulting dataset is
+significantly more interesting than Argoverse 1.1 and validate our intuition that actors scoring highly
+in our heuristic module are more challenging to accurately forecast.
+3.4
+HD Maps
+Each scenario in the three datasets described above shares the same HD map representation. Each
+scenario carries its own local map region, similar to the Waymo Open Motion [12] dataset. This
+is a departure from the original Argoverse datasets in which all scenarios were localized onto two
+city-scale maps—one for Pittsburgh and one for Miami. In the Appendix, we provide examples.
+7
+
+106
+ActorCategory
+ScoredActor
+of Actors
+FocalAgent
+105
+#
+104
+103
+Cyclist
+Bus
+Vehicle
+Pedestrian
+Motorcyclist106
+: of Actors
+104
+#
+102
+100
+Bicycle
+Cyclist
+Bus
+Vehicle
+Static
+Pedestrian
+Unknown
+Construction
+Riderless E104
+Scenarios
+103
+102
+#
+101
+100
+Austin
+Alto
+Miami
+Pittsburgh
+Dearborn
+Palo大
+久久久久
+大0
+2
+4
+6
+8
+10
+12
+14
+Total Interestingness Score
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Miss Rate (K=6)
+Fitted Regression Model
+Bin Centers (1000 Scenarios Each)
+Figure 6: Left: Histogram comparing the distribution of interestingness scores assigned to focal
+agents in both Argoverse 1.1 and 2. Right: Plot showing the relationship between total interestingness
+score and prediction difficulty on the Argoverse 2 test split. We evaluate WIMP [25] over each
+scenario and fit a regression model to the computed miss rate (K=6, 2m threshold).
+Advantages of per-scenario maps include more efficient queries and their ability to handle map
+changes. A particular intersection might be observed multiple times in our datasets, and there could
+be changes to the lanes, crosswalks, or even ground height in that time.
+Lane graph. The core feature of the HD map is the lane graph, consisting of a graph G = (V, E),
+where V are individual lane segments. In the Appendix, we enumerate and define the attributes we
+provide for each lane segment. Unlike Argoverse 1, we provide the actual 3D lane boundaries, instead
+of only centerlines. However, our API provides code to quickly infer the centerlines at any desired
+sampling resolution. Polylines are quantized to 1 cm resolution. Our representation is richer than
+nuScenes, which provides lane geometry only in 2D, not 3D.
+Driveable area. Instead of providing driveable area segmentation in a rasterized format, as we did in
+Argoverse 1, we release it in a vector format, i.e. as 3D polygons. This offers multiple advantages,
+chiefly in compression, allowing us to store separate maps for tens of thousands of scenarios, yet the
+raster format is still easily derivable. The polygon vertices are quantized to 1 cm resolution.
+Ground surface height. Only the sensor dataset includes a dense ground surface height map
+(although other datasets still have sparse 3D height information on polylines). Ground surface height
+is provided for areas within a 5 m isocontour of the driveable area boundary, which we define as
+the region of interest (ROI) [6]. We do so because the notion of ground surface height is ill-defined
+for the interior of buildings and interior of densely constructed city blocks, areas where ground
+vehicles cannot observe due to occlusion. The raster grid is quantized to a 30 cm resolution, a higher
+resolution than the 1 m resolution in Argoverse 1.
+Area of Local Maps. Each scenario’s local map includes all entities found within a 100 m dilation
+in l2-norm from the ego-vehicle trajectory.
+4
+Experiments
+Argoverse 2 supports a variety of downstream tasks. In this section we highlight three different
+learning problems: 3D object detection, point cloud forecasting, and motion forecasting — each
+supported by the sensor, lidar, and motion forecasting datasets, respectively. First, we illustrate the
+challenging and diverse taxonomy within the Argoverse 2 sensor dataset by training a state-of-the-
+art 3D detection model on our twenty-six evaluation classes including “long-tail” classes such as
+stroller, wheel chairs, and dogs. Second, we showcase the utility of the Argoverse 2 lidar dataset
+through large-scale, self-supervised learning through the point cloud forecasting task. Lastly, we
+demonstrate motion forecasting experiments which provide the first baseline for broad taxonomy
+motion prediction.
+4.1
+3D Object Detection
+8
+
+0.14
+Mean
+Argoverse 1
+Proportion of Scenarios
+Argoverse 2
+Mean
+0.12
+Argoverse 1
+Argoverse 2
+0.10
+0.08
+0.06
+0.04
+0.02
+0.00
+0
+2
+4
+6
+8
+10
+12
+Kinematic Score + Social ScoreTable 3: 3d object detection results on the Argoverse 2 Sensor Dataset, taken from the leaderboard
+on Dec 21, 2022. Detectors is the winner of the CVPR 2022 Workshop on Autonomous Driving
+Argoverse 2 3D Object Detection challenge.
+METHOD
+MCDS (↑)
+MAP (↑)
+MATE (↓)
+MASE (↓)
+MAOE (↓)
+CENTERPOINT (OURS)
+0.14
+0.18
+0.49
+0.34
+0.72
+DETECTORS [13]
+0.34
+0.41
+0.40
+0.30
+0.54
+BEVFUSION [32]
+0.37
+0.46
+0.40
+0.30
+0.50
+Regular Vehicle
+Bus
+Pedestrian
+Stop Sign
+Box Truck
+Bollard
+Construction Barrel
+Motorcyclist
+Truck
+Bicyclist
+Mobile Crossing Sign
+Average Metrics
+Motorcycle
+Bicycle
+Articulated Bus
+School Bus
+Truck Cab
+Construction Cone
+Vehicular Trailer
+Sign
+Wheeled Device
+Large Vehicle
+Stroller
+Message Board Trailer
+Dog
+Wheeled Rider
+Wheelchair
+Class Names
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+AP
+Figure 7: Average precision of our 3D object de-
+tection baseline on the validation split of the Sen-
+sor Dataset (Beta). Our experiments showcase
+both our diverse taxonomy and difficult “long-tail”
+classes.
+We provide baseline 3D detection results using
+a state-of-the-art, anchorless 3D object detec-
+tion model – CenterPoint [51].
+Our Center-
+Point implementation takes a point cloud as
+input and crops it to a 200 m × 200 m grid
+with a voxel resolution of [0.1 m, 0.1 m] in the
+xy (bird’s-eye-view) plane and 0.2 m in the z-
+axis. To accommodate our larger taxonomy, we
+include six detection heads to encourage fea-
+ture specialization. Figure 7 characterizes the
+performance of our 3D detection baseline us-
+ing the nuScenes [4] average precision met-
+ric. Our large taxonomy allows us to evaluate
+classes such as “Wheeled Device” (e-Scooter),
+“Stroller”, “Dog”, and “Wheelchair” and we find
+that performance on these categories with strong
+baselines is poor despite significant amounts of
+training data.
+In Table 3, we provide a snapshot of submissions
+to the Argoverse 2 3D Object Detection Leaderboard.
+4.2
+Point Cloud Forecasting
+We perform point cloud forecasting according to the experimental protocol of SPF2 [49] using the
+Argoverse 2 Lidar Dataset. Given a sequence of past scene point clouds, a model is required to predict
+a sequence of future scene point clouds. We take the scene point clouds in the past 1 s (10 Hz) in
+the range image format as input, and then predict the next 1 s of range images. SPFNet predicts two
+output maps at each time step – the first output map is the predicted range values, while the second
+output is a validity mask. Previous point cloud forecasting models were evaluated on smaller datasets
+such as KITTI or nuScenes. To explore how the amount of training data affects the performance, we
+use increasing amounts of data for training the same model architecture, up to the full training set of
+16,000 sequences.
+Evaluation.
+We use three metrics to evaluate the performance of our forecasting model: mean IoU,
+l1-norm, and Chamfer distance. The mean IoU evaluates the predicted range mask. The l1-norm
+measures the average l1 distance between the pixel sets of predicted range image and the ground-
+truth image, which are both masked out by the ground-truth range mask. The Chamfer distance is
+obtained by adding up the Chamfer distances in both directions (forward and backward) between the
+ground-truth point cloud and the predicted scene point cloud which is obtained by back-projecting
+the predicted range image.
+Table 4: Results of point cloud forecasting on the test split of the Lidar Dataset.
+# TRAIN LOGS
+125
+250
+500
+1k
+2k
+4k
+16k
+MEAN IOU (%) (↑)
+55.5
+63.4
+61.7
+65.1
+68.0
+68.4
+70.9
+l1-NORM (↓)
+13.5
+12.5
+11.8
+9.9
+8.9
+8.7
+7.4
+CHAMFER DIST. (↓)
+31.1
+25.9
+22.4
+22.9
+20.5
+18.2
+14.0
+9
+
+Results of SPF2 and Discussion.
+Table 4 contains the results of our point cloud forecasting
+experiments. With increasing training data, the performance of the model grows steadily in all three
+metrics. These results and the works from the self-supervised learning literature [3, 7] indicate
+that a large amount of training data can make a substantial difference. Another observation is that
+the Chamfer distances for predictions on our dataset are significantly higher than predictions on
+KITTI [49]. We conjecture that this could be due to two reasons: (1) the Argoverse 2 Lidar Dataset
+has a much larger sensing range (above 200 m versus 120 m of the KITTI lidar sensor), which tends
+to significantly increase the value of Chamfer distance. (2) the Argoverse 2 Lidar Dataset has a higher
+proportion of dynamic scenes compared with KITTI Dataset.
+4.3
+Motion Forecasting
+We present several forecasting baselines [6] which try to make use of different aspects of the data.
+Those which are trained using the focal agent only and do not capture any social interaction include:
+constant velocity, nearest neighbor, and LSTM encoder-decoder models (both with and without a
+map-prior). We also evaluate WIMP [25] as an example of a graph-based attention method that
+captures social interaction. All hyper-parameters are obtained from the reference implementations.
+Evaluation.
+Baseline approaches are evaluated according to standard metrics. Following [6],
+we use minADE and minFDE as the metrics; they evaluate the average and endpoint L2 distance
+respectively, between the best forecasted trajectory and the ground truth. We also use Miss Rate
+(MR) which represents the proportion of test samples where none of the forecasted trajectories were
+within 2.0 meters of ground truth according to endpoint error. The resulting performance illustrates
+both the community’s progress on the problem and the significant increase in dataset difficulty when
+compared with Argoverse 1.1.
+Table 5: Performance of motion forecasting baseline methods on vehicle-like (vehicle, bus, mo-
+torcyclist) object types from the Argoverse 2 Motion Forecasting (Beta) Dataset. Usage of map
+prior indicates access to map information whereas usage of social context entails encoding other
+actors’ states in the feature representation. Mining intersection (multimodal) scenarios leads to poor
+performance at K=1 for all methods. Constant Velocity models have particularly poor performance
+due to the dataset bias towards kinematically interesting trajectories. Note that modern deep methods
+such as WIMP still have a miss rate of 0.42 at K=6, indicating the increased difficulty of the Argoverse
+2 dataset. Numbers within 1% of the best are in bold.
+K=1
+K=6
+MODEL
+MAP PRIOR
+SOCIAL CONTEXT
+MINADE ↓
+MINFDE ↓
+MR ↓
+MINADE ↓
+MINFDE ↓
+MR ↓
+CONST. VEL. [6]
+7.75
+17.44
+0.89
+-
+-
+-
+NN [6]
+4.46
+11.71
+0.81
+2.18
+4.94
+0.60
+NN [6]
+✓
+6.45
+15.51
+0.84
+4.30
+10.08
+0.78
+LSTM [6]
+3.05
+8.28
+0.85
+-
+-
+-
+LSTM [6]
+✓
+5.07
+12.71
+0.90
+3.73
+9.09
+0.85
+WIMP [25]
+✓
+✓
+3.09
+7.71
+0.84
+1.47
+2.90
+0.42
+Table 6: Motion forecasting results on the Argoverse 2 Motion Forecasting Dataset, taken from
+the online leaderboard on Dec 21, 2022. BANet is the winner of the CVPR 2022 Workshop on
+Autonomous Driving Argoverse 2 Motion Forecasting challenge (#1), and QML and GANet received
+honorable mention (HM) prizes. Entries are sorted below according to Brier-minFDE.
+K=1
+K=6
+METHOD
+MINADE ↓
+MINFDE ↓
+MR ↓
+MINADE ↓
+MINFDE ↓
+MR ↓
+BRIER-MINFDE ↓
+THOMAS (GOHOME SCALAR) [20]
+1.95
+4.71
+0.64
+0.88
+1.51
+0.20
+2.16
+GORELA (W/O ENSEMBLE) [11]
+1.82
+4.62
+0.61
+0.76
+1.48
+0.22
+2.01
+GANET (ENSEMBLE) (HM) [47]
+1.81
+4.57
+0.61
+0.73
+1.36
+0.17
+1.98
+GANET (W/O ENSEMBLE) [47]
+1.77
+4.48
+0.59
+0.72
+1.34
+0.17
+1.96
+QML (HM) [44]
+1.84
+4.98
+0.62
+0.69
+1.39
+0.19
+1.95
+BANET (OPPRED) (#1) [53]
+1.79
+4.61
+0.60
+0.71
+1.36
+0.19
+1.92
+Baseline Results. Table 5 summarizes the results of baselines. For K=1, Argoverse 1 [6] showed that
+a constant velocity model (minFDE=7.89) performed better than NN+map(prior) (minFDE=8.12),
+10
+
+which is not the case here. This further proves that Argoverse 2 is kinematically more diverse and
+cannot be solved by making constant velocity assumptions. Surprisingly, NN and LSTM variants that
+make use of a map prior perform worse than those which do not, illustrating the scope of improvement
+in how these baselines leverage the map. For K=6, WIMP significantly outperforms every other
+baseline. This emphasizes that it is imperative to train expressive models that can leverage map
+prior and social context along with making diverse predictions. The trends are similar to our past 3
+Argoverse Motion Forecasting competitions [43]: Graph-based attention methods (e.g. [25, 31, 37])
+continued to dominate the competition, and were nearly twice as accurate as the next best baseline
+(Nearest Neighbor) at K=6. That said, some of the rasterization-based (e.g. [19]) methods also
+showed promising results. Finally, we also evaluated baseline methods in the context of transfer
+learning and varied object types, the results of which are summarized in the Appendix.
+In Table 6, we provide a snapshot of submissions to the Argoverse 2 Motion Forecasting Leaderboard.
+5
+Conclusion
+Discussion. In this work, we have introduced three new datasets that constitute Argoverse 2. We
+provide baseline explorations for three tasks – 3d object detection, point cloud forecasting and motion
+forecasting. Our datasets provide new opportunities for many other tasks. We believe our datasets
+compare favorably to existing datasets, with HD maps, rich taxonomies, geographic diversity, and
+interesting scenes.
+Limitations. As in any human annotated dataset, there is label noise, although we seek to minimize
+it before release. 3D bounding boxes of objects are not included in the motion forecasting dataset,
+but one can make reasonable assumptions about the object extent given the object type. The motion
+forecasting dataset also has imperfect tracking, consistent with state-of-the-art 3D trackers.
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+[48] Xinshuo Weng, Jianren Wang, Sergey Levine, Kris Kitani, and Nicholas Rhinehart. 4d forecast-
+ing: Sequential forecasting of 100,000 points. In Proceedings of ECCV ’20 Workshops, August
+2020.
+[49] Xinshuo Weng, Jianren Wang, Sergey Levine, Kris Kitani, and Nick Rhinehart. Inverting the
+forecasting pipeline with spf2: Sequential pointcloud forecasting for sequential pose forecasting.
+In Proceedings of (CoRL) Conference on Robot Learning, November 2020.
+[50] Zhenpei Yang, Yuning Chai, Dragomir Anguelov, Yin Zhou, Pei Sun, Dumitru Erhan, Sean
+Rafferty, and Henrik Kretzschmar. Surfelgan: Synthesizing realistic sensor data for autonomous
+driving. In CVPR, June 2020.
+[51] Tianwei Yin, Xingyi Zhou, and Philipp Krahenbuhl. Center-based 3d object detection and
+tracking. In CVPR, June 2021.
+[52] Wei Zhan, Liting Sun, Di Wang, Haojie Shi, Aubrey Clausse, Maximilian Naumann, Julius
+Kummerle, Hendrik Konigshof, Christoph Stiller, Arnaud de La Fortelle, et al. Interaction
+dataset: An international, adversarial and cooperative motion dataset in interactive driving
+scenarios with semantic maps. arXiv preprint arXiv:1910.03088, 2019.
+[53] Chen Zhang, Honglin Sun, Chen Chen, and Yandong Guo. Banet: Motion forecasting with
+boundary aware network, 2022.
+[54] Jannik Zürn, Johan Vertens, and Wolfram Burgard. Lane graph estimation for scene understand-
+ing in urban driving. CoRR, abs/2105.00195, 2021.
+6
+Appendix
+6.1
+Additional Information About Sensor Suite
+In Figure 8, we provide a diagram of the sensor suite used to capture the Argoverse 2 datasets.
+Figure 9 shows the speed distribution for annotated pedestrian 3D cuboids and the yaw distribution.
+Figure 8: Car sensor schematic showing the three coordinate systems: (1) the vehicle frame in the
+rear axle; (2) the camera frame; and the lidar frame.
+6.2
+Additional Information About Motion Forecasting Dataset
+6.2.1
+Interestingness Scores
+Kinematic scoring selects for trajectories performing sharp turns or significant (de)accelerations. The
+map complexity program biases the data set towards trajectories complex traversals of the underlying
+lane graph. In particular, complex map regions, paths through intersections, and lane-changes score
+14
+
+Y ④Ydown
+O Zup0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+0
+50k
+100k
+Argoverse 2
+Waymo Open
+nuScenes
+Argoverse 1
+Speed (m/s)
+Number of 3D pedestrian cuboids
+ (speed > 0.5 m/s)
+−180 −135
+−90
+−45
+0
+45
+90
+135
+2
+5
+1000
+2
+5
+10k
+2
+5
+100k
+2
+5
+Argoverse 1
+Argoverse 2
+Yaw (degrees)
+Number of 3D cuboids
+Figure 9: Left: Number of moving 3D cuboids for pedestrians by speed distribution. We define
+moving objects when the speed is greater than 0.5 m/s. Right: Number of annotated 3D cuboids by
+yaw distribution.
+highly. Social scoring rewards tracks through dense regions of other actors. Social scoring also selects
+for non-vehicle object classes to ensure adequate samples from rare classes, such as motorcycles, for
+training and evaluation. Finally, the autonomous vehicle scoring program encourages the selection of
+tracks that intersect the ego-vehicle’s desired route.
+2019/10
+2019/11
+2019/12
+2020/02
+2020/03
+2020/05
+2020/06
+2020/07
+2020/08
+2020/09
+2020/10
+2020/11
+2020/12
+2021/01
+2021/02
+2021/03
+2021/04
+Year/Month of Submission
+0
+1
+2
+3
+4
+5
+6
+7
+8
+minFDE (K=6)
+State-of-art minFDE (K=6)
+Competition Phase
+Neurips 2019
+CVPR 2020
+None
+CVPR 2021
+Figure 10: MinFDE metric values for submissions on Argoverse 1.1 over time. Individual points
+indicate submissions to the public leader board. Colors indicate specific competition phases. The solid
+black line indicates SOTA performance. The research community made massive gains which have
+plateaued since early 2020. However, we note that the number and diversity of methods performing
+at or near the SOTA continues to grow. Additionally, later competitions sorted the leaderboard by
+“Miss Rate” and probability weighted FDE, and those metrics showed progress. Still, minFDE did
+not improve significantly.
+6.3
+Additional Information About HD Maps
+Examples of HD maps from the Sensor Dataset
+In Figure 12, we display examples of local HD
+maps associated with individual logs/scenarios.
+6.4
+Additional 3D Detection Results
+In Figure 13, we show additional evaluation metrics for our detection baseline.
+15
+
+Figure 11: Histogram of the number of actors (both scored and all types) present in the Motion
+Forecasting Dataset scenarios. The Lidar Dataset is mined by the same criteria and thus follows the
+same distribution.
+MAP ENTITY
+PROVIDED ATTRIBUTES
+TYPE
+DESCRIPTION
+LANE
+SEGMENTS
+IS_INTERSECTION
+BOOLEAN
+WHETHER OR NOT THIS LANE SEGMENT LIES WITHIN AN INTERSECTION.
+LANE TYPE
+ENUMERATED TYPE
+DESIGNATION OF WHICH VEHICLE TYPES MAY LEGALLY UTILIZE THIS LANE FOR TRAVEL.
+LEFT LANE BOUNDARY
+3D POLYLINE
+THE POLYLINE OF THE LEFT BOUNDARY IN THE CITY MAP COORDINATE SYSTEM
+RIGHT LANE BOUNDARY
+3D POLYLINE
+THE POLYLINE OF THE RIGHT BOUNDARY IN THE CITY MAP COORDINATE SYSTEM.
+LEFT LANE MARK TYPE
+ENUMERATED TYPE
+TYPE OF PAINTED LANE MARKING TO THE LEFT OF THE LANE SEGMENT ON THE ROAD.
+RIGHT LANE MARK TYPE
+ENUMERATED TYPE
+TYPE OF PAINTED LANE MARKING TO THE RIGHT OF THE LANE SEGMENT ON THE ROAD.
+LEFT NEIGHBOR
+INTEGER
+THE UNIQUE LANE SEGMENT IMMEDIATELY TO THE LEFT OF SEGMENT, OR NONE.
+RIGHT NEIGHBOR
+INTEGER
+THE UNIQUE LANE SEGMENT IMMEDIATELY TO THE RIGHT OF SEGMENT, OR NONE.
+SUCCESSOR IDS
+INTEGER LIST
+LANE SEGMENTS THAT MAY BE ENTERED BY FOLLOWING FORWARD.
+ID
+INTEGER
+UNIQUE IDENTIFIER
+DRIVABLE
+AREA
+AREA BOUNDARY
+3D POLYGONS
+AREA WHERE IT IS POSSIBLE FOR THE AV TO DRIVE WITHOUT DAMAGING ITSELF
+ID
+INTEGER
+UNIQUE IDENTIFIER
+PEDESTRIAN
+CROSSINGS
+EDGE1, EDGE2
+3D POLYLINES
+ENDPOINTS OF BOTH EDGE ALONG THE PRINCIPAL AXIS, THUS DEFINING A POLYGON.
+ID
+INTEGER
+UNIQUE IDENTIFIER
+GROUND SURFACE HEIGHT
+-
+2D RASTER ARRAY
+RASTER GRID QUANTIZED TO A 30 cm RESOLUTION.
+Table 7: HD map attributes for each Argoverse 2 scenario.
+Average Precision (AP)
+AP =
+1
+101
+�
+t∈T
+�
+r∈R
+pinterp(r)
+(1)
+True Positive Metrics Average Translation Error (ATE)
+ATE = ∥tdet − tgt∥2
+(2)
+Average Scaling Error (ASE)
+ASE = 1 −
+�
+d∈D
+min(ddet, dgt)
+max(ddet, dgt)
+(3)
+Average Orientation Error (AOE)
+AOE = |θdet − θgt|
+(4)
+Composite Detection Score (CDS)
+CDS = mAP ·
+�
+x∈X
+(1 − x)
+(5)
+where X = {mATEunit, mASEunit, mAOEunit}
+16
+
+104
+103
+102
+# 101
+100
+0
+6
+12
+18
+24
+31
+# of Scored Actors104
+Scenarios
+103
+102
+ofs
+#
+101
+0
+50
+100
+150
+200
+250
+# of Total Actors(a) Washington D.C.
+(b) Washington D.C.
+(c) Washington D.C.
+(d) Pittsburgh, PA
+(e) Pittsburgh, PA
+(f) Pittsburgh, PA
+(g) Miami, FL
+(h) Miami, FL
+(i) Miami, FL
+(j) Detroit, MI
+(k) Detroit, MI
+(l) Austin, TX
+Figure 12: Examples of egovehicle (AV) trajectories on local vector maps from the Sensor Dataset
+across several different cities. A 100m × 100m local map region is shown. Crosswalks are indicated
+in purple. Red circles denote the AV pose discretely sampled at 1 Hz for the purposes of illustration.
+Pose is provided at >20 Hz in the dataset, as indicated by the trajectory path indicated by a red line.
+City layouts vary dramatically, e.g. roadways in Miami are usually aligned parallel to a north-south,
+east-west grid, while roadways in Pittsburgh are generally not.
+17
+
+Regular Vehicle
+Bus
+Pedestrian
+Stop Sign
+Box Truck
+Bollard
+Construction Barrel
+Motorcyclist
+Truck
+Bicyclist
+Mobile Crossing Sign
+Average Metrics
+Motorcycle
+Bicycle
+Articulated Bus
+School Bus
+Truck Cab
+Construction Cone
+Vehicular Trailer
+Sign
+Wheeled Device
+Large Vehicle
+Stroller
+Message Board Trailer
+Dog
+Wheeled Rider
+Wheelchair
+Class Names
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+CDS
+Regular Vehicle
+Bus
+Pedestrian
+Stop Sign
+Box Truck
+Bollard
+Construction Barrel
+Motorcyclist
+Truck
+Bicyclist
+Mobile Crossing Sign
+Average Metrics
+Motorcycle
+Bicycle
+Articulated Bus
+School Bus
+Truck Cab
+Construction Cone
+Vehicular Trailer
+Sign
+Wheeled Device
+Large Vehicle
+Stroller
+Message Board Trailer
+Dog
+Wheeled Rider
+Wheelchair
+Class Names
+0.0
+0.5
+1.0
+1.5
+2.0
+ATE
+Regular Vehicle
+Bus
+Pedestrian
+Stop Sign
+Box Truck
+Bollard
+Construction Barrel
+Motorcyclist
+Truck
+Bicyclist
+Mobile Crossing Sign
+Average Metrics
+Motorcycle
+Bicycle
+Articulated Bus
+School Bus
+Truck Cab
+Construction Cone
+Vehicular Trailer
+Sign
+Wheeled Device
+Large Vehicle
+Stroller
+Message Board Trailer
+Dog
+Wheeled Rider
+Wheelchair
+Class Names
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+ASE
+Regular Vehicle
+Bus
+Pedestrian
+Stop Sign
+Box Truck
+Bollard
+Construction Barrel
+Motorcyclist
+Truck
+Bicyclist
+Mobile Crossing Sign
+Average Metrics
+Motorcycle
+Bicycle
+Articulated Bus
+School Bus
+Truck Cab
+Construction Cone
+Vehicular Trailer
+Sign
+Wheeled Device
+Large Vehicle
+Stroller
+Message Board Trailer
+Dog
+Wheeled Rider
+Wheelchair
+Class Names
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+AOE
+Figure 13: 3D object detection performance on the validation split of the Sensor Dataset (Beta).
+Top Row: Composite detection score (left). Average translation error (right) Bottom Row: Average
+scaling error (left), and average orientation error (right). Results are shown on the validation set of
+the Sensor Dataset.
+6.5
+Training Details of SPF2 baseline
+We sample 2-second training snippets (representing 1 second of past and 1 second of future data)
+every 0.5 seconds. Thus, for a training log with 30 second duration, 59 training snippets would be
+sampled. We train the model for 16 epochs by using the Adam optimizer with the learning rate of
+4e − 3, betas of 0.9 and 0.999, and batch size of 16 per GPU.
+6.6
+Additional Motion Forecasting Experiments
+6.6.1
+Transfer Learning
+The results of transfer learning experiments are summarized in Table 8. WIMP was trained and tested
+in different settings with Argoverse 1.1 and Argoverse 2. As expected, the model works best when
+it is trained and tested on the same distribution (i.e. both train and test data come from Argoverse
+1.1, or both from Argoverse 2). For example, when WIMP is tested on Argoverse 2 (6s), the model
+trained on Argoverse 2 (6s) has a minFDE of 2.91, whereas the one trained on Argoverse 1.1 (3s) has
+a minFDE of 6.82 (i.e. approximately 2.3x worse). Likewise, in the reverse setting, when WIMP is
+tested on Argoverse 1.1 (3s), the model trained on Argoverse 1.1 (3s) has a minFDE of 1.14 and the
+one trained on Argoverse 2 (6s) has minFDE of 2.05 (i.e. approximately 1.8x worse). This indicates
+that transfer learning from Argoverse 2 (Beta) to Argoverse 1.1 is more useful than the reverse setting,
+despite being smaller in the number of scenarios. However, the publicly released version of Argoverse
+2 Motion Forecasting (the non-beta 2.0 version) has comparable size with Argoverse 1.1.
+We note that it is a common practice to train and test sequential models on varied sequence length (e.g.
+machine translation). As such, it is still reasonable to expect a model trained with 3s to do well on 6s
+horizon. Several factors may contribute to distribution shift, including differing prediction horizon,
+cities, mining protocols, object types. Notably, however, these results indicate that Argoverse 2 is
+significantly more challenging and diverse than its predecessor.
+18
+
+6.6.2
+Experiment with different object types
+Table 9 shows the results on Nearest Neighbor baseline (without map prior) on different object types.
+As one would expect, the displacement errors in pedestrians are significantly lower than other object
+types. This occurs because they move at significantly slower velocities. However, this does not imply
+that pedestrian motion forecasting is a solved problem and one should rather focus on other object
+types. This instead means that we need to come up with better metrics that can capture that fact lower
+displacement errors in pedestrians can often be more critical than higher errors in vehicles. We leave
+this line of work for future scope.
+Table 8: Performance of WIMP when trained and tested on different versions of Argoverse motion
+forecasting datasets. Training and evaluation is restricted to vehicle-like (vehicle, bus, motorcyclist)
+object types as only vehicles were present in Argoverse 1.1. All the results are for K=6, and prediction
+horizon is specified in parentheses. Notably, the model trained on a 3s horizon performs poorly on the
+longer 6s horizon. ‘Argoverse 2’ below denotes the Argoverse 2 (Beta) Motion Forecasting Dataset.
+Train Split (pred. horizon)
+Test Split (pred. horizon)
+minADE ↓
+minFDE ↓
+MR ↓
+Argoverse 1.1 (3s)
+Argoverse 1.1 (3s)
+0.75
+1.14
+0.12
+Argoverse 2 (6s)
+Argoverse 1.1 (3s)
+1.68
+2.05
+0.27
+Argoverse 1.1 (3s)
+Argoverse 2 (3s)
+0.94
+1.88
+0.26
+Argoverse 1.1 (3s)
+Argoverse 2 (6s)
+4.93
+6.82
+0.77
+Argoverse 2 (6s)
+Argoverse 2 (6s)
+1.48
+2.91
+0.43
+Table 9: Performance of Nearest Neighbor baseline on different object types for K=6. The most
+accurately predicted object type for each evaluation metric is highlighted in bold.
+Object Type
+#Samples
+minADE ↓
+minFDE ↓
+MR ↓
+All
+9955
+2.48
+5.52
+0.66
+Vehicle
+8713
+2.62
+5.87
+0.70
+Bus
+439
+2.69
+5.59
+0.73
+Pedestrians
+677
+0.69
+1.31
+0.17
+Motorcyclist
+39
+2.33
+5.07
+0.61
+Cyclist
+87
+1.48
+2.80
+0.42
+7
+Datasheet for Argoverse 2
+For what purpose was the dataset created?
+Was there a specific task in mind? Was there a
+specific gap that needed to be filled? Please provide a description.
+Argoverse was created to support the global research community in improving the state of the art in
+machine learning tasks vital for self driving. The Argoverse 2 datasets described in this manuscript
+improve upon the initial Argoverse datasets. These datasets support many tasks, from 3D perception
+to motion forecasting to HD map automation.
+The three datasets proposed in this manuscript address different gaps in this space. See the comparison
+charts in the main manuscript for a more detailed breakdown.
+The Argoverse 2 Sensor Dataset has a richer taxonomy than similar datasets. It is the only dataset of
+similar size to have stereo imagery. The 1,000 logs in the dataset were chosen to have a variety of
+object types with diverse interactions.
+The Argoverse 2 Motion Forecasting Dataset also has a richer taxonomy than existing datasets. The
+scenarios in the dataset were mined with an emphasis on unusual behaviors that are difficult to predict.
+The Argoverse 2 Lidar Dataset is the largest Lidar Dataset. Only the concurrent ONCE dataset is
+similarly sized to enable self-supervised learning in lidar space. Unlike ONCE, our dataset contains
+HD maps and high frame rate lidar.
+Who created this dataset (e.g., which team, research group) and on behalf of which entity
+(e.g., company, institution, organization)?
+19
+
+The Argoverse 2 datasets were created by researchers at Argo AI.
+What support was needed to make this dataset? (e.g.who funded the creation of the dataset? If
+there is an associated grant, provide the name of the grantor and the grant name and number, or if it
+was supported by a company or government agency, give those details.)
+The creation of this dataset was funded by Argo AI.
+Any other comments?
+n/a
+COMPOSITION
+What do the instances that comprise the dataset represent (e.g., documents, photos, people,
+countries)?
+Are there multiple types of instances (e.g., movies, users, and ratings; people and
+interactions between them; nodes and edges)? Please provide a description.
+The three constituent datasets of Argoverse 2 have different attributes, but the core instances for each
+are brief “scenarios” or “logs” of 11, 15, or 30 seconds that represent a continuous observation of a
+scene around a self-driving vehicle.
+Each scenario in all three datasets has an HD map that includes lane boundaries, crosswalks, driveable
+area, etc. Scenarios for the Sensor Dataset additionally contain a raster map of ground height at .3
+meter resolution.
+How many instances are there in total (of each type, if appropriate)?
+The Sensor Dataset has 1,000 15 second scenarios.
+The Lidar Dataset has 20,000 30 second scenarios.
+The Motion Forecasting Dataset has 250,000 11 second scenarios.
+Does the dataset contain all possible instances or is it a sample (not necessarily random) of
+instances from a larger set?
+If the dataset is a sample, then what is the larger set? Is the
+sample representative of the larger set (e.g., geographic coverage)? If so, please describe how
+this representativeness was validated/verified. If it is not representative of the larger set, please
+describe why not (e.g., to cover a more diverse range of instances, because instances were withheld
+or unavailable).
+The scenarios in the dataset are a sample of the set of observations made by a fleet of self-driving
+vehicles. The data is not uniformly sampled. The particular samples were chosen to be geographically
+diverse (spanning 6 cities - Pittsburgh, Detroit, Austin, Palo Alto, Miami, and Washington D.C.), to
+include interesting behavior (e.g. cars making unexpected maneuvers), to contain interesting weather
+(e.g. rain and snow), and to contain scenes with many objects of diverse types in motion (e.g. a
+crowd walking, riders on e-scooters splitting lanes between many vehicles, an excavator operating at
+a construction site, etc.).
+What data does each instance consist of?
+“Raw” data (e.g., unprocessed text or images) or
+features? In either case, please provide a description.
+Each Sensor Dataset scenario is 15 seconds in duration. Each scenario has 20 fps video from 7 ring
+cameras, 20 fps video from two forward facing stereo cameras, and 10 hz lidar returns from two
+out-of-phase 32 beam lidars. The ring cameras are synchronized to fire when either lidar sweeps
+through their field of view. Each scenario contains vehicle pose over time and calibration data to
+relate the various sensors.
+Each Lidar Dataset scenario is 30 seconds in duration. These scenarios are similar to those of the
+Sensor Dataset, except that there is no imagery.
+Each Motion Forecasting scenario is 11 seconds in duration. These scenarios contain no sensor data,
+but instead contain tracks of objects such as vehicles, pedestrians, and bicycles. The tracks specify
+the category of each object (e.g. bus or bicycle) as well as their location and heading at a 10 hz
+sampling interval.
+20
+
+The HD map associated with all three types of scenarios contains polylines describing lanes, cross-
+walks, and driveable area. Lanes form a graph with predecessors and successors, e.g. a lane that
+splits can have two successors. Lanes have precisely localized lane boundaries that include paint
+type (e.g. double solid yellow). Driveable area, also described by a polygon, is the area where it is
+possible but not necessarily legal to drive. It includes areas such as road shoulders.
+Is there a label or target associated with each instance? If so, please provide a description.
+Each Sensor Dataset scenario has 3D track annotations for dynamic objects such as vehicles, pedes-
+trians, strollers, dogs, etc. The tracks are suitable as ground truth for tasks such as 3D object detection
+and 3D tracking. The 3D track labels are intentionally held out from the test set. The HD map
+could also be thought of as labels for each instance, and would be suitable as ground truth for lane
+detection or map automation. The vehicle pose data could be considered ground truth labels for visual
+odometry. The lidar depth estimates can act as ground truth for monocular or stereo depth estimation.
+The Lidar Dataset does not have human annotations beyond the HD map. Still, the evolving point
+cloud itself can be considered ground truth for point cloud forecasting.
+Each Motion Forecasting Dataset scenario provides labels specifying which tracks are associated
+with “scored actors”. These tracks exhibit interesting behavior and are guaranteed to be observed
+over the entire duration of each scenario; algorithms will be asked to forecast the future motion for
+these tracks. The future motion of actors in each scenario is intentionally held out in the test set.
+Is any information missing from individual instances?
+If so, please provide a description,
+explaining why this information is missing (e.g., because it was unavailable). This does not include
+intentionally removed information, but might include, e.g., redacted text.
+In the Sensor Dataset, objects are only labeled within 5 meters of the driveable area. For example, a
+person sitting on their front porch will not be labeled.
+In the Sensor Dataset and Motion Forecasting Dataset, instances are not necessarily labeled for the
+full duration of each scenario if the objects move out of observation range or become occluded.
+Z Are relationships between individual instances made explicit (e.g., users’ movie ratings,
+social network links)? If so, please describe how these relationships are made explicit.
+The instances of the three datasets are disjoint. They each carry their own HD map for the region
+around the scenario. These HD maps may overlap spatially, though. For example, many forecasting
+scenarios may take place in the same intersection. If a user of the dataset wanted to recover the
+spatial relationship between scenarios, they could do so through the Argoverse API.
+Are there recommended data splits (e.g., training, development/validation, testing)?
+If so,
+please provide a description of these splits, explaining the rationale behind them.
+We define splits of each dataset. The Sensor Dataset is split 700 / 150 / 150 between train, validation,
+and test. The Lidar Dataset is split 16,000 / 2,000 / 2,000 and the Motion Forecasting Dataset is split
+200,000 / 25,000 / 25,000. In all cases, the splits are designed to make the training dataset as large
+as possible while keeping the validation and test datasets large and diverse enough to accurately
+benchmark models learned on the training set.
+Are there any errors, sources of noise, or redundancies in the dataset? If so, please provide a
+description.
+Every sensor used in the dataset – ring cameras, stereo cameras, and lidar – has noise associated with
+it. Pixel intensities, lidar intensities, and lidar point 3D locations all have noise. Lidar points are
+also quantized to float16 which leads to roughly a centimeter of quantization error. Six degree of
+freedom vehicle pose also has noise. The calibration specifying the relationship between sensors can
+be imperfect.
+The HD map for each scenario can contain noise, both in terms of lane boundary locations and precise
+ground height.
+The 3D object annotations in the Sensor Dataset do not always match the spatial extent and motion of
+an object in the real world. For example, we assume that objects do not change size during a scenario,
+but this could be violated by a car opening its door. 3D annotations for distant objects with relatively
+few pixels and lidar returns are less accurate.
+21
+
+The object tracks in the Motion Forecasting dataset are imperfect and contain errors typical of a
+real-time 3D tracking method. Our expectation is that a motion forecasting algorithm should operate
+well despite this noise.
+Is the dataset self-contained, or does it link to or otherwise rely on external resources (e.g.,
+websites, tweets, other datasets)?
+If it links to or relies on external resources, a) are there
+guarantees that they will exist, and remain constant, over time; b) are there official archival versions
+of the complete dataset (i.e., including the external resources as they existed at the time the dataset
+was created); c) are there any restrictions (e.g., licenses, fees) associated with any of the external
+resources that might apply to a future user? Please provide descriptions of all external resources and
+any restrictions associated with them, as well as links or other access points, as appropriate.
+The data itself is self-hosted, like Argoverse 1 [see https://www.argoverse.org/], and we
+maintain public links to all previous versions of the dataset in case of updates. The data is independent
+of any previous datasets, including Argoverse 1.
+Does the dataset contain data that might be considered confidential (e.g., data that is
+protected by legal privilege or by doctor-patient confidentiality, data that includes the content
+of individuals’ non-public communications)? If so, please provide a description.
+No.
+Does the dataset contain data that, if viewed directly, might be offensive, insulting, threaten-
+ing, or might otherwise cause anxiety? If so, please describe why.
+No.
+Does the dataset relate to people? If not, you may skip the remaining questions in this section.
+Yes, the dataset contains images and behaviors of thousands of people on public streets.
+Does the dataset identify any subpopulations (e.g., by age, gender)? If so, please describe how
+these subpopulations are identified and provide a description of their respective distributions within
+the dataset.
+No.
+Is it possible to identify individuals (i.e., one or more natural persons), either directly or
+indirectly (i.e., in combination with other data) from the dataset? If so, please describe how.
+We do not believe so. All image data has been anonymized. Faces and license plates are obfuscated
+by replacing them with a 5x5 grid, where each grid cell is the average color of the original pixels in
+that grid cell. This anonymization is done manually and is not limited by our 3D annotation policy.
+For example, a person sitting on their front porch 10 meters from the road would not be labeled with
+a 3D cuboid, but their face would still be obscured.
+Does the dataset contain data that might be considered sensitive in any way (e.g., data that
+reveals racial or ethnic origins, sexual orientations, religious beliefs, political opinions or
+union memberships, or locations; financial or health data; biometric or genetic data; forms of
+government identification, such as social security numbers; criminal history)?
+If so, please
+provide a description.
+No.
+Any other comments?
+n/a
+COLLECTION
+How was the data associated with each instance acquired?
+Was the data directly observable
+(e.g., raw text, movie ratings), reported by subjects (e.g., survey responses), or indirectly
+inferred/derived from other data (e.g., part-of-speech tags, model-based guesses for age or language)?
+If data was reported by subjects or indirectly inferred/derived from other data, was the data
+22
+
+validated/verified? If so, please describe how.
+The sensor data was directly acquired by a fleet of autonomous vehicles.
+Over what timeframe was the data collected? Does this timeframe match the creation timeframe
+of the data associated with the instances (e.g., recent crawl of old news articles)? If not, please
+describe the timeframe in which the data associated with the instances was created. Finally, list when
+the dataset was first published.
+The data was collected in 2020 and 2021. The dataset was made public after NeurIPS 2021, in March
+2022.
+What mechanisms or procedures were used to collect the data (e.g., hardware apparatus
+or sensor, manual human curation, software program, software API)?
+How were these
+mechanisms or procedures validated?
+The Argoverse 2 data comes from Argo ‘Z1’ fleet vehicles. These vehicles use Velodyne lidars and
+traditional RGB cameras. All sensors are calibrated by Argo. HD maps and 3D object annotations
+are created and validated through a combination of computational tools and human annotations.
+Object tracks in the Motion Forecasting Dataset are created by a 3D tracking algorithm.
+What was the resource cost of collecting the data? (e.g. what were the required computational
+resources, and the associated financial costs, and energy consumption - estimate the carbon footprint.
+See Strubell et al. for approaches in this area.)
+The data was captured during normal fleet operations, so there was minimal overhead for logging
+particular events. The transformation and post-processing of several terabytes of data consumed an
+estimated 1,000 machine hours. We estimate a Carbon footprint of roughly 1,000 lbs based on the
+CPU-centric workload.
+If the dataset is a sample from a larger set, what was the sampling strategy (e.g., deterministic,
+probabilistic with specific sampling probabilities)?
+The Sensor Dataset scenarios were chosen from a larger set through manual review. The Lidar
+Dataset and Motion Forecasting Dataset scenarios were chosen by heuristics which looked for
+interesting object behaviors during fleet operations.
+Who was involved in the data collection process (e.g., students, crowdworkers, contractors)
+and how were they compensated (e.g., how much were crowdworkers paid)?
+Argo employees and Argo interns curated the data. Data collection and data annotation was done by
+Argo employees. Crowdworkers were not used.
+Were any ethical review processes conducted (e.g., by an institutional review board)?
+If so,
+please provide a description of these review processes, including the outcomes, as well as a link or
+other access point to any supporting documentation.
+No.
+Does the dataset relate to people?
+If not, you may skip the remainder of the questions in this
+section.
+Yes.
+Did you collect the data from the individuals in question directly, or obtain it via third parties
+or other sources (e.g., websites)?
+The data is collected from vehicles on public roads, not from a third party.
+Were the individuals in question notified about the data collection? If so, please describe (or
+show with screenshots or other information) how notice was provided, and provide a link or other
+access point to, or otherwise reproduce, the exact language of the notification itself.
+No, but the data collection was not hidden. The Argo fleet vehicles are well marked and have obvious
+23
+
+cameras and lidar sensors. The vehicles only capture data from public roads.
+Did the individuals in question consent to the collection and use of their data?
+If so, please
+describe (or show with screenshots or other information) how consent was requested and provided,
+and provide a link or other access point to, or otherwise reproduce, the exact language to which the
+individuals consented.
+No. People in the dataset were in public settings and their appearance has been anonymized. Drivers,
+pedestrians, and vulnerable road users are an intrinsic part of driving on public roads, so it is impor-
+tant that datasets contain people so that the community can develop more accurate perception systems.
+If consent was obtained, were the consenting individuals provided with a mechanism to
+revoke their consent in the future or for certain uses? If so, please provide a description, as well
+as a link or other access point to the mechanism (if appropriate)
+n/a
+Has an analysis of the potential impact of the dataset and its use on data subjects (e.g., a data
+protection impact analysis) been conducted? If so, please provide a description of this analysis,
+including the outcomes, as well as a link or other access point to any supporting documentation.
+No.
+Any other comments?
+n/a
+PREPROCESSING / CLEANING / LABELING
+Was any preprocessing/cleaning/labeling of the data done (e.g., discretization or bucketing,
+tokenization, part-of-speech tagging, SIFT feature extraction, removal of instances, processing
+of missing values)? If so, please provide a description. If not, you may skip the remainder of the
+questions in this section.
+Yes. Images are reduced from their full resolution. 3D point locations are quantized to float16.
+Ground height maps are quantized to .3 meter resolution from their full resolution. HD map polygon
+vertex locations are quantized to .01 meter resolution. 3D annotations are smoothed. For the Motion
+Forecasting Dataset, transient 3D tracks are suppressed and object locations are smoothed over time.
+Was the “raw” data saved in addition to the preprocessed/cleaned/labeled data (e.g., to
+support unanticipated future uses)? If so, please provide a link or other access point to the “raw”
+data.
+Yes, but such data is not public.
+Is the software used to preprocess/clean/label the instances available? If so, please provide a
+link or other access point.
+No.
+Any other comments?
+n/a
+USES
+Has the dataset been used for any tasks already? If so, please provide a description.
+Yes, this manuscript benchmarks a contemporary 3D object detection method on the Sensor Dataset
+and a contemporary motion forecasting method on the Motion Forecasting Dataset.
+24
+
+Is there a repository that links to any or all papers or systems that use the dataset?
+If so,
+please provide a link or other access point.
+Yes, the Argoverse 2 API can be found at https://github.com/argoverse/av2-api.
+For the Argoverse 2 datasets, we maintain two leaderboards for 3D Detection [https://eval.ai/
+web/challenges/challenge-page/1710] and Motion Forecasting [https://eval.ai/web/
+challenges/challenge-page/1719].
+For
+the
+Argoverse
+1
+datasets,
+we
+maintain
+four
+leaderboards
+for
+3D
+Tracking
+[https://eval.ai/web/challenges/challenge-page/453/overview],
+3D
+Detection
+[https://eval.ai/web/challenges/challenge-page/725/overview],
+Motion Forecast-
+ing
+[https://eval.ai/web/challenges/challenge-page/454/overview],
+and
+Stereo
+Depth Estimation [https://eval.ai/web/challenges/challenge-page/917/overview].
+Argoverse 1 was also used as the basis for a Streaming Perception challenge [https:
+//eval.ai/web/challenges/challenge-page/800/overview].
+What (other) tasks could the dataset be used for?
+The datasets could be used for research on visual odometry, pose estimation, lane detection, map
+automation, self-supervised learning, structure-from-motion, scene flow, optical flow, time to contact
+estimation, pseudo-lidar, and point cloud forecasting.
+Is there anything about the composition of the dataset or the way it was collected and
+preprocessed/cleaned/labeled that might impact future uses?
+For example, is there anything
+that a future user might need to know to avoid uses that could result in unfair treatment of individuals
+or groups (e.g., stereotyping, quality of service issues) or other undesirable harms (e.g., financial
+harms, legal risks) If so, please provide a description. Is there anything a future user could do to
+mitigate these undesirable harms?
+No.
+Are there tasks for which the dataset should not be used? If so, please provide a description.
+The dataset should not be used for tasks which depend on faithful appearance of faces or license
+plates since that data has been obfuscated. For example, running a face detector to try and estimate
+how often pedestrians use crosswalks will not result in meaningful data.
+Any other comments?
+n/a
+DISTRIBUTION
+Will the dataset be distributed to third parties outside of the entity (e.g., company, institution,
+organization) on behalf of which the dataset was created? If so, please provide a description.
+Yes, the dataset is hosted on https://www.argoverse.org/ like Argoverse 1 and 1.1.
+How will the dataset will be distributed (e.g., tarball on website, API, GitHub)?
+Does the
+dataset have a digital object identifier (DOI)?
+We provide both tar.gz archives and raw files for two of the Argoverse 2 datasets (Motion Forecasting,
+Sensor), but provide only raw files for the Lidar datasets), available via AWS transfer. See https:
+//www.argoverse.org/av2.html#download-link.
+The Argoverse 1 and Argoverse 1.1 were distributed as a series of tar.gz files (See
+https://www.argoverse.org/av1.html#download-link.
+The files are broken up to
+make the process more robust to interruption (e.g. a single 2 TB file failing after 3 days would be
+frustrating) and to allow easier file manipulation (an end user might not have 2 TB free on a single
+drive, and if they do they might not be able to decompress the entire file at once).
+25
+
+When will the dataset be distributed?
+The data was made available for download after NeurIPS 2021, in March 2022.
+Will the dataset be distributed under a copyright or other intellectual property (IP) license,
+and/or under applicable terms of use (ToU)? If so, please describe this license and/or ToU, and
+provide a link or other access point to, or otherwise reproduce, any relevant licensing terms or ToU,
+as well as any fees associated with these restrictions.
+Yes, the dataset was released under the same Creative Commons license as Argoverse 1 – CC BY-
+NC-SA 4.0. Details can be seen at https://www.argoverse.org/about.html#terms-of-use.
+Have any third parties imposed IP-based or other restrictions on the data associated with
+the instances?
+If so, please describe these restrictions, and provide a link or other access point
+to, or otherwise reproduce, any relevant licensing terms, as well as any fees associated with these
+restrictions.
+No.
+Do any export controls or other regulatory restrictions apply to the dataset or to individual
+instances? If so, please describe these restrictions, and provide a link or other access point to, or
+otherwise reproduce, any supporting documentation.
+No.
+Any other comments?
+n/a
+MAINTENANCE
+Who is supporting/hosting/maintaining the dataset?
+Argo AI
+How can the owner/curator/manager of the dataset be contacted (e.g., email address)?
+The Argoverse team responds through the Github page for the Argoverse 2 API: https://github.
+com/argoverse/av2-api/issues.
+The Argoverse team responds through the Github page for the Argoverse 1 API: https://github.
+com/argoverse/argoverse-api/issues. It currently contains 2 open issues and 126 closed
+issues.
+For privacy concerns, contact information can be found here: https://www.argoverse.org/
+about.html#privacy
+Is there an erratum? If so, please provide a link or other access point.
+No.
+Will the dataset be updated (e.g., to correct labeling errors, add new instances, delete
+instances)? If so, please describe how often, by whom, and how updates will be communicated to
+users (e.g., mailing list, GitHub)?
+It is possible that the constituent Argoverse 2 datasets are updated to correct errors. This was the
+case with Argoverse 1 which was incremented to Argoverse 1.1. Updates will be communicated on
+Github and through our mailing list.
+If the dataset relates to people, are there applicable limits on the retention of the data
+associated with the instances (e.g., were individuals in question told that their data would be
+retained for a fixed period of time and then deleted)?
+If so, please describe these limits and
+explain how they will be enforced.
+26
+
+No.
+Will older versions of the dataset continue to be supported/hosted/maintained? If so, please
+describe how. If not, please describe how its obsolescence will be communicated to users.
+Yes.
+We still host Argoverse 1 even though we have declared it “deprecated”.
+See
+https://www.argoverse.org/av1.html#download-link.
+We will use the same warn-
+ing if we ever deprecate Argoverse 2. Note: Argoverse 2 does not deprecate Argoverse 1. They are
+independent collections of datasets.
+If others want to extend/augment/build on/contribute to the dataset, is there a mechanism for
+them to do so? If so, please provide a description. Will these contributions be validated/verified? If
+so, please describe how. If not, why not? Is there a process for communicating/distributing these
+contributions to other users? If so, please provide a description.
+Yes. For example, the streaming perception challenge was built by CMU researchers who added new
+2D object annotations to Argoverse 1.1 data. The Creative Commons license we use for Argoverse 2
+ensures that the community can do the same thing without needing Argo’s permission.
+We do not have a mechanism for these contributions/additions to be incorporated back into the ‘base’
+Argoverse 2. Our preference would generally be to keep the ‘base’ dataset as is, and to give credit to
+noteworthy additions by linking to them as we have done in the case of the Streaming Perception
+Challenge (see link at the top of this Argoverse page https://www.argoverse.org/tasks.html).
+Any other comments?
+n/a
+Environmental Impact Statement. Amount of Compute Used: We estimate 2,000 CPU and 500
+GPU hours were used in the collection of the dataset and the performance of baseline experiments.
+27
+
diff --git a/xtAyT4oBgHgl3EQfnvhT/content/tmp_files/load_file.txt b/xtAyT4oBgHgl3EQfnvhT/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,1413 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf,len=1412
+page_content='Argoverse 2: Next Generation Datasets for Self-Driving Perception and Forecasting Benjamin Wilson∗†,1, William Qi∗†, Tanmay Agarwal∗†, John Lambert†, Jagjeet Singh†, Siddhesh Khandelwal2, Bowen Pan†,3, Ratnesh Kumar†, Andrew Hartnett†, Jhony Kaesemodel Pontes†, Deva Ramanan†,4, Peter Carr†, James Hays†,1 1Georgia Tech, 2UBC, 3MIT, 4CMU Abstract We introduce Argoverse 2 (AV2) — a collection of three datasets for perception and forecasting research in the self-driving domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The annotated Sensor Dataset con- tains 1,000 sequences of multimodal data, encompassing high-resolution imagery from seven ring cameras, and two stereo cameras in addition to lidar point clouds, and 6-DOF map-aligned pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Sequences contain 3D cuboid annotations for 26 object categories, all of which are sufficiently-sampled to support training and evaluation of 3D perception models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The Lidar Dataset contains 20,000 sequences of unlabeled lidar point clouds and map-aligned pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' This dataset is the largest ever collection of lidar sensor data and supports self-supervised learning and the emerging task of point cloud forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Finally, the Motion Forecasting Dataset contains 250,000 scenarios mined for interesting and challenging interactions be- tween the autonomous vehicle and other actors in each local scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Models are tasked with the prediction of future motion for “scored actors" in each scenario and are provided with track histories that capture object location, heading, velocity, and category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' In all three datasets, each scenario contains its own HD Map with 3D lane and crosswalk geometry — sourced from data captured in six distinct cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We believe these datasets will support new and existing machine learning research problems in ways that existing datasets do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' All datasets are released under the CC BY-NC-SA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='0 license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 1 Introduction In order to achieve the goal of safe, reliable autonomous driving, a litany of machine learning tasks must be addressed, from stereo depth estimation to motion forecasting to 3D object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' In recent years, numerous high quality self-driving datasets have been released to support research into these and other important machine learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Many datasets are annotated “sensor” datasets [4, 45, 39, 40, 24, 33, 18, 14, 41, 36] in the spirit of the influential KITTI dataset [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The Argoverse 3D Tracking dataset [6] was the first such dataset with “HD maps” — maps containing lane-level geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Also influential are self-driving “motion prediction” datasets [12, 22, 34, 4, 52] — containing abstracted object tracks instead of raw sensor data — of which the Argoverse Motion Forecasting dataset [6] was the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' In the last two years, the Argoverse team has hosted six competitions on 3D tracking, stereo depth estimation, and motion forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We maintain evaluation servers and leaderboards for these tasks, Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' †Work completed while at Argo AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 35th Conference on Neural Information Processing Systems (NeurIPS 2021) Track on Datasets and Benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='00493v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='CV] 2 Jan 2023 as well as 3D detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The leaderboards collectively contain thousands of submissions from four hundred teams1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We also maintain the Argoverse API and have addressed more than one hundred issues2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' From these experiences we have formed the following guiding principles to guide the creation of the next iteration of Argoverse datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Bigger isn’t always better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Self-driving vehicles capture a flood of sensor data which is logisti- cally difficult to work with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Sensor datasets are several terabytes in size, even when compressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If standard benchmarks grow further, we risk alienating much of the academic community and leaving progress to well-resourced industry groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' For this reason, we match but do not exceed the scale of sensor data in nuScenes [4] and Waymo Open [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Make every instance count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Much of driving is boring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Datasets should focus on the difficult, interesting scenarios where current forecasting and perception systems struggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Therefore we mine for especially crowded, dynamic, and kinematically unusual scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Diversity matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Training on data from wintertime Detroit is not sufficient for detecting objects in Miami — Miami has 15 times the frequency of motorcycles and mopeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Behaviors differ as well, so learned pedestrian motion behavior might not generalize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Accordingly, each of our datasets are drawn from six diverse cities — Austin, Detroit, Miami, Palo Alto, Pittsburgh, and Washington D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' — and different seasons, as well, from snowy to sunny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Map the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' HD maps are powerful priors for perception and forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Learning-based methods that found clever ways to encode map information [31] performed well in Argoverse competitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' For this reason, we augment our HD map representation with 3D lane geometry, paint markings, crosswalks, higher resolution ground height, and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Self-supervise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Other machine learning domains have seen enormous success from self-supervised learning in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Large-scale lidar data from dynamic scenes, paired with HD maps, could lead to better representations than current supervised approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' For this reason, we build the largest dataset of lidar sensor data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Fight the heavy tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Passenger vehicles are common, and thus we can assess our forecasting and detection accuracy for cars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' However, with existing datasets, we cannot assess forecasting accuracy for buses and motorcycles with their distinct behaviors, nor can we evaluate stroller and wheel chair detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Thus we introduce the largest taxonomy to date for sensor and forecasting datasets, and we ensure enough samples of rare objects to train and evaluate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' With these guidelines in mind we built the three Argoverse 2 (AV2) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Below, we highlight some of their contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The 1,000 scenario Sensor dataset has the largest self-driving taxonomy to date – 30 categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 26 categories contain at least 6,000 cuboids to enable diverse taxonomy training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The dataset also has stereo imagery, unlike recent self-driving datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The 20,000 scenario Lidar dataset is the largest dataset for self-supervised learning on lidar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The only similar dataset, concurrently developed ONCE [36], does not have HD maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The 250,000 scenario Motion Forecasting Dataset has the largest taxonomy – 5 types of dynamic actors and 5 types of static actors – and covers the largest mapped area of any such dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We believe these datasets will support research into problems such as 3D detection, 3D tracking, monocular and stereo depth estimation, motion forecasting, visual odometry, pose estimation, lane detection, map automation, self-supervised learning, structure from motion, scene flow, optical flow, time to contact estimation, and point cloud forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 2 Related Work The last few years have seen rapid progress in self-driving perception and forecasting research, catalyzed by many high quality datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Sensor datasets and 3D Object Detection and Tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' New sensor datasets for 3D object detection [4, 45, 39, 40, 24, 33, 18, 14, 41, 36] have led to influential detection methods such as 1This count includes private submissions not posted to the public leaderboards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='com/argoverse/argoverse-api 2 anchor-based approaches like PointPillars [27], and more recent anchor-free approaches such as AFDet [16] and CenterPoint [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' These methods have led to dramatic accuracy improvements on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' In turn, these improvements have made isolation of object-specific point clouds possible, which has proven invaluable for offboard detection and tracking [42], and for simulation [8], which previously required human-annotated 3D bounding boxes [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' New approaches explore alternate point cloud representations, such as range images [5, 2, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Streaming perception [29, 21] introduces a paradigm to explore the tradeoff between accuracy and latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' A detailed comparison between the AV2 Sensor Dataset and recent 3D object detection datasets is provided in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Motion Forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' For motion forecasting, the progress has been just as significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' A transition to attention-based methods [28, 38, 37] has led to a variety of new vector-based representations for map and trajectory data [15, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' New datasets have also paved the way for new algorithms, with nuScenes [4], Lyft L5 [22], and the Waymo Open Motion Dataset [12] all releasing lane graphs after they proved to be essential in Argoverse 1 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Lyft also introduced traffic/speed control data, while Waymo added crosswalk polygons, lane boundaries (with marking type), speed limits, and stop signs to the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' More recently, Yandex has released the Shifts [34] dataset, which is the largest (by scenario hours) collection of forecasting data available to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Together, these datasets have enabled exploration of multi-actor, long-range motion forecasting leveraging both static and dynamic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Following upon the success of Argoverse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1, we position AV2 as a large-scale repository of high- quality motion forecasting scenarios - with guarantees on data frequency (exactly 10 Hz) and diversity (>2000 km of unique roadways covered across 6 cities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' This is in contrast to nuScenes (reports data at just 2 Hz) and Lyft (collected on a single 10 km segment of road), but is complementary to Waymo Open Motion Dataset (employs a similar approach for scenario mining and data configuration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Complementary datasets are essential for these safety critical problems as they provide opportunities to evaluate generalization and explore transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' To improve ease of use, we have also designed AV2 to be widely accessible both in terms of data size and format — a detailed comparison vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' other recent forecasting datasets is provided in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Broader Problems of Perception for Self-Driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Aside from the tasks of object detection and motion forecasting, new, large-scale sensor datasets for self-driving present opportunities to explore dozens of new problems for perception, especially those that can be potentially solved via self- supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' A number of new problems have been recently proposed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' real-time 3D semantic segmentation in video has received attention thanks to SemanticKITTI [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' HD map automation [54, 30] and HD map change detection [26] have received additional attention, along with 3D scene flow and pixel-level scene simulation [50, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Datasets exist with unique modalities such as thermal imagery [10, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Our new Lidar Dataset enables large-scale self-supervised training of new approaches for freespace forecasting [23] or point cloud forecasting [48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 3 The Argoverse 2 Datasets 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 Sensor Dataset The Argoverse 2 Sensor Dataset is the successor to the Argoverse 1 3D Tracking Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' AV2 is larger, with 1,000 scenes, up from 113 in Argoverse 1, but each AV2 scene is also richer – there are 23x as many non-vehicle, non-pedestrian cuboids in AV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The constituent 30 s scenarios in the Argoverse 2 Sensor Dataset were manually selected by the authors to contain crowded scenes with under-represented objects, noteworthy weather, and interesting behaviors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', cut ins and jaywalking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Each scenario is fifteen seconds in duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Table 1 compares the AV2 Sensor Dataset with a selection of self-driving datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Figures 1, 2, and 3 plot how the scenarios of AV2 compare favorably to other datasets in terms of annotation range, object diversity, object density, and scene dynamism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The most similar sensor dataset to ours is the highly influential nuScenes [4] – both datasets have 1,000 scenarios and HD maps, although Argoverse is unique in having ground height maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' nuScenes contains radar data while AV2 contains stereo imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' nuScenes has a large taxonomy – twenty-three object categories of which ten have suitable data for training and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Our dataset contains thirty object categories of which twenty-six are well sampled enough for training and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' nuScenes spans two cities, while our proposed dataset spans six.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 3 Table 1: Comparison of the Argoverse 2 Sensor and Lidar datasets with other sensor datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Name # Scenes Cities Lidar?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' # Cameras Stereo HD Maps?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' # Classes # Evaluated Classes Argoverse 1 [6] 113 2 ✓ 7 ✓ ✓ 15 3 KITTI [17] 22 1 ✓ 2 ✓ 3 3 nuScenes [4] 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='000 2 ✓ 6 ✓ 23 10 ONCE [36] 581 – ✓ 7 5 3 Waymo Open [45] 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='150 3 ✓ 5 4 4 Argoverse 2 Sensor 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='000 6 ✓ 9 ✓ ✓ 30 26 Argoverse 2 Lidar 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='✓ ✓ Regular vehicle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='Pedestrian ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
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+page_content='Animal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='10k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='100k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='10M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='Argoverse 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='Argoverse 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='nuScenes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='ONCE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='Waymo Open ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='Number of 3D cuboids ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='Figure 1: Number of annotated 3D cuboids per category for Argoverse 1 3D Tracking,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Argoverse 2 Sensor Dataset,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' nuScenes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' ONCE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' and Waymo Open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The nuScenes annotation rate is 2 Hz, compared to 10 Hz for Argoverse, but that does not account for the relative increase in object diversity in Argoverse 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Sensor Suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Lidar sweeps are collected at 10 Hz, along with 20 fps imagery from 7 cameras positioned to provide a fully panoramic field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' In addition, camera intrinsics, extrinsics and 6-DOF ego-vehicle pose in a global coordinate system are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Lidar returns are captured by two 32-beam lidars, spinning at 10 Hz in the same direction, but separated in orientation by 180°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The cameras trigger in-sync with both lidars, leading to a 20 Hz frame-rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The seven global shutter cameras are synchronized to the lidar to have their exposure centered on the lidar sweeping through their fields of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' In the Appendix, we provide a a schematic figure illustrating the car sensor suite and its coordinate frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Lidar synchronization accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' In AV2, we improve the synchronization of cameras and lidars significantly over Argoverse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Our synchronization accuracy is within [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='39, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='39] ms, which compares favorably to the Waymo Open Dataset, which is reported as [−6, 7] ms [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The AV2 Sensor Dataset contains 10 Hz 3D cuboid annotations for objects within our 30 class taxonomy (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Cuboids have track identifiers that are consistent over time for the 0 50 100 150 200 250 0 50k 100k 150k 200k 250k Waymo Open Argoverse 2 nuScenes Argoverse 1 ONCE Range (m) Number of 3D cuboids 0 50 100 150 200 250 300 0 500 1000 1500 2000 2500 Waymo Open Argoverse 2 nuScenes Argoverse 1 ONCE Number of 3D cuboids Number of lidar frames Figure 2: Left: Number of annotated 3D cuboids by range in the Argoverse 2 Sensor Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' About 14% of the Argoverse 2 cuboids are beyond 75 m – Waymo Open, nuScenes, and ONCE have less than 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Right: Number of 3D cuboids per lidar frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Argoverse 2 has an average of 75 3D cuboids per lidar frame – Waymo Open has an average of 61, nuScenes 33, and ONCE 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 4 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 0 20k 40k 60k 80k 100k Waymo Open Argoverse 2 Argoverse 1 nuScenes ONCE Number of different categories Number of lidar frames 5 10 15 20 25 0 20k 40k 60k 80k Argoverse 2 Waymo Open Argoverse 1 nuScenes Speed (m/s) Number of 3D vehicle cuboids (speed > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 m/s) Figure 3: Left: Number of annotated categories per lidar frame in the Argoverse 2 Sensor Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Per scene, Argoverse 2 is about 2× more diverse than Argoverse 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='3× more diverse than Waymo Open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Right: Speed distribution for the vehicle category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We consider only moving vehicles with speeds greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Argoverse 2 has about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='3× more moving vehicles than Waymo Open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' About 28% of the vehicles in Argoverse 2 are moving with an average speed of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='27 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We did not compare against the ONCE dataset because it does not provide tracking information for the 3D cuboids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' same object instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Objects are annotated if they are within the “region of interest” (ROI) – within five meters of the mapped “driveable” area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' All faces and license plates, whether inside vehicles or outside of the driveable area, are blurred extensively to preserve privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Sensor Dataset splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We randomly partition the dataset with train, validation, and test splits of 700, 150, and 150 scenarios, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='2 Lidar Dataset The Argoverse 2 Lidar Dataset is intended to support research into self-supervised learning in the lidar domain as well as point cloud forecasting [48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Because lidar data is more compact than the full sensor suite, we can include double-length scenarios (30 s instead of 15 s), and far more – 20,000 instead of 1,000 – equating to roughly 40x as many driving hours, for 5x the space budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The AV2 Lidar Dataset is mined with the same criteria as the Forecasting Dataset (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='2) to ensure that each scene is interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' While the Lidar Dataset does not have 3D object annotations, each scenario carries an HD map with rich, 3D information about the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Our dataset is the largest such collection to date with 20,000 thirty second sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The only similar dataset, concurrently released ONCE [36], contains 1 M lidar frames compared to 6 M lidar frames in ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Our dataset is sampled at 10 Hz instead of 2 Hz, as in ONCE, making our dataset more suitable for point cloud forecasting or self-supervision tasks where point cloud evolution over time is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Lidar Dataset splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We randomly partition the dataset with train, validation, and test splits of 16,000, 2,000, and 2,000 scenarios, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='3 Motion Forecasting Dataset Motion forecasting addresses the problem of predicting future states (or occupancy maps) for dynamic actors within a local environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Some examples of relevant actors for autonomous driving include: vehicles (both parked and moving), pedestrians, cyclists, scooters, and pets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Predicted futures generated by a forecasting system are consumed as the primary inputs in motion planning, which conditions trajectory selection on such forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Generating these forecasts presents a complex, multi-modal problem involving many diverse, partially-observed, and socially interacting agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' However, by taking advantage of the ability to “self-label” data using observed ground truth futures, motion forecasting becomes an ideal domain for application of machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Building upon the success of Argoverse 1, the Argoverse 2 Motion Forecasting dataset provides an updated set of prediction scenarios collected from a self-driving fleet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The design decisions enumerated below capture the collective lessons learned from both our internal research/development, 5 Table 2: Comparison between the Argoverse 2 Motion Forecasting dataset and other recent motion forecasting datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Hyphens "-" indicate that attributes are either not applicable, or not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We define “mined for interestingness” to be true if interesting scenarios/actors are mined after data collection, instead of taking all/random samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' † Public leaderboard counts as retrieved on Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 27, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' ARGOVERSE [6] INTER [52] LYFT [22] WAYMO [12] NUSCENES [4] YANDEX [34] OURS # SCENARIOS 324k 170k 104k 41k 600k 250k # UNIQUE TRACKS 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='7M 40k 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='4M 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='6M 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='4M 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='9M AVERAGE TRACK LENGTH 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='48 s 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='8 s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='8 s 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='04 s 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='16 s TOTAL TIME 320 h 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 h 1118 h 574 h 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 h 1667 h 763 h SCENARIO DURATION 5 s 25 s 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 s 8 s 10 s 11 s TEST FORECAST HORIZON 3 s 3 s 5 s 8 s 6 s 5 s 6 s SAMPLING RATE 10 Hz 10 Hz 10 Hz 10 Hz 2 Hz 5 Hz 10 Hz # CITIES 2 6 1 6 2 6 6 UNIQUE ROADWAYS 290 km 2 km 10 km 1750 km 2220 km AVG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' # TRACKS PER SCENARIO 50 79 75 29 73 # EVALUATED OBJECT CATEGORIES 1 1 3 3 1 2 5 MULTI-AGENT EVALUATION × ✓ ✓ ✓ × ✓ ✓ MINED FOR INTERESTINGNESS ✓ × ✓ × × ✓ VECTOR MAP ✓ × × ✓ ✓ × ✓ DOWNLOAD SIZE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='8 GB 22 GB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='4 TB 48 GB 120 GB 58 GB # PUBLIC LEADERBOARD ENTRIES† 194 935 23 18 3 as well as feedback from more than 2,700 submissions by nearly 260 unique teams3 across 3 competitions [43]: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Motion forecasting is a safety critical system in a long-tailed domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Consequently, our dataset is biased towards diverse and interesting scenarios containing different types of focal agents (see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Our goal is to encourage the development of methods that ensure safety during tail events, rather than to optimize the expected performance on “easy miles”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' There is a “Goldilocks zone” of task difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Performance on the Argoverse 1 test set has begun to plateau, as shown in Figure 10 of the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Argoverse 2 is designed to increase prediction difficulty incrementally, spurring productive focused research for the next few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' These changes are intended to incentivize methods that perform well on extended forecast horizons (3 s → 6 s), handle multiple types of dynamic objects (1 → 5), and ensure safety in scenarios from the long tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Future Argoverse releases could continue to increase the problem difficulty by reducing observation windows and increasing forecasting horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Usability matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Argoverse 1 benefited from a large and active research community—in large part due to the simplicity of setup and usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Consequently, we took care to ensure that existing Argoverse models can be easily ported to run on Argoverse 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' In particular, we have prioritized intuitive access to map elements, encouraging methods which use the lane graph as a strong prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' To improve training and generalization, all poses have also been interpolated and resampled at exactly 10 Hz (Argoverse 1 was approximate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The new dataset includes fewer, but longer and more complex scenarios;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' this ensures that total dataset size remains large enough to train complex models but small enough to be readily accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 Data Representation The dataset consists of 250,000 non-overlapping scenarios (80/10/10 train/val/test random splits) mined from six unique urban driving environments in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' It contains a total of 10 object types, with 5 from each of the dynamic and static categories (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Each scenario includes a local vector map and 11 s (10 Hz) of trajectory data (2D position, velocity, and orientation) for all tracks observed by the ego-vehicle in the local environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The first 5 s of each scenario is denoted as the observed window, while the subsequent 6 s is denoted as the forecasted horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Within each scenario, we mark a single track as the “focal agent”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Focal tracks are guaranteed to be fully observed throughout the duration of the scenario and have been specifically selected to maximize interesting interactions with map features and other nearby actors (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' To evaluate multi-agent forecasting, we also mark a subset of tracks as “scored actors” (as shown in Figure 5), with guarantees for scenario relevance and minimum data quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 3This count includes private submissions not posted to the public leaderboards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 6 Figure 4: Object type and geographic histograms for the Motion Forecasting Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Left: Histogram of object types over the “focal” and “scored” categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Center: Histogram of object types over all tracks present in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The fine grained distinctions between different static object types (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Construction Cone vs Riderless Bicycle) are unique among forecasting datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Right: Histogram of metropolitan areas included in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Figure 5: Visualization of a few interesting scenarios from the Motion Forecasting Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The scenarios demonstrate a mix of the various object types (Vehicle, Pedestrian, Bus, Cyclist, or Motor- cyclist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The ego-vehicle is indicated in green, the focal agent is purple, and scored actors are orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Other un-scored tracks are shown in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Object positions are captured at the last timestep of the observed history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' For visualization purposes the full 5 s history and 6 s future are rendered for the focal agent, while only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 s of future are shown for the other scored actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Left shows a pedestrian crossing in front of the ego-vehicle, while center and right depict a motorcyclist weaving through traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='2 Mining Interesting Scenarios The source data for Argoverse 2 was drawn from fleet logs tagged with annotations consistent with interesting or difficult-to-forecast events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Each log was trimmed to 30 s and run through an interestingness scoring module in order to bias data selection towards examples from the long-tail of the natural distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We employ heuristics to score each track in the scene across five dimensions: object category, kinematics, map complexity, social context, and relation to the ego-vehicle (details in Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The final scenarios are generated by extracting non-overlapping 11 s windows where at least one candidate track is fully observed for the entire duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The highest scoring candidate track is denoted as the “focal agent”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' all other fully observed tracks within 30 m of the ego-vehicle are denoted as “scored actors”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The resulting dataset is diverse, challenging, and still right-sized for widespread use (see the download size in Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' In Figure 6, we show that the resulting dataset is significantly more interesting than Argoverse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 and validate our intuition that actors scoring highly in our heuristic module are more challenging to accurately forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='4 HD Maps Each scenario in the three datasets described above shares the same HD map representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Each scenario carries its own local map region, similar to the Waymo Open Motion [12] dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' This is a departure from the original Argoverse datasets in which all scenarios were localized onto two city-scale maps—one for Pittsburgh and one for Miami.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' In the Appendix, we provide examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 7 106 ActorCategory ScoredActor of Actors FocalAgent 105 # 104 103 Cyclist Bus Vehicle Pedestrian Motorcyclist106 : of Actors 104 # 102 100 Bicycle Cyclist Bus Vehicle Static Pedestrian Unknown Construction Riderless E104 Scenarios 103 102 # 101 100 Austin Alto Miami Pittsburgh Dearborn Palo大 久久久久 大0 2 4 6 8 10 12 14 Total Interestingness Score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='0 Miss Rate (K=6) Fitted Regression Model Bin Centers (1000 Scenarios Each) Figure 6: Left: Histogram comparing the distribution of interestingness scores assigned to focal agents in both Argoverse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Right: Plot showing the relationship between total interestingness score and prediction difficulty on the Argoverse 2 test split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We evaluate WIMP [25] over each scenario and fit a regression model to the computed miss rate (K=6, 2m threshold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Advantages of per-scenario maps include more efficient queries and their ability to handle map changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' A particular intersection might be observed multiple times in our datasets, and there could be changes to the lanes, crosswalks, or even ground height in that time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Lane graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The core feature of the HD map is the lane graph, consisting of a graph G = (V, E), where V are individual lane segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' In the Appendix, we enumerate and define the attributes we provide for each lane segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Unlike Argoverse 1, we provide the actual 3D lane boundaries, instead of only centerlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' However, our API provides code to quickly infer the centerlines at any desired sampling resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Polylines are quantized to 1 cm resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Our representation is richer than nuScenes, which provides lane geometry only in 2D, not 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Driveable area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Instead of providing driveable area segmentation in a rasterized format, as we did in Argoverse 1, we release it in a vector format, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' as 3D polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' This offers multiple advantages, chiefly in compression, allowing us to store separate maps for tens of thousands of scenarios, yet the raster format is still easily derivable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The polygon vertices are quantized to 1 cm resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Ground surface height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Only the sensor dataset includes a dense ground surface height map (although other datasets still have sparse 3D height information on polylines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Ground surface height is provided for areas within a 5 m isocontour of the driveable area boundary, which we define as the region of interest (ROI) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We do so because the notion of ground surface height is ill-defined for the interior of buildings and interior of densely constructed city blocks, areas where ground vehicles cannot observe due to occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The raster grid is quantized to a 30 cm resolution, a higher resolution than the 1 m resolution in Argoverse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Area of Local Maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Each scenario’s local map includes all entities found within a 100 m dilation in l2-norm from the ego-vehicle trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 4 Experiments Argoverse 2 supports a variety of downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' In this section we highlight three different learning problems: 3D object detection, point cloud forecasting, and motion forecasting — each supported by the sensor, lidar, and motion forecasting datasets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' First, we illustrate the challenging and diverse taxonomy within the Argoverse 2 sensor dataset by training a state-of-the- art 3D detection model on our twenty-six evaluation classes including “long-tail” classes such as stroller, wheel chairs, and dogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Second, we showcase the utility of the Argoverse 2 lidar dataset through large-scale, self-supervised learning through the point cloud forecasting task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Lastly, we demonstrate motion forecasting experiments which provide the first baseline for broad taxonomy motion prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 3D Object Detection 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='14 Mean Argoverse 1 Proportion of Scenarios Argoverse 2 Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='12 Argoverse 1 Argoverse 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='00 0 2 4 6 8 10 12 Kinematic Score + Social ScoreTable 3: 3d object detection results on the Argoverse 2 Sensor Dataset, taken from the leaderboard on Dec 21, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Detectors is the winner of the CVPR 2022 Workshop on Autonomous Driving Argoverse 2 3D Object Detection challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' METHOD MCDS (↑) MAP (↑) MATE (↓) MASE (↓) MAOE (↓) CENTERPOINT (OURS) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='72 DETECTORS [13] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='54 BEVFUSION [32] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='50 Regular Vehicle Bus Pedestrian Stop Sign Box Truck Bollard Construction Barrel Motorcyclist Truck Bicyclist Mobile Crossing Sign Average Metrics Motorcycle Bicycle Articulated Bus School Bus Truck Cab Construction Cone Vehicular Trailer Sign Wheeled Device Large Vehicle Stroller Message Board Trailer Dog Wheeled Rider Wheelchair Class Names 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='7 AP Figure 7: Average precision of our 3D object de- tection baseline on the validation split of the Sen- sor Dataset (Beta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Our experiments showcase both our diverse taxonomy and difficult “long-tail” classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We provide baseline 3D detection results using a state-of-the-art, anchorless 3D object detec- tion model – CenterPoint [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Our Center- Point implementation takes a point cloud as input and crops it to a 200 m × 200 m grid with a voxel resolution of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 m, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 m] in the xy (bird’s-eye-view) plane and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='2 m in the z- axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' To accommodate our larger taxonomy, we include six detection heads to encourage fea- ture specialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Figure 7 characterizes the performance of our 3D detection baseline us- ing the nuScenes [4] average precision met- ric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Our large taxonomy allows us to evaluate classes such as “Wheeled Device” (e-Scooter), “Stroller”, “Dog”, and “Wheelchair” and we find that performance on these categories with strong baselines is poor despite significant amounts of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' In Table 3, we provide a snapshot of submissions to the Argoverse 2 3D Object Detection Leaderboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='2 Point Cloud Forecasting We perform point cloud forecasting according to the experimental protocol of SPF2 [49] using the Argoverse 2 Lidar Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Given a sequence of past scene point clouds, a model is required to predict a sequence of future scene point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We take the scene point clouds in the past 1 s (10 Hz) in the range image format as input, and then predict the next 1 s of range images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' SPFNet predicts two output maps at each time step – the first output map is the predicted range values, while the second output is a validity mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Previous point cloud forecasting models were evaluated on smaller datasets such as KITTI or nuScenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' To explore how the amount of training data affects the performance, we use increasing amounts of data for training the same model architecture, up to the full training set of 16,000 sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We use three metrics to evaluate the performance of our forecasting model: mean IoU, l1-norm, and Chamfer distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The mean IoU evaluates the predicted range mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The l1-norm measures the average l1 distance between the pixel sets of predicted range image and the ground- truth image, which are both masked out by the ground-truth range mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The Chamfer distance is obtained by adding up the Chamfer distances in both directions (forward and backward) between the ground-truth point cloud and the predicted scene point cloud which is obtained by back-projecting the predicted range image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Table 4: Results of point cloud forecasting on the test split of the Lidar Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' # TRAIN LOGS 125 250 500 1k 2k 4k 16k MEAN IOU (%) (↑) 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='4 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='7 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='0 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='4 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='9 l1-NORM (↓) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='4 CHAMFER DIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' (↓) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='9 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='4 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='9 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='0 9 Results of SPF2 and Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Table 4 contains the results of our point cloud forecasting experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' With increasing training data, the performance of the model grows steadily in all three metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' These results and the works from the self-supervised learning literature [3, 7] indicate that a large amount of training data can make a substantial difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Another observation is that the Chamfer distances for predictions on our dataset are significantly higher than predictions on KITTI [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We conjecture that this could be due to two reasons: (1) the Argoverse 2 Lidar Dataset has a much larger sensing range (above 200 m versus 120 m of the KITTI lidar sensor), which tends to significantly increase the value of Chamfer distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' (2) the Argoverse 2 Lidar Dataset has a higher proportion of dynamic scenes compared with KITTI Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='3 Motion Forecasting We present several forecasting baselines [6] which try to make use of different aspects of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Those which are trained using the focal agent only and do not capture any social interaction include: constant velocity, nearest neighbor, and LSTM encoder-decoder models (both with and without a map-prior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We also evaluate WIMP [25] as an example of a graph-based attention method that captures social interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' All hyper-parameters are obtained from the reference implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Baseline approaches are evaluated according to standard metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Following [6], we use minADE and minFDE as the metrics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' they evaluate the average and endpoint L2 distance respectively, between the best forecasted trajectory and the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We also use Miss Rate (MR) which represents the proportion of test samples where none of the forecasted trajectories were within 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='0 meters of ground truth according to endpoint error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The resulting performance illustrates both the community’s progress on the problem and the significant increase in dataset difficulty when compared with Argoverse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Table 5: Performance of motion forecasting baseline methods on vehicle-like (vehicle, bus, mo- torcyclist) object types from the Argoverse 2 Motion Forecasting (Beta) Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Usage of map prior indicates access to map information whereas usage of social context entails encoding other actors’ states in the feature representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Mining intersection (multimodal) scenarios leads to poor performance at K=1 for all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Constant Velocity models have particularly poor performance due to the dataset bias towards kinematically interesting trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Note that modern deep methods such as WIMP still have a miss rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='42 at K=6, indicating the increased difficulty of the Argoverse 2 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Numbers within 1% of the best are in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' K=1 K=6 MODEL MAP PRIOR SOCIAL CONTEXT MINADE ↓ MINFDE ↓ MR ↓ MINADE ↓ MINFDE ↓ MR ↓ CONST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' VEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' [6] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='75 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='89 NN [6] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='46 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='81 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
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+page_content='85 LSTM [6] ✓ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='07 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
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+page_content='85 WIMP [25] ✓ ✓ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='09 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
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+page_content='42 Table 6: Motion forecasting results on the Argoverse 2 Motion Forecasting Dataset, taken from the online leaderboard on Dec 21, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' BANet is the winner of the CVPR 2022 Workshop on Autonomous Driving Argoverse 2 Motion Forecasting challenge (#1), and QML and GANet received honorable mention (HM) prizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Entries are sorted below according to Brier-minFDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' K=1 K=6 METHOD MINADE ↓ MINFDE ↓ MR ↓ MINADE ↓ MINFDE ↓ MR ↓ BRIER-MINFDE ↓ THOMAS (GOHOME SCALAR) [20] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
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+page_content='16 GORELA (W/O ENSEMBLE) [11] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
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+page_content='01 GANET (ENSEMBLE) (HM) [47] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
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+page_content='98 GANET (W/O ENSEMBLE) [47] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='77 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
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+page_content='96 QML (HM) [44] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='84 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
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+page_content='95 BANET (OPPRED) (#1) [53] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='79 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='71 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
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+page_content='19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='92 Baseline Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Table 5 summarizes the results of baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' For K=1, Argoverse 1 [6] showed that a constant velocity model (minFDE=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='89) performed better than NN+map(prior) (minFDE=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='12), 10 which is not the case here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' This further proves that Argoverse 2 is kinematically more diverse and cannot be solved by making constant velocity assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Surprisingly, NN and LSTM variants that make use of a map prior perform worse than those which do not, illustrating the scope of improvement in how these baselines leverage the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' For K=6, WIMP significantly outperforms every other baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' This emphasizes that it is imperative to train expressive models that can leverage map prior and social context along with making diverse predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The trends are similar to our past 3 Argoverse Motion Forecasting competitions [43]: Graph-based attention methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' [25, 31, 37]) continued to dominate the competition, and were nearly twice as accurate as the next best baseline (Nearest Neighbor) at K=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' That said, some of the rasterization-based (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' [19]) methods also showed promising results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Finally, we also evaluated baseline methods in the context of transfer learning and varied object types, the results of which are summarized in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' In Table 6, we provide a snapshot of submissions to the Argoverse 2 Motion Forecasting Leaderboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 5 Conclusion Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' In this work, we have introduced three new datasets that constitute Argoverse 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We provide baseline explorations for three tasks – 3d object detection, point cloud forecasting and motion forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Our datasets provide new opportunities for many other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We believe our datasets compare favorably to existing datasets, with HD maps, rich taxonomies, geographic diversity, and interesting scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' As in any human annotated dataset, there is label noise, although we seek to minimize it before release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 3D bounding boxes of objects are not included in the motion forecasting dataset, but one can make reasonable assumptions about the object extent given the object type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The motion forecasting dataset also has imperfect tracking, consistent with state-of-the-art 3D trackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' References [1] Jens Behley, Martin Garbade, Andres Milioto, Jan Quenzel, Sven Behnke, Cyrill Stachniss, and Jurgen Gall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' SemanticKITTI: A dataset for semantic scene understanding of lidar sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' In ICCV, October 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' [2] Alex Bewley, Pei Sun, Thomas Mensink, Drago Anguelov, and Cristian Sminchisescu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Range conditioned dilated convolutions for scale invariant 3d object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' In Conference on Robot Learning, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' [3] Tom B Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Language models are few-shot learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' arXiv preprint arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='14165, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' [4] Holger Caesar, Varun Bankiti, Alex H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Lang, Sourabh Vora, Venice Erin Liong, Qiang Xu, Anush Krishnan, Yu Pan, Giancarlo Baldan, and Oscar Beijbom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' nuScenes: A Multimodal Dataset for Autonomous Driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' In CVPR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' [5] Yuning Chai, Pei Sun, Jiquan Ngiam, Weiyue Wang, Benjamin Caine, Vijay Vasudevan, Xiao Zhang, and Dragomir Anguelov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' To the point: Efficient 3d object detection in the range image with graph convolution kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' In CVPR, June 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' [6] Ming-Fang Chang, John Lambert, Patsorn Sangkloy, Jagjeet Singh, Slawomir Bak, Andrew Hartnett, De Wang, Peter Carr, Simon Lucey, Deva Ramanan, and James Hays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Argoverse: 3D Tracking and Forecasting With Rich Maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
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+page_content=' Accessed: 08-27-2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
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+page_content=' Interaction dataset: An international, adversarial and cooperative motion dataset in interactive driving scenarios with semantic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' arXiv preprint arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='03088, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' [53] Chen Zhang, Honglin Sun, Chen Chen, and Yandong Guo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Banet: Motion forecasting with boundary aware network, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
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+page_content=' Lane graph estimation for scene understand- ing in urban driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' CoRR, abs/2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
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+page_content=' 6 Appendix 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 Additional Information About Sensor Suite In Figure 8, we provide a diagram of the sensor suite used to capture the Argoverse 2 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Figure 9 shows the speed distribution for annotated pedestrian 3D cuboids and the yaw distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Figure 8: Car sensor schematic showing the three coordinate systems: (1) the vehicle frame in the rear axle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' (2) the camera frame;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' and the lidar frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='2 Additional Information About Motion Forecasting Dataset 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 Interestingness Scores Kinematic scoring selects for trajectories performing sharp turns or significant (de)accelerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The map complexity program biases the data set towards trajectories complex traversals of the underlying lane graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' In particular, complex map regions, paths through intersections, and lane-changes score 14 Y ④Ydown O Zup0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 0 50k 100k Argoverse 2 Waymo Open nuScenes Argoverse 1 Speed (m/s) Number of 3D pedestrian cuboids (speed > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 m/s) −180 −135 −90 −45 0 45 90 135 2 5 1000 2 5 10k 2 5 100k 2 5 Argoverse 1 Argoverse 2 Yaw (degrees) Number of 3D cuboids Figure 9: Left: Number of moving 3D cuboids for pedestrians by speed distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We define moving objects when the speed is greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Right: Number of annotated 3D cuboids by yaw distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' highly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Social scoring rewards tracks through dense regions of other actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Social scoring also selects for non-vehicle object classes to ensure adequate samples from rare classes, such as motorcycles, for training and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Finally, the autonomous vehicle scoring program encourages the selection of tracks that intersect the ego-vehicle’s desired route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 2019/10 2019/11 2019/12 2020/02 2020/03 2020/05 2020/06 2020/07 2020/08 2020/09 2020/10 2020/11 2020/12 2021/01 2021/02 2021/03 2021/04 Year/Month of Submission 0 1 2 3 4 5 6 7 8 minFDE (K=6) State-of-art minFDE (K=6) Competition Phase Neurips 2019 CVPR 2020 None CVPR 2021 Figure 10: MinFDE metric values for submissions on Argoverse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Individual points indicate submissions to the public leader board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Colors indicate specific competition phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The solid black line indicates SOTA performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The research community made massive gains which have plateaued since early 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' However, we note that the number and diversity of methods performing at or near the SOTA continues to grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Additionally, later competitions sorted the leaderboard by “Miss Rate” and probability weighted FDE, and those metrics showed progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Still, minFDE did not improve significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='3 Additional Information About HD Maps Examples of HD maps from the Sensor Dataset In Figure 12, we display examples of local HD maps associated with individual logs/scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='4 Additional 3D Detection Results In Figure 13, we show additional evaluation metrics for our detection baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 15 Figure 11: Histogram of the number of actors (both scored and all types) present in the Motion Forecasting Dataset scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The Lidar Dataset is mined by the same criteria and thus follows the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' MAP ENTITY PROVIDED ATTRIBUTES TYPE DESCRIPTION LANE SEGMENTS IS_INTERSECTION BOOLEAN WHETHER OR NOT THIS LANE SEGMENT LIES WITHIN AN INTERSECTION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' LANE TYPE ENUMERATED TYPE DESIGNATION OF WHICH VEHICLE TYPES MAY LEGALLY UTILIZE THIS LANE FOR TRAVEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' LEFT LANE BOUNDARY 3D POLYLINE THE POLYLINE OF THE LEFT BOUNDARY IN THE CITY MAP COORDINATE SYSTEM RIGHT LANE BOUNDARY 3D POLYLINE THE POLYLINE OF THE RIGHT BOUNDARY IN THE CITY MAP COORDINATE SYSTEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' LEFT LANE MARK TYPE ENUMERATED TYPE TYPE OF PAINTED LANE MARKING TO THE LEFT OF THE LANE SEGMENT ON THE ROAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' RIGHT LANE MARK TYPE ENUMERATED TYPE TYPE OF PAINTED LANE MARKING TO THE RIGHT OF THE LANE SEGMENT ON THE ROAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' LEFT NEIGHBOR INTEGER THE UNIQUE LANE SEGMENT IMMEDIATELY TO THE LEFT OF SEGMENT, OR NONE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' RIGHT NEIGHBOR INTEGER THE UNIQUE LANE SEGMENT IMMEDIATELY TO THE RIGHT OF SEGMENT, OR NONE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' SUCCESSOR IDS INTEGER LIST LANE SEGMENTS THAT MAY BE ENTERED BY FOLLOWING FORWARD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' ID INTEGER UNIQUE IDENTIFIER DRIVABLE AREA AREA BOUNDARY 3D POLYGONS AREA WHERE IT IS POSSIBLE FOR THE AV TO DRIVE WITHOUT DAMAGING ITSELF ID INTEGER UNIQUE IDENTIFIER PEDESTRIAN CROSSINGS EDGE1, EDGE2 3D POLYLINES ENDPOINTS OF BOTH EDGE ALONG THE PRINCIPAL AXIS, THUS DEFINING A POLYGON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' ID INTEGER UNIQUE IDENTIFIER GROUND SURFACE HEIGHT 2D RASTER ARRAY RASTER GRID QUANTIZED TO A 30 cm RESOLUTION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Table 7: HD map attributes for each Argoverse 2 scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Average Precision (AP) AP = 1 101 � t∈T � r∈R pinterp(r) (1) True Positive Metrics Average Translation Error (ATE) ATE = ∥tdet − tgt∥2 (2) Average Scaling Error (ASE) ASE = 1 − � d∈D min(ddet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' dgt) max(ddet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' dgt) (3) Average Orientation Error (AOE) AOE = |θdet − θgt| (4) Composite Detection Score (CDS) CDS = mAP · � x∈X (1 − x) (5) where X = {mATEunit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' mASEunit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' mAOEunit} 16 104 103 102 # 101 100 0 6 12 18 24 31 # of Scored Actors104 Scenarios 103 102 ofs # 101 0 50 100 150 200 250 # of Total Actors(a) Washington D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' (b) Washington D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' (c) Washington D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' (d) Pittsburgh, PA (e) Pittsburgh, PA (f) Pittsburgh, PA (g) Miami, FL (h) Miami, FL (i) Miami, FL (j) Detroit, MI (k) Detroit, MI (l) Austin, TX Figure 12: Examples of egovehicle (AV) trajectories on local vector maps from the Sensor Dataset across several different cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' A 100m × 100m local map region is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Crosswalks are indicated in purple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Red circles denote the AV pose discretely sampled at 1 Hz for the purposes of illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Pose is provided at >20 Hz in the dataset, as indicated by the trajectory path indicated by a red line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' City layouts vary dramatically, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' roadways in Miami are usually aligned parallel to a north-south, east-west grid, while roadways in Pittsburgh are generally not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 17 Regular Vehicle Bus Pedestrian Stop Sign Box Truck Bollard Construction Barrel Motorcyclist Truck Bicyclist Mobile Crossing Sign Average Metrics Motorcycle Bicycle Articulated Bus School Bus Truck Cab Construction Cone Vehicular Trailer Sign Wheeled Device Large Vehicle Stroller Message Board Trailer Dog Wheeled Rider Wheelchair Class Names 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
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+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='6 CDS Regular Vehicle Bus Pedestrian Stop Sign Box Truck Bollard Construction Barrel Motorcyclist Truck Bicyclist Mobile Crossing Sign Average Metrics Motorcycle Bicycle Articulated Bus School Bus Truck Cab Construction Cone Vehicular Trailer Sign Wheeled Device Large Vehicle Stroller Message Board Trailer Dog Wheeled Rider Wheelchair Class Names 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='0 ATE Regular Vehicle Bus Pedestrian Stop Sign Box Truck Bollard Construction Barrel Motorcyclist Truck Bicyclist Mobile Crossing Sign Average Metrics Motorcycle Bicycle Articulated Bus School Bus Truck Cab Construction Cone Vehicular Trailer Sign Wheeled Device Large Vehicle Stroller Message Board Trailer Dog Wheeled Rider Wheelchair Class Names 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='2 ASE Regular Vehicle Bus Pedestrian Stop Sign Box Truck Bollard Construction Barrel Motorcyclist Truck Bicyclist Mobile Crossing Sign Average Metrics Motorcycle Bicycle Articulated Bus School Bus Truck Cab Construction Cone Vehicular Trailer Sign Wheeled Device Large Vehicle Stroller Message Board Trailer Dog Wheeled Rider Wheelchair Class Names 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 AOE Figure 13: 3D object detection performance on the validation split of the Sensor Dataset (Beta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Top Row: Composite detection score (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Average translation error (right) Bottom Row: Average scaling error (left), and average orientation error (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Results are shown on the validation set of the Sensor Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 Training Details of SPF2 baseline We sample 2-second training snippets (representing 1 second of past and 1 second of future data) every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='5 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Thus, for a training log with 30 second duration, 59 training snippets would be sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We train the model for 16 epochs by using the Adam optimizer with the learning rate of 4e − 3, betas of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='9 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='999, and batch size of 16 per GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='6 Additional Motion Forecasting Experiments 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 Transfer Learning The results of transfer learning experiments are summarized in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' WIMP was trained and tested in different settings with Argoverse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 and Argoverse 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' As expected, the model works best when it is trained and tested on the same distribution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' both train and test data come from Argoverse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1, or both from Argoverse 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' For example, when WIMP is tested on Argoverse 2 (6s), the model trained on Argoverse 2 (6s) has a minFDE of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='91, whereas the one trained on Argoverse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 (3s) has a minFDE of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='82 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' approximately 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='3x worse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Likewise, in the reverse setting, when WIMP is tested on Argoverse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 (3s), the model trained on Argoverse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 (3s) has a minFDE of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='14 and the one trained on Argoverse 2 (6s) has minFDE of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='05 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='8x worse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' This indicates that transfer learning from Argoverse 2 (Beta) to Argoverse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 is more useful than the reverse setting, despite being smaller in the number of scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' However, the publicly released version of Argoverse 2 Motion Forecasting (the non-beta 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='0 version) has comparable size with Argoverse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We note that it is a common practice to train and test sequential models on varied sequence length (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' machine translation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' As such, it is still reasonable to expect a model trained with 3s to do well on 6s horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Several factors may contribute to distribution shift, including differing prediction horizon, cities, mining protocols, object types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Notably, however, these results indicate that Argoverse 2 is significantly more challenging and diverse than its predecessor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 18 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='2 Experiment with different object types Table 9 shows the results on Nearest Neighbor baseline (without map prior) on different object types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' As one would expect, the displacement errors in pedestrians are significantly lower than other object types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' This occurs because they move at significantly slower velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' However, this does not imply that pedestrian motion forecasting is a solved problem and one should rather focus on other object types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' This instead means that we need to come up with better metrics that can capture that fact lower displacement errors in pedestrians can often be more critical than higher errors in vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We leave this line of work for future scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Table 8: Performance of WIMP when trained and tested on different versions of Argoverse motion forecasting datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Training and evaluation is restricted to vehicle-like (vehicle, bus, motorcyclist) object types as only vehicles were present in Argoverse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' All the results are for K=6, and prediction horizon is specified in parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Notably, the model trained on a 3s horizon performs poorly on the longer 6s horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' ‘Argoverse 2’ below denotes the Argoverse 2 (Beta) Motion Forecasting Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Train Split (pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' horizon) Test Split (pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' horizon) minADE ↓ minFDE ↓ MR ↓ Argoverse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 (3s) Argoverse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 (3s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='12 Argoverse 2 (6s) Argoverse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 (3s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='68 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='27 Argoverse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 (3s) Argoverse 2 (3s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='94 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='26 Argoverse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 (3s) Argoverse 2 (6s) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='93 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='77 Argoverse 2 (6s) Argoverse 2 (6s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='48 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='43 Table 9: Performance of Nearest Neighbor baseline on different object types for K=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The most accurately predicted object type for each evaluation metric is highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Object Type #Samples minADE ↓ minFDE ↓ MR ↓ All 9955 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='48 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='66 Vehicle 8713 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='62 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='70 Bus 439 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='69 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='73 Pedestrians 677 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='69 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='17 Motorcyclist 39 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='33 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='61 Cyclist 87 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='48 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='42 7 Datasheet for Argoverse 2 For what purpose was the dataset created?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Was there a specific task in mind?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Was there a specific gap that needed to be filled?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Please provide a description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Argoverse was created to support the global research community in improving the state of the art in machine learning tasks vital for self driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The Argoverse 2 datasets described in this manuscript improve upon the initial Argoverse datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' These datasets support many tasks, from 3D perception to motion forecasting to HD map automation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The three datasets proposed in this manuscript address different gaps in this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' See the comparison charts in the main manuscript for a more detailed breakdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The Argoverse 2 Sensor Dataset has a richer taxonomy than similar datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' It is the only dataset of similar size to have stereo imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The 1,000 logs in the dataset were chosen to have a variety of object types with diverse interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The Argoverse 2 Motion Forecasting Dataset also has a richer taxonomy than existing datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The scenarios in the dataset were mined with an emphasis on unusual behaviors that are difficult to predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The Argoverse 2 Lidar Dataset is the largest Lidar Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Only the concurrent ONCE dataset is similarly sized to enable self-supervised learning in lidar space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Unlike ONCE, our dataset contains HD maps and high frame rate lidar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Who created this dataset (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', which team, research group) and on behalf of which entity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', company, institution, organization)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 19 The Argoverse 2 datasets were created by researchers at Argo AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' What support was needed to make this dataset?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='who funded the creation of the dataset?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If there is an associated grant, provide the name of the grantor and the grant name and number, or if it was supported by a company or government agency, give those details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=') The creation of this dataset was funded by Argo AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Any other comments?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' n/a COMPOSITION What do the instances that comprise the dataset represent (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', documents, photos, people, countries)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Are there multiple types of instances (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', movies, users, and ratings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' people and interactions between them;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' nodes and edges)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Please provide a description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The three constituent datasets of Argoverse 2 have different attributes, but the core instances for each are brief “scenarios” or “logs” of 11, 15, or 30 seconds that represent a continuous observation of a scene around a self-driving vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Each scenario in all three datasets has an HD map that includes lane boundaries, crosswalks, driveable area, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Scenarios for the Sensor Dataset additionally contain a raster map of ground height at .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='3 meter resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' How many instances are there in total (of each type, if appropriate)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The Sensor Dataset has 1,000 15 second scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The Lidar Dataset has 20,000 30 second scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The Motion Forecasting Dataset has 250,000 11 second scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Does the dataset contain all possible instances or is it a sample (not necessarily random) of instances from a larger set?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If the dataset is a sample, then what is the larger set?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Is the sample representative of the larger set (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', geographic coverage)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please describe how this representativeness was validated/verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If it is not representative of the larger set, please describe why not (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', to cover a more diverse range of instances, because instances were withheld or unavailable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The scenarios in the dataset are a sample of the set of observations made by a fleet of self-driving vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The data is not uniformly sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The particular samples were chosen to be geographically diverse (spanning 6 cities - Pittsburgh, Detroit, Austin, Palo Alto, Miami, and Washington D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='), to include interesting behavior (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' cars making unexpected maneuvers), to contain interesting weather (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' rain and snow), and to contain scenes with many objects of diverse types in motion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' a crowd walking, riders on e-scooters splitting lanes between many vehicles, an excavator operating at a construction site, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' What data does each instance consist of?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' “Raw” data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', unprocessed text or images) or features?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' In either case, please provide a description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Each Sensor Dataset scenario is 15 seconds in duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Each scenario has 20 fps video from 7 ring cameras, 20 fps video from two forward facing stereo cameras, and 10 hz lidar returns from two out-of-phase 32 beam lidars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The ring cameras are synchronized to fire when either lidar sweeps through their field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Each scenario contains vehicle pose over time and calibration data to relate the various sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Each Lidar Dataset scenario is 30 seconds in duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' These scenarios are similar to those of the Sensor Dataset, except that there is no imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Each Motion Forecasting scenario is 11 seconds in duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' These scenarios contain no sensor data, but instead contain tracks of objects such as vehicles, pedestrians, and bicycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The tracks specify the category of each object (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' bus or bicycle) as well as their location and heading at a 10 hz sampling interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 20 The HD map associated with all three types of scenarios contains polylines describing lanes, cross- walks, and driveable area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Lanes form a graph with predecessors and successors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' a lane that splits can have two successors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Lanes have precisely localized lane boundaries that include paint type (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' double solid yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Driveable area, also described by a polygon, is the area where it is possible but not necessarily legal to drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' It includes areas such as road shoulders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Is there a label or target associated with each instance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please provide a description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Each Sensor Dataset scenario has 3D track annotations for dynamic objects such as vehicles, pedes- trians, strollers, dogs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The tracks are suitable as ground truth for tasks such as 3D object detection and 3D tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The 3D track labels are intentionally held out from the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The HD map could also be thought of as labels for each instance, and would be suitable as ground truth for lane detection or map automation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The vehicle pose data could be considered ground truth labels for visual odometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The lidar depth estimates can act as ground truth for monocular or stereo depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The Lidar Dataset does not have human annotations beyond the HD map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Still, the evolving point cloud itself can be considered ground truth for point cloud forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Each Motion Forecasting Dataset scenario provides labels specifying which tracks are associated with “scored actors”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' These tracks exhibit interesting behavior and are guaranteed to be observed over the entire duration of each scenario;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' algorithms will be asked to forecast the future motion for these tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The future motion of actors in each scenario is intentionally held out in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Is any information missing from individual instances?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please provide a description, explaining why this information is missing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', because it was unavailable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' This does not include intentionally removed information, but might include, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', redacted text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' In the Sensor Dataset, objects are only labeled within 5 meters of the driveable area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' For example, a person sitting on their front porch will not be labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' In the Sensor Dataset and Motion Forecasting Dataset, instances are not necessarily labeled for the full duration of each scenario if the objects move out of observation range or become occluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Z Are relationships between individual instances made explicit (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', users’ movie ratings, social network links)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please describe how these relationships are made explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The instances of the three datasets are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' They each carry their own HD map for the region around the scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' These HD maps may overlap spatially, though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' For example, many forecasting scenarios may take place in the same intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If a user of the dataset wanted to recover the spatial relationship between scenarios, they could do so through the Argoverse API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Are there recommended data splits (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', training, development/validation, testing)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please provide a description of these splits, explaining the rationale behind them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We define splits of each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The Sensor Dataset is split 700 / 150 / 150 between train, validation, and test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The Lidar Dataset is split 16,000 / 2,000 / 2,000 and the Motion Forecasting Dataset is split 200,000 / 25,000 / 25,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' In all cases, the splits are designed to make the training dataset as large as possible while keeping the validation and test datasets large and diverse enough to accurately benchmark models learned on the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Are there any errors, sources of noise, or redundancies in the dataset?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please provide a description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Every sensor used in the dataset – ring cameras, stereo cameras, and lidar – has noise associated with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Pixel intensities, lidar intensities, and lidar point 3D locations all have noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Lidar points are also quantized to float16 which leads to roughly a centimeter of quantization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Six degree of freedom vehicle pose also has noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The calibration specifying the relationship between sensors can be imperfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The HD map for each scenario can contain noise, both in terms of lane boundary locations and precise ground height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The 3D object annotations in the Sensor Dataset do not always match the spatial extent and motion of an object in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' For example, we assume that objects do not change size during a scenario, but this could be violated by a car opening its door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 3D annotations for distant objects with relatively few pixels and lidar returns are less accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 21 The object tracks in the Motion Forecasting dataset are imperfect and contain errors typical of a real-time 3D tracking method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Our expectation is that a motion forecasting algorithm should operate well despite this noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Is the dataset self-contained, or does it link to or otherwise rely on external resources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', websites, tweets, other datasets)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If it links to or relies on external resources, a) are there guarantees that they will exist, and remain constant, over time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' b) are there official archival versions of the complete dataset (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', including the external resources as they existed at the time the dataset was created);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' c) are there any restrictions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', licenses, fees) associated with any of the external resources that might apply to a future user?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Please provide descriptions of all external resources and any restrictions associated with them, as well as links or other access points, as appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The data itself is self-hosted, like Argoverse 1 [see https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='argoverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='org/], and we maintain public links to all previous versions of the dataset in case of updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The data is independent of any previous datasets, including Argoverse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Does the dataset contain data that might be considered confidential (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', data that is protected by legal privilege or by doctor-patient confidentiality, data that includes the content of individuals’ non-public communications)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please provide a description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Does the dataset contain data that, if viewed directly, might be offensive, insulting, threaten- ing, or might otherwise cause anxiety?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please describe why.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Does the dataset relate to people?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If not, you may skip the remaining questions in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Yes, the dataset contains images and behaviors of thousands of people on public streets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Does the dataset identify any subpopulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', by age, gender)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please describe how these subpopulations are identified and provide a description of their respective distributions within the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Is it possible to identify individuals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', one or more natural persons), either directly or indirectly (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', in combination with other data) from the dataset?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please describe how.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We do not believe so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' All image data has been anonymized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Faces and license plates are obfuscated by replacing them with a 5x5 grid, where each grid cell is the average color of the original pixels in that grid cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' This anonymization is done manually and is not limited by our 3D annotation policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' For example, a person sitting on their front porch 10 meters from the road would not be labeled with a 3D cuboid, but their face would still be obscured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Does the dataset contain data that might be considered sensitive in any way (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', data that reveals racial or ethnic origins, sexual orientations, religious beliefs, political opinions or union memberships, or locations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' financial or health data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' biometric or genetic data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' forms of government identification, such as social security numbers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' criminal history)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please provide a description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Any other comments?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' n/a COLLECTION How was the data associated with each instance acquired?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Was the data directly observable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', raw text, movie ratings), reported by subjects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', survey responses), or indirectly inferred/derived from other data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', part-of-speech tags, model-based guesses for age or language)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If data was reported by subjects or indirectly inferred/derived from other data, was the data 22 validated/verified?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please describe how.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The sensor data was directly acquired by a fleet of autonomous vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Over what timeframe was the data collected?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Does this timeframe match the creation timeframe of the data associated with the instances (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', recent crawl of old news articles)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If not, please describe the timeframe in which the data associated with the instances was created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Finally, list when the dataset was first published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The data was collected in 2020 and 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The dataset was made public after NeurIPS 2021, in March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' What mechanisms or procedures were used to collect the data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', hardware apparatus or sensor, manual human curation, software program, software API)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' How were these mechanisms or procedures validated?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The Argoverse 2 data comes from Argo ‘Z1’ fleet vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' These vehicles use Velodyne lidars and traditional RGB cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' All sensors are calibrated by Argo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' HD maps and 3D object annotations are created and validated through a combination of computational tools and human annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Object tracks in the Motion Forecasting Dataset are created by a 3D tracking algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' What was the resource cost of collecting the data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' what were the required computational resources, and the associated financial costs, and energy consumption - estimate the carbon footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' See Strubell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' for approaches in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=') The data was captured during normal fleet operations, so there was minimal overhead for logging particular events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The transformation and post-processing of several terabytes of data consumed an estimated 1,000 machine hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We estimate a Carbon footprint of roughly 1,000 lbs based on the CPU-centric workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If the dataset is a sample from a larger set, what was the sampling strategy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', deterministic, probabilistic with specific sampling probabilities)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The Sensor Dataset scenarios were chosen from a larger set through manual review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The Lidar Dataset and Motion Forecasting Dataset scenarios were chosen by heuristics which looked for interesting object behaviors during fleet operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Who was involved in the data collection process (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', students, crowdworkers, contractors) and how were they compensated (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', how much were crowdworkers paid)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Argo employees and Argo interns curated the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Data collection and data annotation was done by Argo employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Crowdworkers were not used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Were any ethical review processes conducted (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', by an institutional review board)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please provide a description of these review processes, including the outcomes, as well as a link or other access point to any supporting documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Does the dataset relate to people?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If not, you may skip the remainder of the questions in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Did you collect the data from the individuals in question directly, or obtain it via third parties or other sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', websites)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The data is collected from vehicles on public roads, not from a third party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Were the individuals in question notified about the data collection?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please describe (or show with screenshots or other information) how notice was provided, and provide a link or other access point to, or otherwise reproduce, the exact language of the notification itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' No, but the data collection was not hidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The Argo fleet vehicles are well marked and have obvious 23 cameras and lidar sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The vehicles only capture data from public roads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Did the individuals in question consent to the collection and use of their data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please describe (or show with screenshots or other information) how consent was requested and provided, and provide a link or other access point to, or otherwise reproduce, the exact language to which the individuals consented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' People in the dataset were in public settings and their appearance has been anonymized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Drivers, pedestrians, and vulnerable road users are an intrinsic part of driving on public roads, so it is impor- tant that datasets contain people so that the community can develop more accurate perception systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If consent was obtained, were the consenting individuals provided with a mechanism to revoke their consent in the future or for certain uses?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please provide a description, as well as a link or other access point to the mechanism (if appropriate) n/a Has an analysis of the potential impact of the dataset and its use on data subjects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', a data protection impact analysis) been conducted?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please provide a description of this analysis, including the outcomes, as well as a link or other access point to any supporting documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Any other comments?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' n/a PREPROCESSING / CLEANING / LABELING Was any preprocessing/cleaning/labeling of the data done (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', discretization or bucketing, tokenization, part-of-speech tagging, SIFT feature extraction, removal of instances, processing of missing values)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please provide a description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If not, you may skip the remainder of the questions in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Images are reduced from their full resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 3D point locations are quantized to float16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Ground height maps are quantized to .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='3 meter resolution from their full resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' HD map polygon vertex locations are quantized to .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='01 meter resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 3D annotations are smoothed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' For the Motion Forecasting Dataset, transient 3D tracks are suppressed and object locations are smoothed over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Was the “raw” data saved in addition to the preprocessed/cleaned/labeled data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', to support unanticipated future uses)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please provide a link or other access point to the “raw” data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Yes, but such data is not public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Is the software used to preprocess/clean/label the instances available?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please provide a link or other access point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Any other comments?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' n/a USES Has the dataset been used for any tasks already?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please provide a description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Yes, this manuscript benchmarks a contemporary 3D object detection method on the Sensor Dataset and a contemporary motion forecasting method on the Motion Forecasting Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 24 Is there a repository that links to any or all papers or systems that use the dataset?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please provide a link or other access point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Yes, the Argoverse 2 API can be found at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='com/argoverse/av2-api.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' For the Argoverse 2 datasets, we maintain two leaderboards for 3D Detection [https://eval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='ai/ web/challenges/challenge-page/1710] and Motion Forecasting [https://eval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='ai/web/ challenges/challenge-page/1719].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' For the Argoverse 1 datasets, we maintain four leaderboards for 3D Tracking [https://eval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='ai/web/challenges/challenge-page/453/overview], 3D Detection [https://eval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='ai/web/challenges/challenge-page/725/overview], Motion Forecast- ing [https://eval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='ai/web/challenges/challenge-page/454/overview], and Stereo Depth Estimation [https://eval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='ai/web/challenges/challenge-page/917/overview].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Argoverse 1 was also used as the basis for a Streaming Perception challenge [https: //eval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='ai/web/challenges/challenge-page/800/overview].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' What (other) tasks could the dataset be used for?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The datasets could be used for research on visual odometry, pose estimation, lane detection, map automation, self-supervised learning, structure-from-motion, scene flow, optical flow, time to contact estimation, pseudo-lidar, and point cloud forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Is there anything about the composition of the dataset or the way it was collected and preprocessed/cleaned/labeled that might impact future uses?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' For example, is there anything that a future user might need to know to avoid uses that could result in unfair treatment of individuals or groups (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', stereotyping, quality of service issues) or other undesirable harms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', financial harms, legal risks) If so, please provide a description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Is there anything a future user could do to mitigate these undesirable harms?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Are there tasks for which the dataset should not be used?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please provide a description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The dataset should not be used for tasks which depend on faithful appearance of faces or license plates since that data has been obfuscated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' For example, running a face detector to try and estimate how often pedestrians use crosswalks will not result in meaningful data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Any other comments?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' n/a DISTRIBUTION Will the dataset be distributed to third parties outside of the entity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', company, institution, organization) on behalf of which the dataset was created?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please provide a description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Yes, the dataset is hosted on https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='argoverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='org/ like Argoverse 1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' How will the dataset will be distributed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', tarball on website, API, GitHub)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Does the dataset have a digital object identifier (DOI)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We provide both tar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='gz archives and raw files for two of the Argoverse 2 datasets (Motion Forecasting, Sensor), but provide only raw files for the Lidar datasets), available via AWS transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' See https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='argoverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='org/av2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='html#download-link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The Argoverse 1 and Argoverse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 were distributed as a series of tar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='gz files (See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='argoverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='org/av1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='html#download-link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The files are broken up to make the process more robust to interruption (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' a single 2 TB file failing after 3 days would be frustrating) and to allow easier file manipulation (an end user might not have 2 TB free on a single drive, and if they do they might not be able to decompress the entire file at once).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 25 When will the dataset be distributed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The data was made available for download after NeurIPS 2021, in March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Will the dataset be distributed under a copyright or other intellectual property (IP) license, and/or under applicable terms of use (ToU)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please describe this license and/or ToU, and provide a link or other access point to, or otherwise reproduce, any relevant licensing terms or ToU, as well as any fees associated with these restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Yes, the dataset was released under the same Creative Commons license as Argoverse 1 – CC BY- NC-SA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Details can be seen at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='argoverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='org/about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='html#terms-of-use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Have any third parties imposed IP-based or other restrictions on the data associated with the instances?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please describe these restrictions, and provide a link or other access point to, or otherwise reproduce, any relevant licensing terms, as well as any fees associated with these restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Do any export controls or other regulatory restrictions apply to the dataset or to individual instances?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please describe these restrictions, and provide a link or other access point to, or otherwise reproduce, any supporting documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Any other comments?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' n/a MAINTENANCE Who is supporting/hosting/maintaining the dataset?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Argo AI How can the owner/curator/manager of the dataset be contacted (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', email address)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The Argoverse team responds through the Github page for the Argoverse 2 API: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' com/argoverse/av2-api/issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The Argoverse team responds through the Github page for the Argoverse 1 API: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' com/argoverse/argoverse-api/issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' It currently contains 2 open issues and 126 closed issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' For privacy concerns, contact information can be found here: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='argoverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='org/ about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='html#privacy Is there an erratum?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please provide a link or other access point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Will the dataset be updated (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', to correct labeling errors, add new instances, delete instances)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please describe how often, by whom, and how updates will be communicated to users (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', mailing list, GitHub)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' It is possible that the constituent Argoverse 2 datasets are updated to correct errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' This was the case with Argoverse 1 which was incremented to Argoverse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Updates will be communicated on Github and through our mailing list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If the dataset relates to people, are there applicable limits on the retention of the data associated with the instances (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=', were individuals in question told that their data would be retained for a fixed period of time and then deleted)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please describe these limits and explain how they will be enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 26 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Will older versions of the dataset continue to be supported/hosted/maintained?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please describe how.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If not, please describe how its obsolescence will be communicated to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We still host Argoverse 1 even though we have declared it “deprecated”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='argoverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='org/av1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='html#download-link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We will use the same warn- ing if we ever deprecate Argoverse 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Note: Argoverse 2 does not deprecate Argoverse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' They are independent collections of datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If others want to extend/augment/build on/contribute to the dataset, is there a mechanism for them to do so?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please provide a description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Will these contributions be validated/verified?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please describe how.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If not, why not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Is there a process for communicating/distributing these contributions to other users?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' If so, please provide a description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' For example, the streaming perception challenge was built by CMU researchers who added new 2D object annotations to Argoverse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='1 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' The Creative Commons license we use for Argoverse 2 ensures that the community can do the same thing without needing Argo’s permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' We do not have a mechanism for these contributions/additions to be incorporated back into the ‘base’ Argoverse 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Our preference would generally be to keep the ‘base’ dataset as is, and to give credit to noteworthy additions by linking to them as we have done in the case of the Streaming Perception Challenge (see link at the top of this Argoverse page https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='argoverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='org/tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content='html).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Any other comments?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' n/a Environmental Impact Statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' Amount of Compute Used: We estimate 2,000 CPU and 500 GPU hours were used in the collection of the dataset and the performance of baseline experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}
+page_content=' 27' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfnvhT/content/2301.00493v1.pdf'}